RAMA 11 Recueil des Resum´ es´ RAMA11.pdfLa Direction Gen´ ´erale de la Recherche Scientiﬁque et du Developpement Technologique´ L’Universite Djillali Liabes, Sidi´ Bel Abbes

RAMA 11Recueil des Resumes

Sidi Bel Abbes, 21–24 novembre 2019

La Direction Generale de laRecherche Scientifique et duDeveloppement Technologique

L’Universite Djillali Liabes, SidiBel Abbes

organisent la

Onzieme Rencontre d’Analyse Mathematiques etApplications

RAMA 11SBA, 21–24 novembre 2019

Cette rencontre est une manifestation periodique. Elle se propose de promouvoirla recherche en mathematique et de stimuler et encourager la cooperation scien-tifique internationale en offrant un espace d’echange du savoir mathematique enanalyse et au-dela.

Cette manifestation est organisee, essentiellement, par le laboratoire de Mathematiques(LDM) et avec l’aide des laboratoires Statistiques et processus stochastiques (LSPS)et le laboratoire d’Analyse et Controle des Equations aux Derivees Partielles (LACEDP)Conferenciers Boumedienne Abdellaoui (Tlemcen)

Kais Ammari (Monastir)Cherif Amrouche (Pau)El Hadi Ait Dads (Marrakech)Gilles Carbou (Pau)Khalid Ezzinbi (Marrakech)Toufik Hmidi (Rennes)Claude Lobry (Nice)Keddour Lemrabet (BabEzouar)Abdelghani Ouahab (Adrar)Ludovic Rifford (Nice)Arnaud Rougirel (Poitiers)Tewfik Sari (Montpellier)Guy Vallet (Pau)

Presidents d’honneurHafid Aourag, Directeur General, DGRSDTMourad Meghachou, Recteur, Universite SBAMustapha Lakrib, Doyen de la Faculte des Sciences Exactes, SBAFethallah Tebboune, Recteur, Universite de Saida

Comite de pilotageAissa Aibeche, Universite Setif 1Abdelkader Bouyakoub, Universite Oran 1Abderrahmane Yousfate, Universite SBA

Comite d’organisationMouffak Benchohra, Chafi Boudekhil, Abbes Benaissa, Abdelkader Gheriballah,Mustapha Mechab, Abderrahmane Oumansour, Fatine Aliouane, Soumia Benakreti.Comite scientifique

Dalila Azzam-Laouir (Jijel)Amina Angelika Bouchentouf (Sidi Bel Abbes)Sidi Mohamed Bouguima (Tlemcen)Djamel Chacha (Ouargla)Lahcene Mezrag (M’Sila)Mohamed Morsli (Tizi-Ouzou)Boubaker-Khaled Sadallah (ENS Kouba)

Themes

EDP, EDO, Analyse Fonctionnelle, Controle

Algebre, Geometrie

Probabilite et Statistique

Applications (Biomathematiques, Mathematiques Financieres, Intelligence Arti-ficielle, etc.)

ContactUniversite Djilali Liabes, Sidi Bel AbbesEmail: [email protected]://www.univ-sba.dz/rama11

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Contents

Conferences 1Nonlocal problem with non local gradient term: existence and non exis-

tence of positive solution (Boumedienne Abdellaoui) . . . . . . . . . . 1Compact Almost Automorphic Solutions for some Nonlinear Integral

Equation (El Hadi Ait Dads) . . . . . . . . . . . . . . . . . . . . . . . 2How to use observability inequalities to solve some inverse problems

for evolution equations (Kais Ammari) . . . . . . . . . . . . . . . . . 2Elliptic Problems in Smooth and Non Smooth Domains (Cherif Amrouche,

Mohand Moussaoui, Huy Hoang Nguyen) . . . . . . . . . . . . . . . . 3Domain Walls pinning in ferromagnetic nanowires (Abdelkader Al Sayed,

Gilles Carbou, David Sanchez) . . . . . . . . . . . . . . . . . . . . . . . 3A new existence theory for periodic solutions to evolution equations

(Khalil Ezzinbi) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Time periodic and quasi-periodic solutions for the generalized quasi-

geostrophic equations. (Taoufik Hmidi) . . . . . . . . . . . . . . . . . 6Impedance d’une bande mince en elasticite lineaire isotrope (Keddour

Lemrabet) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6L’Analyse Non Standard et l’evolution des mathematiques contempo-

raines (Claude Lobry) . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Some deterministic and random fixed point theorems and applications

(Abdelghani Ouahab) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Sur quelques problemes de geometrie sous-riemannienne (Ludovic Rifford) 11Equations fractionnaires de Schrodinger et de Delsarte (Arnaud Rougirel) 12Modelisation mathematique de la competition des especes microbienne

dans le chemostat (Tewfik Sari) . . . . . . . . . . . . . . . . . . . . . . 12On the stochastic ∆p problem (Guy Vallet) . . . . . . . . . . . . . . . . . . 14

Communications 15Global Behavior of the Solutions to a Class of Nonlinear, Singular Sec-

ond Order ODE (Mama Abdelli,Alain Haraux) . . . . . . . . . . . . . 15Some properties of generalized Newton transformations (Mohammed Ab-

delmalek) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Stochastic Approximation procedure under α-mixing data ( Nabila AIANE,

Abdelnasser DAHMANI ) . . . . . . . . . . . . . . . . . . . . . . . . . 21On the p-harmonic and p-biharmonic maps (Ahmed Mohammed Cherif ) . 23Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

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Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24p-Harmonic maps and conformal vector fields . . . . . . . . . . . . 24Liouville type theorems for p-biharmonic maps . . . . . . . . . . . . 25

A recursive kernel estimates of the functional modal regression underergodic dependence condition (Fatima Zohra Ardjoun, Larbi Ait Hen-nani, Ali Laksaci ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Using refined descriptive sampling in integration (Leila BAICHE, Meg-douda OURBIH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Mathematical analysis of a B-cell chronic lymphocytic leukemia modelwith immune response (Youcef Belgaid, Mohamed Helal, Ezio Ventiruno) 34

About a Nonlinear Caputo-Hadamard Fractional Differential Equationwith Hadamard integral boundary conditions in Banach Spaces(Maamar Benbachir, Abdallatif Boutiara ) . . . . . . . . . . . . . . . . . 36

On the regularity of the solution of non-homogeneous Burgers equation(Yassine Benia, Boubaker-Khaled Sadallah ) . . . . . . . . . . . . . . . . 38

Second order optimality in three-stage design for estimating a productof two means (Zohra Benkamra, Mounir Tlemcani ) . . . . . . . . . . . 42

Global dynamics of an age structured alcoholism model (Soufiane Bentout,Salih Djilali ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Ultrametric q-difference equations and q-Wronskian (Benharrat Belaidi,Rabab Bouabdelli, Abdelbaki Boutabaa) . . . . . . . . . . . . . . . . . . 50

Quadratic error of the conditional mode function in the local linear esti-mation for functional data (Wahiba Bouabsa ) . . . . . . . . . . . . . . 54

The eikonal equation: some applications, existence and uniqueness re-sults (Rachida Boudjerada) . . . . . . . . . . . . . . . . . . . . . . . . . 58

Tail probabilities and complete convergence for weighted sequences ofLNQD random variables with application to first-order autoregres-sive processes model (Boulenoir Zouaouia ) . . . . . . . . . . . . . . . 61

AVERAGING FOR DIFFERENTIAL INCLUSIONS (Amel Bourada, MustaphaLakrib ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Integer/decimal parts decomposition of a standard Gaussian distribu-tion (Bourada Sofiane, Tlemcani Mounir ) . . . . . . . . . . . . . . . . . 67

Terminal Value Problem for Differential Equations with Hilfer–KatugampolaFractional Derivative (Mouffak Benchohra, Soufyane Bouriah ) . . . . . 70

Kahler-Golden manifolds (Habib Bouzir , Beldjilali Gherici) . . . . . . . . . 74On the p-harmonic and p-biharmonic maps (Ahmed Mohammed Cherif ) . 77Two-dimensional inverse heat conduction problem in quarter plane. In-

tegral approach (L. Chorfi, A. Bel-Hadj Hassin) . . . . . . . . . . . . . 82Asymptotic result of the conditional estimator of cumulative distribu-

tion in functional spatial data case (Hamza Daoudi, Boubaker Mechab) 84Modele d’evolution de la transmission de la maladie de Chagas (Naba-

hats Dib, Rabah Labbas , Tewfik Mahjoub, Ahmed Medeghri ) . . . . . . 87Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Presentation du Modele . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Construction du modele: . . . . . . . . . . . . . . . . . . . . . . . . . 88

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Transformation du systeme precedent en une equation differentielleabstraite: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Groups whose proper subgroups are (locally π−finite)-by-(locally nilpo-

tent) (Amel Dilmi, Nadir Trabelsi) . . . . . . . . . . . . . . . . . . . . . 92Recursive conditional hazard function estimator with functional station-

ary ergodic data (Amina GOUTAL) . . . . . . . . . . . . . . . . . . . 94Power Utility Maximization in Levy market model (Malika HAMMAD) . 97Attractors for a Nonautonomous Reaction-Diffusion Equation with De-

lay (Hafidha Harraga, Mustapha Yebdri) . . . . . . . . . . . . . . . . . 99Etude de la fonction quantile pour des donnees tronquees aleatoirement

a gauche et a covariables fonctionnels (Nacera Helal , Elias OULDSAID ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Optimal exploitation of a fishery under conservation of fish species (MeryemHELLAL ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Estimation d’un Melange de Modeles GARCH Periodiques (Rokia Hemis,Hafida Guerbyenne, Faycal Hamdi) . . . . . . . . . . . . . . . . . . . . 109

Reconstruction de la solution de l’equation de la chaleur avec des deplacements(Hisao Fujita Yashima,Narimene Achour) . . . . . . . . . . . . . . . . . 112

Clustering des series temporelles (Yamina Khemal Bencheikh, Welid Grimes) 116Quelque surfaces minimales r dans l’espace de Lorentz-Heisenberg H1

3( Hamid KHIAR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Regular generalized Roumieu ultradistributions ( Fatima Zohra KOR-BAA , Khaled BENMERIEM ) . . . . . . . . . . . . . . . . . . . . . . . 120

On the analysis of unreliable Markovian multiserver queue M/M/cwith retrials. ( Faiza LIMAM-BELARBI, Meriem ELHADDAD) . . . . 124

Estimation for infinite variance moving average process ( Tawfiq FawziMami) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Asymptotic result of the conditional estimator of cumulative distribu-tion in functional spatial data case (Hamza Daoudi, Boubaker Mechab) 130

A New Error Estimate on Uniform Norm of Schwarz Algorithm for El-liptic Quasi-Variational Inequalities with Nonlinear Source Terms(Allaoua Mehri , Samira Saadi) . . . . . . . . . . . . . . . . . . . . . . . 133

Numerical analysis of the Bloch-Torrey equation in deforming media:Application to cardiac diffusion MRI ( Imen Mekkaoui) . . . . . . . . 137

Study of Generalized finite operators and orthogonality ( Hadia Mes-saoudene , Nadia! Mesbah) . . . . . . . . . . . . . . . . . . . . . . . . . 141

Optimal Control Problems for a Semilinear Evolution System with Infi-nite Delay (Fatima Zahra MOKKEDEM) . . . . . . . . . . . . . . . . 143

Estimation of extremal index for heavy tailed ARMAX process ( HakimOuadjed ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

Special values of generalized multiple Hurwitz zeta function at non-positive integers ( Boualem SADAOUI) . . . . . . . . . . . . . . . . . 149

Stability and Bifurcations in 2D Spatiotemporal Discrete Systems ( M. L.Sahari, A. K. Taha, L. Randriamihamison) . . . . . . . . . . . . . . . . . 152

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TH-surfaces dans les espaces 3-dimensionnel euclidien E3 et lorentzienE3

1 (Bendehiba Senoussi) . . . . . . . . . . . . . . . . . . . . . . . . . . 155Compatibilite des structures riemanniennes et des structures de Jacobi

(Yacine Aıt Amrane, Zeglaoui Ahmed) . . . . . . . . . . . . . . . . . . . 158On the regular vortex patch topic for the planar Boussinesq system with

fractional dissipation ( Mohamed Zerguine) . . . . . . . . . . . . . . . 163

Posters 169Functional local linear estimate of the conditional cumulative distribu-

tion function (Khadidja Abdelhak) . . . . . . . . . . . . . . . . . . . . 169Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172On the conditional distribution function estimate in the single functional

index model under censored data (Fatima Akkal) . . . . . . . . . . . 172Numerical solution for time-fractional partial differential equation (N.

Attia, D. Seba, A. Nour) . . . . . . . . . . . . . . . . . . . . . . . . . . 175Estimation of discrete semi-Markov process (Chafiaa Ayhar ) . . . . . . . . 179Mean-field reflected delayed Backward Stochastic Differential Equations

with jumps (Khalida Bachir Cherif ) . . . . . . . . . . . . . . . . . . . . 182A model of competition between plasmid-bearing and plasmid-free in a

chemostat with external lethal inhibitor (Bar Bachir, Mohamed Dellal) 184Least Squares Estimation in Periodic Restricted Expar(p) Models (Sabah

Becila, Mouna Merzougui) . . . . . . . . . . . . . . . . . . . . . . . . . 187Growth of solutions of a class of linear differential equations near a sin-

gular point (Samir Cherief ) . . . . . . . . . . . . . . . . . . . . . . . . 190On the local linear estimation of the nonparametric robust regression for

dependent functional data (Souheyla Chemikh) . . . . . . . . . . . . . 194Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196Well-posedness of sea-ice model (CHATTA. Sofiane ) . . . . . . . . . . . . 198

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

Model 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199Model 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

Bibliographie 200On the resolution of nonlinear fractional elliptic problem (Sara Dob ) . . . 201Spectral element methods a priori and a posteriori error estimates for a

penalized unilateral obstacle problem (Bochra Djeridi) . . . . . . . . 203Solvability for fractional Sturm-Liouville boundary value problems with

p-Laplacian operator in Banach spaces (Choukri Derbazi) . . . . . . . 205Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205EXISTENCE, UNIQUENESS, COMPACTNESS OF THE SOLUTION SET,

AND DEPENDENCE ON A PARAMETER FOR AN IMPULSIVEPERIODIC BOUNDARY VALUE PROBLEM (Khelifa Daoudi ) . . . . 207

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

Uniqueness of solutions . . . . . . . . . . . . . . . . . . . . . . . . . 208Existence of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

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Positive solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209Estimation of the jump-diffusion parameter models using real data (Frihi

Zahrate El Oula) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210Analyze the S&P 500 historical data . . . . . . . . . . . . . . . . . . . . . 211Maximum-likelihood estimation of jump-diffusion models parameters . 211Comparison between the theoretical and the empirical distribution of

the S&P 500 log returns . . . . . . . . . . . . . . . . . . . . . . . . . . 211Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213The Model And The Estimates . . . . . . . . . . . . . . . . . . . . . . . . . 214Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

Linear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215Nonlinear model (Exponential model) . . . . . . . . . . . . . . . . . 215

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215Some p-summing operators and their conjugates (FERRADI Athmane) . . 216Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219The perfect mixing and the logistic equation (ELBETCH Bilel ) . . . . . . 219

Le modele mathematique . . . . . . . . . . . . . . . . . . . . . . . . . . . 220Resultats obtenus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

Cas de trois sites avec dispersion non symetrique . . . . . . . . . . 221Cas de n sites avec dispersion symetrique . . . . . . . . . . . . . . . 221

Stability of The Lame System With A Time Delay Condition of FractionalType (GAOUAR Soumia) . . . . . . . . . . . . . . . . . . . . . . . . . 223

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224On a Markovian multiserver vacation queueing system with waiting

servers and impatient customers (Abdelhak Guendouzi) . . . . . . . . 225Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

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Conferences

Nonlocal problem with non local gradient term: existence andnon existence of positive solution

Boumedienne AbdellaouiUniversity of Tlemcen

Resume: In this talk we analyze the existence and regularity of positive solutionto problem

(−∆)su = µ(x)D2s (u) + λ f (x) , in Ω,

u = 0 , in RN \Ω,(1)

where s ∈ (12 , 1), Ω is a bounded domain and λ > 0 is a real parameter. Here f is

a nonnegative functions that belongs to a suitable Lebesgue space, µ ∈ L∞(Ω).The operator (−∆)s is the classical fractional Laplacian of order s defined by

(−∆)su(x) := aN,s P.V.∫

RN

u(x)− u(y)|x− y|N+2s dy, s ∈ (0, 1), (2)

where

aN,s := 22s−1π−N2

Γ(N+2s2 )

|Γ(−s)| ,

is the normalization constant so that the identity

(−∆)su = F−1(|ξ|2sFu), ξ ∈ RN, s ∈ (0, 1),

holds in S(RN), the class of Schwartz function.The term D2

s is a nonlocal “gradient square” term given by

D2s (u) =

aN,s

2

∫RN

|u(x)− u(y)|2|x− y|N+2s dy .

Depending on the real parameter λ > 0, we derive existence and non-existenceresults.

The proof of our existence result relies on sharp Calderon-Zygmund type reg-ularity results for the fractional Poisson equation with low integrability data. Wealso obtain existence results for related problems involving different nonlocal dif-fusion terms.

The talk is a part of the following papers:

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B. Abdellaoui, I. Peral, Towards a deterministic KPZ equation with fractional diffu-sion: The Stationary case, to appear in Nonlinearity 2018.

B. Abdellaoui, A. Fernandez, Nonlinear fractional Laplacian problems with nonlocal“gradient terms”, to appear in Proceedings of the Royal Society of Edinburgh A2020.

Compact Almost Automorphic Solutions for some NonlinearIntegral Equation

El Hadi Ait DadsUniversity Marrakesh, Morocco

In this work, we study the existence of compact almost automorphic solu-tions for a class of integral equations with a time-dependent delay and a history-dependent delay. An application to a blowflies model and a transmission linesmodel is carried out to support the theoretical finding.Key words: Almost periodic and almost automorphic solutions, integral equations, neu-tral equation, state dependent delay, Nicholson’s model, Lossless transmission lines.

2010 Mathematics Subject Classification: 47H10,47H30, 54H25.

How to use observability inequalities to solve some inverseproblems for evolution equations

Kais AmmariUniversity of Monastir, Tunisia

We survey some of our recent results on inverse problems for evolution equa-tions. The goal is to provide an unified approach to solve various types of evo-lution equations. The inverse problems we consider consist in determining un-known coefficients from boundary measurements by varying initial conditions.Based on observability inequalities, and a special choice of initial conditions weprovide uniqueness and stability estimates for the recovery of volume and bound-ary lower order coefficients in wave and heat equations.

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Elliptic Problems in Smooth and Non Smooth DomainsCherif Amrouche, Mohand Moussaoui, Huy Hoang Nguyen

University of Pau, France

We are interested here in questions related to the regularity of solutions ofelliptic problems with Dirichlet or Neumann boundary condition (see ([1]). Forthe last 30 years, many works have been concerned with questions when Ω is aLipschitz domain. We give here some complements for the case of the Laplacian(see [3]), the Bilaplacian ([2], [6]) and the operator div(A∇) (see ([5]), when A isa matrix or a function, and we extend this study to obtain other regularity resultsfor domains having an adequate regularity. Using the duality method, we willthen revisit the work of Lions-Magenes [4], concerning the so-called very weaksolutions, when the data are less regular. Thanks to the inter-polation theory, itpermits us to extend the classes of solutions and then to obtain new results ofregularity.Keywords: Elliptic problems, Lipschitz domains, regularityAMS Classification: 35C15, 35J25, 35J40References

[1] C. Amrouche, M. Moussaoui, H.H. Nguyen. Laplace equation in smoothor non smooth domains. Work in Progress.

[2] B.E.J. Dahlberg, C.E. Kenig, J. Pipher, G.C. Verchota. Area integral esti-mates for higher order elliptic equations and systems. Ann. Inst. Fourier, 47, no.5, 1425–1461, (1997).

[3] D. Jerison, C.E. Kenig. The Inhomogeneous Dirichlet Problem in LipschitzDomains, J. Funct. Anal. 130, 161–219, (1995).

[4] J.L. Lions, E. Magenes. Probl‘emes aux limites non-homog‘enes et applica-tions, Vol. 1, Dunod, Paris, (1969).

[5] J. Necas. Direct methods in the theory of elliptic equations. Springer Mono-graphs in Mathematics. Springer, Heidelberg, (2012).

[6] G.C. Verchota The biharmonic Neumann problem in Lipschitz domains.Acta Math. 194 no. 2, 217–279, (2005).

Domain Walls pinning in ferromagnetic nanowiresAbdelkader Al Sayed, Gilles Carbou, David Sanchez


Ferromagnetic nanowires have promising applications in data storage. Insuch devices, the information is encoded by Domain Walls which are thin zonesof magnetization reversal. The magnetization behavior is described by the nonlinear Landau-Lifschitz model. In this talk we investigate the stability of domainwall configurations. In particular we highlight pinning effects of notches, bendsand junctions of wires.

Key words: ferromagnetic materials, domain walls, Landau-Lifschitz equation, stability.

2010 Mathematics Subject Classification: .

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IntroductionIn [Sc], promising applications of ferromagnetic nanowires in the domain of data storageare highlighted. Domain walls formation in such devices allows bits encoding, and wallsmotion induced by spin current injection makes data reading faster than in classical de-vices. This storage process is low power-consuming, non volatile and avoid mechanicalstress. It also allows data processing in spintronic devices (see [AXC]).

For data storage applications, the stability of walls positions is crucial to ensure thenon volatility of the storage since an undesired wall motion can deteriorate the informa-tion. As it is proved in [CL], walls configurations in straight nanowires are stable but notasymptotically stable, so that both chirality and position of walls are not fixed. In addi-tion, (see [CL2]) in finite length nanowire, walls configurations are unstable. Therefore, astronger control of walls positions is indispensable.

In this talk, we aim to study stabilising effects of the wire geometry, that is by bends,junctions or notches patterned on the wire. We will deal with one dimensional models offerromagnetic nanowires. these models are justified by asymptotic process in [ACL] and[BCL]. The wire is represented by the interval I ⊂ R. The magnetisation is modelled by avector field called magnetic moment: m : (t, x) ∈ R+

t × Ix 7→ m(t, x). The ferromagneticmaterial is supposed to be saturated so that we assume that the norm of m is equal to 1:

|m(t, x)| = 1 in R+ × I. (3)

The variations of m are described by the Landau-Lifschitz equation:∂m∂t

= −m× h(m)− αm× (m× h(m)) on R+ × I,

h(m) = ∂xxm +a′

a∂xm− 1

2(m2e2 + m3e3) + ha,

(4)

where × is the cross product in R3, (e1, e2, e3) is the canonical basis in R3, (m1, m2, m3)are the coordinates of m. We denote by a : I −→ R∗+ the area of the wire section. Thepositive number α is called damping coefficient, and ha ∈ R3 is the applied magneticfield.

Notched nanowiresAt first, we will consider an infinite-length nanowire with one notch, that is we considerEquation (4) with I = R and with a : R −→ R satisfying

a = 1 outside [−l0, l0],

a is even and non decreasing on [0, l0],

0 < a1 ≤ a(x) ≤ 1,

(5)

i.e. the notch is restricted to the domain [−l0, l0] ⊂ R, where l0 > 0 is fixed. We remarkthat the model is invariant by rotation around the wire axis, that is: if m satisfies (4), thenRϕm is also a solution of (4), where:

Rϕ =

1 0 00 cos ϕ − sin ϕ0 sin ϕ cos ϕ

.

4

We aim to prove the stability of solutions representing a magnetisation switching, calledDomain Wall. We obtain the following theorem:

Theorem 1 We assume that ha = 0. There exists a stationary solution m0 for (4), satisfying thesaturation constraint (3) and tending to −e1 (resp. +e1) when x tends to −∞ (resp.+∞). Thissolution is stable and asymptotically stable modulo rotations around the wire axis, that is: for allε > 0, there exists η > 0 such that for all solution m for (3)-(4) satisfying ‖m(0, ·)−m0‖H1(R) ≤η, then

• ∀t ≥ 0, ‖m(t, ·)−m0‖H1(R) ≤ ε,

• there exists ϕ∞ such that ‖m(t, ·)− Rϕ∞ m0‖H1(R) −→ 0 when t −→ 0.

In case of non vanishing small applied field, we establish also that the magnetisationswitching remains close to the notch, since we can prove the existence of a static solutionclose to the m0 constructed in Theorem 43. We remark that without notch, a small ap-plied field induces a translation of the Domain wall moves along the wire with a velocityproportional to the applied field. These results are established in [CS].

Junctions of wiresIn [AC1] and [AC2], we deal with junctions of two or three straight wires. In case of twowires forming a bend and in case of the junction of a finite wire on a infinite length wire,we prove the existence of configurations presenting a domain wall at the junction. In ad-dition, these configurations are asymptotically stable and remain pinned at the junctionin presence of an small applied field.bf References

(AD) D. A. Allwood, Gang Xiong, M. D. Cooke, C. C. Faulkner, D. Atkinson, N.Vernier, R. P. Cowburn, Submicrometer Ferromagnetic NOT Gate and Shift Register, Science296 (2002).

(AC1) A. Al Sayed and Gilles Carbou, Walls in infinite bent Ferromagnetic Nanowires,Ann. Fac. Sci. Toulouse Math. (6) 27 (2018), no. 5, 897-924.

(AC2) A. Al Sayed, Gilles Carbou and Stephane Labbe, In preparation.(ACL) A. Al Sayed, Gilles Carbou and Stephane Labbe, Asymptotic model for twisted,

bent wires with electric current, Z. Angew. Math. Phys. 70 (2019), no. 1, Art. 6, 15 pp.(BCL) S. Bokoch, G. Carbou and S. Labbe, Circuits of ferromagnetic nano wires, preprint.(CL) Gilles Carbou and Stephane Labbe, Stability for static walls in ferromagnetic nanowires,

Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 273–290.(CL2) Gilles Carbou and Stephane Labbe, Stabilization of Walls for Nano-Wires of Finite

Length, ESAIM Control Optim. Calc. Var. 18 (2012), no. 1, 1–21.(CS) Gilles Carbou and David Sanchez, Stabilization of walls in notched ferromagnetic

nanowires, preprint.(Sc) S. P. Parkin, M. Hayashi and L. Thomas, Magnetic Domain-Wall Racetrack Memory,

Science, 320 (2008), 190–194.

5

A new existence theory for periodic solutions to evolutionequations

Khalil EzzinbiUniversity of Marrakesh, Morocco

This paper deals with a new existence theory for periodic solutions to a broadclass of evolution equations. We first establish new fixed point theorems for affinemaps in locally convex spaces and ordered Banach spaces. Our new fixed pointresults extend, encompass and complement a number of well-known theoremsin the literature, including the famous Chow and Hale fixed point theorem. Withthese obtained fixed point results, we investigate the existence of periodic solu-tions for some class of nonhomogeneous linear systems in Banach spaces withlack of compactness. Some illustrative examples are also given.Key words: Affine maps; Weak topology; Locally convex spaces; Fixed point Theorems;Semigroups; Evolution equations; Functional differential equations.2010 Mathematics Subject Classification: 47H10,47H30, 54H25.

Time periodic and quasi-periodic solutions for the generalizedquasi-geostrophic equations.

Taoufik HmidiUniversity of Rennes, France

We shall discuss in the first part the existence of periodic solutions for the gen-eralized quasi-geostrophic equation. At the linear level this allows to constructinvariant torus filled by quasi-periodic solutions and we intend in the second partof this lecture to analyze their nonlinear stability. The construction of Cantor fam-ilies of small amplitude quasi-periodic solutions relies on different tools comingfrom bifurcation theory, KAM reduction and Nash-Moser scheme.

Impedance d’une bande mince en elasticite lineaire isotropeKeddour Lemrabet

USTHB, Algiers

SoientΩδ = R2 × ]0, δ[ , Γδ = R2 × δ , Σ = R2 × 0 .

On considere le probleme aux limites :divσ(u) = f dans Ωδ,σ(u).n = g sur Γδ,u = ϕ sur Σ,

6

ou l’inconnue f , g et ϕ sont donnees et l’inconnue u est le deplacement. Letenseur des deformations σ(u) est lie au tenseurs des contraintes linearise (2ε(u) =(∇u +∇u)) par les coefficients de Lame λ et µ par la loi lineaire

σ(u) = λtrε(u)Id + 2µε(u).

On a note n la normale au bord,orientee vers l’interieur de Ωδ et δ > 0 est unparametre destine a tendre vers 0.

L’operateur d’impedance de la bande mince Ωδ associe aux donnees f , g et ϕla trace de la composante normale de la contrainte sur Σ

Tδ ( f , g, ϕ) = [σ(u).n]Σ .

On note VT, Vn, la composante tangentielle et la composante normale du vecteurV.

On pose

V (ξ, z) =∫

R2eiξ.xV (x, z) dx.

La solution de l’equation homogene

divσ(u) = 0 dans Ωδ

est donnee paru = ∆ζ − 2κ∇ (∇.ζ)

ou ζ est solution de∆2ζ = 0

et2κ =

λ + µ

λ + 2µ

On montre alors en utilisant cette ecriture de Kelvin-Mindlin de la solution dusysteme de l’elasticite que

uT (z, ξ) =

[a+T − αz

(iξ.a+T + |ξ| a+n

)(z, ξ)

iξ|ξ|

]ez|ξ|

+

[a−T + αz

(iξ.a−T − |ξ| a

−n)(z, ξ)

iξ|ξ|

]e(δ−z)|ξ|

un (z, ξ) =[a+n − αz

(iξ.a+T + |ξ| a+n

)](z, ξ) ez|ξ|

+[a−n − αz

(iξ.a−T − |ξ| a

−n)]

(z, ξ) e(δ−z)|ξ|

et

1µ(σ.n)T (z, ξ) =

|ξ| a+T (z, ξ) + a+n (z, ξ) iξ

−α (1 + 2z |ξ|)(

iξ.a+T (z, ξ)+ |ξ| a+n (z, ξ)

)iξ|ξ|

ez|ξ|

[(− |ξ| a−T (z, ξ) + a−n (z, ξ) iξ

+α (1− 2z |ξ|)(iξ.a−T (z, ξ)− |ξ| a−n (z, ξ)

) iξ|ξ|

)]e(δ−z)|ξ|

7

(σ.n)n (z, ξ) =

2µ |ξ| a+n (z, ξ) +(λ (1− α)

−2µα (1 + z |ξ|)

)(iξ.a+T (z, ξ)

+ |ξ| a+n (z, ξ)

) ez|ξ|

−2µ |ξ| a−n (z, ξ) +(λ (1− α)

−2µα (1− z |ξ|)

)(iξ.a−T (z, ξ)− |ξ| a−n (z, ξ)

) e(δ−z)|ξ|

avec α = λ+µ(λ+3µ)

.

Les conditions aux limites σ(u).n = g sur Γδ et u = ϕ sur Σ se traduisent parun systeme lineaire qui permet de calculer les coefficients a±T (z, ξ) et a±n (z, ξ) .

Les coefficients a±T (0, ξ) et a±n (0, ξ) permettent de donner une expression ex-plicite de l’operateur d’impedance Tδ ( f , g, ϕ) en fonction de Tδ ( f , g, ϕ).

L’Analyse Non Standard et l’evolution des mathematiquescontemporaines

Claude LobryUniversity of Nice, France

L’Analyse Non Standard (ANS) est une technique mathematique qui permet defaire des calculs avec des nombres ε > 0 infiniment petits. Inventee par Robinsondans les annees 1960, elle a ete, a quelques exceptions pres, au mieux, accueil-lie sans enthousiasme, au pire, rejetee par la communaute mathematique. Parmices exceptions il faut mentionner l’ecole 0rano- Strasbourgeoise qui se distinguaa la fin des annees 1970 dans l’etude des systemes differentiels singulierementperturbes et l’invention des ”solutions-canard”, maintenant consacrees. En undemi-siecle les choses ont evolue. De rejetee qu’elle etait l’ANS est devenueune methode consideree comme legitime et meme preconisee par de tres grandsmathematiciens comme M. Gromov ou T. Tao. Comment expliquer cette evolution?

Nous tenterons de le faire en replacant l’ANS dans le contexte plus large del’evolution des mathematiques au cours du XXeme siecle et, encore plus large-ment, de l’evolutions des techniques de calcul de la Renaissance a nos jours.Nous verrons comment l’ANS s’inscrit dans les bouleversements de la pratiquemathematique que la revolution informatique impose.

8

Some deterministic and random fixed point theorems andapplications

Abdelghani OuahabUniversity of Adrar

In this talk, we present some old and new fixed point theorems which candirectly applied to differential equations and inclusions. We also establish a ran-dom multivalued version of a Krasnosel’skii type fixed point theorem for thesum B + G, where B is a linear or nonlinear random operator and G is a ran-dom multivalued operator. We hope that our synthetic approach will be a usefulcontribution to fixed point theory, including nonlinear analysis and game theory.Key words: Fixed point, multivalued map, random variable, matrix, random differential

inclusions, Nash equilibrium.

2010 Mathematics Subject Classification: 47H10,47H30, 54H25.The problem of solving equations or multivalued equations belongs to the most im-

portant questions in mathematics. Practically, any part of mathematics deals with equa-tions. Therefore, we can consider algebraic, differential, integral, analytical and manyother types of equations. In our lecture we shall concentrate on topological approach toequations, multivalued equations in deterministic and random cases. What do we meanby equations or multivalued equations ?

Let X be an arbitrary nonempty set, A : X → X be a given operator and F : X →P(X) be a given multifunction.

By the equation or multivalued equation (with the left hand side f or F) we shallunderstand the following question: is it true that there exists an element x ∈ X such that

A(x) = x. (6)

andx ∈ F(x).

So let us assume that there exists some measurable space (Ω,F ) and f : Ω × X → Xand F. For such a operator A and multivalued operator , we shall consider the followingrandom equation and inclusions

A(ω, x) = x. (7)

andx ∈ F(ω, x).

ReferencesJ. Andres and L. Gorniewicz, Topological Fixed Point Principles for Boundary Value Prob-

lems, Kluwer, Dordrecht, 2003.T. N. Anh, Random equations and applications to general random fixed point theo-

rems, New Zealand J. Math. 41 (2011), 17–24.A. Arunchai and S. Plubtieng, Random fixed point theorem of Krasnosel’skii type for

the sum of two operators, Fixed Point Theory Appl. 2013 (2013), No. 142, 10pp.I. Beg and N. Shahzad, Some random approximation theorems with applications,

Nonlinear Anal. 35 (1999), 609–616.

9

I. Beg and N. Shahzad, Random fixed points of weakly inward operators in conicalshells, J. App. Math. Stochastic Anal. 8 (1995), 261–264.

A. T. Bharuch-Reid, Random Integral Equations, New York, Academic Press, 1972.A. T. Bharuch-Reid, Fixed point theorems in probabilistic analysis, Bull. Amer. Math.

Soc. 82 (1976), 641–657.M. Boriceanu, Krasnosel’skii-type theorems for multivalued operators, Fixed Point

Theory 9 (2008), 35–45.P. V. Chu’o’ng, Random versions of Kakutani-Ky Fan’s fixed point theorems, J. Math.

Anal. Appl. 82 (1981), 473–490.S. Djebali, L. Gorniewicz and A. Ouahab, Solutions Sets for Differential Equations and

Inclusions, De Gruyter Series in Nonlinear Analysis and Applications 18, De Gruyter,Berlin, 2013.

J. Garcia-Falset, Existence of fixed points for the sum of two operators, Math. Nachr.12 (2010), 1726–1757.

J. Garcia-Falset and O. Muniz-Perez, Fixed point theory for 1-set weakly contractiveand pseudocontractive mappings. Appl. Math. Comput. 219 (2013), 6843–6855.

L. Gorniewicz, Topological Fixed Point Theory of Multi-valued Mappings, Mathematicsand its Applications, 495, Kluwer, Dordrecht, 1999.

J. R. Graef, J. Henderson, and A. Ouahab, Impulsive Differential Inclusions, A Fixed PointApproach, De Gruyter Series in Nonlinear Analysis and Applications Vol. 20, De Gruyter,Berlin, 2013.

J. R. Graef, J. Henderson and A. Ouahab, Multivalued versions of a Krasnosel’skiitype fixed point theorem, J. Fixed Point Theory Appl., to appear.

O. Hans, Random operator equations, Proc. 4th Berkeley Sympos. Math. Statist. andProb., Univ. California Press, Berkeley, Calif., (1961), II, 185–202.

O. Hans and A. Spacek, Random fixed point approximation by differentiable trajec-tories. 1960 Trans. 2nd Prague Conf. Information Theory pp. 203–213, Publ. HouseCzechoslovak Acad. Sci., Prague, Academic Press, New York.

S. Hu and N. S. Papageorgiou, Handbook of Multivariate Analysis, Vol.I: Theory, Mathe-matics and its Applications, Vol. 419, Kluwer, Dordrecht, 1997.

S. Itoh, Random fixed point theorems with an application to random differentialequations in Banach spaces, J. Math. Anal. Appl. 67 (1979), 261–273.

S. Itoh, Nonlinear random equations with monotone operators in Banach spaces,Math. Ann. 236 (1978), 133–146.

M. Kisielewicz, Differential Inclusions and Optimal Control, Kluwer, Dordrecht, 1991.M. A. Krasnosel’skii, Some problems of nonlinear analysis, Amer. Math. Soc. Transl.

Ser. (2) 10 (1958), 345–409.K. Kuratowski and C. Ryll-Nardzewski, A general theorem on selectors, Bull. Acad.

Polon. Sci. Ser. Sci. Math. Astronom. Phys. 13 (1965), 397–403.F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical

mechanis, in: Fractals and Fractional Calculus in Continuum Mechanics (A. Carpinteriand F. Mainardi, Eds.), pp. 291-348, Springer-Verlag, Wien, 1997

A. Mukherjea, Random Transformations of Banach Spaces, Ph. D. Dissertation, WayneState Univ., Detroit, Michigan, 1968.

W. Padgett and C. Tsokos, Random Integral Equations with Applications to Life Sci-ence and Engineering, Academic Press, New York, 1976.

N. S. Papageorgiou, Random fixed point theorems for measurable multifunctions inBanach spaces, Proc. Amer. Math. Soc. 97 (1986), 507–514.

10

A. I. Perov, On the Cauchy problem for a system of ordinary differential equations,Pribliz. Met. Reshen. Differ. Uravn. 2 (1964), 115–134. (in Russian).

I. R. Petre, A multivalued version of Krasnosel’skii’s theorem in generalized Banachspaces, An. St. Univ. “Ovidius” Constanta Ser. Mat. 22 (2014), 177–192.

I. A. Rus, The theory of a metrical fixed point theorem: theoretical and applicativerelevances, Fixed Point Theory 9 (2008), 541–559.

L. E. Rybinski, Random fixed points and viable random solutions of functional-differentialinclusions, J. Math. Anal. Appl. 142 (1989), 53–61,

V. M. Sehgal and S. P. Singh, On random approximations and a random fixed pointtheorem for set valued mappings, Proc. Amer. Math. Soc. 95 (1985), 91–94.

N. Shahzad, Some general random coincidence point theorems, New Zealand J. Math.33 (2004), 95–103.

N. Shahzad, Random fixed point results for continuous pseudo-contractive randommaps, Indian J. Math. 50 (2008), 263–271.

N. Shahzad, and L. A. Khan, Random fixed points for 1-set-contractive random mapsin Frechet spaces, J. Math. Anal. Appl. 231 (1999), 68–75.

M. L. Sinacer, J. J. Nieto, and A. Ouahab, Random fixed point theorem in generalizedBanach space and applications, Random Oper. Stoch. Equ. 24 (2016), 93–112.

A. Skorohod, Random Linear Operators, Reidel, Boston, 1985.K. K. Tan and Z. X. Yuan, On deterministic and random fixed points, Proc. Amer.

Math. Soc. 119 (1993), 849–856.E. Tarafdar, P. Watson, and X. Z. Yuan, Jointly measurable selections of condensing

Caratheodory set-valued mappings and its applications to random fixed points, NonlinearAnal. 28 (1997), 39–48.

H. D. Thang and P. T. Anh, Random fixed points of completely random operators,Random Oper. Stoch. Equ. 21 (2013), 1–20.

R. S. Varga, Matrix Iterative Analysis, 2nd revised and expanded, Springer Series inComputational Mathematics, Springer, Berlin, 2000.

A. Viorel, Contributions to the Study of Nonlinear Evolution Equations, Ph.D. thesis,Babes-Bolyai University Cluj-Napoca Department of Mathematics, 2011.

T. Xiang and R. Yuan, A class of expansive-type Krasnosel’skii fixed point theorems,Nonlinear Anal. 71 (2009), 3229–3239.

Sur quelques problemes de geometrie sous-riemannienneLudovic Rifford

University of Nice, France

La geometrie sous-riemannienne constitue une generalisation tres naturellede la geometrie riemannienne dans laquelle on cherche a connecter des pointsde maniere optimale via des courbes dites horizontales. Apres avoir introduit lecontexte, nous presenterons les quelques problemes fondamentaux de geometriesous-riemannienne. Etant en lien avec de nombreux domaines tres en vogueactuellement, ces problemes sont l’objet de tres nombreuses recherches.

11

Equations fractionnaires de Schrodinger et de DelsarteArnaud Rougirel

University of Poitiers, France

Apres une breve introduction aux derivees fractionnaires, je proposerai une clas-sification des EDP lineaires avec derivee temporelle fractionnaire d’ordre inferieura 1, suivant leur caractere parabolique ou hyperbolique. Cette classification per-mettra de donner deux equations de Schrodinger fractionnaires, qui sont dis-tinctes de celle habituellement etudiee. La derniere partie de cet expose sera con-sacree aux equations de Delsarte, et en particulier a leur version fractionnaire quigeneralise l’equation de transport.

Modelisation mathematique de la competition des especesmicrobienne dans le chemostat

Tewfik SariIRSTEA, Montpellier, France

Le chemostat joue un role important en tant que modele mathematique enbiologie. Dans sa forme la plus simple, c’est un modele ou les populations micro-biennes sont en competition pour le nutriment disponible. Il est utilise commepoint de depart pour les modeles de traitement des eaux usees. Ce modele preditqu’a l’equilibre, au plus une population evite l’extinction. C’est le theoreme del’exclusion competitive. Cependant, la coexistence de populations en competitionest frequemment rencontree dans la nature. Pour expliquer cela, plusieurs meca-nismes de coexistence ont ete envisages dans la litterature : la densite dependancedes fonctions de croissance, la floculation des especes, les entrees periodiques,non constantes, les modeles avec plusieurs substrats, ... Le but de l’expose estde donner les principaux resultats de cette theorie obtenus dans les travaux derecherches et les theses realises dans le cadre du reseau euro-mediterraneen TREA-SURE (http://www.inra.fr/treasure).

Key words: Almost periodic and almost automorphic solutions, integral equations,neutral equation, state dependent delay, Nicholson’s model, Lossless transmission lines.

2010 Mathematics Subject Classification: 47H10,47H30, 54H25.ReferencesOuvrages :

1. J. Harmand, C. Lobry, A. Rapaport, T. Sari. The Chemostat: Mathematical Theory ofMicroorganism Cultures, 2017, Wiley-ISTE The Chemostat Volume 1

2. J. Harmand, C. Lobry, A. Rapaport, T. Sari. Le Chemostat: Theorie mathematiquede la culture continue de micro-organismses, 2016, ISTE. Le Chemostat Volume 1

3. H.L. Smith , P. Waltman , The Theory of the Chemostat: Dynamics of MicrobialCompetition, Cambridge University Press, 1995.

Articles :B. Bar, T. Sari. The operating diagram for a model of competition in a chemostat with

an external lethal inhibitor (2019). Discrete and Continuous Dynamical System - Series B,https://doi.org/10.3934/dcdsb.2019203

12

R. Fekih-Salem, T. Sari. Properties of the chemostat model with aggregated biomassand distinct removal rates (2019). SIAM Journal on Applied Dynamical Systems, 18,481–509. https://doi.org/10.1137/18M1171801

Y. Daoud, N. Abdellatif, T. Sari, J. Harmand. Steady state analysis of a syntrophicmodel: The effect of a new input substrate concentration (2018). Math. Model. Nat.Phenom, 13, 31. https://doi.org/10.1051/mmnp/2018037

M. Dellal, M. Lakrib, T. Sari. The operating diagram of a model of two competitors ina chemostat with an external inhibitor (2018). Math. Biosci., 302, 27-45.https://doi.org/10.1016/j.mbs.2018.05.004

Z. Khedim, B. Benyahia, B. Cherki, T. Sari, J. Harmand. Effect of control parameterson biogas production during the anaerobic digestion of protein-rich substrates (2018).Applied Mathematical Modeling. 61, 351-376. https://doi.org/10.1016/j.apm.2018.04.020

T. Sari, M.J. Wade (2017). Generalised approach to modelling a three-tiered microbialfood-web, Mathematical Biosciences. 291, 21–37.https://doi.org/10.1016/j.mbs.2017.07.005

R. Fekih-Salem, C. Lobry, T. Sari (2017). A density-dependent model of competitionfor one resource in the chemostat. Mathematical Biosciences. 286, 104–122.https://doi.org/10.1016/j.mbs.2017.02.007

R. Fekih-Salem, A. Rapaport, T. Sari. Emergence of coexistence and limit cycles inthe chemostat model with flocculation for a general class of functional responses (2016).Appl. Math. Model. 40, 7656–7677. https://doi.org/10.1016/j.apm.2016.03.028

T. Sari, J. Harmand (2016). A model of a syntrophic relationship between two micro-bial species in a chemostat including maintenance. Mathematical Biosciences. 275, 1–9.https://doi.org/10.1016/j.mbs.2016.02.008

N. Abdellatif, R. Fekih-Salem, T. Sari (2016). Competition for a single resource andcoexistence of several species in the chemostat. Math. Biosci. Eng. 13, 631–652.https://doi.org/10.3934/mbe.2016012

M.J. Wade, J. Harmand, B. Benyahia, T. Bouchez, S. Chaillou, B. Cloez, J.-J. Godon,B. Moussa Boudjemaa, A. Rapaport, T. Sari, R. Arditi, C. Lobry (2016). Perspectives inmathematical modelling for microbial ecology. Ecological Modelling. Vol 321, 64-74.https://doi.org/10.1016/j.ecolmodel.2015.11.002

B. Benyahia, T. Sari, B. Cherki, J. Harmand. Anaerobic membrane bioreactor mod-eling in the presence of Soluble Microbial Products (SMP) - the Anaerobic Model AM2b(2013). Chemical Engineering Journal, 228, 1011-1022. https://doi.org/10.3182/20110828-6-IT-1002.01195

T. Sari (2013). Competitive Exclusion for Chemostat Equations with Variable Yields,Acta Applicandae Mathematicae, 123, 201-219. https://doi.org/10.1007/s10440-012-9761-8

R. Fekih-Salem, J. Harmand, C. Lobry, A. Rapaport, T. Sari (2013). Extension of thechemostat model with flocculation, Journal of Mathematical Analysis and Applications,397, 292-306. https://doi.org/10.1016/j.jmaa.2012.07.055

T. Sari, M. El Hajji, J. Harmand (2012). The mathematical analysis of a syntrophicrelationship between two microbial species in a chemostat, Mathematical Biosciences andEngineering, 9, 627-645. https://doi.org/10.3934/mbe.2012.9.627

B. Benyahia, T. Sari, B. Cherki, J. Harmand (2012). Bifurcation and stability analysisof a two step model for monitoring anaerobic digestion processes. Journal of ProcessControl, 22, 1008-1019. https://doi.org/10.1016/j.jprocont.2012.04.012

13

On the stochastic ∆p problemGuy Vallet


We will be interested in that talk by the following non linear Stochastic PartialDifferential Equation (SPDE):

du− ∆p(.)udt = h(., u)dW in Ω× (0, T)× Dwithu = 0 on Ω× (0, T)× ∂D, u(0, .) = u0 in L2(D),where (Ω, F, P) is a probability space and D ⊂ Rd is a bounded domain.

This SPDE is with a multiplicative noise and is related to a monotone operator of∆p−type. We will go back on classical methods of well-posedness for such prob-lems and adapt them to the situation of p−laplace operator when p is a functionof t ∈ (0, T), x ∈ D and ω ∈ Ω.

Keywords: SPDE, monotone, variable exponent.

14

Communications

Global Behavior of the Solutions to a Class of Nonlinear,Singular Second Order ODE

Mama Abdelli,Alain HarauxMustapha Stambouli University of Mascara, Univ Paris 06

The initial value problem and global properties of solutions are studied for thescalar second order ODE: (|u′|lu′)′ + c|u′|αu′ + d|u|βu = 0, where α, β, l, c, d arepositive constants. In particular, existence, uniqueness and regularity as well asoptimal decay rates of solutions to 0 are obtained depending on the various pa-rameters, and the oscillatory or non-oscillatory behavior is elucidated .Key words: Second order scalar ODE, existence of solution, oscillatory solutions, Decayrate.

2010 Mathematics Subject Classification: 34A34, 34C10, 34D05, 34E99 .

IntroductionWe consider the scalar second order ODE(

|u′|lu′)′

+ c|u′|αu′ + d|u|βu = 0, (8)

where α, β, c, d are positive constants and l ≥ 0.In the special case l = 0 and d = 1 we find the simpler equation

u′′ + c|u′|αu′ + |u|βu = 0. (9)

The solutions of (53) are global for t ≥ 0 and both u and u′ decay to 0 as t → ∞. Thisequation was studied in [3] by the second author who used some modified energy func-tion to estimate the rate of decay. In addition, he showed that if α > β

β+2 all non-trivial

solutions are oscillatory and if α < ββ+2 they are non-oscillatory.

We use some techniques from [3] to estimate the energy decay of the solutions to (359)and to show that all non-trivial solutions of (359) are oscillatory for α > β(l+1)+l

β+2 and

α > l, non-oscillatory for α < β(l+1)+lβ+2 and β > l. One major difference with [2] is that

here we have to establish well-posedness in a regularity class compatible with the pres-

ence of the singular leading term(|u′|lu′

)′in the equation. This singularity prevents u

to have a second derivative at all points where u′ vanishes for any non-trivial solution.We show that such points are isolated, which allow us to generalize the methods of [2],sometimes by using the density of regular points after proving identities or inequalitiesoutside the singularities.

15

The initial value problem for Equation (359)Remark 1 If u 6≡ 0, at any point t0 where u(t0) 6= 0 and u′(t0) = 0, the second derivativeu′′(t0) does not exist. At least when α > l − 1. Indeed for t 6= t0 and |t− t0| < ε we have

u′′(t) = − cl + 1

|u′|α−lu′(t)− d|u|βu(l + 1)|u′(t)|l ,

hence, as t→ t0, u′′(t) has a constant sign and

|u′′(t)| → +∞ as t→ t0.

Theorem 2 Let (u0, u1) ∈ R2. The problem (359) has a global solution satisfying

u ∈ C1(R+), |u′|lu′ ∈ C1(R+) and u(0) = u0, u′(0) = u1.

Corollary 1 Let u ∈ C1(J) be any solution of (359) with |u′|lu′ ∈ C1(J), u 6≡ 0. Then for eachcompact interval K ⊂ J the set F = t ∈ K, u(t) = 0 is finite.

Corollary 2 Let u ∈ C1(J) be any solution of (359) with |u′|lu′ ∈ C1(J), u 6≡ 0. Then for eachcompact interval K ⊂ J the set G = t ∈ K, u′(t) = 0 is finite.

Energy estimates for equation (359)We define the energy associated to the solution of the problem by the following formula

E(t) =l + 1l + 2

|u′|l+2 +d

β + 2|u|β+2. (10)

By multiplying equation (359) by u′, we obtain that on any interval where u is C2, E(t) isC1 with

ddt

E(t) = −c|u′|α+2 ≤ 0. (11)

In particular (337) holds, whenever u′(t) 6= 0.Now let t0 be such that u′(t0) = 0. As a consequence of Corollary 2 there exists ε > 0such that u ∈ C2((t0, t0 + ε] ∪ [t0 − ε, t0)). Integrating (337) over (τ, t),

E(t)− E(τ) = −c∫ t

τ|u′(s)|α+2 ds, t0 < τ ≤ t ≤ t0 + ε,

By letting τ → t0, we obtain

E(t)− E(t0) = −c∫ t

t0

|u′(s)|α+2 ds,

In particular we obtain that E is right-differentiable at t0 with right-derivative equal to−c|u′|α+2. A similar calculation on the left allows to conclude that E is differentiable at t0and finally (337) is true at any point.

Theorem 3 Assuming α > l, there exists a positive constant η such that if u is any solution of(359) with E(0) 6= 0

lim inft→+∞

tl+2α−l E(t) ≥ η. (12)

Moreover,

16

(i) If α ≥ β(1+l)+lβ+2 , then there is a constant C(E(0)) depending on E(0) such that

∀t ≥ 1, E(t) ≤ C(E(0))t−l+2α−l ,

(ii) If α < β(1+l)+lβ+2 , then there is a constant C(E(0)) depending on E(0) such that

∀t ≥ 1, E(t) ≤ C(E(0))t−(α+1)(β+2)

β−α .

Oscillatory behavior of solutionsTheorem 4 Assume that

α >β(l + 1) + l

β + 2or

α =β(l + 1) + l

β + 2and c < c0 = (β + 2)

( (β + 2)(l + 1)d(β + 1)(l + 2)

) β+1β+2

.

Then, any solution u(t) of (359) which is not identically 0 changes sign on each interval (T, ∞)and the same thing is true for u′(t) .

Theorem 5 Assume

α <β(l + 1) + l

β + 2

Then any solution u(t) of (359) which is not identically 0 has a finite number of zeros on (0, ∞).Moreover, for t large, u′(t) has the opposite sign to that of u(t) and u′′(t) has the same sign asu(t).

Theorem 6 Assume that

α =β(l + 1) + l

β + 2; c ≥ c0 = (β + 2)

( (β + 2)(l + 1)d(β + 1)(l + 2)

) β+1β+2

.

Then any solution u(t) of (359) which is not identically 0 has at most one zero on (0, ∞).

References1. M. Abdelli and A. Benaissa, Energy decay of solutions of a degenerate Kirchhoff equation

with a weak nonlinear dissipation, Nonlinear Analysis 69 (2008), 1999-2008.2.A. Haraux, Asymptotics for some nonlinear O.D.E of the second order, Nonlinear Anal-

ysis 10 (1986), no 12, 1347-1355. 95 (2005), 297-321.3.A. Haraux, Sharp decay estimates of the solutions to a class of nonlinear second order

ODE’s, Analysis and Applications, 9 (2011), 49-69.4.A. Haraux, Slow and fast decay of solutions to some second order evolution equations, J.

Anal. Math. 95 (2005), 297-321.

17

Some properties of generalized Newton transformationsMohammed Abdelmalek

ESM Tlemcen

Dans ce travail on donne une condition de la transversalite de deux sous varietesriemanniennes en codiemsnion arbitraire. Cette condition est donnee par l’ellipticitedes transformations de Newton generalisees.

Key words: Les transformations de Newton generalisees, ellipticite, transversalite, fonc-tions symetriques

2010 Mathematics Subject Classification: .53A07

Introduction

1993 Rosenberg [11] a montrer le theoreme :Soient Σn−1 une sous variete strictement convexe dans un hyperplan Π de

Rn+1, et Mn une hypersurface compacte plongee dans Rn+1 a courbure moyenned’ordre superieure Hr constante non nulle et bordee par Σn−1. Si Mn est trans-verse a Π le long du bord Σn−1, alors Mn est inclue dans l’un des demi espaces deRn+1 determine par Π, et Mn herite toutes les symetries de Σn−1. En particuliersi Σn−1 est une sphere, alors Mn est une calotte spherique.

Comme consequence, il a obtenu que :Les seules hypersurfaces compactes, plongees dans l’espace Euclidien Rn+1, a

courbure moyenne d’ordre superieur Hr (2 ≤ r ≤ n) constante, et de bord spheriquesont les disques du plan (avec Hr = 0), et les calottes spheriques (avec Hr 6= 0).

En 2006, Alias, Lopez et Malacarne [3] ont etendus le resultat precedent au casdes hypersurfaces d’une variete riemannienne, ils ont obtenus le :

Soient Σn−1 une sous-variete compacte strictement convexe de dimension (n− 1)de l’hyperplan Π ⊂ Rn+1 et ψ : Mn → Rn+1 une hypersurface compacte etplongee de bord Σn−1.

Supposons que pour tout 2 ≤ r ≤ n, la courbure moyenne d’ordre r de Mn estconstante non nulle. Alors Mn herite toutes les symmetries de Σn−1. En partic-ulier si le bord Σn−1 est une sphere de dimension (n− 1), alors Mn est une calottespherique.

Une question qui ce pose est :Est-il possible d’etendre les resultats precedent au cas de la codimension superieure

?Dans notre travail nous allons resoudre partiellement ce probleme en mon-

trant que Mn et Pn sont transverses si les transformations de Newton generaliseessont definies positives.

Main results

Hypotheses :

18

Ψ : Mn −→ Mn+q une sous variete connexe, compacte et orientee du bordΣn−1 = Ψ (∂Mn).

Pn une sous variete totalement embilique de Mn+q de codimension q .(e1, ..., en−1) une base orthonormale locale dans Σn−1. ν le champ de vecteurs

sortant unite normal de l’inclusion ∂Mn ⊂ Mn et η le champ de vecteurs normalunite de l’inclusion Σn−1 ⊂ Pn.(

ξ1, ..., ξq)

une base orthonormale orthogonale a Pn, et(

N1, ..., Nq)

une baseorthonormale orthgonale a Mn.

On considere les operateurs de la seconde forme fondamentale :

AΣ, Aξ1 , ..., Aξp , AN1 , ..., ANq ,

correspondent aux inclusions Σn−1 ⊂ Pn, Pn ⊂ Mn+q et Mn ⊂ Mn+q.Les GNT

Tu = Tu

(AN1 , ..., ANq

), Tv = Tv

(AΣ, Aξ1 |Σ, ..., Aξq |Σ

)et

Tu = Tu

(AN1 |Σ, ..., ANq |Σ

)Proposition : Soit A = (A1|Σ, ..., Aq|Σ), avec Aα|Σ = ρα AΣ + µα I . Posons σu =

σu(A). Alors pour tout multi-indice u ∈Nq, nous avons :

σu =1

(n− 1− |u|)! ∑l≤u

(|l|l

)ρlµu−l

(n− 1− |l|

u− l

)σ|l| (AΣ) (13)

Avec (|l|l

)=|l|!

l1!...lq!

Proposition : Soit A = (A1|Σ, ..., Aq|Σ) ,ou Aα|Σ = ρα AΣ + µα I, et A =

(AΣ, µ1 I, ..., µq I). Pour un multi indice u ∈Nq, posons σu = σu(A) et σu = σu(

A).Alors

:σu = ∑

l≤u

(|l|l

)ρlσ(|l|,u−l). (14)

Proposition : Soit A = (A1, ..., Aq) et A = (A1|Σ, ..., Aq|Σ). Alors

σu = σu + ∑α

Cασα[(u) + ∑α,β

∑α]β](0)≤w≤u

(−1)|w|−|v|+1

×(|u| − |w|

u− w

)B>α Au−wBβσα[β[(w)

ou B>α = (〈Aαν, e1〉, ..., 〈Aαν, en−1〉) et Cα = 〈Aαν, ν〉.On utilisant les resultat precedents, nous obtenons :Proposition : Soient Mn+q une variete riemanienne de dimension (n + q),

et Pn ⊂ Mn+q une sous variete totalement embilique de Mn+q de dimension n.Notons par Σn−1 ⊂ Pn une hypersurface compacte de Pn de dimension (n− 1).

19

Soit Ψ : Mn −→ Mn+q une sous variete connexe, compacte et orientee de Mn+q

de bord Σn−1 = Ψ (∂M) . Alors le long du bord ∂M, nous avons

〈Tuν, ν〉 = σu(A1|Σ, ..., Aq|Σ). (15)

Si l’inclusion Pn ⊂ Mn+q est totallement geodesique, alors :

〈Tuν, ν〉 = ρuσ|u| (AΣ)

Et nous avons :Corollaire : Sous les hypothese du corollaire (2) les sous varietes Mn et Pn

sont transverses le long de ∂M a condition que pour certain multi-indice u delongueur 1 ≤ |u| ≤ n− 1, la tranformation de Newton generalisee Tu soit definiepositive sur Mn .

Conclusion

Le resultat precedent nous donne une reponse partiel au probleme. La question quise pose est comment peut-ont generaliser la deuxieme partie du premier theoreme deH.Rosenberg pour puis repondre au conjecture de la calotte spherique en codimensionsuperieure.

References(ABN) M. Abdelmalek, M. Benalili, K. Niedziałomski, Geometric Configu-

ration of Riemannian Submanifolds of arbitrary Codimension, J. Geom. (2017),doi:10.1007/s00022-017-0374-2.

(ADM) L.J. Alias, J. H. S. de Lira, J. M. Malacarne, Constant higher-order meancurvature hypersurfaces in Riemannian spaces, J. Inst. Math. Jussieu 5(4). (2006),527–562.

(AM) L.J. Alıas, J. Melendez, Hypersurfaces with constant higher order meancurvature in Euclidean space, Geom. Dedic. 182 (2016), 117–131.

(AW) K. Andrzejewski, P. Walczak, The Newton transformations and new in-tegral formulae for foliated manifolds, Ann. Glob. Anal.Geom. 37 (2010), 103–111.

(CL) L. Cao, H. Li, r-minimal submanifolds in space forms, Ann Glob AnalGeom. 32 (2007), 311 341..

(R) H. Rosenberg, Hypersurfaces of constant curvature in space forms, Bull.Sc. Math. 117 (1993), 211–239

(RW) V. Rovenski, P. G. Walczak, Integral formulae on foliated symmetricspaces, Math. Ann. 352 (2012), 223–237.

20

Stochastic Approximation procedure under α-mixing dataNabila AIANE, Abdelnasser DAHMANI

University A.MIRA Bejaia, Center of Tamenghest

In this work we consider the linear ill posed problem described by an oper-ator equation in Hilbert space where Ax = u, the second member is measuredwith α-mixing errors. To solve this problem we propose a stochastic procedureof Robbins-Monro type which converges almost completely to the exact solution.To check the validity of our results, we consider some numerical examples.

Key words: stochastic approximation; ill-posed problems; α-mixing.

2010 Mathematics Subject Classification: 45B05, 62L20, 65N20.

IntroductionWe consider in this work a linear ill-posed problem described by an operator equationwhere the second member is measured with α-mixing errors. We estimate the solutionusing Robbins-Monro algorithm [robbins-1951] and show the almost complete conver-gence (a.co) of this algorithm specifying the rate of convergence. The Robbins-Monroalgorithm is a recursive procedure which is originally an algorithm for the estimationof zero of function f whose evaluation is contaminated with noise [blum-1954,dippon-2006]. Its form is:

Xn+1 = Xn − an( f (Xn)− ξn) (16)

with arbitrary X1 and (an)n∈N∗ is a sequence of positive numbers that converge towardszero when n→ ∞, (ξn)n∈N∗ is a sequence of random variable and f (Xn)− ξn is the noisyobservation of f at Xn.

Robbins & Monro [robbins-1951] showed that if the sequence (an)n∈N∗ satisfies thefollowing conditions,

an > 0,+∞

∑n=1

an = ∞ and+∞

∑n=1

a2n < ∞ (17)

so the algorithm converges towards an exact solution in probability when n→ ∞.

Statement of problemLet (Ω,F , P) be a probability space, H a separable Hilbert space and A an injective linearoperator which is bounded from H to H. Without lost of generality, we assume that‖A‖ ≤ 1.

Let us consider the following operator equation:

Ax = u. (18)

In practice, the second member of Eq (327) is the result of measurements, and it usuallyknown just approximately [kaipio-2004,tarantola-2005].

21

When carrying out n independent experiments, we obtain a sample u1, u2, ..., unwhich has the following property

ui = uex + ξi

where uex represents the unknown exact value of the second member of Eq (327) (uex =AXex) and (ξn)n∈N∗ a sequence of identically distributed and algebraically α-mixing withrate b > 1, defined on the probabilistic space (Ω,F , P) and taking values into the Hilbertspace H.

In addition, given that AH is not closed. Consequently, the problem (327) is an ill-posed problem [ramm-2005,tikhonov-1977]

The main objective here is to estimate the solution of (327) by using the stochasticapproximation procedure [robbins-1951] defined in our work by:

Xn+1 = Xn − an [AXn − uex − ξn] (19)

for which we show the convergence almost completely (a.co) to element Xex.

Main resultsLemma 1 Assume (19) and for 1 ≤ i, j ≤ n, then

Xn+1 − Xex =n

∏i=1

(I − ai A) (X1 − Xex) +n

∑i=1

n

∏j=i+1

n

∏j=i+1

(I − aj A

)aiξi (20)

with X1 arbitrary element andn

∏j=n+1

n

∏j=n+1

(I − aj A

)= I

Lemma 2 Let i ∈N∗ and under the condition

inf∥∥∥∥exp

(u(

I2− A

))∥∥∥∥ ; u ∈ R+

< 1 (21)

we have

limn→+∞

∥∥∥∥∥ n

∏i=1

(I − ai A) (X1 − Xex)

∥∥∥∥∥ = 0 for ai =ai

where a is a positive constant (22)

Theorem 7 Assume that (ξn)n are identically distributed and are algebraically α-mixing withrate b > 1. Assume also (21) and if∃ p > 2, ∃ θ > 2 and M > 0 such that

∀t > M, P(‖ξ1‖ > t) ≤ t−p and s− (b+1)p

b+pn = o

(n−θ

)where

s2n =

n

∑i=1

∑n

∑∑j=1

∣∣Cov(‖ξi‖ ,∥∥ξ j∥∥)∣∣

then we have

Xn+1 − Xex = O(√

n−2s2n ln n

)a.co. (23)

22

ConclusionIn this work, we have estimated the solution of linear stochastic ill-posed problem by us-ing Robbins-Monro algorithm and have showing for it the almost complete convergenceto the exact solution.References

Blum J.R., Approximation Methods Which Converge with Probability one, J. Ann.Math.Statist., 1954, 25, 382–386.

Dippon J.,Walk M., The Averaged Robbins-Monro Method for Linear Problems inBanach Space, Journal of Theoretical Probability., 2006, 19(1), 166–189.

Kaipio J., Somersalo E.,Computational and statistical methods for inverse problems,J. NewYork: Springer., 2004.

Ramm A.G., Inverse problems Mathematical and Analytical Techniques with Appli-cations to Engineering, J. Springer., 2005, 218, 856–859.

Robbins H., Monro S., Stochastic Approximation Method, J. Discrete Math., 1951,22(12), 400–407.

Tarantola A., Inverse problem Theory and Methods for Model Parameter Estimation,J. SIAM., 2005, 59, 3769–3776.

Tikhonov A.N., Arsenin V.Y, Solution for Ill-Posed Problems, J. New York: Wiley., 1977,61, 3893–3901.

On the p-harmonic and p-biharmonic mapsAhmed Mohammed Cherif

University of Mascara

In this paper, we study the existence of p-harmonic maps into Riemannianmanifolds admitting a conformal vector field. We also prove a Liouville typetheorem for p-biharmonic maps. Key words: p-Harmonic maps, p-Biharmonic maps,

Conformal vector fields.2010 Mathematics Subject Classification: 53C43, 58E20, 53A30.

IntroductionWe give some definitions. Let (M, g) be a Riemannian manifold. By RM we denote theRiemannian curvature tensor of (M, g). Then RM is defined by:

RM(X, Y)Z = ∇MX∇M

Y Z−∇MY ∇M

X Z−∇M[X,Y]Z, (24)

where ∇M is the Levi-Civita connection with respect to g, and X, Y, Z ∈ Γ(TM). Thedivergence of (0, p)-tensor α on M is defined by:

(divM α)(X1, ..., Xp−1) = (∇Mei

α)(ei, X1, ..., Xp−1), (25)

where X1, ..., Xp−1 ∈ Γ(TM), and ei is an orthonormal frame. Given a smooth functionλ on M, the gradient of λ is defined by:

g(gradM λ, X) = X(λ), (26)

23

where X ∈ Γ(TM) (for more details, see for example [?]).A vector field ξ on a Riemannian manifold (M, g) is called a conformal if Lξ g = 2 f g,for some smooth function f on M, where Lξ g is the Lie derivative of the metric g withrespect to ξ, that is:

g(∇MX ξ, Y) + g(∇M

Y ξ, X) = 2 f g(X, Y), X, Y ∈ Γ(TM). (27)

The function f is then called the potential function of the conformal vector field ξ. If ξ isconformal with constant potential function f , then it is called homothetic, while f = 0 itis Killing (see [?], [?], [?]).If ϕ : (M, g)→ (N, h) is a smooth map between two Riemannian manifolds, its p-energyis defined by

Ep(ϕ; D) =1p

∫D|dϕ|pvg (p ≥ 2). (28)

where D is a compact subset of M. The p-energy functional (227) includes as a specialcase (p = 2) the energy functional, whose critical points are the usual harmonic maps (see[?]). We say that ϕ is a p-harmonic map if it is a critical point of the p-energy functional,that is to say, if it satisfies the Euler-Lagrange equation of the functional (227), that is,

τp(ϕ) ≡ divM(|dϕ|p−2dϕ) = 0. (29)

In particular, we note that every harmonic map with constant energy density is p-harmonicfor all p ≥ 2 (for more details on the concept of p-harmonic maps see [?, ?, ?]). Let τ(ϕ)the tension field of ϕ given by:

τ(ϕ) = traceg∇dϕ = ∇ϕei dϕ(ei)− dϕ(∇M

eiei), (30)

where∇M is the Levi-Civita connection of (M, g),∇ϕ denote the pull-back connection onϕ−1TN and ei is an orthonormal frame on (M, g) (see [?], [?], [?]). Then ϕ is p-harmonicif and only if (see [?]):

|dϕ|p−2τ(ϕ) + (p− 2)|dϕ|p−3dϕ(gradM |dϕ|) = 0. (31)

Main results

p-Harmonic maps and conformal vector fieldsProposition 1 Let (M, g) be a compact orientable Riemannian manifold without boundary, and(N, h) a Riemannian manifold admitting a conformal vector field ξ with potential function f > 0at any point. Then, any p-harmonic map ϕ from (M, g) to (N, h) is constant.

Since the identity map of a compact Riemannian manifold is always a p-harmonicmap (p ≥ 2), from Proposition 10 we get the following result:

Corollary 3 Let (M, g) be a compact orientable Riemannian manifold without boundary, and ξ aconformal vector field with potential function f on M. Then, the set of annulation of the functionf is never empty on M.

From Proposition 10, we deduce:

24

Corollary 4 [?] Let (M, g) be a compact orientable Riemannian manifold without boundary, and(N, h) a Riemannian manifold admitting a homothetic vector field ξ with homothetic constantk 6= 0. Then, any harmonic map ϕ from (M, g) to (N, h) is constant.

In the case of non-compact Riemannian manifold, we obtain the following result:

Theorem 8 Let (M, g) be a complete non-compact Riemannian manifold, and (N, h) a Rieman-nian manifold admitting a conformal vector field ξ with potential function f > 0 at any point. Ifϕ : (M, g) −→ (N, h) is p-harmonic map, satisfying:∫

M

|dϕ|p−2|ξ ϕ|2f ϕ

vg < ∞, (32)

then ϕ is constant.

From Theorem 46, we deduce:

Theorem 9 Let (M, g) be a complete non-compact Riemannian manifold, and (N, h) a Rie-mannian manifold admitting a homothetic vector field ξ with homothetic constant k 6= 0. Ifϕ : (M, g) −→ (N, h) is p-harmonic map, satisfying:∫

M|dϕ|p−2|ξ ϕ|2vg < ∞, (33)


Remark 2 Proposition 10 and Theorem 46 remains true if the potential function f < 0 on N(consider the conformal vector field ξ = −ξ).

Liouville type theorems for p-biharmonic mapsConsider a smooth map ϕ : (M, g) −→ (N, h) between Riemannian manifolds and letp ≥ 2. A natural generalization of p-harmonic maps is given by integrating the square ofthe norm of τp(ϕ). More precisely, the p-bienergy functional of ϕ is defined by

E2,p(ϕ; D) =12

∫D|τp(ϕ)|2vg. (34)

A map is called p-biharmonic, if it is a critical point of the p-bienergy functional (147)over any compact subset D of M.

Remark 3 Note that in [?] and [?], the terminology ”p-biharmonic maps” was used as the criticalpoints of the functional

∫M |τ(ϕ)|pvg, so there is no link between the concept introduced in this

paper, and the notion of p-biharmonicity that was used, introduced and studied by X. Cao, Y.Han, and Y. Luo.

Theorem 10 Let ϕ : (M, g) → (N, h) be a smooth map between Riemannian manifolds, D acompact subset of M and let ϕtt∈(−ε,ε) be a smooth variation with compact support in D. Then

ddt

E2,p(ϕt; D)∣∣∣t=0

= −∫

Dh(τ2,p(ϕ), v) vg, (35)

where τ2,p(ϕ) ∈ Γ(ϕ−1TN) given by

τ2,p(ϕ) = −|dϕ|p−2 traceg RN(τp(ϕ), dϕ)dϕ− traceg∇ϕ|dϕ|p−2∇ϕτp(ϕ)

−(p− 2) traceg∇ < ∇ϕτp(ϕ), dϕ > |dϕ|p−4dϕ. (36)

25

Remark 4 Let ei be an orthonormal frame on (M, g), then:

traceg RN(τp(ϕ), dϕ)dϕ = RN(τp(ϕ), dϕ(ei))dϕ(ei),

traceg∇ϕ|dϕ|p−2∇ϕτp(ϕ) = ∇ϕei |dϕ|p−2∇ϕ

ei τp(ϕ)− |dϕ|p−2∇ϕ

∇Mei ei

τp(ϕ),

traceg∇ < ∇ϕτp(ϕ), dϕ > |dϕ|p−4dϕ = ∇ϕei < ∇

ϕτp(ϕ), dϕ > |dϕ|p−4dϕ(ei)

− < ∇ϕτp(ϕ), dϕ > |dϕ|p−4dϕ(∇Mei

ei).


Theorem 11 Let ϕ : (M, g) → (N, h) be a smooth map between Riemannian manifolds. Then,ϕ is p-biharmonic if and only if:


−(p− 2) traceg∇ < ∇ϕτp(ϕ), dϕ > |dϕ|p−4dϕ = 0.

Example 1 Let M the manifold R2\(0, 0) ×R equipped with the Riemannian metric g =dx2

1 + dx22 + dx2

3, and let N the manifold R2 equipped with the Riemannian metric h = dy21 + dy2

2.

The map ϕ : (M, g) −→ (N, h), (x1, x2, x3) 7−→ (√

x21 + x2

2, x3), is p-biharmonic if and only ifp = 4.

Remark 5 The previous examples prove the following results. 1) If the energy density of a smoothmap ϕ : (M, g) −→ (N, h) is constant, then there is no equivalence between the biharmonicityof ϕ and the p-biharmonicity of ϕ, for p 6= 2. 2) There are p-biharmonic maps that are neitherp-harmonic nor harmonic, with p 6= 2.

Theorem 12 Let (M, g) be a compact orientable Riemannian manifold without boundary, and(N, h) a Riemannian manifold with non-positive sectional curvature. Then, every p-biharmonicmap from (M, g) to (N, h) is p-harmonic.

In the case of non-compact Riemannian manifold, we have the following result:

Theorem 13 Let (M, g) be a complete Riemannian manifold, (N, h) be a Riemannian manifoldwith non-positive sectional curvature and p ≥ 2. Then, every p-biharmonic map ϕ : (M, g) →(N, h) satisfying: ∫

M|dϕ|p−2|τp(ϕ)|2vg < ∞,

∫M|dϕ|p−2vg = ∞,

is p-harmonic map.

If p = 2, we arrive at the following corollary:

Corollary 5 [?] Let (M, g) be a complete Riemannian manifold with infinite volume i.e.∫

M vg =∞, (N, h) a Riemannian manifold with non-positive sectional curvature. Then, every biharmonicmap ϕ : (M, g)→ (N, h) with finite bienergy, i.e.

∫M |τ(ϕ)|2vg < ∞, is harmonic map.

References P. Baird, J. C. Wood, Harmonic morphisms between Riemannain manifolds,Clarendon Press Oxford, 2003.

P. Baird, S. Gudmundsson, p-Harmonic maps and minimal submanifolds, Math. Ann.294 (1992), 611-624.

26

B. Bojarski, and T. Iwaniec, p-Harmonic equation and quasiregular mappings, Partialdifferential equations (Warsaw, 1984), 25-38, Banach Center Publ., vol. 19. PWN, Warsaw,1987.

X. Cao and Y. Luo, On p-biharmonic submanifolds in nonpositively curved mani-folds, Kodai Math. J. 39 (2016), 567-578.

J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J.Math. 86 (1964), 109-160.

A. Fardoun, On equivariant p-harmonic maps, Ann.Inst. Henri. Poincare, 15 (1998),25-72.

L. Greco, and A. Verde, A regularity property of p-harmonic functions. Ann. Acad.Sci. Fenn. Math., 25, (2000), 317-23.

Y. Han, Y. Luo, Several results concerning nonexistence of proper p-biharmonic mapsand Liouville type theorems, arXiv: 1801.05181v1.

G. Y. Jiang, 2-Harmonic maps between Riemannian manifolds, Annals of Math.,China, 7A(4)(1986), 389-402.

W. Kuhnel, H. Rademacher, Conformal vector fields on pseudo-Riemannian spaces,Differential Geometry and its Applications 7 (1997), 237-250.

A. Mohammed Cherif, Some results on harmonic and bi-harmonic maps, Interna-tional Journal of Geometric Methods in Modern Physics, 14 (2017).

A. Mohammed Cherif, M. Djaa, Geometry of Energy and Bienergy Variations be-tween Riemannian Manifolds, Kyungpook Math. J. 55 (2015), 715-730.

N. Nakauchi, H. Urakawa, S. Gudmundsson, Biharmonic maps into a Riemannianmanifold of non-positive curvature. Geom. Dedicata 169 (2014), 263-272.

O’Neil, Semi- Riemannian Geometry, Academic Press, New York, 1983.Y. Xin, Geometry of harmonic maps, Fudan University, 1996.K. Yano and T. Nagano, The de Rham decomposition, isometries and affine transfor-

mations in Riemannian space, Japan. J. Math., 29 (1959), 173-184.S. T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure

Appl. Math. 28 (1975), 201-228.

A recursive kernel estimates of the functional modal regressionunder ergodic dependence condition

Fatima Zohra Ardjoun, Larbi Ait Hennani, Ali Laksaciy

Djillali Liabes University,University of technology france, Djillali Liabes Univer-sity

In this paper, we consider an alternative estimator of the conditional modewhen the explanatory variable takes values in a semi-metric space. This alter-native estimate is based in a recursive kernel method. Under the ergodicity hy-pothesis, we quantify the asymptotic properties of this estimate, by giving thealmost complete convergence rate. The asymptotic normality of this estimate isalso given.Key words: Functional data, Conditional mode Conditional distribution, recursive esti-

mation, ergodic condition.

2010 Mathematics Subject Classification: 60G42, 62F12, 62G20, 62G05.

27

Introduction

Usually, the time series data is modeled by using the mixing conditions, our main aim inthis paper is to analyze the functional time series data under an alternative dependency.More precisely, we consider the problem of the nonparametric functional analysis underthe ergodicity assumption. This dependence structure covers several case does not sat-isfy the usual mixing structures. On the other hand, the ergodicity condition is one of aprincipal postulate of statistical physics. The litterateur in egrodic functional data is stilllimited. This problem has been initiated by Laib and Louani [LS2011]. They consider theproblem of functional estimation for nonparametric regression operators under the er-godicity condition. Among the lot of papers concerning the nonparametric recursive es-timation or modelization of variable taking values in infinite dimensional spaces, we onlyrefer to the papers by Roussas et al. [RT1989], Thiam [TB2006], Amiri et al. [BFV2014],Ling and Xu [LX2012] Ferraty et al. [FL2010], Ramsay and Silverman [RS2005, Ferratyand Vieu [FV2006].

Recursive conditional mode estimation

Let Zi = (Xi, Yi)i=1,...n be a F ×R-valued measurable strictly stationary process, definedon a probability space (Ω, A, P), where F is a semi-metric space, d denoting the semi-metric. Assume there exists a regular version of the conditional probability of Y given X.Assume that for a given x there is a compact subset S = [θ − ξ, θ + ξ], ξ > 0, such thatthe conditional density of Y given X = x has an unique mode θ on S. In the remainder ofthe paper x is fixed in F and Nx denotes a neighborhood of x. Let f x be the conditionaldensity of the variable Y given X = x. We define the kernel estimator f x of f x as follows:

f x(y) =

n

∑i=1

K(

a−1i d(x, Xi)

)b−1

i H(

b−1i (y−Yi)

)n

∑i=1

K(

a−1i d(x, Xi)

) , (37)

where K and H are kernels and an (resp. bn) is a sequence of positive real numbers. Notethat we can treat the classical kernel method studied by Ferraty et al. (2006) as particularcase of the present study by taking ai = an and bi = bn; for all i = 1, . . . , n. Now, a naturalextension of the kernel estimator θ of the conditional mode θ to the functional frameworkis given by:

θ = arg supy∈S

f x(y). (38)

Notation and hypothesis

In order to establish our asymptotic results we need the following hypotheses:

28

(H1)

(i) The function φ(x, r) := (X ∈ B(x, r)) is such that φ(x, r) > 0, ∀ r > 0(ii) For all i = 1, . . . , n there ∃ a deterministic function φi(x, ·) /

almost surely 0 < (Xi ∈ B(x, r)|Fi−1) ≤ φi(x, r), ∀ r > 0, φi(x, r)→ 0as r → 0

.(iii) For all sequence (ri)i=1,...n > 0, ∑ni=1(Xi∈B(x,ri)|Fi−1)

∑ni=1 φ(x,ri)

→ 1 a.co.

...

(H5) The sequences an and bn such thati) There exists γ > 0 such that limn→∞ nγbn = ∞ and supi

(bnbi

)< C,

ii) limn→∞log n

nbnψn(x) = 0 where ψn(x) = n−1 (∑ni=1 φ(x,ai))

2

∑ni=1 φi(x,ai)

iii)∑n

i=1

(hβ

i φi(x,ai))

hβn ∑n

i=1 φ(x,ai)= O(hβ

n) with hi ∈ ai, bi and β ∈ β1, β2

Main resultsThe following Theorem gives the almost-complete convergence (a.co.) of θ.

Theorem 14 Under the hypotheses (H1)-(H5) we have

θ − θ = O(

aβ1/2n

)+ O

(bβ2/2

n

)+ O

(√log n

nbnψn(x)

)1/2

a.co.

Now, we study the asymptotic normality of θ. To, do that we replace the conditions(H1)-(H5) the following hypotheses,

(N1) there exists functions βx(·) and δ(·) such that

∀s ∈ [0, 1], limn→∞

∑ni=1 φ(x, sai)

∑ni=1 φ(x, ai)

= βx(s)

and

limn→∞

b3n

∑ni=1 b−3

i πi(x, sai)

∑ni=1 φ(x, sai)

= δ(s) In probability.

where πi(x, r) = (Xi ∈ B(x, r)|Fi−1)

(N2) The conditional density f x satisfies (H2). Moreover ∀(y1, y2) ∈ S× S, ∀(x1, x2) ∈Nx × Nx

| f x(j)1 (y1)− f x(j)

2 (y2)| ≤ Cx

(d(x1, x2)

βj1 + |y1 − y2|β

j2

)where β

j1, β

j2 > 0 for j = 1, 2.

(N3) The Kernel K is a differentiable function supported on [0, 1] satisfies (H3). Itsderivative K′ such that

K2(1)δ(1)−∫ 1

0(K2(u))′δ(u)βx(u)du > 0 and K(1)−

∫ 1

0K′(u)βx(u)du 6= 0.

29

(N4) The kernel H is of class C2, of compact support, satisfies∫H′2(t)dt and

∫|t|β

ji H(t)dt < ∞, i, j = 1, 2.

(N5) The sequences an and bn satisfy (H5i) and such that :

nb3n ϕn(x)log n

→ ∞,

and √b3

nnϕn(x)

n

∑i=1

(aβ1

1i + bβ1

2i

)πi(x, ai)→ 0 as n→ ∞ In probability.

Main resultsThe asymptotic normality of θ is given in the following Theorem.

Theorem 15 Under the hypotheses (N1)-(N5) we have(nb3

n ϕn(x)σ2(x)

)1/2

(θ − θ) −→ N(0, 1) as n→ ∞ (39)

where

σ2(x) =f x(θ)

(K2(1)δ(1)−

∫ 10 (K

2(s))′δ(s)βx(s)ds)

(f x(2)(θ)

(K(1)−

∫ 10 K′(s)βx(s)ds

))2

∫H′2(t)dt.

Conclusionthis work treat the almost complete convergence and asymptotic normality of the recur-sive estimate wich is a new method for estimating the condition mode under the func-tional ergodicity condition.References

Amiri A., Crambes Ch., and Thiam B., Recursive estimation of nonparametric regres-sion with functional covariate, Comput. Statist. Data Anal., 2014, 69, 154–172.

Ferraty F., Vieu, P., Nonparametric Functional Data Analysis: Theory and Practice,Springer Series in Statistics, 2006.

Ferraty F., Laksaci A., Tadj A., Vieu P., (2010). Rate of uniform consistency for non-parametric estimates with functional variables, J. Statist. Plann. and Inf., 2010, 140, 335–352.

Laıb N. Louani, D., Strong consistency of the regression function estimator for func-tional stationary ergodic data , J. Statist. Plann. and Inf., 2011, 141, 359–372.

Ling N., Xu, Q., Asymptotic normality of the conditional density estimation in thesingle index model for functional time series data , Stat. Probab. Lett., 2012, 82, 2235–2243.

Roussas G.R., Tran L.T., Asymptotic normality of the recursive kernel regression esti-mate under dependence conditions , Ann. Statist., 1989, 20, 98–120.

Ramsay J. O., Silverman B. W., Applied functional data analysis; Methods and casestudies. Springer, New York, 2005.

Thiam B. Recursive estimation of functional, university versailies, 2006.

30

Using refined descriptive sampling in integrationLeila BAICHE, Megdouda OURBIH

University of Bejaia, Centre Universitaire de Tipaza

In Monte Carlo simulation, Latin hypercube sampling (LHS) is a well knownvariance reduction technique for independent random variable vectors. In this ar-ticle we present the refind decriptif sampling with dependent variables (RDSD).We used the Iman and Conover algorithm which is based on rank correlation ina sample of observations of the input variables.

Key words: Monte Carlo Methods; Variance reduction; Estimation.

2010 Mathematics Subject Classification: 91G60, 62G05, 62E10.

IntroductionLet us suppose having a system with no mathematical solution and its stochastic be-

havior depends on a random vector X ∈ RK having K independent components X1, ...XK,which we refer to as input variables. A logical model is built in a simulation study andused as a vehicle for experimentation. Then, experiments are carried out on the modeland unknown parameters of the output random variables of interest are estimated. Veryoften, we want to estimate the expected value of some measures of performance of thesystem being studied, given by the function h(X) denoted by Y ∈ R. Thus, the problemis to find the best estimator of the integral I such that I =

∫[0,1[K h(X)dX which is the

expected value of h(X).The Simple Random Sampling (SRS) (Fishman, 1997), is the traditional technique to

evaluate multidimensional integrals. This approach is frequently used when the numer-ical methods are too difficult to implement.

Monte Carlo (MC) integration has the advantage of applicability but it generates es-timation errors that are commonly obtained via independent and identically distributedsampling. Apart from the MC method, we can mention other methods, for instance,Quasi Monte Carlo ( QMC ) (Keller, 2008), Latin Hypercube Sampling ( LHS ) (Mckay,1979)and (Loh, 1996), Descriptive Sampling ( DS ) (Saliby, 1990) and Refined DescriptiveSampling (RDS) (Tari, 2006).

Descriptive Sampling and Latin Hypercube Sampling are both based on a randompermutation of the input numbers but select their values differently. The LHS is com-pared to MC estimates, for instance, (Stein, 1987) has conducted an application to aprinter actuator where he concluded that LHS is more accurate than SRS. In (Saliby, 1997),the author has proved that DS has a smaller variance than LHS. The contribution of DS isdemonstrated in several empirical comparisons. For example (Saliby and Pacheco, 2002)where the authors studied the efficiency of six Monte Carlo simulation sampling meth-ods namely QMC (Halton, Sobol and Faure numeric sequences), DS, LHS and SRS intwo finance applications. In this study, DS and LHS have shown the best results. There-fore, both DS and LHS outperformed QMC and SRS methods leading to more accurateestimates in a simulation study.

Tari and Dahmani ( 2006 ) introduce RDS as a method of stratifying on all the inputdimensions simultaneously. It was proposed to make DS safe by removing its possible

31

sampling bias, efficient by producing estimates with lower variance and convenient byremoving the need to determine in advance the sample size. This approach is consideredas a variance reduction technique similar to LHS.

An empirical efficiency of RDS over DS and SRS is proved by several comparisons,for instance, on a flow shop system (Tari and Dahmani, 2005a) and a production system(Tari and Dahmani, 2005b). We can then deduce that RDS outperforms DS and SRS, QMCand LHS methods (Ourbih, 2009). Then, to value RDS method and make it more attrative,this paper gives a theoretical contribution.

In this paper, RDS as the best sampling method is examined and compared to SRSwith respect to the mean of the output random variable, when the inputs variables areindependent. In Section 2, the SRS estimator is described. In section 3, we describedRefined Descriptive Sampling method. Section 4 gives an example of how to use RDS. Insection 5, RDS is presented in K dimension and finally, section 6 proves asymptoticallythat the variance of RDS estimator is lower than that of SRS, for any function h havingfinite second moment.

The estimator of SRSMonte Carlo methods are usually used for high-dimensional problems. That is, N valuesof the input random vector, X1, X2, ..., XN are generated in some manner such that theintegral I = E(Y) can be estimated by

ISRS = I(Y1, Y2, ..., YN) =1N

N

∑i=1

Yi

The Estimator Of RDSIn the jth sub-run, using RDS the integral I is estimated as follows

Ij = I(

Y j1, Y j

2, ..., Y jpj

)=

1pj

pj

∑i=1

Y ji

and in the simulation experiment defined by m sub-runs, the integral I is estimatedby the following IRDS estimator defined by the average of the sub-runs estimates

IRDS = T( I1, I2, .., Im) =1N

m

∑j=1

pj × Ij

=1N

m

∑j=1

pj

∑i=1

Y ji ,

and where the arguments Y ji = h(Xj

i), i = 1, ..., pj constitute the jth descriptivesample of size pj of Y denoted by Yj and the arguments Yj, j = 1, ..., m constitute arefined descriptive sample of size N of the output Y observed through simulation.

32

The asymptotic variance of IRDS

If Eh2 < +∞ and if ∃ pj → +∞ then, the asymptotic variance of IRDS is given by

Var( IRDS) = Var( ISRS) +1

N2

m

∑j=1

(pj − 1)[K(Eh)2 −K

∑k=1

∫[0,1[

(gk(x))2dFk(x) + O(

p−1j

)]

Theorem 16 If Eh2 < +∞ and if ∃ pj → +∞ then,

Var( IRDS) ≤ Var( ISRS)

ConclusionWe have proved asymptotically, in case of independent input variables, that the varianceof RDS estimator of the mean of the output variable, is less than that of SRS, when thefunction h has a finite second moment.References

Fishman, G.S.Monte Carlo: concepts, algorithms and applications.Spriger-verlag, 1997.Handcock, S. M. (1991). On cascading Latin Hypercube designs and additive models

for experiments. Communications in Statistics: Theory and Methods, 20, 417–439.Keller, A., Heinrich, S.and Niederreiter, H. (2008). Monte Carlo and Quasi Monte

Carlo methods. Springer (Berlin).Loh, W.L. (1996) On Latin hypercube sampling. Ann. Stat., 24, 2058–2080.Mckay, M.D., Conover, W.J. and Beckman, R.J. (1984). A comparison of three methods

for selecting values of input variables. Comput. Optim. Appl., 21, 169–175.Saliby, E. (1990) Descriptive sampling : a better approach to monte carlo simulation.

J. Operat. Res. Soc, 41, 1133–1142.Saliby, E. and Pacheco, F. (2002). An empirical evaluation of sampling methods in risk

analysis simulation: Quasi Monte Carlo, Descriptive Sampling, and Latin Hypercubesampling. Proceedings of the 2002 Winter Simulation Conference. E. Yucesan, Chen. L, J.Snowdon and J.M Charnes eds, 2, 1606–1610.

Stein, M. (1987). Large sample properties of simulation using Latin hypercube sam-pling. Technometrics, 29, 143–151.

Tari, M. and Dahmani, A. (2005). Flow shop simulator using different sampling meth-ods. Operational Research An International Journal, 5, 261–272.

Tari, M. and Dahmani, A. (2005). The three phase discrete event simulation usingsome sampling methods”, International Journal of Applied Mathematics Statistics, 3, 37–48

Tari, M. and Dahmani, A. (2006). Refined descriptive sampling : a better approach tomonte carlo simulation. Simulation Modeling Practice and Theory, 14, 143–160.

33

Mathematical analysis of a B-cell chronic lymphocytic leukemiamodel with immune response

Youcef Belgaid, Mohamed Helal, Ezio Ventiruno

Univ Sidi-Bel-Abbes, Universite de Torino

A B-cell chronic lymphocytic leukemia has been modeled via a highly nonlinearsystem of ordinary differential equations. We consider the rather important theo-retical question of the equilibria existence. Under suitable assumptions all modelpopulations are shown to coexist.Key words: Leukemia, Descartes rule, Local stability.


Introduction

B-cell chronic lymphocytic leukemia (B-CLL) has been undertaken by means of a highlynonlinear mathematical model based on ordinary differential equations. In this short pa-per we would reconsider the model to tackle one issue that is still missing in the analysisof [seema]. Specifically, we consider a rather important theoretical question, namely theissue of the equilibria existence of the mentioned model. In this paper we fill the gap,by providing a proof showing that all the model populations can always coexist, undersuitable and meaningful assumptions.

Main results

Mathematical model

The model is fully described in [seema], thus is expressed by the set nonlinear system ofordinary differential equations stated below:

dBdt

= bB + (r− db)B− dBN BN − dBTBT (40)

dNdt

= bN − dN N − dNBNB (41)

dTdt

= bT − dTT − dTBTB + kaTHBL

s + BL TH (42)

dTH

dt= bTH − dTH TH + aTH

BL

s + BL TH. (43)

The parameters are all assumed to be positive and their meaning is defined in Table1. Specific assumptions on some of the parameters are

dBT 1, r > dB. (44)

34

*Conditions We need to impose the following conditions :

dTH B∗L + sdTH > aTH B∗L, B∗L[bTdTH + kaTH bTH ] + sbTdTH ≥ aTH B∗L. (45)

bBbN > dBNdNBB∗2N∗2, (dT + dTBB∗)∆2 + bNdBTkaTH

T∗HN∗

sLB∗L

(s + B∗L)2> bNdBT

B∗

N∗dTBT∗. (46)

Coexisting EquilibriumThe case L = 2

Theorem 17 The coexistence equilibrium E∗(B∗, N∗, T∗, T∗H) of the system (40)-(356) with L =2 exists unconditionally and it is feasible if conditions (45) are satisfied.

The case L = 3Theorem 18 The coexistence equilibrium E∗(B∗, N∗, T∗, T∗H) of the system (40)-(356) in thecase L = 3 exists unconditionally. Once again, for it to be feasible, conditions (45) need to besatisfied. Multiple roots are possible, arising possibly through saddle-node bifurcations.

The general case L ≥ 4Theorem 19 The coexistence equilibrium E∗(B∗, N∗, T∗, T∗H) of the system (40)-(356) in thegeneral case L ≥ 4 exists unconditionally. Once again, for it to be feasible, conditions (45) needto be satisfied.

Stability AnalysisIn this section we investigate the local stability of the coexistence equilibrium, in theparticular case L = 2 and in the general one L ≥ 3.

The case L = 2We have the following result:

Theorem 20 For L = 2, The coexistence equilibrium E∗(B∗, N∗, T∗, T∗H) of the system (40)-(356) is locally asymptotically stable if b0 > 0, b1 > 0, b2 > 0, b3 > 0, where these coefficientsare defined in the proof.

The case L ≥ 3Theorem 21 For L ≥ 3, The coexistence equilibrium E∗(B∗, N∗, T∗, T∗H) of the system (40)-(356) is locally asymptotically stable if conditions (46) hold.

ReferencesY.Belgaid, M.Helal, E.Vonturino, Mathematical analysis of a B-cell chronic lympho-

cytic leukemia model with immune response, Applied Mathematics and Nonlinear Sci-ences, Accepted 2019.

35

Nanda, S., dePillis, L., Radunskaya, A., (2013) B cell chronic lymphocytic leukemia -a model with immune response, Discrete and Continuous Dynamical Systems, B, 18(4),1053-1076.

Perko, L., (1991) Differential Equations and Dynamic Systems, Springer Verlag.

About a Nonlinear Caputo-Hadamard Fractional DifferentialEquation with Hadamard integral boundary conditions in

Banach SpacesMaamar Benbachir, Abdallatif Boutiara

University Saad Dahlab , University of Ghardaia

The aim of this talk is to give sufficient conditions witch guarantee existencesolution of boundary value problem including a fractional order differential equa-tion involving the Caputo-Hadamard fractional derivative. We conclude by ap-plying Boyd-Wong Contraction Principle, two illustrative examples are presented.The boundary conditions introduced in this work are of quite general nature andreduce to many special cases by fixing the parameters involved in the conditions.Key words: Fractional differential equation, fractional integral conditions, Hadamardfractional integral, Caputo-Hadamard derivative, Fixed point theorems.

2010 Mathematics Subject Classification:26A33; 34B25; 34B15.

IntroductionDifferential equations of fractional order have recently proved to be valuable tools in themodeling of many phenomena in various fields of science and engineering. There arenumerous applications in viscoelasticity, electrochemistry, control theory, porous media,electromagnetism, etc.In 2008, Benchohra et al. studied the existence and uniqueness of solutions of the follow-ing nonlinear fractional differential equations:

CDαy(t) = f (t, y(t)), for each t ∈ J := [0, T], 0 < α < 1.ay(0) + byT) = c

where CDα is the caputo fractional derivative of order α, f : [0, T]×R→ R is a givencontinuous function, and a, b, c are real constants with a + b 6= 0.In 2019, A. Ardjouni et al. discussed the existence and uniqueness of positive solutionsof the following nonlinear fractional differential equation with integral boundary condi-tions:

Dα1 y(t) = f (t, y(t)), for each t ∈ J := [1, e], 0 < α < 1.

y(1) = b∫ e

1 y(s)ds + d,

where Dα1 is the Caputo-Hadamard fractional derivative of order 0 < α ≤ 1, λ ≥ 0,

d > 0 and f : [1, e]× [0, ∞)→ [0, ∞) is a given continuous function.

36

In this work, we concentrate on the following boundary value problem,of nonlinearfractional differential equation with fractional integral as well as integer and fractionalderivative:

CHDr

1+x(t) = f (t, x(t)), t ∈ J := [1, T], 0 < r ≤ 1. (47)

with fractional boundary conditions:

αx(1) + βx(T) = λIqx(η) + δ, q ∈ (0, 1] (48)

where CHDr

1+ denote the Caputo-Hadamard fractional derivative and Iq denotes the stan-dard Hadamard fractional integral. Throughout this paper, we always assume that 0 <r ≤ 1, 0 < q ≤ 1 f : [1, T] × R → R is continuous. α, β, λ, δ are real constants, andη ∈ (1, T).

Main resultsDefinition 1 Assume that E is a Banach space and T : E → E is a mapping. If there exists acontinuous nondecreasing function ψ : R+ → R+ such that ψ(0) = 0 and ψ(ε) < ε for allε > 0 with the property:

‖Tx− Ty‖ ≤ ψ(‖x− y‖), ∀x, y ∈ E.

then, we say that T is a nonlinear contraction.

Theorem 22 (Boyd-Wong Contraction Principle)Suppose that B is a Banach space and T : B→ B is a nonlinear contraction. Then T has a uniquefixed point in B.

Theorem 23 Assume that f : [1, T]×R → R are continuous functions and H > 0 satisfyingthe condition

| f (t, x)− f (t, y)| ≤ |x− y|H + |x− y| , for t ∈ J, x, y ∈ R. (49)

Then the fractional BVP (47)-(48) has a unique solution on J .

ExampleWe consider the problem for Caputo-Hadamard fractional differential equations of theform:

CHD

23 x(t) = f (t, x(t)), (t, x) ∈ ([1, e], R+),

x(1) + x(e) = 12

(I

12 x(2)

)+ 3

4 .(50)

Here

r =23

, q =12

, α = 1, β = 1,

δ =34

, λ =12

, η = 2, T = e.

37

With

f (t, y(t)) =1

t2 + 4cosx, t ∈ [1, e]

Clearly, the function f is continuous.For each x ∈ R+ and t ∈ [1, e], we have

| f (t, x(t))− f (t, y(t))| ≤ 14|x− y|

Hence, the hypothesis (H1) is satisfied with L = 14 .

Further,

M :=(log T)r

Γ(r + 1)+|λ|(log η)r+q

|Λ|Γ(r + q + 1)+|β|(log T)r

|Λ|Γ(r + 1)' 2.0286

andLM ' 0.5071 < 1.

Therefore, by the conclusion of Theorem 3, It follows that the problem (50) has a uniquesolution defined on [1, e].

Conclusion

Though the technique applied to establish the existence results for the problem at hand isa standard one, yet its exposition in the present framework is new. An illustration to thepresent work is also given by presenting an example. Our result is new and generalizesome available results on the topic.References

A. Ardjouni, A. Djoudi, Positive solutions for nonlinear Caputo-Hadamard fractionaldifferential equations with integral boundary conditions, Open J. Math. Anal. 2019, 3(1),62-69

M. Benchohra, S. Hamani and S.K. Ntouyas, Boundary value problems for differen-tial equations with fractional order, Surveys in Mathematics and its Applications vol, 3(2008), 1-12.

On the regularity of the solution of non-homogeneous Burgersequation

Yassine Benia, Boubaker-Khaled Sadallah

University of Algiers, E.N.S. kouba

We consider Cauchy–Dirichlet problem for Burgers equation, and we give anew regularity result of the solution in anisotropic Sobolev spaces. // Key words:Regularity, Burgers equation, Anisotropic Sobolev space.

2010 Mathematics Subject Classification: 35K58, 35Q35.

38

IntroductionThis article is concerned with the regularity of the solution of the semilinear parabolicproblem

∂tu(t, x) + u(t, x)∂xu(t, x)− ν∂2xu(t, x) = f (t, x) (t, x) ∈ R,

u(0, x) = ψ(x) x ∈ Γ0,u(t, 0) = u(t, 1) = 0 t ∈ (0, T),

(51)

in the rectangle R = (0, T) × I with I = (0, 1) and Γ0 = 0 × I, T is finite and ν isa positive constant; ψ ∈ H3(Γ0) ∩ H1

0(Γ0) and f ∈ H1,2(R) are given functions, whereH3(Γ0), H1

0(Γ0) are usual Sobolev spaces and H1,2(R) is the anisotropic Sobolev spacedefined by

H1,2(R) =

u ∈ L2(R) : ∂tu ∈ L2(R), ∂xu ∈ L2(R), ∂2xu ∈ L2(R)

.

In previous works (see [benia1, benia2]) we have studied Burgers equation ∂tu + u∂xu−∂2

xu = f (with Dirichlet boundary conditions) in a non rectangular domain Ω ⊂ R2.When the right-hand side f lies in the Lebesgue space L2(Ω), and the initial condition isin the space H1

0(Γ0), we have established the existence of a unique solution in H1,2(Ω).

Main resultsOur main result is as follows:

Theorem 24 Let f ∈ H1,2(R) and ψ ∈ H3(Γ0) ∩ H10(Γ0). Asumme that f and ψ satisfy the

compatibility condition f|Γ0+ ψ′′ ∈ H1

0(Γ0). Then Problem (359) admits a unique solution u liesin

H2,4(R) =

u ∈ L2(R) : ∂it∂

jxu ∈ L2(R), 2i + j ≤ 4

,

We know (see [benia1]) that Problem (359) admits a unique solution u ∈ H1,2(R).Then to prove Theorem 43 we have to obtain the regularity u ∈ H2,4(R).

We construct approximate solutions to (359) in the form

un(t, x) =n

∑j=1

cj(t)ej(x), (t, x) ∈ R,

where (ej)j∈N? are solutions of Dirichlet problem−e

′′j = λjej, j ∈N∗,

ej(0) = ej(1) = 0.

(ej)j∈N? is an orthonormal basis in L2(R).Consider the approximate problem

1∫0

∂tunej dx +

1∫0

un∂xunej dx + ν

1∫0

∂xun∂xej dx =

1∫0

f ej dx,

un(0, x) = u0n, x ∈ (0, 1),

(52)

the sequence u0n will be chosen to converge in H3(Γ0) ∩ H10(Γ0) to ψ.

39

In [benia1], we have proved that (336) is equivalent to a system of n uncoupled linearordinary differential equations, and for every j = 1, · · · , n, there exists on the interval(0, T), a unique regular solution cj.

All constants (Ki)1≤i≤3, (Ci)1≤i≤8 and C are independent of n.We prove the following result using the same method as in [benia2].

Lemma 3 There exists a positive constant K1 such that

‖un‖2L2(I) + ‖∂tun‖2

L2(I) + ‖∂xun‖2L2(I) ≤ K1.


T∫0

‖∂s∂xun(s)‖2L2(I) ds ≤ K2


t∫0

‖∂s∂2xun(s)‖2

L2(I) ds ≤ K3.

In [benia1], we have proved that the approximation un converges to the unique solutionu ∈ H1,2(R) of Problem (359).

Proposition 2 Under the hypotheses of Theorem 43, the solution of Problem (359) is in H2(R)and ∂t∂

2xu ∈ L2(R).

Proof 1 of Proposition 110.Observe that Lemma 4 and 5 imply that the solution of Problem (359)satisfies ∂t∂xu ∈ L2(R) and ∂t∂

2xu ∈ L2(R). So, it is enough to prove that ∂2

t u ∈ L2(R).Differentiating (359), taking L2-norms and integrating the obtained equation with respect to t,we get

t∫0

‖∂2t u(s)‖2

L2(I) ds ≤ν

t∫0

‖∂s∂2xu(s)‖2

L2(I) ds +t∫

0

‖∂tu(s)∂xu(s)‖2L2(I) ds

+

t∫0

‖u(s)∂s∂xu(s)‖2L2(I) ds +

t∫0

‖∂t f (s)‖2L2(I) ds.

(53)

We need to estimate the terms of the right-hand side in (53), we have

‖∂tu∂xu‖2L2(I) ≤ ‖∂tu‖2

L4(I)‖∂xu‖2L4(I)

≤ C(‖∂tu‖2

L2(I) + ‖∂t∂xu‖12L2(I)‖∂tu‖

32L2(I)

)×(‖∂xu‖2

L2(I) + ‖∂2xu‖

12L2(I)‖∂xu‖

32L2(I)

),

(54)

and‖u∂t∂xu‖2

L2(I) ≤ ‖u‖2L∞(I)‖∂t∂xu‖2

L2(I)

≤ C(‖u‖2

L2(I) + ‖∂xu‖2L2(I)

)‖∂t∂xu‖2

L2(I).(55)

40

Uning the previous estimates, Lemma 4 and the fact that u ∈ H1,2(R), we deduce that

t∫0

‖∂2t u(s)‖2

L2(Γ0)ds ≤ C. (56)

where C is a constant independent of n and t.

Lemma 6 Assume that s1, s2 and s are real numbers such that s1, s2 ≥ s ≥ 0. If u ∈ Hs1(R)and v ∈ Hs2(R) then uv ∈ Hs(R) where s < s1 + s2 − 1.

This lemma is a special case of Theorem 7.5, [Behzadan].

Proof 2 of Theorem 43Recall that f is given in H1,2(R). So, ∂t f ∈ L2(R).Let v = ∂tu and g = v∂xv− v∂xu−u∂xv+ ∂t f . From Lemma 5, we deduce v ∈ L∞(0, T, H1

0(I)).Then v ∈ L∞(R), consequently, v∂xv ∈ L2(R). On the other hand, u ∈ H2(R) implies thatv∂xu ∈ L2(R), and choosing s1 = 2, s2 = 0 in Lemma 6, we obtain u∂xv ∈ L2(R). Finally, weget g ∈ L2(R).

Differentiating (359) with respect to t we deduce

∂tv + v∂xu + u∂xv− ν∂2xv = ∂t f ,

then∂tv + v∂xv− ν∂2

xv = g.

Observe that v is a solution of the problem∂tv + v∂xv− ν∂2

xv = g (t, x) ∈ R,v(0, x) = f|Γ0

+ ψ′′ x ∈ Γ0,v(t, 0) = v(t, 1) = 0 t ∈ (0, T),

(57)

where (according to the hypothesis of Theorem 43) f|Γ0+ ψ′′ ∈ H1

0(Γ0). Consequently, by themain result of [benia1] v is in H1,2(R).

On the other hand, from (359) we have

ν∂4xu = ∂t∂

2xu + 3∂xu∂2

xu + u∂3xu− ∂2

x f , (58)

as all the terms of the right-hand side in (58) are in L2(R), ∂4xu is in L2(R), we deduce that

u ∈ H2,4(R) which is the optimal regularity of the solution u.

ReferencesA. Behzadan, M. Holst, Multiplication in Sobolev spaces, revisited. Available as

arXiv:1512.07379 [math.AP], 2015.Y. Benia, B-K. Sadallah, Existence of solutions to Burgers equations in domains that

can be transformed into rectangles, Electron. J. Diff. Eq., 157(2016), 1-13.Y. Benia, B-K. Sadallah, Existence of solutions to Burgers equations in a non-parabolic

domain, Electron. J. Diff. Eq., 20(2018), 1-13.C. I. Byrnes, D. S. Gilliam and V. I. Shubov, Boundary control for a viscous Burgers’

equation, in H. T. Banks, R. H. Fabiano, and K. Ito (Eds.), Identification and Control forSystems Governed by Partial Differential Equations, SIAM, (1993), 171-185.

O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Ural’ceva, Linear and Quasi linear Equa-tions of Parabolic Type, Translations of AMS, Vol. 23, 1968.

41

Second order optimality in three-stage design for estimating aproduct of two means

Zohra Benkamra, Mounir TlemcaniUniversite des sciences et de la technologie d’Oran - Mohamed-Boudiaf

For estimating a product of two means by allocation from independent Bernoullipopulations, we consider a three-stage design to achieve a lower bound for thevariance. The asymptotic analysis is based on the rate of convergence in thestrong law of large numbers which is upper limited by the central limit theorem.Key words: Sequential allocation, Second order, Lower bound, Asymptotic optimality,

Strong law of large numbers, Rate of convergence, Two-stage, Three-stage, Sampling,Product of Bernoulli proportions

2010 Mathematics Subject Classification: 62L10, 62L12

IntroductionAssume that two independent random variables X1, X2 are observable from Bernoullipopulations P1,P2 with probabilities of success p1, p2 respectively. In non linear estima-tion, the problem of estimating the product p = p1 p2 has many applications in reliabilityengineering (for example a series system of two components i = 1, 2 with reliabilitypi, i = 1, 2), cf. [?, ?]. Because designers are generally risk averse, a crucial objective ofsuch problems is the reduction of the variance incurred by the sampling method leadingto a discrete optimization problem under the constraint of a total sample size T fixed.Assuming T is large enough, sequential allocation procedures are good alternatives toderive asymptotically optimal partitions M1 + M2 = T in the sense that the producedvariance reaches a lower bound when T goes to infinity and Mi is the sample size takento estimate pi from observations in the population Pi for i = 1, 2. When these coefficientsare unknown as in practice, since they depend themselves on the unknown proportionsp1, p2, adaptive design can be made sequentially, based on accruing data. The focus inthis paper is on second order efficiency of 3-stage designs for estimating the product oftwo Bernoulli proportions under the constraint of a total and fixed sample size T largeenough.

Asymptotics of the sample variance induced by alloca-tionLet Mi the sample size allocated to estimate the proportion pi, i = 1, 2. The maximumlikelihood estimator (M.L.E) for the product p = p1 p2 is p = p1 p2, where pi is the sample

proportion, pi =SiMiMi

, and SiMi counts the number of success in Mi Bernoulli trials in thepopulation Pi.

Assuming independence within and across populations, and using the fact that pi isan unbiased estimator of pi, the sample variance of p can be written as

Var( p) = p2[(

1 +c2

1M1

)(1 +

c21

M1

)− 1]

, (59)

42

where ci is the coefficient of variation of the Bernoulli population Pi, i.e., ci =√

1pi− 1.

So, the scaled variance Var( p)p2 can be written as

Var( p)p2 =

(c1

√1 + c2

2T + c2

√1 + c2

1T

)2

T+

RT

, (60)

where

R =

(M1c2

√1 + c2

1T −M2c1

√1 + c2

2T

)2

M1M2.

It follows that the exact optimality condition is R = 0 in the case of two proportions andis explicit, i.e.

M1

M2=

c1

√1 + c2

2T

c2

√1 + c2

1T

, (61)

which can be solved analytically when c1 and c2 are known. So, the exact lower boundfor the scaled variance is

Q =

(c1

√1 + c2

2T + c2

√1 + c2

1T

)2

T. (62)

When T is large enough, one can perform a Taylor expansion of Q and R, and obtain,for T → ∞, :

Q =(c1 + c2)

2

T+

c1c2 (c1 + c2)2

T2 − c1c2 (c1 − c2)2 (c1 + c2)

2

4T3 + ...,

RT

=M1c2 −M2c1

M2M1

M1c2

(c2

1T + 1

)−M2c1

(c2

2T + 1

)T

+c1c2 (c1 − c2)

2 (c1 + c2)2

4T3 + ....

Then, the conditionM1

M2=

c1

c2(63)

becomes an asymptotic optimality criteria for the scaled variance Var( p)/p2 to achievethe second order lower bound:

Q2 =(c1 + c2)

2

T

(1 +

c1c2

T

), (64)

in the sense that for all M1, M2 satisfying M1 + M2 = T,

Var( p)p2 ≥ Q2 + o

(1

T2

), as T → ∞. (65)

Remark that the proportionality condition (63) with the constraint M1 + M2 = T give anexplicit solution to the optimal design:

M1 = Tc1

c1 + c2and M2 = T −M1,

which is the starting idea to construct sequential schemes to solve the problem when c1and c2 are unknowns. As a first theoretical rate of convergence result, we have:

43

Theorem 25 Assume M1 + M2 = T → ∞ and that

M1

T=

c1

c1 + c2+ o

(1

Tα

), (66)

for a real α ≥ 0; thenVar( p)

p2 = Q2 + o(

1Tβ

), (67)

where β = min (2α + 1, 2 + α) .

One-stage sequential design

Recall that the coefficients of variation are unknowns in practice. Starting from equation(66) in Theorem 93, one can choose

M1

T=

c2

c1 + c2, M2 = T −M1, (68)

where, for example, ciL is the M.L.E of ci obtained by a first stage sampling with samplesize L ≤ T/2 in each population Pi. Thus, the strong law of large numbers (S.L.L.N)yields with probability one:

Mi

T→ ci

c1 + c2, as L→ ∞.

If we consider only two-stage design and choose for example L according to [?], then Lis a function of T and must satisfy some conditions like: (i) L→ ∞ and (ii) L = o(T) asT → ∞. A good candidate is L = Θ

(√T)

as T → ∞. A result based on the S.L.L.N ofMarcinkiewicz gives the following Lemma.

Lemma 7 Assume 0 < pi < 1 for i = 1, 2 and let ciL the M.L.E of ci with sample size L, i.e.,

ciL =

√L

SiL− 1.

Let Mi given by the relation (68). Then,

Mi

T=

ci

c1 + c2+ o

(1Lq

)(69)

with probability one, for any q, 0 ≤ q < 1/2.

Remark 6 It should be pointed that the exponent q = 1/2 is not allowed in the previous Lemmasince the rate of convergence in the S.L.L.N is upper limited by the central limit theorem whichgives

L−1/2 SiL/L− pi√pi (1− pi)

→ N (0, 1) , as L→ ∞,

44

Two-stage sequential designFor example, if one proceeds the sequential design in two stages with L ≤ T/2, L → ∞and L = o(T) as T → ∞, then it is well known that the second stage completes the firstone as follows: Mi is the nearest integer to

M1 = min

max(

L, Tc1L

c1L + c2L

), T − L

, M2 = T − M1.

Since L/T → 0 as T → ∞, then for large T, one obtains MiT = ciL

c1L+c2L, and the S.L.L.N

prescribes the rate of convergence as given by Lemma 20. Hence, if one chooses L =√

Tin the first stage length, then Lq = T

q2 and Theorem 93 will provide at most a weaker

rate of convergence of the scaled variance to the second order lower bound, as shown forexample in the case of almost sure convergence.

Proposition 3 Let M1, M2 given by the two-stage design with a first stage length L =√

T →∞, then

Var( p)p2 = Q2 + o

(1

Tβ

), (70)

for all β, 0 ≤ β < 3/2.

A three-stage sequential designwe propose an intermediate stage in order to obtain more than 3/2 order in the rate ofconvergence (70). The idea is to update the M.L.E (s) of the coefficients of variation ciL by

the more accurate one ciMi=

√Mi

SiMi− 1, based on the new sample size Mi which has the

advantage to be at the same order of T, as T → ∞. We are then conducted to propose thefollowing three-stage design, let [x] denotes the nearest integer to the real number x. Thethree-stage design is as follows:

1st stage: Sample L =[√

T/2]

from each population and evaluate ciL =√

LSiL− 1, i =

1, 2; and

M1 = min

max

L,[

T2

ciL

c1L + c2L

],[

T2

]− L

, M2 =

[T2

]− M1.

2nd stage: Sample Mi − L more observations in Pi and recalculate ciMi=

√Mi

SiMi− 1, i =

1, 2;

3rd stage: Sample Mi − Mi more in population Pi where

M1 = min

max

M1,

[T

c1M1

c1M1+ c2M2

], T − M2

, M2 = T −M1.

Theorem 26 (Almost sure convergence) Let M1, M2 given by the three-stage design with afirst stage length L =

√T/2→ ∞, then

Var( p)p2 = Q2 + o

(1

Tγ

), (71)

with probability one, for all γ, 0 ≤ γ < 2.

45

ConclusionA more detailed analysis can be found in [?] where authors considered several propor-tions with Monte-Carlo simulation. Our aim is to extend key ideas presented in thisabstract to more interesting situations as in the case of the one parameter exponentialfamily and more generally in a Bayesian framework with cost per unit observation.References

Benkamra, Z., Terbeche, M., Tlemcani, M., 2015. Nearly second order three-stagedesign for estimating a product of several Bernoulli proportions, Journal of StatisticalPlanning and Inference Volume 167, December 2015, Pages 90-101

Benkamra, Z., Terbeche, M., Tlemcani, M., 2012. An allocation scheme for estimatingthe reliability of a parallel-series system. Adv. Decis. Sci. 2012, Art. ID 289035, 14 pp.

Benkamra, Z., Terbeche, M., Tlemcani, M., 2013. Bayesian sequential estimation ofthe reliability of a parallel-series system. Appl. Math. Comput. 219 (23), 10842–10852.

Rekab, K., 1992. A sampling scheme for estimating the reliability of a series system.IEEE. Trans. Reliability. 42, 287–291.

Rekab, K., 1992. A nearly optimal 2-stage procedure. Comm. Statist. Theory Methods21, 197–201.

Global dynamics of an age structured alcoholism modelSoufiane Bentout, Salih Djilali

Centre Universitaire Belhadj Bouchaib , universite Hassiba Benbouali

In this paper, we consider an age-structured alcoholism model, where theglobal behavior of the system is investigated. It is proved that the system has twoequilibriums, the alcohol-free equilibrium and the endemic equilibrium. Further,the global stability of those equilibriums investigated under a critical values ofthe basic reproduction number R0, where for R0 ≤ 1 the alcohol-free equilibriumis globally stable and for R0 > 1 the system is persistent and endemic equilib-rium is globally stable. Furthermore, the impact of the susceptible drinkers rateand the repulse rate of the recovers to alcoholics were investigated. The obtainedresults are tested numerically.

Key words: Age-structured; Alcohol model; Uniform Persistence; Uniform Persistence..

2010 Mathematics Subject Classification: 92D25 · 92D30

IntroductionIn the past two decades, it has been considered that alcohol consumption is harmful tohealth and sometimes fatal, chemically, alcohol is defined as organic compounds thatcontain hydroxyl groups. It is a toxic substance found in spirits drinks that include beer,wine and distilled liquor. Those chemicals are chemically known as ethanol containingtwo atoms of carbon and one hydroxyl group, and alcohol can be considered as a toxicsubstance because of its ability to dissolve fat, as it works to dissolve the fat in the mem-branes of cells, which destroys the structure of cells and kill them, and in this way alcohol

46

is very effective in sterilization. Ethanol is less toxic than other alcohol, and when it issufficiently diluted, it causes effects on the brain that some people would like to get, but itshould be known that alcohol intake, whatever its quantity and concentration, cannot besafe completely and without risk. Many people in many parts of the world drink alcoholat social events, but drinking alcohol has many risks and health and social consequences.Although, the majority of chronic alcohol-related illnesses result from frequent and long-term use of alcohol, drinking in small amounts or in sporadic intervals also carries manyrisks, and science has found that there is no safe minimum for drinking alcohol. On theother hand, many researchers gives an increasing interest in studying the impact of usingof alcohol in population and focus on the manner of alcohol spreading in our societies.

Main resultsBased on the discussion used in the previous section we consider the following alco-holism model with age of alcoholism ”a”

dS(t)dt = Λ− S(t)L(t)− (µ + α)S(t),

∂A(t,a)∂t + ∂A(t,a)

∂a = − (µ + a1 + δ(a)) A(t, a),

dR(t)dt =

∫ +∞0 δ(a)A(t, a)da− (µ + a2 + ρ) R(t),

(72)

where

L(t) =∫ +∞

0β(a)A(t, a)da,

which represents the contact function, and the boundary and initial conditions

A(t, 0) = αS(t) + ρR(t) + S(t)L(t),A(0, a) = A0(a) ∈ L1

+((0,+∞), R), S(0) = s0 > 0, R(0) = r0 > 0.(73)

The parameters Λ is the entering flux into the susceptible class S (New born). δ(a)is the transfer rate from alcoholics with the age a to recovered individuals. r is the birthrate of the susceptible population, m is the natural death rate which is constant for allthe considered populations, a1, a2 are the death rates of excessive drinking, respectively.The coefficient α is the rate of susceptible drinkers S(t) develop into alcoholics because ofsome of their own reasons, such as losses of earnings, unemployment or family problems,so on. ρ is the transfer rate from alcoholics to recovered individuals, µ is the natural deathrate which is constant for all the considered populations. In this model, we can see thatthe incidence rates S(t)L(t) like a new alcoholics peoples.

For the biologically relevant of the model parameters we make the following assump-tions

1. Λ, µ, α, a1, a2 ≥ 0.

2. β ∈ CBU (R+, R+) (where CBU (R+, R+) is the set of all bounded and uniformlycontinuous function from R+ to R+ with β(a) ≤ β for all a > 0.

3. δ ∈ L∞+ (R+), with essential upper bound ‖δ‖∞.

47

4. δ(a) > δ0 for all a > 0.

We set Ω(a) = e−(µa+a1a+∫ a

0 δ(s)ds), by using the characteristic method we get,

A(t, a) =

(αS(t− a) + ρR(t− a) + S(t− a)L(t− a))Ω(a), t > a ≥ 0,A0(a− t) Ω(a)

Ω(a−t) , a > t ≥ 0(74)

Next, we define the basic reproduction number R0(the average of the new alcoholicsindividuals )

R0 = S0B +ρ

µ + a1 + ρδ− αS0

α + µB. (75)

where B =∫ ∞

0 β(a)Ω(a)da and δ =∫ ∞

0 δ(a)Ω(a)da.

Remark 7 We can find the formula of the basic reproduction number R0 using the renewal pro-cess, which is the spectral radius of the next generation matrix (see [Diekmann]).

Global stability of the alcohol-free equilibriumFirst, we show the local stability of the alcohol-free equilibrium E0.

Theorem 27 If R0 < 1, then the alcohol-free equilibrium is stable and unstable for R0 > 1.

In the following theorem, we prove the global stability of the alcohol-free equilibrium.

Theorem 28 The alcohol-free equilibrium E0 is globally asymptotically stable whenever R0 ≤ 1.

Proof Defining the functional H(z) = z− 1− ln(z) ≥ 0, and consider the followingLyapunov function

V(t) = V1(t) + V2(t) + V2(t),

where

V1(t) = H( S

S0

)+∫ ∞

0 ψ(a)A(t, a)da + ρµ+a2+ρ R(t),

V2(t) =(

R0 +αS0 Bα+µ

) ∫ ∞t αS(τ)dτ,

V3(t) = αS0 Bα+µ

∫ ∞t (ρR(τ) + L(τ)S(τ)) dτ,

Existence of the endemic equilibrium and persistenceAn endemic equilibrium (S∗, A∗(a), R∗) verifies the following system

Λ− (µ + α)S∗ − S∗L∗ = 0,A∗(a) = A∗(0)Ω(a),

R∗ = A∗(0)∫ ∞

0 δ(a)Ω(a)daµ+a2+ρ ,

A∗(0) = αS∗ + ρR∗ + S∗L∗,

the last equation of the system gives

48

A∗(0) = αS∗ + ρR∗ + S∗L∗,

= αΛ

µ + α + L∗+ ρ

A∗(0)∫ ∞

0 δ(a)Ω(a)daµ + a2 + ρ

+Λ

µ + α + L∗,

by a straightforward computation, we get

A∗(0)2(

1− ρδ

µ + a2 + ρ

)B +

((1− ρδ

µ + a2 + ρ)(µ + α)− BΛ

)A∗(0)− αΛ = 0, (76)

it’s clear that if 1− ρδµ+a2+ρ ≤ 0, then the equation (76) has not roots.

We assume that 1 > ρδµ+a2+ρ , then we calculate ∆

∆ =

((1− ρδ

µ + a2 + ρ

)(µ + α)− BΛ

)2

+ 4αΛ(

1− ρδ

µ + a2 + ρ

)B > 0,

we get two roots:

A∗1(0) =−(

1− ρδµ+a2+ρ

)(µ + α) + BΛ−

√∆

2(

1− ρδµ+a2+ρ

)B

< 0,

A∗2(0) =−(

1− ρδµ+a2+ρ

)(µ + α) + BΛ +

√∆

2(

1− ρδµ+a2+ρ

)B

> 0.

Proposition 4 If(

1 ≥ ρδµ+a2+ρ

), then system (72)-(73) has an endemic positive equilibrium

E∗(S∗, A∗(a), R∗) with

S∗ =Λ

µ + α + A∗(0)2B, (77)

A∗(a) = A∗2(0)Ω(a), (78)

R∗ =δA∗(0)2

µ + a2 + ρ. (79)

Next ,we suppose ∫ ∞

0A0(a)Ξ1(a)da > 0, (80)

where

Ξ1(a) =∫ ∞

0β(a + t)

Ω(a + t)Ω(a)

dt,

and also we assume, ∫ ∞

0A0(a)Ξ2(a)da > 0, (81)

with

Ξ2(a) =∫ ∞

0δ(a + t)

Ω(a + t)Ω(a)

dt.

49

Remark 8 The goal of the hypothesis (80) ensures that the initial condition is chosen in such away that future alcohol will occur.

We define a persistence function ρ : X → R+ as follows

ρ(Φ(t, x)) = C(t) = αS(t) + ρR(t) + S(t)L(t),

ConclusionThe impact of the susceptible drinkers rate has been established where a negative impactis noticed on the basic reproduction number, figure ?? shows this result. In the real life,this result means that the susceptible drinkers rate is a spread reducer of the alcoholuse. The basic reproduction rate R0 is increasing in ρ (the repulse rate) which means thatthis rate can be considered as an accelerator of the spread of the alcohol use. On the otherhand, the effect of the variable α on the endemic equilibrium state is investigated, where ithas been considered that this rate has a varying impact on the endemic equilibrium statewhere it has a negative impact on the susceptible density equilibrium S∗ and positiveimpact on both the alcoholic density equilibrium state A∗(α, a) and the recovered densityequilibrium R∗. Figure ?? shows this effect in details. The importance of studying thiseffect in the real life is for determining a useful strategy for controlling the spread ofthe alcohol use. The health education is one of the most effective ways in controllingthe spread of using alcohol in our communities where this strategy touch mostly thesusceptible drinkers and the repulse recovers to alcoholics.References

S. Bentout, T.M. Touaoula, Global analysis of an infection age model with a class of nonlin-ear incidence rates, J.Math. Anal.Appl., 434(2) (2016) 1211-1239.

S. Djilali, T. M. Touaoula and S. E. H. Miri, A heroin epidemic model: very general nonlinear incidence, treat-age, and global stability, Acta Applicandae Mathematicae, 152 (2017)171–194.

H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Amer.Math. Soc., Providence, 2011.

Ultrametric q-difference equations and q-WronskianBenharrat Belaidi, Rabab Bouabdelli, Abdelbaki Boutabaa

Abdel Hamid Ibn Badis University, Blaise Pascal University

Let K be an ultrametric complete and algebraically closed field and let q be anelement of K which is not a root of unity and is such that |q| = 1. In this article,we establish some inequalities linking the growth of generalized q-wronskians ofa finite family of elements of K[[x]] to the growth of the ordinary q-wronskian ofthis family of power series.We then apply these results to study some q-difference equations with coefficientsin K[x]. Specifically, we show that the solutions of such equations are rationalfunctions.

Key words: ultrametric, p-adic, q-difference equation, q-wronskian, Algebra

2010 Mathematics Subject Classification:Primary 12J25 .

50

IntroductionFor every prime number p, we denote by Qp the field of p-adic numbers and by Cp thecompletion of an algebraic closure of Qp, (cf. [1] for further details). More generally, inthe sequel, K is a complete ultrametric algebraically closed field.

Given R > 0, we denote by d(0, R−) and d(0, R) the disks: x ∈ K / |x| < R andx ∈ K / |x| ≤ R respectively. We denote by A(K) the K-algebra of entire functionsin K and by M(K) the field of meromorphic functions in K. In the same way, wedenote by A(d(0, R−)) the K-algebra of analytic functions inside the disk d(0, R−) andbyM(d(0, R−)) the field of meromorphic functions in d(0, R−).

For every r ∈]0, R[ we define a multiplicative norm | |(r) on A(d(0, R−)) by | f |(r) =supn≥0 |an|rn for every function f (x) = ∑n≥0 anxn of A(d(0, R−)). We extend this toM(d(0, R−)) by setting | f |(r) = |g|(r)/|h|(r) for every element f = g/h ofM(d(0, R−)),(cf. [5]). Let q be an element of K which is not a root of unity and is such that |q| = 1.In this work, we will first prove some inequalities linking the growth of a generalizedq-Wronskian to the growth of the ”ordinary” q-Wronskian.

We then apply this result to study some q-difference equations and show that:If a linear q-difference equation (E) with coefficients in K[x] has a complete system of solutionsconsisting of elements ofM(K), then any solution of (E) is a rational function.

This work has its origins in the articles [3] and [4] where it is established that, ingeneral, a differential equation with coefficients in K[x] could admit transcendental en-tire solutions. This study is continued in [2], where J. P. Bezivin gets rationality criteriafor solutions of some p-adic differential equations. Here, we study some q-differenceequations and show that several types of such equations have no solution except rationalfunctions. The method used is based on a comparison of the growth of q-Wronskians andclosely follows the one used in [2].

Main resultsFor n ∈N∗, we set [n] = (qn − 1)/(q− 1) and [n]! = ∏n

i=1[i], (we agree that [0]! = 1. Fork ∈N such that k ≤ n, we set[

nk

]= [n]!/([k]!)([n− k]!). (82)

We easily check that:

[n + 1

k

]=

[n

k− 1

]+ qk

[nk

]. (83)

We finally define the operators σq and Dq in K[[x]] by:

σq( f )(x) = f (qx) and Dq( f )(x) = (σq − Id)( f )(x)/(q− 1)x. (84)

The operator Dq is an endomorphism of the K-vector space K[[x]]. The operator σq is anautomorphism of the K-algebra K[[x]] and we have σ−1

q = σ1q. For k ∈ N∗, we denote

by σkq ( f ) (resp. Dk

q( f )) the application K times of the operator σq (resp. Dq) to the formalpower series f . We agree that σ0

q = D0q = Id, where Id is the identity mapping in K[[x]].

Some properties of these operators are summarized in the following Lemma:

51

Lemma 8 i) σq = (q− 1)xDq + Id, Dq = (1/q)D(1/q) σq,ii) Dk

q σ`q = qk`σ`

q Dkq, ∀k, ` ∈N,

iii) Dqx− xDq = σq, and Dqx− qxDq = Id,iv) Dq( f g) = (Dq f )(σqg)+ f (Dqg), ∀ f , g ∈ K[[x]], v) Dq(( f /g)) = (gDq f − f Dqg)/gσqg,

∀ f , g ∈ K[[x]], vi) Dnq ( f g)(x) = ∑n

k=0

[nk

]Dk

q( f )σkq Dn−k

q (g)(x), ∀ f , g ∈ K[[x]].

Let f1, · · · , fs, (s ≥ 1), be elements of K[[x]] and let k1, · · · , ks ∈N.

Definition 2 We call q-wronskian (or ordinary q-wronskian) of f = ( f1, · · · , fs) and we denote

by Wq( f ) the determinant of the matrix (Djq( fi))1≤i≤s,0≤j≤s−1.

Definition 3 We call generalized q-wronskian of f = ( f1, · · · , fs) relatively to k = (k1, · · · , ks)

and we denote by Wq( f ; k) the determinant of the matrix (Dk jq ( fi))1≤i≤s,1≤j≤s.

Remark 9 1) The ordinary q-wronskian of f = ( f1, · · · , fs) is equal to the generalized q-wronskian Wq( f ; ks) of f relatively to ks = (0, 1, ..., s− 1).2) More generally, let kj = (0, · · · , j, · · · , s) = (0, · · · , j − 1, j + 1, · · · , s) for every j ∈0, · · · , s. If we consider the usual derivation D = d/dx, we obtain a family of (usual) gen-eralized wronksians W( f ; kj) of f = ( f1, · · · , fs), for 0 ≤ j ≤ s . And we easily check :DW( f ; ks) = W( f ; k(s−1)).

Now, consider the family of generalized q-wronksians Wq( f ; kj) of f for 0 ≤ j ≤ s. Wesee that, for DqWq( f ; ks), we do not have an expression as simple as the one above. However,the following lemma allows us to express DqWq( f ; ks) as a combination of all the generalizedq-wronksians Wq( f ; kj) of f for 0 ≤ j ≤ s− 1.

Lemma 9 With the notations above, we have:Dq(Wq( f ; ks)) = ∑s−1

j=0 [(q− 1)x]s−1−jWq( f ; kj).

Let s ≥ 2 and let f1, · · · , fs be elements of K[[x]], linearly independent over K. Letus set f = ( f1, · · · , fs) and kj = (0, · · · , j, · · · , s) for 0 ≤ j ≤ s. Let us also set g =

( f1, · · · , fs−1), ì = (0, · · · , i, · · · , s− 1) for 0 ≤ i ≤ s− 1 and `(i,s−1) = (0, · · · , i− 1, i +1, · · · , s− 2, s) for 0 ≤ i ≤ s− 2. Recall that Wq( f ; kj) is the generalized q-wronskian of frelatively to kj, for 0 ≤ j ≤ s. In the same way, Wq(g; ì) is the generalized q-wronskian ofg relatively to ì, for 0 ≤ i ≤ s− 1. Finally, Wq(g; `(i,s−1)) is the generalized q-wronskianof g relatively to `(i,s−1), for 0 ≤ i ≤ s− 2. In the following lemma, the q-derivative ofthe q-wronskian is given by an expression which is better suited for the comparison ofthe growth of q-wronskians.

Lemma 10 With the notations above, we have for s ≥ 2:

i)Wq( f ;kj)

Wq( f ;ks)=

Wq(g;`j)

Wq(g;`(s−1))

Wq( f ;k(s−1))

Wq( f ;ks)− Wq(g;`(j,s−1))

Wq(g;`(s−1)), ∀ 0 ≤ j ≤ s− 2;

ii)Wq( f ;k(s−1))

Wq( f ;ks)=

Wq(g;`(s−1))

σqWq(g;`(s−1))

DqWq( f ;ks)

Wq( f ;ks)+ (∑s−2

i=0 [(q− 1)x]s−1−i Wq(g;`(i,s−1))

σqWq(g;`(s−1))).

Here, we only consider the case |q| = 1. Indeed the case |q| 6= 1 is more difficult andwill be treated later. Hence, from now on, we make this assumption: q is an element ofK which is not a root of unity and is such that |q| = 1.

52

GROWTH OF THE q-WRONKSIANSIn the following result, we give inequalities linking the growth of generalized q-wronskiansof a family of analytic functions to that of the of ordinary q-wronskian of this family offunctions.

Theorem 29 Let s be an integer ≥ 1 and let f1, · · · , fs be s elements of A(d(0, R−)). Letk1, · · · , ks be integers≥ 0. Let k = (k1, · · · , ks), and ks = (0, 1, · · · , s− 1). For every ρ ∈]0, R[,we have:i) |Wq( f ; k)|(ρ) ≤ |Wq( f ; ks)|(ρ)/ρk1+k2+···+ks− s(s−1)

2 .Particularly, for kj = (0, · · · , j, · · · , s), j = 0, · · · , s, we have:ii) |Wq( f ; kj)|(ρ) ≤ |Wq( f ; ks)|(ρ)/ρs−j.

Remark 10 The property |q| = 1 is used when it is stated that, for a meromorphic function ϕ, wehave |σq(ϕ)|(ρ) = |ϕ|(ρ), which is not true in general if |q| 6= 1. So, generalizing our resultsto any |q| is not at all clear and would require a deep change in the method of proof.

Now, we extend the result of Theorem 1 to meromorphic functions.

Corollary 6 Let f1, · · · , fs, be elements ofM(d(0, R−)) and let k1, · · · , ks be integers ≥ 0. Letf = ( f1, · · · , fs), k = (k1, · · · , ks) and ks = (0, · · · , s− 1). Then, we have for every ρ ∈]0, R[:

|Wq( f ; k)|(ρ) ≤|Wq( f ; ks)|(ρ)

ρ(k1+···+ks)− s(s−1)2

.

The following result gives an algebraic property of the q-wronskians of polynomialsor rational functions. Recall that if P(x), Q(x) are polynomials, then the algebraic degreeof the rational function R(x) = P(x)/Q(x) is dega R = deg P− deg Q.

Theorem 30 Let f1, · · · , fs, s ≥ 1, be elements of A(K), f = ( f1, · · · , fs) and ks =

(0, · · · , s− 1). Suppose that the q-Wronskian Wq( f , ks) is a nonzero polynomial. Then f1, · · · , fsare polynomials.

Corollary 7 Let L be a field and let q be a nonzero element of L different from any root of unity.Let Q1, · · · , Qs, s ≥ 1, be elements of L(x) linearly independent over L. Let Q = (Q1, · · · , Qs),ks = (0, · · · , s− 1) and k = (k1, · · · , ks), where k1, · · · , ks are integers ≥ 0. Let d1, d2 be thealgebraic degrees of the rational functions Wq(Q; ks) and Wq(Q; k) respectively. Then we have:

d2 ≤ d1 +s(s−1)

2 − (k1 + · · ·+ ks).

Remark 11 The previous result does not extend toM(K). Indeed, let g be a non-polynomialentire function and let h be an entire function such that Dqh = gσqg. Let f1 = 1/g, andf2 = h/g. We see that f1, f2 are non-rational meromorphic functions while the q-wronskian off1, f2 is equal to 1.

Theorem 31 Let P0, · · · , Ps, s ≥ 1, be elements of K[x] such that Ps 6= 0. Suppose that theequation: (E) PsDs

qy + · · ·+ P1Dqy + P0y = 0 has a complete system of solutions in A(K).Then every entire solution of (E) is a polynomial.

We can now generalize the above result toM(K):

Theorem 32 Let P0, · · · , Ps, s ≥ 1, be elements of K[x] such that Ps 6= 0. Suppose that theequation: (E) PsDs

qy + · · ·+ P1Dqy + P0y = 0 has a complete system of solutions inM(K).Then every solution of (E) is a rational function.

53

ConclusionAdopting q-Wronskien and q-derivation definition we obtain remarkable results on q-difference equations which were completed by applications on number theory.References

1. Y. AMICE, Les nombres p-adiques, P.U.F., 1975. 2. J.-P. BEZIVIN, Wronskien etequations differentielles p-adiques, Acta Arithmetica 158 no 1 (2013), pp.61–pp.78

3. A. BOUTABAA, On some p-adic functional equations, Lecture Notes in Pure andApplied Mathematics no 192 (1997), pp.61–pp.78

4. A. BOUTABAA, A note on p-adic linear differential equations, J. of Number theoryno 87 (2001), pp.301–pp.305

5.A. ESCASSUT, Analytic elements in p-adic analysis,W.S.P.C. Singapore, 1995.

Quadratic error of the conditional mode function in the locallinear estimation for functional data

Wahiba BouabsaSidi Bel Abbes University

In this paper we investigate the asymptotic mean square error and the rates ofconvergence of the estimator based on the local linear method of the conditionalmode function. Under some general conditions, the expressions of the bias andvariance are given. The efficiency of our estimator is evaluated through a simula-tion study. We proved, theoretically and on the scope of a simulation study, thatour proposed estimator has better performance than the estimator based on thestandard kernel method.

Key words: Nonparametric local linear estimation,mean squared error, conditional

mode function, functional variable.


IntroductionThe problem of statistic overelaborate in the modeling of functional random variableshas recognized increasing holdings in late literature (see for instance for nonparamet-ric context Niang(2002), Niang(2003), Masry(2005), Ferraty(2004), Attouch(2013), andBouabsa(2017), Bouabsa(2018).Many multivariate statistical techniques, concerning parametric models, have been ex-tended to functional data and a good review on this topic can be found in Ramsay andSilverman (2005) or Bosq (2000). Recently, new studies have been carried out in order topropose nonparametric methods taking into account functional data. For a more com-prehensive review on this subject the reader is referred to Ferraty and Vieu (2006) and toFerraty and Vieu (2002) for specialized monographs.

In this functional part, the prime results about the conditional mode estimation getby [Ferraty2006a]. They founded in the i.i.d. case the almost complete convergence of the

54

kernel estimator. The dependent case was established by [Ferraty2005], and in the caseof both cases i.i.d. and strong mixing, [Ezzahrioui2006b], [Ezzahrioui2006a] have estab-lished the asymptotic normality of the kernel estimator of the conditional mode. An im-portant collection of statistical tools for nonparametric prediction of functional variableswas presented in the monograph of Ferraty(2006b). Lately, the convergence in Lp norm ofthe condition al mode function in the independent case was stated by Niang(2007), thenthe nonparametric estimation of the conditional mode using the k-Nearest Neighbors forindependent functional data was deal by Attouch(2013), the conditional mode by the k-NN method for dependence case was studied by Attouch(2018).However, it is well known that a local polynomial smoothing procedure has many advan-tages over the kernel method (see, Fan and Yao (2003) and Fan and Gijbels (1996), etc.).In particular, the former method has better properties, in terms of bias estimation. Thelocal linear smoothing in the functional data setting has been considered by many au-thors. The first results on the regression function were established in Ba‘yllo and Grane(2009), Boj et al. (2010), Berlinet et al. (2011) and El methni and Rachdi (2011). Otherworks have been realized on this subject, for example Barrientos-Marin et al. (2010) de-veloped a smoothing local linear estimation of the regression operator for independentdata. Moreover, Demongeot et al. (2010) established the almost complete consistency oflocal linear estimator of the conditional density when the explanatory variable is func-tional and the observations are i.i.d. The mean squared error of the last estimator wasstudied by Rachdi et al. (2014). The asymptotic properties (almost complete convergenceand convergence in mean square, with rates) of the local linear estimator of the condi-tional cumulative distribution were established by Demongeot et al. (2014).This work deals with the functional nonparametric estimation of the hazard and/or theconditional hazard function. Historically, this function was first introduced by Watsonand Leadbetter (1964). Since then, several results have been added by many authors. Forexample Roussas (1989). States that there is extensive literature on nonparametric esti-mation of the conditional hazard function using a wide variety of methods. This functionis important in a variety of fields such as Medicine, Reliability, Survival Analysis or Seis-mology, etc.

Main results

Model and estimator

Let us consider a sequence (Xi; Yi)i ≥ 1 of independent and identically random pair ac-cording to the distribution of the pair (X, Y), all of them defined on the same probabilityspace (Ω, A, P) and taking their values in a space F ×R, where (F , d) is a semi-metricspace.

We suppose that F ×R is endowed with the product -algebra of the Borel σ- algebrasB(F ) and B(R) on F and on R respectively. For a fixed x ∈ F , we denote by Fx theconditional cumulative distribution function (cdf) of Y given (X = x) and we

suppose that Fx is absolutely continuous with respect to the Lebesgue measure withRadon-Nikodym derivative f x, which is the conditional probability density function(pdf) of Y given (X = x). Accordingly, the conditional mode function (cmf) of Y , given X= x, is

55

f x(θ) = supy∈S

f x(y). (85)

We suppose that f x(.) has a only mode, noted by θ(x) assumed uniquely defined in thecompact set S which is given by

f x(θ(x)) = supy∈S

f x(y). (86)

θ(x) is a kernel estimator of the conditional mode which given as the random variableθ(x) that maximizes the kernel estimator f x(.) of f x(.).

f x(θ(x)) = supy∈S

f x(y). (87)

By the fast functional local modeling (cf. Fan (1992)), the conditional cumulativedistribution function Fx(y) is estimated as the argmax value of a in the optimizationproblem, for each n ≥ 1, the following equation

Fx(y) = arg min(a;b)∈R2

n

∑i=1

(G(h−1G (y−Yi))− a− b β(Xi; x))2K(h−1

K δ(x; Xi)) (88)

where β(.; .) and δ(.; .) are locating functions defined from F 2 into R, such that:

∀ξ ∈ F ; β(ξ; ξ) = 0 and d(.; .) = |δ(.; .)|

and where the function K: kernel function, G: distribution function (df) and h = hK :=hK,n and hG = hG,n are suites of positive real numbers, as n goes to infinity goes to zero .Clearly, the estimator a, given by (88), can be explicitly written as follows:

Fx(y) =∑

1≤i,j≤nvij(x)G(h−1

G (y−Yi))

∑1≤i,j≤n

vij(x)∀y ∈ IR. (89)

where:vij(x) = βi(βi − β j)K(h−1

K δ(x, Xi)) with: βi = β(Xi, x) and the convention 0/0 = 0Below a differentiability assumption of G(.), we can get the conditional density estima-tion function as follows:

f x(y) =∑

1≤i,j≤nvij(x)G(1)(h−1

G (y−Yi))

hG ∑1≤i,j≤n

vij(x)∀y ∈ IR.

where G(1) is the derivative of G.

We need more notations and clear assumptions given below.

56

Notations and assumptionsHere and below, x (resp. y) will denote a fixed point in (F (resp.R), Nx (resp. Ny) willdenote a fixed neighborhood of a fixed point x (resp. of y) and φx(r1; r2) = P(r2 <σ(X; x) < r1). Then, we assume that our nonparametric model satisfies the followingconditions:

(H1) For any r > 0, φx(r) := φx(−r; r) > 0. There exists a function χx(.) such that:

∀t ∈ (−1, 1), limhK→0

φx(t hK; hK)

φx(hK)= χx(t)

(H2) The bandwidths hK and hG satisfylimn→∞

hK = 0, limn→∞

hG = 0 width limn→∞

n h(j)G φx(hK) = ∞, for j=0,1.

Main results: Mean Squared ConvergenceTheorem 33 Under assumptions (H1)-(H2), we obtain

E[ f x(y)− f x(y)] = B2f ;G(x; y)h4

G + B2f ;K(x; y)h4

K +V f

HK(x; y)n hGφx(hK)

+ o(h4G)+ o(h4K)+ o(1

n hGφx(hK))

(90)

ReferencesAttouch, M., Bouabsca, W., and Chiker el mozoaur, Z.,2018. The k-nearest neighbors

estimation of the conditional mode for functional data under dependency. InternationalJournal of Statistics & EconomicsDM, 19-1, 48-60.

Bouabsca, W. and Attouch, M., 2013. The k-nearest neighbors estimation of the con-ditional mode for functional data. Rev. Roumaine Math. Pures Appl,58, 4, 393-415.

Baıllo, A. and Grane, A., 2009. Local linear regression for functional predictor andscalar response. Journal of Multivariate Analysis, 100, 102-111. Barrientos-Marin, J., Ferraty,F. and Vieu, P., 2010. Locally modelled regression and functional data, Journal of Nonpara-metric Statistics, 22(5), 617-632. Boj, E., Delicado, P. and Fortiana, J. 2010. Distance-basedlocal linear regression for functional predictors, Computational Statistics and Data Analy-sis, 54, 429-437. Bosq, D. (2000). Linear Processes in Function Spaces: Theory and applica-tions, Vol. 1 of Lecture Notes in Statistics, pringer Science+Business Media, LLC, Inc.,233 Spring Street, New York,NY10013, USA. Fan, J. and Gijbels, I. 1996. Local PolynomialModelling and its Applications, Vol. 1 of Monographs on Statistics and Applied Probability,66,(Chapman & Hall, 2-6 Boundary Row, London SE1 8HN, UK).

Ferraty, F., Rabhi, A. and Vieu, P. 2008. Estimation non-parametrique de la fonction dehasard avec variable explicative fonctionnelle, Revue de Mathematiques Pures et Appliquees,53, 1-18.

Ferraty, F. and Vieu, P. 2006. Nonparametric Functional Data Analysis, Vol. 1 of SpringerSeriesin Statistics, Springer Science+Business Media, Inc., (233 Spring Street, New York,NY10013,USA).

57

The eikonal equation: some applications, existence anduniqueness results

Rachida BoudjeradaUSTHB

In this work, we study the eikonal equation by giving first some applicationsto this partial differential equation in its evolutive form and in the case of sta-tionary equations. Existence and uniqueness results are available in the case ofLipschitz solutions. Basing on a new BV estimate and a finite speed propagationproperty of this equation, we give global existence result for the eikonal equationin one space dimension with BV initial data.

Key words: Eikonal equation, front propagation, dislocations dynamics, discontinuousviscosity.

35A01, 74G25, 35F20, 35F21, 70H20, 35Q74

IntroductionThe eikonal equation is a first order Hamilton-Jacobi equation of the form

∂tu(x, t) = c(x, t)|∇u(x, t)| in Rn × (0,+∞)

u(x, 0) = u0(x) in Rn.(91)

This equation describes the evolution of interfaces with a non-signed normal velocity c,using the level-set method.An interesting motivation of our work, is the application to dislocation dynamics. Dislo-cations are linear defects in crystals, where theirs dynamics is one of the mean explana-tion physics of plastic deformation in crystals under exterior constraints. The modelingof this problem was established by R. Monneau and its collaborators in the paper [1].They obtain a non-local eikonal equation, with a non-local velocity given by

c(x, t) = c0(·, t) ? u(·, t)(x).

The convolution is taken in space and c0, is a kernel depending on the elastic propertiesof the crystal.We can also see the eikonal equation in its stationary form given by

|∇u(x)| = c(x) in Rn

u(x) = φ(x) on Γ,

where Γ ⊂ Rn. It is of general interest in fields such as computational geometric, multi-phase flow, path planing, ect. In the particular case when the speed function is a constant,usually equal to one, then the solution of the eikonal equation will represent the shortestdistance from a point x to the zero distance to the curve, given by Γ, that if φ(x) = 0.Many algorithms are used for computation of solutions of eikonal equation. There existother applications of the eikonal equation in geometric optic and image processing.

58

As the Hamilton-Jacobi equations, the natural framework to define solutions for theeikonal equation is the viscosity solutions theory introduced by Crandall and Lions in80′s (see [2], [5]). In this context, existence and uniqueness results are available only forthe case of Lipschitz velocity and initial data. In the general case, global existence anduniqueness results are difficult to obtain. We have considering in our work, the followingproblem

∂tu(x, t) = c(x, t)|∂xu(x, t)| in R× (0, T)

u(x, 0) = u0(x) in R,(92)

with the assumptions(H1) u0 ∈ L∞(R) ∩ BV(R)

(H2) c ∈ L∞(R× (0, T)) ∩ L∞((0, T); BV(R)).

Where BV(R), is the space of functions of bounded variation defined by

BV(R) = f ∈ L1loc(R) | TV( f ) < +∞,

with VT( f ) is the total variation, defined by

TV( f ) = sup∫

Rf (x)φ′(x)dx; φ ∈ C1

c (R) and ‖φ‖L∞(R) ≤ 1

.

Before giving our main result, let us recall the definition of discontinuous viscosity solu-tion for the eikonal equation (91).

Definition 4 (Discontinuous viscosity solutions)Assume that c and u0 are locally bounded in Rn × (0, T) and Rn, respectively.

(i) An upper semi-continuous function u (resp. a lower semi-continuous function v ) is a sub-solution (resp. super-solution) of (91), if u(x, 0) ≤ (u0)?(x) (resp. v(x, 0) ≥ (u0)?(x)) andfor all (x0, t0) ∈ Rn × (0, T) and all function test φ ∈ C1(Rn × (0, T)) such that u− φ has amaximum (resp. v− φ a minimum) at point (x0, t0), we have:

∂tφ(x0, t0)− c?(x0, t0)|∇φ(x0, t0)| ≤ 0

(resp. ∂tφ(x0, t0)− c?(x0, t0)|∇φ(x0, t0)| ≥ 0).

(ii) A locally bounded function w is a discontinuous viscosity solution of (91), if its upper semi-continuous envelope w? (resp. its lower semi-continuous envelope w?), is a viscosity sub-solution(resp. a viscosity super-solution).

Main resultsBefore giving our result, let us recall the semi-limits defined by Barles and Perthame [3]as follows

u(x, t) = lim sup? uε(x, t) = lim supε−→0

(y,s)−→(x,t)

uε(y, s) (93)

andu(x, t) = lim inf? uε(x, t) = lim inf

ε−→0(y,s)−→(x,t)

uε(y, s) (94)

59

Using a new BV estimate and finite speed propagation of the solution, we have obtainedin [4], a global existence result of weak discontinuous viscosity solution, given in thefollowing Theorem

Theorem 34 (Global existence result of problem 92)Assume that (H1) and (H2) are satisfied. Then, there exists a discontinuous viscosity sub-solution u and a discontinuous super-solution u of (92) in L∞(R× (0, T))∩ L∞((0, T); BV(R),satisfying for all t ∈ (0, T), the following equality

u(·, t) = u(·, t) except at most on a countable set in R. (95)

Moreover, if the initial data u0 is non-decreasing then, problem (92), admits a discontinuousviscosity solution in sense of Definition 4.

To prove this result, we use a parabolic regularization of the eikonal equation and weprove Lipschitz estimate on the approximate solution and a BV estimate. These esti-mates allows us to pass to the limit when parameters go to zero. Using stability result ondiscontinuous viscosity solutions [2], we obtain a sub- and super solution that we iden-tify on R× (0, T) expect a countable set, using fine properties of functions of boundedvariations in one dimensional space.Recently, this new BV estimate was used to prove global existence result of coupled non-local systems modeling dynamics of dislocations densities [6].

ConclusionWe have obtained global existence of weak discontinuous viscosity solution result for theeikonal equation with unsigned velocity and BV initial data. But the uniqueness remainsan open problem. Numerical schemes are also interesting problem for this equation.References

1. O. Alvarez, P. Hoch, Y. Le Bouar, and R. Monneau, Dislocation dynamics: short-time existence and uniqueness of the solution, Arch. Ration. Mech. PP. 449-504.

2. G.Barles, Solutions de viscosite des equations de Hamilton-Jacobi, Mathematicsand Applications. Springer-Verlag, 1994.

3. G. Barles and B. Perthame, Exit time problems in optimal control and vanishingviscosity method, SIAM J. Control Optim., 26 (1988), pp. 1133–1148.

4. R. Boudjerada and A. El Hajj, Global existence results for eikonal equation with BVinitial data, NoDEA Nonlinear Differential Equations Appl., 22 (2015), pp. 947-978

5. M. G. Crandall and P.-L. Lions, On existence and uniqueness of solutions of Hamilton-Jacobi equations, Nonlinear Anal., 10 (1986), pp. 353-370.

6. H. R. V. El Hajj, Ahmad; Ibrahim, Global bv solution for a non-local coupled systemmodeling the dynamics of dislocation densities, Journal of Differential Equations, (2017).

60

Tail probabilities and complete convergence for weightedsequences of LNQD random variables with application to

first-order autoregressive processes modelBoulenoir Zouaouia

University djilali liabes

In this work, a new concentration inequality and complete convergence ofweighted sums for arrays of row-wise linearly negative quadrant dependent (LNQD,in short) random variables has been established, we also obtained a result deal-ing with complete convergence of first-order autoregressive processes with iden-tically distributed LNQD innovations.

Key words: Autoregressive process, complete convergence, tail probabilities, LNQD se-quence, weighted sums.

2000 Mathematics Subject Classification: 60B, 60F05, 60F17, 60F15, 60G10.

IntroductionLe concept de convergence complete d’une suite de variables aleatoires a ete introduitpar [?] comme suit. Une suite Xn, n ≥ 1 de variables aleatoires converge completementvers la constante C si

∞

∑n=1

P(|Xn − C| > ε) < ∞ pour tout ε > 0.

Par le lemme de Borel-Cantelli, cela implique Xn → C presque sure (a.s.), et l’implicationreciproque est vraie si les Xn, n ≥ 1 sont independantes. [?] ont prouve que la suitedes moyens arithmetiques des variables aleatoires independantes et identiquement dis-tribuees (i.i.d.) convergent completement vers la valeur estimee si la variance des sommesest finie. [?] a prouve l’inverse.

La convergence complete pour une suite de variables aleatoires joue un role im-portant dans le domaine des theoremes limite dans la theorie des probabilites et statis-tiques mathematiques. les conditions d’independance et distribution identique de vari-ables aleatoires sont fondamentales dans les resultats donnes par Bernoulli, Borel et Kol-mogorov. Par la suite, plusieurs tentatives ont ete faites pour assouplir les conditionsfortes. On rappele que la matrice Xni, 1 ≤ i ≤ n, n ≥ 1 de variables aleatoiresest dite domine stochastiquement dominee par une variable aleatoire positive X (ecritXni ≺ X) s’il existe une constante C > 0 telle que

P(|Xni| > t) ≤ CP(X > t) ∀ t > 0 , n ≥ 1, 1 ≤ i ≤ n. (96)

Main resultsSoit Xni, 1 ≤ i ≤ n, n ≥ 1 une suite de variables aleatoires definies sur un espace deprobabilite fixe (Ω,z, P) et cn, n ≥ 1 est une suite strictement croissante de constantes

positives. Notons Sn =n∑

i=1Xni et B2

n =n∑

i=1E(X2

ni) pour tout 1 ≤ i ≤ n et n ≥ 1.

61

Soit: Xni = X1,ni + X2,ni + X3,ni, 1 ≤ i ≤ n, et n ≥ 1 tel que X1,n1, X1,n2, . . . , X1,nn sontdes variables aleatoires bornees ∀n ≥ 1, avec

X1,ni = −anIxni<−an + XniI|Xni |≤an + anIXni>anX2,ni = (Xni − an)IXni>anX3,ni = (Xni + an)IXni<−an

(97)

Theorem 35 Soit Xni; 1 ≤ i ≤ n, n ≥ 1 est un vercteur ligne de variables aleatoires iden-tiquement distribuees LNQD et Xp,ni, 1 ≤ i ≤ n, n ≥ 1, p = 2, 3, definies par (97).

On suppose qu’il existe un τ > 0 qui verifie sup|µ|≤τ

E(eµX11) ≤ Aτ < ∞ ou Aτ > 0,n

∑i=1

b2ni =

O((log n)−1).Alors ∀ε > 0 et µ ∈ (0, τ] on a:

P

(|

n

∑i=1

bni(Xp,ni −E(Xp,ni)) |≥ ε

)≤ Φ(µ, ε, τ, a)

1n

a2 log n

, p = 2, 3. (98)

ou: Φ(µ, ε, τ, a) = 2a+1aae−aDD′Aτ

µ2+aKa1(E(X11)2)a/2ε2 en choisissant a > 2.

Theorem 36 . Soit Xni, 1 ≤ i ≤ n, n ≥ 1 un vecteur ligne de variables aleatoires identique-ment distribuees LNQD telle que E(Xni) = 0 avec E|X11|γ+1 < ∞ ∀γ ≥ 1, on suppose quebni, 1 ≤ 1 ≤ n, n ≥ 1 est un vecteur de constantes qui verifie :

max1≤i≤n

|bni| = O(c−δn ), 0 < cn ∞, ∀δ > 0, et, anc−δ

n ≤ 1, (99)

etn

∑i=1

b2ni = O

(1

log(n)

). (100)

Alorsn

∑i=1

bniXni converge completement vers zero quand n tend vers l’infini.

Application au modele auto-regessif d’ordre unOn considere une serie chronologique autoregressive de premier ordre AR(1) definie par:

Xn+1 = θXn + ςn+1; n = 1, 2, . . . . (101)

ou ςn, n ≥ 0 est une suite de variables aleatoires LNQD identiquement distribueesavec: ς0 = X0 = 0 et 0 < E(ς4

k) < ∞, k = 1, 2, . . . et θ est un parametre avec: |θ| < 1.On peut ecrire Xn+1 dans l’equation (101) comme suit:

Xn+1 = θn+1X0 + θnς1 + θn−1ς2 + . . . + ςn+1. (102)

On estime θ par la methode de moindres carres:

θn =

n

∑j=1

XjXj−1

n

∑j=1

X2j−1

. (103)

62

D’apres les equations (101) et (103), on obtient:

θn − θ =

n

∑j=1

ς jXj−1

n

∑j=1

X2j−1

. (104)

Theorem 37 Si ςn, n ≥ 1 est une suite de variables aleatoires LNQD identiquement dis-

tribuees tel que |ς1|4 < α, alors ∀R ∈ R , ε >E(ς2

1)R2 et 0 < β < α

eα−α−1 on a:

P

(|

n

∑j=1

(ς2j −E(ς2

j )) |≥ n(R2 ε−E(ς21))

)≤ 2 exp−β

n(R2 ε−E(ς21))

2

36+ 2

Φ(ε, τ, a)n

a2+1

(105)ou :

Φ(ε, τ, a) = 92a+1aae−aDDAτ

µa+2Ka1(E(ς4

1))a/2(R2 ε−E(ς2

1))2

.

Theorem 38 Sous les conditions du theoreme 37, on a pour tout (E(ς21))

1/2

R2 < ε positive:

P(√

n|θn − θ| > R) ≤ 2 exp−βn(R2 ε−E(ς2

1))2

36+ 2

Φ(ε, τ, a)na/2+1

+ exp−12

n(T1 − nε2)2

T2 (106)

ou: T1 = E(X21) < ∞ et T2 = E(X4

1) < ∞.

Corollary 8 La suite (θn)n∈N defini dans (104) converge completement vers le parametre θ duprocessus auto-regressif d’ordre 1.

Remark 12 L’inegalite (106), donnent la possibilite de construire un intervalle de confiance pourle parametre du processus auto-regressif du premier ordre. Pour R est large, tels que R = ε

√n,

qui suit:

limn→+∞

Un = 2exp−β

(nε4 −E(ς21))

2n36

+ 2

φ(ε, τ, a)na/2+1 + exp

−1

2n(T1 − nε2)2

T2

= 0.

Ce qui signifie que, pour un niveau $ donne, on peut trouver un entier naturel n$ tel que :

∀n ≥ n$ on a Un ≤ $. (107)

Par consequent,P(|θn$ − θ| < ε) ≥ 1− $ (108)

Ce qui signifie que le parametre du processus auto-regressif du premier ordre appartient a l’intervalleinclusif du centre θn$ et du rayon ε avec une probabilite superieure ou egale a 1− $.

63

ReferencesHsu, P. L. and Robbins, H.. Complete convergence and the law of large numbers. Proceed-

ings of the National Academy of Sciences, USA, Numero: 25–31, 1947.Erdos, P.On a theorem of Hsu and Robbins.Ann. Math. Statist, Numero: 286–291, 1949.Wang, X. and Rao, M. B. and Yang, X. Convergence rates on strong laws of large numbers

for arrays of rowwise independent elements.Stochastic Anal. Appl, Numero: 105–132, 1993.Ahmed, S. E. and Antonini, R. G. and Volodin, A. On the rate of complete convergence

for weighted sums of arrays of Banach space valued random elements with application to movingaverage processes.Statist. Probab. Lett., Numero: 185–194, 2002.

AVERAGING FOR DIFFERENTIAL INCLUSIONSAmel Bourada, Mustapha Lakrib

Mustapha Stambouli University, Djilali liabes University

We consider ordinary differential inclusions and we state some averaging re-sults for these inclusions. Our results are proved under weaker conditions thanthe results in the literature.

Key words: Ordinary differential inclusions, averaging method.2010 Mathematics Subject Classification: 34A60, 34C29.

IntroductionDans ce travaille nous montrons et discutons quelques resultats de moyennisation pourcertaines inclusions differentielles ordinaires perturbees sous des conditions plus faiblesque celles de la litterature.

Il est consacre a la justification de la methode de moyennistation pour les IDO rapi-dement oscillantes qui s’ecrivent sous la forme:

x ∈ εF (t, x) , x(0) = x0. (109)

Nous exposons notre contribution qui est basee essentiellement sur l’affaiblissementdes conditions de regularite sur la fonction multivoque F. Nous discutons alors de cesconditions et de leur role dans l’obtention des resultats proposes.

Main resultsNous considerons des inclusions differentielles ordinaires et nous affirmons une certainemoyenne pour ces inclusions . Nos resultats sont prouves dans des conditions plusfaibles que les resultats de la litterature.

Soit U un ensemble ouvert de Rd, x0 ∈ U et F : R+ × U → Conv(Rd) une fonctionmultivoque.

Soit ε > 0 un petit parametre.

x(t) ∈ F(

tε, x(t)

), x(0) = x0 (110)

sur l’intervalle de temps fini t ∈ [0, L].on considere les hypotheses suivantes :

64

(H1) Pour tous x ∈ U, la fonction multivoque F(., x) : R+ → Conv(Rd) est mesurable.

(H2) Pour tous t ∈ R+, la fonction multivoque F(t, .) : U→ Conv(Rd) est continue.

(H3) La continuete de la fonction multivoque F on la seconde variable est uniforme enrespectant la premiere.

(H4) Il existe une fonction localement Lebesgue integrable b : R+ → R+ et une con-stante B > 0 telle que

‖F(t, x)‖ ≤ b(t), ∀t ∈ R+, ∀x ∈ U

avec

limT→∞

1T

∫ T

0b(t)dt = B

(H5) Pour tous x ∈ U il existe une limite

limT→∞

1T

∫ T

0F(τ, x)dτ =: F(x)

(H6) Il existe une constante λ > 0 telle que pour les fonctions continues u, v : R+ → Uet t1, t2 ∈ R+, t1 ≤ t2,

ρ

(∫ t2

t1

F(τ, u(τ))dτ,∫ t2

t1

F(τ, v(τ))dτ

)≤ λ

∫ t2

t1

|u(τ)− v(τ)|dτ. (111)

On consiere le probleme (110) avec le probleme moyennise

y(t) ∈ F(y(t)), y(0) = x0 (112)

Theorem 39 Soient F : R+ ×U → Conv(Rd) une fonction multivoque et x0 ∈ U. Supposonsque les conditions (H1 )-(H5) sont satisfaites. Alors, pour tous L > 0 et µ > 0, il existe ε0 =ε0(x0, L, µ) > 0 tels que, pour tout ε ∈]0, ε0] et toues solutions xε de (110) definie sur [0, L], ilexiste une solution y de (112) tel que y est definie sur [0, L] et verifie |xε(t) − y(t)| < µ pourtout t ∈ [0, L].

Nous rappelons que sous des hypotheses (H2)-(H3) il est seulement possible d’obtenirunilaterale approximations, soit le rapprochement des solutions aux problemes (110) parceux de la moyenne de probleme (112). l’approximation inverse est, en generale, fausse,meme pour les equations differentielles ordinaires, comme il est montre dans l’exemplesuivant:

Exemple On considere le probleme a valeur initiale suivant

x(t) =√

x(t) + sin(tε) x(0) = 0. (113)

le probleme moyennise associe est

y(t) =√

y(t) y(0) = 0 (114)

Il n’y a aucune solution de probleme (113) approxime la solution triviale y(t) ≡ 0 duprobleme moyennise (114).

Si le probleme (112) admet une solution unique on a le resultat interesant suivant.

65

Theorem 40 Soient F : R+ ×U → Conv(Rd) une fonction multivoque et x0 ∈ U. Supposonsque les conditions (H1 )-(H5) sont verifiees. Supposons que le probleme a valeur initiale (112)admet une solution unique. Soit y la solution (unique) de (112). Alors pour tous L > 0 tel que yest definie sur [0, L] et tous µ > 0, il existe ε0 = ε0(x0, L, µ) > 0 tels que, pour tous ε ∈]0, ε0],tous solution xε de (110) est definie sur [0, L] et verifie |xε(t)− y(t)| < µ pour t ∈ [0, L].

Si dans le theoreme 39 on considere les hypotheses (H1),(H4)-(H6) la conclusion dutheoreme 39 reste vrai (voir (ii) dans le theoreme 41 si dessous) et en outre nous obtenonsle deuxieme resultat principale qui etablit la proximite de solutions des problemes (110)et (112) dans des intervalles de temps finis.

Theorem 41 Soient F : R+ ×U → Conv(Rd) une fonction multivoque et x0 ∈ U. Supposonsque les conditions (H1 ), (H4)-(H6) sont verifie. Alors, pour tous L > 0 et µ > 0, il existeε0 = ε0(x0, L, µ) > 0 tels que, pour tous ε ∈]0, ε0] les conditions suivantes sont verifies:

(i) pour toutes solutions xε de (110) qui est definie sur [0, L], il existe une solution y de (112)tel que y est definie sur [0, L] et satisfait |xε(t)− y(t)| < µ pour tout t ∈ [0, L]

(ii) pour toute solution y de (112) qui est definie sur [0, L], il existe une solution xε de (110) telque xε est definie sur [0, L] et satisfait |xε(t)− y(t)| < µ pour tous t ∈ [0, L].

La preuve du theoreme 40 est basee sur le resultat du theoreme 39 et la propriete decontinuete des solutions des inclusions differentielles ordinaires.

ConclusionNous demontrons un resultat de moyennisation pour les IDO. Les conditions que noussupposons sont plus generales que celles imposees dans la litterature, en particulier lesresultats de Plotnikov sur la justification de la methode de moyennisation dans le cadredes inclusions differentielles ordinaires.

En fait, au lieu de la condition de Lipschitz sur F, nous supposons que c’est sonintegrale qui verifie une condition de Lipschitz et au lieu de la condition d’uniformebornitude de la fonction f nous supposons qu’elle est bornee par une fonction admettantune moyenn.References

J. P. Aubin and A. Cellina, Differential Inclusions, Springer-Verlag, Berlin, 1984.A. Bourada, R. Guen, M. Lakrib and K. Yadi, Some Averaging Results for Ordinary

Differential Inclusions, Disc. Math. Diff. Incl., Cont. and Opt. 35 (2015) pp. 47-63.K. Deimling, Multivalued Differential Equations, De Gruyter, Berlin, 1992.S. Klymchuk, A. Plotnikov and N. Skripnik, Overview of V. A. Plotnikov’s research on

averaging of differential inclusions, Physica D: Nonlinear Phenomena, 241, No. 22 (2012),pp. 1932-1947.

M. Lakrib, On the averaging method for differential equations with delay, Electron. J.Diff. Eqns. 2002, No. 65 (2002), pp. 1-16.

M. Lakrib, Stroboscopie et moyennisation dans les equations differentielles fonctionnelles aretard. These de Doctorat en Mathematiques, Universite de Haute Alsace, Mulhouse,2004 (http://tel.archives-ouvertes.fr/tel-00444149/fr).

M. Lakrib, An averaging theorem for ordinary differential inclusions, Bull. Belg. Math.Soc. Simon Stevin 16, No. 1 (2009), pp. 13-29.

66

N. V. Skripnik, Averaging of impulsive differential inclusions with Hukuhara deriva-tive, Neliniini Koliv, 10, No. 3 (2007), 416-432 (Russian); translation in Nonlinear Oscil. 10 ,No. 3 (2007), pp. 422-43.

G. V. Smirnov, Introduction to the Theory of Differential Inclusions, Graduate Studies inMath. 4, American Math. Soc. Providence, 2002.

Integer/decimal parts decomposition of a standard Gaussiandistribution

Bourada Sofiane, Tlemcani MounirUniversite Oran1 Ahmed Benbella, Universite des sciences et de la technologie d’Oran,

Mohamed-Boudiaf

In this work, we focus on the random character of the digits representing aGaussian random variable, namely its integer part and its decimal part. At firstsight we are interested in the probability law that governs this integer part. Forthe decimal part it will be seen that for relatively large values of the standarddeviation, this continuous random variable on (0, 1) can be considered to be uni-form, the key in establishing this result is the summation formula of Poisson,approximate formulas are established on probabilistic considerations includinga piece-wise linear approximation of the distribution function of a reduced cen-tered Gaussian.

Key words: Integer part, decimal part, Gaussian distribution, approximation by decom-position

2010 Mathematics Subject Classification: 60G50,60E07,62G07

IntroductionIn the classical theory of probability, known also as the arithmetic of probability distri-bution e.g. [?], authors deal with representation formulas for a random variable X asthe sum of two independent random variables Y and Z having computationally simplecharacteristic functions ΦY, ΦZ in such a way that the characteristic function of X can berecovered easily by ΦX = ΦYΦZ. In this paper, we study the random character of thedigits representing a Gaussian random variable, namely its integer part and its decimalpart. The result provides impressively simple and accurate approximation formula forthe standard normal variable and are promising for other distributions as the exponentialone which is very useful in reliability and engineering.

Regression relation between a standard normal distri-bution and its integer partLet X a standard normal random variable (r.v) whose distribution function is denoted byΦ. The r.v Z = bXc the integer part of X has a discrete distribution function defined by

67

PZ(n) = Φ(n + 1)−Φ(n), n ∈ Z, which inherit the discrete version symmetry propertyof a normal distribution,

PZ(−n) = PZ(n− 1) (115)

Since the probability density function (p.d.f ) ϕ of X is decreasing on the positive half-line,one has by the mean value theorem, for all n ∈N,

PZ(n) = Φ(n + 1)−Φ(n) =1√2π

∫ n+1

ne−

x22 dx < ϕ(n).

It follows that Z admits moments at any order,

E(Zs) = ∑n∈Z

nsP(Z = n) =∞

∑n=0

nsP(Z = n) +∞

∑n=1

(−n)sP(Z = −n),

where convergence of the two series comes from the standard following limit resultlimn→+∞ nr · ϕ(n) = 0, ∀ r ≥ 0. In particular, for s = 1, 2 one may obtain

E(Z) = −+∞

∑n=0

P(Z = n) = −Φ(0) = −12

, V(Z) = E(Z2)−E(Z)2 =+∞

∑n=0

(2n2 + 2n)P(Z = n)+14

Approximately, it can be shown that V(Z) ∼= 1.0833, i.e. σZ ∼= 1, 0408.By conditional cumulative distribution function (cdf ) of X which can be written as

P(X < x|Z = n) =P(n ≤ X < x)

P(Z = n)= 1[n;n+1[

Φ(x)−Φ(n)P(Z = n)

(116)

one can easily deduce the conditional expectation

E(X|Z = n) =ϕ(n)− ϕ(n + 1)

P(Z = n). (117)

We have, thanks to conditional expectation properties and the fact that X is centered,

cov(X, Z) = E(XZ) = EE(XZ|Z) = E(ZE(X|Z)) (118)

which gives

cov(X, Z) = ∑n∈Z

nE(X|Z = n)P(Z = n)

= ∑n∈Z

n(ϕ(n)− ϕ(n + 1)) ∼= 1, (119)

On the other hand, we have the correlation coefficient

ρ =cov(X, Z)

σZ · σX∼= 0, 9607

which suggests strongly that X ∼= a + b · Z with the regression coefficients

b = ρ · σX

σZ=

cov(X, Z)σ2

Z, a = E(X)− b · E(Z)

or numerically,X ∼= 0, 461538 + 0, 923076 · Z (120)

It follows, thanks to the exact expression of E(X|Z = n) given by (117), that

P(Z = n) ∼=ϕ(n)− ϕ(n + 1)

a + b · n = B(n) (121)

which yields effectively a good approximation of PZ(n) (See some numerical valuesin table(2)).

68

Decimal part of a standard GaussianTo improve (120), we can see that the slope of this Regression Line is close to 1 and thatfor Z = 0 one has X ∼= 1

2 which is in fact the expected value of a uniform variable Udefined as the decimal part of X, as U = X− Z in [0; 1[.

Proposition 5 The pdf of the random variable U is given by

fU(u) = ∑n∈Z

ϕ(n + u) ∀u ∈ [0; 1[ (122)

and one hasE(U) =

12

, V(U) ∼=112

Proof 3 This follows from the conditional cdf

P(U < u|Z = n) = P(X− Z < u|Z = n) =Φ(n + u)−Φ(n)

P(Z = n)

which yields the conditional pdf of U by derivation,

f (u|Z = n) =ϕ(n + u)P(Z = n)

(123)

and the proposition follows thanks to total probabilities formula. The expectation and varianceresults also follow from E(U) = E(X)− E(Z) and V(U) = V(X)− 2cov(X, Z) + V(Z) ∼=0, 0833 ∼= 1

12 .

A rigorous proof of this proposition can be done using discrete Fourier transformand Poisson summation formula. As a result, the decimal part of X behaves closely as arandom uniform variable U ([0; 1]).

Approximation of a standard Gaussian by decomposi-tionIt follows now from the two previous sections and assuming independence between Zand U, that one can propose the approximation formula for the cdf fonctions

Φ(x) ∼= FU+Z(x).

Indeed, for all x > 0, one has

P(U + Z < x) = Φ([x]) + (x− [x]) · P(Z = [x])

which is not a first order Taylor expansion of Φ(x) around [x] the integer part of x. This isactually a simple and acceptable approximation of Φ which can be easily used to remem-ber (or recalculate) values in a standard normal distribution TABLES as in the literature(see Table [FUV] and Table (4)) in a very simple way. In order to evaluate the precision,let us denote n = bxc and consider for all x ∈ [n; n + 1[,

ψn(x) = Φ(x)−Φ(n)− (x− n) · P(Z = n),

69

We have

ψ′n(x) = ϕ(x)− P(Z = n) = 0⇒ x = xn =

√ln

12πP2(Z = n)

,

in addition, ψ′′n (x) = −x · ϕ(x) < 0 for x > 0 , hence ψn has a unique maximum atx = xn. Moreover ψn(n) = ψn(n + 1) = 0 and ψ′n(x) = ϕ(n) − P(Z = n) > 0, yieldψn(x) ≥ 0, for all x ∈ [n; n + 1[.

In summary,

0 ≤ Φ(x)−Φ(n)− (x− n) · P(Z = n) ≤ ψn(xn)

and by relation Φ(−x) = 1−Φ(x), we conclude that one can use the approximation forall x ∈ R.References

A-L.J Luis An Approximation to the Probability Normal Distribution and its Inverse ,Ingenierıa Investigacion y Tecnologıa, Volume 16, Issue 4, October–December 2015, Pages605-611

J. Hoffmann-Jørgensen et al. Strong Decomposition of Random Variables, J TheorProbab (2007) 20: 211–220, DOI 10.1007/s10959-007-0061-6

A. A. Kulikovaa, Yu. V. Prokhorovb ( Distribution of the fractional parts of randomvectors: The Gaussian case. I, Teor. Veroyatnost. i Primenen., 2003, Volume 48, Issue 2,Pages 399–402 R.J.G. Wilms Fractional parts of random variables : limit theorems and in-finite divisibility, Department of Mathematics and Computer Science, Phd Thesis 1 (1994)

Terminal Value Problem for Differential Equations withHilfer–Katugampola Fractional Derivative

Mouffak Benchohra, Soufyane BouriahDjillali Liabes University, Hassiba Benbouali University

We present in this work the existence results and uniqueness of solutions fora class of boundary value problems of terminal type for fractional differentialequations with the Hilfer–Katugampola fractional derivative. The reasoning ismainly based upon different types of classical fixed point theory such as the Ba-nach contraction principle and Krasnoselskii’s fixed point theorem. We illustrateour main findings, with a particular case example to show the applicability of ouroutcomes.

Key words: Hilfer–Katugampola fractional derivative; terminal value problem; exis-

tence; uniqueness; fixed point.2010 Mathematics Subject Classification:26A33.

IntroductionIt is well known [LAKSH] that the comparison principle for initial value problems of or-dinary differential equations is a very useful tool in the study of qualitative and quanti-tative theory. Recently, attempts have been made to study the corresponding comparison

70

principle for terminal value problems (TVP) [HALL].Motivated by the works above, we establish in this paper existence and uniqueness re-sults to the terminal value problem of the following Hilfer–Katugampola type fractionaldifferential equation:(

ρDα,βa+ y

)(t) = f

(t, y(t),

(ρDα,β

a+ y)(t))

, for each , t ∈ (a, T], a > 0 (124)

y(T) = c ∈ R, (125)

where ρDα,βa+ is the Hilfer–Katugampola fractional derivative (to be defined below) of or-

der α ∈ (0, 1) and type β ∈ [0, 1] and f : (a, T] ×R×R → R is a given function. Toour knowledge, no papers on terminal value problem for implicit fractional differentialequations exist in the literature, in particular for those involving the Hilfer–Katugampolafractional derivative.

PreliminariesIn this part, we present notations and definitions that we will use throughout this paper.Let 0 < a < T, J = [a, T]. By C(J, R) we denote the Banach space of all continuousfunctions from J into R with the norm:

‖y‖∞ = sup|y(t)| : t ∈ J.

We consider the weighted spaces of continuous functions:

Cγ,ρ(J) =

y : (a, T]→ R :(

tρ − aρ

ρ

)γ

y(t) ∈ C(J, R)

, 0 ≤ γ < 1,

and

Cnγ,ρ(J) =

y ∈ Cn−1(J) : y(n) ∈ Cγ,ρ(J)

, n ∈N,

C0γ,ρ(J) = Cγ,ρ(J),

with the norms

‖y‖Cγ,ρ = supt∈J

∣∣∣∣( tρ − aρ

ρ

)γ

y(t)∣∣∣∣

and

‖y‖Cnγ,ρ

=n−1

∑k=0‖y(k)‖∞ + ‖y(n)‖Cγ,ρ .

Consider the space Xpc (a, b), (c ∈ R, 1 ≤ p ≤ ∞) of those complex-valued Lebesgue

measurable functions f on [a, b] for which ‖ f ‖Xpc< ∞, where the norm is defined by:

‖ f ‖Xpc=

(∫ b

a|tc f (t)|p dt

t

) 1p

, (1 ≤ p < ∞, c ∈ R).

In particular, when c = 1p , the space Xp

c (a, b) coincides with the Lp(a, b) space: Xp1p(a, b) =

Lp(a, b).

71

Definition 5 (KATU1) (Katugampola fractional integral).Let α ∈ R+, c ∈ R and g ∈ Xpc (a, b).

The Katugampola fractional integral of order α is defined by:

(ρ Iαa+g) (t) =

∫ t

asρ−1

(tρ − sρ

ρ

)α−1 g(s)Γ(α)

ds, t > a, ρ > 0,

where Γ(·) is the Euler gamma function defined by: Γ(α) =∫ ∞

0 tα−1e−tdt, α > 0.

Definition 6 (KATU1) (Katugampola fractional derivative).Let α ∈ R+ \N and ρ > 0. The Katugampola fractional derivative ρDα

a+ of order α is defined by:

(ρDαa+g) (t) = δn

ρ (ρ In−α

a+ g)(t)

=

(t1−ρ d

dt

)n ∫ t

asρ−1

(tρ − sρ

ρ

)n−α−1 g(s)Γ(n− α)

ds, t > a, ρ > 0,

where n = [α] + 1 and δnρ =

(t1−ρ d

dt

)n.

Lemma 11 (OLIV) Let α > 0, and 0 ≤ γ < 1. Then, ρ Iαa+ is bounded from Cγ,ρ(J) into Cγ,ρ(J).

Main resultsTheorem 42 Let γ = α+ β− αβ where 0 < α < 1 and 0 ≤ β ≤ 1; let f : (a, T]×R×R→ R

be a function such that f (·, y(·), u(·)) ∈ C1−γ,ρ(J) for any y, u ∈ C1−γ,ρ(J).If y ∈ Cγ

1−γ,ρ(J), then y satisfies Equations (124) and (125) if and only if y is the fixed point ofthe operator N : C1−γ,ρ(J)→ C1−γ,ρ(J) defined by:

Ny(t) = M(

tρ−aρ

ρ

)γ−1+ 1

Γ(α)

∫ ta

(tρ−sρ

ρ

)α−1sρ−1g(s)ds, t ∈ (a, T] (126)

where:

M :=(

Tρ − aρ

ρ

)1−γ[

c− 1Γ(α)

∫ T

a

(Tρ − sρ

ρ

)α−1

sρ−1g(s)ds

]and g : (0, T]→ R be a function satisfying the functional equation:

g(t) = f (t, y(t), g(t)).

Clearly, g ∈ C1−γ,ρ(J). In addition, by Lemma Re f lem5, Ny ∈ C1−γ,ρ(J).

Suppose that the function f : (a, T]×R×R → R is continuous and satisfies the condi-tions:

(H1) The function f : (a, T]×R×R→ R is such that:

f (·, u(·), v(·)) ∈ Cβ(1−α)1−γ,ρ for any u, v ∈ C1−γ,ρ(J).

(H2) There exist constants K > 0 and 0 < L < 1 such that:

| f (t, u, v)− f (t, u, v)| ≤ K|u− u|+ L|v− v|,

for any u, v, u, v ∈ R and t ∈ (a, T].

72

Now, we state and prove our existence result for Equations (124) and (125) based onBanach’s fixed point.

Theorem 43 Assume (H1) and (H2) hold. If:

KΓ(γ)Γ(α + γ)(1− L)

(Tρ − aρ

ρ

)α

<12

, (127)

then the Equations (124) and (125) has unique solution in Cγ1−γ,ρ(J) ⊂ Cα,β

1−γ,ρ(J).

Theorem 44 Assume (H1) and (H2) hold. If:

KΓ(γ)Γ(α + γ)(1− L)

(Tρ − aρ

ρ

)α

< 1, (128)

then Equations (124) and (125) have at least one solution.

An ExampleConsider the following terminal value problem:

12 D

12 ,01+ y(t) =

2 + |y(t)|+∣∣∣∣ 1

2 D12 ,00+ y(t)

∣∣∣∣108e−t+3

(1 + |y(t)|+

∣∣∣∣ 12 D

12 ,00+ y(t)

∣∣∣∣) +ln(√

t + 1)

3√√

t− 1, t ∈ (1, 2] (129)

y(2) = c ∈ R. (130)

Set:

f (t, u, v) =2 + u + v

108e−t+3(1 + u + v)+

ln(√

t + 1)3√

t, t ∈ (1, 2], u, v ∈ [0,+∞).

We have:

Cβ(1−α)1−γ,ρ ([1, 2]) = C0

12 , 1

2([1, 2]) =

h : (1, 2]→ R :

√2(√

t− 1) 1

2 h ∈ C([1, 2])

,

with γ = α = ρ = 12 and β = 0. Clearly, the function f ∈ C 1

2 , 12([1, 2]). Hence condition

(H1) is satisfied.For each u, u, v, v ∈ R and t ∈ (1, 2] :

| f (t, u, v)− f (t, u, v)| ≤ 1108e−t+3 (|u− u|+ |v− v|)

≤ 1108e

(|u− u|+ |v− v|) .

Therefore, (H2) is verified with K = L = 1108e .

The condition:

KΓ(γ)Γ(α + γ)(1− L)

(Tρ − aρ

ρ

)α

≈ 0.0055 < 1,

is satisfied with with T = 2 and a = 1. It follows from Theorem 44 that Equations (129)

and (130) have a solution in the space C1212 , 1

2([1, 2]).

73

ConclusionWe have provided sufficient conditions ensuring the existence and uniqueness of so-lutions to a class of terminal value problem for differential equations with the Hilfer–Katugampola type fractional derivative. The arguments are based on the classical Ba-nach contraction principle, and the Krasnoselskii’s fixed point theorem. An example isincluded to show the applicability of our results.

ReferencesKatugampola, U. A new approach to a generalized fractional integral. Appl. Math.

Comput. 2011, 218, 860–865.Lakshmikantham, V.; Leela, S. Differential and Integral Inequalities; Academic Press:

New York, NY, USA, 1969; Volume I.Hallam, T.G. A comparison principle for terminal value problems in ordinary differ-

ential equations. Trans. Am. Math. Soc. 1972, 169, 49–57.Oliveira, D.S.; de Oliveira, E.C. Hilfer-Katugampola fractional derivative. Comput.

Appl. Math. 2018, 37, 3672–3690.

Kahler-Golden manifoldsHabib Bouzir , Beldjilali Gherici


In this work, we discuss some geometric properties of almost complex Goldenstructure (i.e. a polynomial structure with the structure polynomial Q(X) = X2−X + 3

2 I) and we introduce such some new classes of almost Hermitian Goldenstructures. We give a concrete examples.

Key words: Golden manifold, almost complex Golden structure, almost Hermitian

Golden structures.


IntroductionTo equip a space with a structure leads to the production of a new mathematical objectand consequently to contribute to the development of science. Manifolds equipped withcertain differential-geometric structures are richer and more practical spaces, they havebeen studied widely in differential geometry. Indeed, D. Chinea and C. Gonzalez ob-tained a classification of the (2n + 1)-dimensional almost contact metric manifold basedon U(n) × I representation theory, which is an analogy of the classification of the 2n-dimensional almost Hermitian manifolds established by A. Gray and H. M. Hervella .Being inspired by the Golden ratio, the notion of Golden manifold M was defined in[GHR] by a tensor field Φ on M satisfying Φ2 = Φ + I. The authors studied some prop-erties of this manifold and they showed that Φ is an automorphism of the tangent bundleTM and its eigenvalues are φ = 1+

√5

2 and 1− φ. There are also several recent works inthis direction. And in the same article [GHR], they introduced the notion of complex

74

Golden structure as a tensor Φc of type (1, 1) satisfies Φ2c = Φc − 3

2 I and its eigenvalues

are φc =1+i√

52 and 1− φc.

In this work, rely on the relationship between the almost complex structure J and thealmost complex Golden structure Φc given in [GHR], we extract the geometric tools forthe almost hermitian Golden structure and we use them to define certain new classes.

Aiming at our purpose, we organize this work as follows:Section 2 is devoted to the background of the almost complex Golden structure and wegive some new and important properties such as Riemannian metric which is compatiblewith the structure, the fundamental 2-form and others.In Section 3 we establish an important proposition that allows us to state our main theo-rem concerning the classes of almost Hermitian Golden structures.The last sections is devoted to building a concrete example.

Almost complex Golden manifold

The complex Golden ratio section φc is the root of the polynomial equation x2− x+ 32 = 0,

i.e., φc =1+i√

52 where i2 = −1 and the secod root denoted by φ∗c , satisfies φ∗c = 1−i

√5

2 =1− φc is his conjugate.

Definition 7 (GHR, GY) . Let M be a C∞ differentiable manifold of an even dimension andlet I be the identity (1, 1) tensor field. A tensor field F of type (1, 1) on M is said to define apolynomial structure if F satisfies the algebraic equation

Q(X) = Xn + anXn−1 + ... + a2X + a1 I = 0,

where Fn−1(p), Fn−2(p), ..., F(p) and I are linearly independent for every p ∈ M. The polyno-mial Q(X) is called the structure polynomial.

Definition 8 (GHR) . A non-null tensor field Φc of type (1, 1) and of class C∞ satisfying theequation

Φ2c = Φc −

32

I, (131)

is called an almost complex Golden structure on M of even dimensional.

A straightforward computation yields:

Proposition 6 1. The eigenvalues of an almost complex Golden structure Φc are the complexGolden ratio φc and φ∗c = 1− φc.

2. An almost complex Golden structure Φc is an isomorphism on the tangent space of themanifold, Tp M, for every p ∈ M.

3. It follows that Φc is invertible and its inverse Φ−1c given by

Φ−1c =

−23

(Φc − I) .

75

Proposition 7 (GHR) .If J is an almost complex structure on M, then

Φc =12

(I +√

5J)

, (132)

is an almost complex Golden structure. Conversely, if Φc is an almost Golden structure on Mthen

J =1√5(2Φc − I) , (133)

is an almost complex structure on M.

Main resultsProposition 8 For a twin pair Φc, J, on an even dimensional manifold M with any free linearconnection ∇, one has

4NΦc = 5NJ and 2∇Φc =√

5∇J. (134)

Definition 9 An almost Hermitian Golden structure is a pair (Φc, g) where Φc is an almostcomplex Golden structure and g is a Riemannian metric,with

g(ΦcX, ΦcY) =32

g(X, Y), (135)

or equivalenty,g(ΦcX, Y) + g(X, ΦcY) = g(X, Y). (136)

The Riemannian metric (135) is called Φc-compatible and the triple (M, Φc, g) is an almost Her-mitian Golden manifold.

Proposition 9 The operator J is a g-anti-symmetric endomorphism if and only if the associatedalmost complex Golden structure (132) is so.

Definition 10 Let (M, Φc, g) be an almost Hermitian Golden manifold. Set

Ω(X, Y) =1√5

(2g(X, ΦcY)− g(X, Y)

),

for all X, Y vectors fields on M. Ω is a 2-form on M and it is called ”fundamental 2-form”.

Lemma 12 For an almost Hermitian Golden structure (Φc, g), we have:

1. g((∇XΦc)Y, Z

)= −g

(Y, (∇XΦc)Z

),

2.(∇XΦc

)ΦcY = (I −Φc)

(∇XΦc

)Y,

3. g((∇XΦc)ΦcY, Z

)= g

((∇XΦc)Y, ΦcZ

),

for all vectors fields X, Y, Z on M where ∇ denotes the Levi-Civita connexion.

Theorem 45 Let (M, Φc, g) be an almost Hermitian Golden manifold and ∇ denotes the Rie-mannian connection of g. The following conditions are equivalent:

76

(a) ∇Φc = 0

(b) ∇Ω = 0

(c) NΦc ≡ 0 et dΩ = 0

Definition 11 Let (M, Φc, g) be an almost Hermitian Golden manifold. (M, Φc, g) is said to be:

1. Hermitian Golden (HG) manifold if and only if NΦc = 0

2. locally conformal Golden (l.c.G) manifold if there exists a closed one-form η such that:

dΩ = η ∧Ω.

3. Kahler-Golden (KG) manifold if and only if NΦc = 0 and dΩ = 0 or equivalently,∇Φc =0.

4. Nearly Golden (NG) manifold if and only if (∇XΦc)X = 0.

5. Quasi Golden (QG) manifold if and only if (∇XΦc)Y + (∇ΦcXΦc)ΦcY = 0.

ReferencesD. Chinea and C. Gonzalez, A classification of almost contact metric manifolds, Ann.

Mat. Pura Appl. (4) 156 (1990), 15-36A. Gray, L. M. Hervella, The sixteen classes of almost Hermitian manifolds and their

linear imvarients, Ann. Mat. Pura Appl. 123 (1980), 35-58.M. Crasmareanu , C.E. Hretcanu, Golden differential geometry, Chaos, Solitons & Frac-

tals. 38 no. 5 (2008), 1124-1146. doi: 10.1016/j.chaos.2008.04.007.S. I. Goldberg, K. Yano, Polynomial structures on manifolds, Kodai Math. Sem. Rep.

22 (1970), 199-218

On the p-harmonic and p-biharmonic mapsAhmed Mohammed Cherif


In this paper, we study the existence of p-harmonic maps into Riemannian mani-folds admitting a conformal vector field. We also prove a Liouville type theoremfor p-biharmonic maps.Key words: p-Harmonic maps, p-Biharmonic maps, Conformal vector fields.2010 Mathematics Subject Classification: 53C43, 58E20, 53A30.

IntroductionWe give some definitions. Let (M, g) be a Riemannian manifold. By RM we denote theRiemannian curvature tensor of (M, g). Then RM is defined by:

RM(X, Y)Z = ∇MX∇M

Y Z−∇MY ∇M

X Z−∇M[X,Y]Z, (137)

77

where ∇M is the Levi-Civita connection with respect to g, and X, Y, Z ∈ Γ(TM). Thedivergence of (0, p)-tensor α on M is defined by:

(divM α)(X1, ..., Xp−1) = (∇Mei

α)(ei, X1, ..., Xp−1), (138)

where X1, ..., Xp−1 ∈ Γ(TM), and ei is an orthonormal frame. Given a smooth functionλ on M, the gradient of λ is defined by:

g(gradM λ, X) = X(λ), (139)

where X ∈ Γ(TM) (for more details, see for example [ON]).A vector field ξ on a Riemannian manifold (M, g) is called a conformal if Lξ g = 2 f g,for some smooth function f on M, where Lξ g is the Lie derivative of the metric g withrespect to ξ, that is:

g(∇MX ξ, Y) + g(∇M

Y ξ, X) = 2 f g(X, Y), X, Y ∈ Γ(TM). (140)

The function f is then called the potential function of the conformal vector field ξ. If ξ isconformal with constant potential function f , then it is called homothetic, while f = 0 itis Killing (see [BW], [WH], [yano]).If ϕ : (M, g)→ (N, h) is a smooth map between two Riemannian manifolds, its p-energyis defined by

Ep(ϕ; D) =1p

∫D|dϕ|pvg (p ≥ 2). (141)

where D is a compact subset of M. The p-energy functional (227) includes as a specialcase (p = 2) the energy functional, whose critical points are the usual harmonic maps (see[ES]). We say that ϕ is a p-harmonic map if it is a critical point of the p-energy functional,that is to say, if it satisfies the Euler-Lagrange equation of the functional (227), that is,

τp(ϕ) ≡ divM(|dϕ|p−2dϕ) = 0. (142)

In particular, we note that every harmonic map with constant energy density is p-harmonicfor all p ≥ 2 (for more details on the concept of p-harmonic maps see [BG,BI,ali]). Letτ(ϕ) the tension field of ϕ given by:

τ(ϕ) = traceg∇dϕ = ∇ϕei dϕ(ei)− dϕ(∇M

eiei), (143)

where ∇M is the Levi-Civita connection of (M, g), ∇ϕ denote the pull-back connectionon ϕ−1TN and ei is an orthonormal frame on (M, g) (see [BW], [ES], [YX]). Then ϕ isp-harmonic if and only if (see [BG]):

|dϕ|p−2τ(ϕ) + (p− 2)|dϕ|p−3dϕ(gradM |dϕ|) = 0. (144)

Main results

p-Harmonic maps and conformal vector fieldsProposition 10 Let (M, g) be a compact orientable Riemannian manifold without boundary,and (N, h) a Riemannian manifold admitting a conformal vector field ξ with potential functionf > 0 at any point. Then, any p-harmonic map ϕ from (M, g) to (N, h) is constant.

78

Since the identity map of a compact Riemannian manifold is always a p-harmonicmap (p ≥ 2), from Proposition 10 we get the following result:

Corollary 9 Let (M, g) be a compact orientable Riemannian manifold without boundary, and ξ aconformal vector field with potential function f on M. Then, the set of annulation of the functionf is never empty on M.

From Proposition 10, we deduce:

Corollary 10 (cherif) Let (M, g) be a compact orientable Riemannian manifold without bound-ary, and (N, h) a Riemannian manifold admitting a homothetic vector field ξ with homotheticconstant k 6= 0. Then, any harmonic map ϕ from (M, g) to (N, h) is constant.

In the case of non-compact Riemannian manifold, we obtain the following result:

Theorem 46 Let (M, g) be a complete non-compact Riemannian manifold, and (N, h) a Rie-mannian manifold admitting a conformal vector field ξ with potential function f > 0 at anypoint. If ϕ : (M, g) −→ (N, h) is p-harmonic map, satisfying:∫

M

|dϕ|p−2|ξ ϕ|2f ϕ

vg < ∞, (145)



Theorem 47 Let (M, g) be a complete non-compact Riemannian manifold, and (N, h) a Rie-mannian manifold admitting a homothetic vector field ξ with homothetic constant k 6= 0. Ifϕ : (M, g) −→ (N, h) is p-harmonic map, satisfying:∫

M|dϕ|p−2|ξ ϕ|2vg < ∞, (146)


Remark 13 Proposition 10 and Theorem 46 remains true if the potential function f < 0 on N(consider the conformal vector field ξ = −ξ).

Liouville type theorems for p-biharmonic mapsConsider a smooth map ϕ : (M, g) −→ (N, h) between Riemannian manifolds and letp ≥ 2. A natural generalization of p-harmonic maps is given by integrating the square ofthe norm of τp(ϕ). More precisely, the p-bienergy functional of ϕ is defined by

E2,p(ϕ; D) =12

∫D|τp(ϕ)|2vg. (147)

A map is called p-biharmonic, if it is a critical point of the p-bienergy functional (147)over any compact subset D of M.

Remark 14 Note that in [CL] and [HL], the terminology ”p-biharmonic maps” was used as thecritical points of the functional

∫M |τ(ϕ)|pvg, so there is no link between the concept introduced

in this paper, and the notion of p-biharmonicity that was used, introduced and studied by X. Cao,Y. Han, and Y. Luo.

79

Theorem 48 Let ϕ : (M, g) → (N, h) be a smooth map between Riemannian manifolds, D acompact subset of M and let ϕtt∈(−ε,ε) be a smooth variation with compact support in D. Then

ddt

E2,p(ϕt; D)∣∣∣t=0

= −∫

Dh(τ2,p(ϕ), v) vg, (148)

where τ2,p(ϕ) ∈ Γ(ϕ−1TN) given by


−(p− 2) traceg∇ < ∇ϕτp(ϕ), dϕ > |dϕ|p−4dϕ. (149)

Remark 15 Let ei be an orthonormal frame on (M, g), then:

traceg RN(τp(ϕ), dϕ)dϕ = RN(τp(ϕ), dϕ(ei))dϕ(ei),

traceg∇ϕ|dϕ|p−2∇ϕτp(ϕ) = ∇ϕei |dϕ|p−2∇ϕ

ei τp(ϕ)− |dϕ|p−2∇ϕ

∇Mei ei

τp(ϕ),

traceg∇ < ∇ϕτp(ϕ), dϕ > |dϕ|p−4dϕ = ∇ϕei < ∇

ϕτp(ϕ), dϕ > |dϕ|p−4dϕ(ei)

− < ∇ϕτp(ϕ), dϕ > |dϕ|p−4dϕ(∇Mei

ei).


Theorem 49 Let ϕ : (M, g) → (N, h) be a smooth map between Riemannian manifolds. Then,ϕ is p-biharmonic if and only if:


−(p− 2) traceg∇ < ∇ϕτp(ϕ), dϕ > |dϕ|p−4dϕ = 0.

Example 2 Let M the manifold R2\(0, 0) ×R equipped with the Riemannian metric g =dx2

1 + dx22 + dx2

3, and let N the manifold R2 equipped with the Riemannian metric h = dy21 + dy2

2.

The map ϕ : (M, g) −→ (N, h), (x1, x2, x3) 7−→ (√

x21 + x2

2, x3), is p-biharmonic if and only ifp = 4.

Remark 16 The previous examples prove the following results. 1) If the energy density of asmooth map ϕ : (M, g) −→ (N, h) is constant, then there is no equivalence between the bihar-monicity of ϕ and the p-biharmonicity of ϕ, for p 6= 2. 2) There are p-biharmonic maps that areneither p-harmonic nor harmonic, with p 6= 2.

Theorem 50 Let (M, g) be a compact orientable Riemannian manifold without boundary, and(N, h) a Riemannian manifold with non-positive sectional curvature. Then, every p-biharmonicmap from (M, g) to (N, h) is p-harmonic.

In the case of non-compact Riemannian manifold, we have the following result:

Theorem 51 Let (M, g) be a complete Riemannian manifold, (N, h) be a Riemannian manifoldwith non-positive sectional curvature and p ≥ 2. Then, every p-biharmonic map ϕ : (M, g) →(N, h) satisfying: ∫

M|dϕ|p−2|τp(ϕ)|2vg < ∞,

∫M|dϕ|p−2vg = ∞,

is p-harmonic map.

80

If p = 2, we arrive at the following corollary:

Corollary 11 (NUG) Let (M, g) be a complete Riemannian manifold with infinite volume i.e.∫M vg = ∞, (N, h) a Riemannian manifold with non-positive sectional curvature. Then, every

biharmonic map ϕ : (M, g) → (N, h) with finite bienergy, i.e.∫

M |τ(ϕ)|2vg < ∞, is harmonicmap.

References[BW] P. Baird, J. C. Wood, Harmonic morphisms between Riemannain manifolds,

Clarendon Press Oxford, 2003.[BG] P. Baird, S. Gudmundsson, p-Harmonic maps and minimal submanifolds, Math.

Ann. 294 (1992), 611-624.[BI] B. Bojarski, and T. Iwaniec, p-Harmonic equation and quasiregular mappings,

Partial differential equations (Warsaw, 1984), 25-38, Banach Center Publ., vol. 19. PWN,Warsaw, 1987.

[CL] X. Cao and Y. Luo, On p-biharmonic submanifolds in nonpositively curved man-ifolds, Kodai Math. J. 39 (2016), 567-578.

[ES] J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds,Amer. J. Math. 86 (1964), 109-160. [ali] A. Fardoun, On equivariant p-harmonic maps,Ann.Inst. Henri. Poincare, 15 (1998), 25-72. [GV] L. Greco, and A. Verde, A regularityproperty of p-harmonic functions. Ann. Acad. Sci. Fenn. Math., 25, (2000), 317-23.

[HL] Y. Han, Y. Luo, Several results concerning nonexistence of proper p-biharmonicmaps and Liouville type theorems, arXiv: 1801.05181v1.

[Jiang] G. Y. Jiang, 2-Harmonic maps between Riemannian manifolds, Annals ofMath., China, 7A(4)(1986), 389-402.

[WH] W. Kuhnel, H. Rademacher, Conformal vector fields on pseudo-Riemannianspaces, Differential Geometry and its Applications 7 (1997), 237-250.

[cherif] A. Mohammed Cherif, Some results on harmonic and bi-harmonic maps,International Journal of Geometric Methods in Modern Physics, 14 (2017).

[MD] A. Mohammed Cherif, M. Djaa, Geometry of Energy and Bienergy Variationsbetween Riemannian Manifolds, Kyungpook Math. J. 55 (2015), 715-730. [NUG] N.Nakauchi, H. Urakawa, S. Gudmundsson, Biharmonic maps into a Riemannian man-ifold of non-positive curvature. Geom. Dedicata 169 (2014), 263-272.

[ON] O’Neil, Semi- Riemannian Geometry, Academic Press, New York, 1983.[YX] Y. Xin, Geometry of harmonic maps, Fudan University, 1996.[yano] K. Yano and T. Nagano, The de Rham decomposition, isometries and affine

transformations in Riemannian space, Japan. J. Math., 29 (1959), 173-184.[pa] S. T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure

Appl. Math. 28 (1975), 201-228.

4V

81

Two-dimensional inverse heat conduction problem in quarterplane. Integral approachL. Chorfi, A. Bel-Hadj Hassin

Badji Mokhtar University, El Manar University

We consider a two-dimensional inverse heat conduction problem in the do-mainΩ = x > 0, y > 0 . Our aim is to determine the temperature f (y, t) = u(0, y, t)from the measured data g(y, t) = u(1, y, t), where u(x, y, t) is a solution of theheat equation ut = ∆u in Ω× [0, T] with initial value u(x, y, 0) = 0 and boundaryvalue u(x, 0, t) = 0. This is a severely ill-posed problem.Key words: Inverse problem, Heat kernel, Integral equation.

2010 Mathematics Subject Classification: 35K05, 47A52.

Direct problemThe direct problem is set as follows: given the source f , determine u which satisfies thesystem

ut(x, y, t) = uxx(x, y, t) + uyy(x, y, t), x > 0, y > 0, t > 0, (150a)u(0, y, t) = f (y, t), y ≥ 0, t > 0, (150b)u(x, y, 0) = 0, x > 0, y ≥ 0, (150c)u(x, 0, t) = 0, x > 0, t ≥ 0, (150d)

u(x, y, t)|x→+∞bounded, y > 0, t > 0. (150e)

For the reconstruction of the solution we use the integral method ([Chap]). The Green’sfunction for the heat equation in the upper half-plane has the form

E((x, y, t), (ξ, η, τ)) = E0(x− ξ, y− η, t− τ)− E0(x− ξ, y + η, t− τ) (151)

where E0(x, y, t) is the fundamental solution of the 2D heat equation as given by

E0(x, y, t) =

1

4πt e−x2+y2

4t for t > 0,0 for t ≤ 0.

(152)

Then we can seek the solution of the direct problem (150a)-(150d) as the potential ([Gunt)

u(x, y, t) =∫ t

0

∫ +∞

0E(x, y− η, t− τ)q(η, τ)dηdτ, (153)

with a density q. According to the trace properties of the potential, the condition (150b)holds if the following equation is satisfied

f (y, t) = u(0, y, t) =∫ t

0

∫ +∞

0E(0, y− η, t− τ)q(η, τ)dηdτ, ∀(y, t) ∈ R∗+× (0, T). (154)

Now, we use the sinus-Fourier transform in y

f (w, t) =∫ +∞

0f (y, t) sin(wy)dy. (155)

82

Figure 1: The approximate interior temperature g(y, T) = u(1, y, T) with the twomethods (Integral approach) and (Finite Difference Mehod) at time T = 3.

to get the integral equation of first kind:∫ t

0K(w, t− τ)q(w, τ)dτ = f (w, t) (156)

with the kernel K(w, t) = α√t

exp (−w2t). Let now introduce the space

H = C([0, T], H1(R+)

)∩ C1([0, T], L2(R+))

equipped with its natural norm.

Theorem 52 Assume that f ∈ H and f (·, 0) = 0. Then equation (156) admits a unique solutionq and we have the following estimate, for 0 < t ≤ T,

‖q(·, t)‖L2(]0,+∞[) ≤ c2√

T‖ f ‖H. (157)

Figure 9 shows a numerical simulation with the boundary data f (y, t) = 0.25yt exp(t−y), y > 0.

Inverse problem. Integral equationAssume that u(1, y, t) = g(y, t) is given, find the boundary condition u(0, y, t) = f (y, t).It is well known that such problem (IHCP) is an ill-posed problem because small errorsin the data induce large errors in the computed temperature.

Using the representation (153), the potential q must satisfy the equation

f (y, t) = u(0, y, t) =∫ t

0

∫ +∞

0E(1, y− η, t− τ)q(η, τ)dηdτ, ∀(y, t) ∈ R∗+× (0, T). (158)

Using Fourier-sinus transform, we obtain the equation∫ t

0K(w, t− τ)q(w, τ)dτ = g(w, t) (159)

83

Figure 2: Exact and approximate solution at x = 0 (left) with δ = 0; (right) withδ = 0.01 .

with the smooth kernelK(w, t) =

α√t

exp(−1/4t− w2t).

It’s a Volterra integral equation of the first kind, so it is ill-posed, which need a regular-ization. For this we use the Tikhonov regularization.

Numerical examplesWe shall show some numerical examples to illustrate the validity of the method withexact data δ = 0 or perturbed data with δ = 0.01, δ is the level noise. In the Figure weshow the reconstruction of the exact boundary condition f (y, t) = 0.25ytete−0.2(y−5)2

(1 +sin(πy)).References

[Chap]R. Chapko. On the numerical solution of direct and inverse problems for theheat equation in a semi-infnite region. Journal of Computational and Applied Mathematics108 (1999) 41-55.

[Gar-Has016] M. Garshasbi and F. Hassani: Boundary temperature reconstruction inan inverse heat conduction problem using boundary integral equation method. Bulletinof the Iranian Mathematical Society, 42, no. 5, 1039–1057 (2016)

[Gunt] R.B. Guenther, J.W. Lee. Partial Differential Equations of Mathematical Physicsand Integral Equations. Prentice-Hall, Englewood Cliffs, 1988.

Asymptotic result of the conditional estimator of cumulativedistribution in functional spatial data case

Hamza Daoudi, Boubaker MechabIbn Khaldoun University, Djillali Liabes University

The purpose of the present paper is to investigate by the local linear methoda nonparametric estimate of the conditional quantile of scalar response variablegiven a functional variable when the observations are spatially dependent. Themain goal is to establish the almost complete convergence with rate of this esti-mate under some general conditions.Key words: Spatial functional data, Local linear estimation, Conditional function, Strongly

84

mixing process.

2010 Mathematics Subject Classification: 62G05, 62G20.

Introduction

The conditional quantile arise in a variety of fields including econometrics, epidemiology,environmental science and many others. There is a large literature on the estimation ofthis function for independent as well as for dependent mixing data. In recent years, therehas been a considerable interest in functional data analysis. We refer to Ferraty and Vieu[FV2006], Ramsay and Silverman [RS2002]. In view of the importance of the estimate bythe linear local method, the local polynomial sitting has been recognized to have superiorbias properties than the kernel method (see Fan and Gijbels [FG1996] for more details).Works in the context of the functional non spatial data as well as the functional spatialdata are very limited and key references of this topic can be found in Barrientos-Marin etal. [BFV2010], Demongeot et al. [DLMR2010], Chouaf and Laksaci[CL2012].

Presentation of the spatial data

In order to define the spatial functional version of the local linear estimator of the con-ditional distribution function, let Zi = (Xi, Yi) for i ∈ ZN and N ≥ 1, be (F × R)-valued measurable and strictly stationary spatial process, defined on a probability space(Ω,A, P), where (F , d) is a semimetric space. Moreover, a point i = (i1, ..., iN) ∈ ZN willbe referred to as a site.On the other hand, we assume that the process Zi = (Xi, Yi), under study, is observedover a rectangular domain:

In =

i = (i1, ..., iN) ∈ ZN , 1 ≤ ik ≤ nk, k = 1, ..., N

where n = (n1, ..., nN) ∈ ZN .All along the paper, when no confusion is possible, we denote by C and C’ any genericpositive constants and we will write:

n→ ∞ if mink=1,...,N

nk → ∞ and |nj/nk| < C

for all j, k ∈ 1, ..., N. This kind of design is known as an asymptotically increasing do-main, which allows the area of observations to become larger but keeping a minimumdistance between observation sites. Throughout this paper, we denote by ℵx a fixedneighborhood of a fixed point x ∈ F , and by § a fixed compact subset of R. Assumethat the Zi ’s are identically distributed to Z = (X, Y) and that there exists a regularversion of the conditional probability of Y given X. Then, let Fx be the conditional dis-tribution of the variable Y given X = x. Moreover, we suppose that Fx has a continuousprobability density f x with respect to (w.r.t.) the Lebesgue’s measure over R.Then, in what follows, we mainly study the local linear estimation of the functions Fx

and f x when the functional random field (Zi = (Xi, Yi), i ∈ ZN) satisfies the following

85

mixing condition:

There exists a function ϕ (t) ↓ 0 as t→ ∞, such that∀ E, E

′subsets of NN with finite cardinals

α(B (E) , B

(E′))

= supB∈B(E), C∈B(E′)

|P (B ∩ C)−P (B)P (C)|

≤ ψ(

Card (E) , Card(

E′))

ϕ(

dist(

E, E′))

,

(160)

where B (E) (respectively,B(E′))denotes the Borel σ-field generated by (Zi,i ∈ E) (re-

spectively, (Zi, i ∈ E′)), Card(E) (respectively, Card(E′)) is the cardinality number of E(respectively, E′),dist(E, E′) is the Euclidean distance between E and E′ and ψ : Z2 → R+

is a symmetric positive nondecreasing function in each variable such that:

∀n, m ∈ Z, ψ(n, m) ≤ C min(n, m). (161)

Remark that, when (217) holds with ψ ≡ 1 or N = 1, then the random field Zi = (Xi, Yi)is called strongly mixing. We also assume that the process Z satisfies the following mixingcondition:

∞

∑i=1

ik ϕ(i) < ∞, k > 0. (162)

Model and hypotheses

Here, we adopt the fast functional local modeling, that is, the conditional cumulativedistribution function Fx is estimated by a where the couple (a, b) is obtained by theoptimization rule:

(a, b) = arg min(a,b)∈R2

∑i∈In

(H(h−1

H (y−Yi))− a− bβ(Xi, x))2 K(h−1

H δ(x, Xi)) (163)

where β(., .) and δ(., .) are locating functions defined from F 2 into R, such that:

∀ξ ∈ F , β(ξ, ξ) = 0 and d(., .) = |δ(., .)|

and where the function K is a kernel, H is a distribution function and hK = hK,n(respectively,hH = hH,n) is a sequence of positive real numbers which converges to 0 when n → ∞.Clearly, the estimator a, given by (291), can be explicitly written as follows:

Fx(y) =

∑i,j∈In

i 6=j

Wij(x)H(hH−1(y−Yj))

∑i,j∈In

i 6=j

Wij(x), ∀y ∈ R (164)

whereWij(x) = β(Xi, x)(β(Xi, x)− β(Xj, x))K(h−1

K δ(x, Xi))K(h−1K δ(x, Xj))

86

Main results

We first give the hypotheses that are necessary in deriving the almost-complete conver-gence (a.co.) of the functional locally modeled estimator of Fx(y).

Theorem 53 Under some assumptions, and (217)-(219), we have:

supy∈S|Fx(y)− Fx(y)| = O

(hb1

K + hb2H

)+ O

((log n

n φx(hK)

) 12)

a.co. (165)

Conclusion

This work presents the study of the almost complete convergence of the estimator of thedistribution function by the linear local method in the spatial context in the presence of afunctional explanatory variable.References

[BFV2010] Barrientos-Marin J., Ferraty F., Vieu P., Locally Modelled Regression andFunctional Data, Journal of Nonparametric Statistics., 2010, 22(5), 617–632.

[CL2012] Chouaf A., Laksaci A., On the functional local linear estimate for spatialregression. Stat. Risk. Model., 2012, 29(3), 189–214.

[DLMR2010] Demongeot J., Laksaci A., Madani F., RachdiM., Local linear estimationof the conditional density for functional data, C. R., Math., Acad. Sci. Paris, 2010, 348,931–934.

[FG1996] Fan J., Gijbels I., Local Polynomial Modelling and its Applications. London,Chapman & Hall, 1996.

[FV2006] Ferraty F., Vieu, P., Nonparametric Functional Data Analysis: Theory andPractice, Springer Series in Statistics, 2006. [RS2002] Ramsay J. O., Silverman B. W., Ap-plied functional data analysis; Methods and case studies. Springer-Verlag, New York,2002.

Modele d’evolution de la transmission de la maladie de Chagas

Nabahats Dib, Rabah Labbas , Tewfik Mahjoub, Ahmed MedeghriUniversite de Tlemcen,Universite du Havre, Universite de Tlemcen, Universite de

Mostaganem

Le systeme biologique etudie est decrit par un systeme d’equations de reaction-diffusion sur un domaine forme de 2 sous-domaines contigus representant unezone refuge et une zone tampon. Sur la frontiere commune aux sous domainessont posees des conditions d’assymetrie. Des conditions de Dirichlet sont poseessur le reste de la frontiere. Cette premiere etape consiste a transformer ce systemeen une equation differentielle abstraite Key words: Probleme de transmission, reaction-

diffusion, semi groupe, operateur sectoriel.

87

IntroductionLa maladie de Chagas, ou trypanosomiase americaine, est une parasitose infectant lesmammiferes d’une vingtaine de pays d’Amerique centrale et du sud. Son agent causal estTrypanosoma cruzi, un protozoaire flagelle vehicule par des punaises hematophages, ap-partenant a la sous-famille des Triatominae. Chez l’homme, la maladie se caracterise parune phase aigue de quelques semaines, suivie d’une phase latente asymptomatique pou-vant durer plusieurs dizaines d’annees et se terminant, dans certains cas, par une phasechronique conduisant rapidement a la mort[?]. Le probleme considere est l’etude duprocessus de re-infestation des villages par les especes de triatomines venant des foretsavoisinantes, bien qu’une lutte insecticide y soit menee. Certaines especes, comme Tri-atoma dimidiata, s’adaptent, en effet, aussi bien a une vie sylvatique (pleine foret) qu’aune vie domestique (a l’interieur du village) ou peri-domestique[?].

Presentation du Modele

Construction du modele:Bien que le cycle de vie des triatomines se compose de sept stades de developpement : unstade ”oeufs”, cinq stades larvaires et un stade ”adultes”, on considere le developpementde l’oeuf au cinquieme stade larvaire comme un seul stade, qu’on appellera stade juvenile.On note J(t, x, y) la densite de cette classe de la population au temps t > 0 et au point(x, y) et A(t, x, y) la densite des adultes (on raisonnera en dim 2 spatialement). Le cyclede vie des triatomines est decrit dans [?] ( figure 1).

Figure 3: Representation schematique du cycle de vie utilise dans le modele deTriatomine.

Le village sera considere comme un pave ]− l, 0[×]0, 1[ note Ω−. Sur ce dernier, lesdensites des juveniles et des adultes sont notees respectivement J−(t, x, y) et A−(t, x, y).

Entre ce village et la foret, il existe une zone tampon sous forme d’un autre pave]0, L[×]0, 1[ qu’on notera Ω+.Sur celle-ci, les densites des juveniles et des adultes sontnotees respectivement J+(t, x, y) et A+(t, x, y).

A l’exterieur de ces deux habitats on supposera pour simplifier qu’aux bords

]− l, L[×0 et ]− l, L[×1

88

on a des barrieres naturelles hostiles aux triatomines et donc a partir desquelles il n’y apas de diffusion.

Figure 4: Densite de population dans deux habitats.

Les triatomines sont attirees vers le village pour leur repas (la lumiere est aussi unfacteur attirant); cette attirance est plus forte lorsque ces triatomines se trouvent dans lazone tampon Ω+. D’ou les conditions biaisees de flux sur l’interface Γ = ∂Ω− [?]

pdj−∂J−∂ν

= (1− p)dj+∂J+∂ν

pda−∂A−∂ν

= (1− p)da+∂A+

∂ν

La diffusion des triatomines peut alors se modeliser par des equations de reaction-diffusion du type

∂J−∂t

= dj−∆J− + ωj(t)sj(t)J− + fa(t)sa(t)A−∂A−

∂t= da−∆A− + (1−ωj(t))sj(t)J− + sa(t)A−

sur Ω−

∂J+∂t

= dj+∆J+ + ωj(t)sj(t)J+ + fa(t)sa(t)A+

∂A+

∂t= da+∆A+ + (1−ωj(t))sj(t)J+ + sa(t)A+

sur Ω+

avec les conditions initiales:

(CI)

J−(0, x, y) = J0−(x, y), A−(0, x, y) = A0

−(x, y) sur Ω−J+(0, x, y) = J0

+(x, y), A+(0, x, y) = A0+(x, y) sur Ω+

les conditions de transmission suivantes:

89

(CT)

pdj−

∂J−∂ν

= (1− p)dj+∂J+∂ν

pda−∂A−∂ν

= (1− p)da+∂A+

∂ν(J−A−

)=

(J+A+

) sur Γ = 0×]0, 1[ ,

(preservant la continuite des triatomines a l’interface; On peut aussi considerer le cas denon continuite des densites et considerer la continuite des diffusions) et par exemple lesconditions aux limites :

(CL1)

(

J−A−

)= 0 sur −l×]0, 1[

∂

∂ν

(J+A+

)= 0 sur L×]0, 1[

traduisant simplement le fait que le flux des triatomines (juveniles et adultes) venant d’uncote est nul et que de l’autre cote les densites sont donnees (on peut considerer d’autrescas). On supposera aussi les conditions aux limites nulles sur les densites sur les bordsdes deux zones hostiles aux triatomines:

(CL2)

J+(., x, 0) = J+(., x, 1) = 0A+(., x, 0) = A+(., x, 1) = 0

x ∈]0, L[

J−(., x, 0) = J−(., x, 1) = 0A−(., x, 0) = A−(., x, 1) = 0

x ∈]− l, 0[

Transformation du systeme precedent en une equation differentielleabstraite:

L’objectif est d’ecrire ce systeme sous une forme abstraite en suivant la methode deLabbas[?]

dj− = da− = d−, dj+ = da+ = d+.

Posons:

pdj− = µ−, (1− p)dj+ = µ+,pda− = α−, (1− p)da+ = α+,

V(t, x, y) =(

J(t, x, y)A(t, x, y)

)sur Ω = Ω− ∪Ω+, t > 0

V− = V|Ω− =

(J−A−

), V+ = V|Ω+

=

(J+A+

),

P− + B(t) =

(dj−∆ 0

0 da−∆

)+

(ωj(t)sj(t)I fa(t)sa(t)I

(1−ωj(t))sj(t)I sa(t)I

)P+ + B(t) =

(dj+∆ 0

0 da+∆

)+

(ωj(t)sj(t)I fa(t)sa(t)I

(1−ωj(t))sj(t)I sa(t)I

).

90

On definit alors l’operateur L suivant (qui agit par rapport a l’espace)

LV(t, ., .) = L(

J(t, ., .)A(t, ., .)

)

=

(

dj−∆ 00 da−∆

)(J−(t, ., .)A−(t, ., .)

)sur Ω−(

dj+∆ 00 da+∆

)(J+(t, ., .)A+(t, ., .)

)sur Ω+

=

(

dj−∆J−(t, ., .)da−∆A−(t, ., .)

)sur Ω−(

dj+∆J+(t, ., .)da+∆A+(t, ., .)

)sur Ω+

avec

D(L) =

W = (u, v) ∈ [Lp (Ω)]2 : W− = (u−, v−) ∈[W2,p (Ω−)

]2

W+ = (u+, v+) ∈[W2,p (Ω+)

]2et W−, W+ verifiant (CT), (CL1),(CL2)

.

On adoptera la notation vectorielle operationnelle classique suivante

V(t, x, y) = V(t)(x, y).

Alors le probleme precedent s’ecrit sous la forme d’un systeme de reaction-diffusion ab-strait suivant:

V ′(t) = LV(t) + B(t)V(t), t > 0

V(0) = V0 =

(J0(0, ., .)A0(0, ., .)

),

(166)

pose dans l’espace de Banach UMD E = [Lp (Ω)]2 qu’on norme par exemple par∥∥∥∥(uv

)∥∥∥∥E= max

(‖u‖Lp(Ω) , ‖v‖Lp(Ω)

),

ConclusionApres avoir transforme notre modele en une equation differentielle abstraite, nous etudieronsalors l’existence et l’unicite de la solution par application des outils de l’analyse fonction-nelle, la theorie des operateurs sectoriels et les espaces H infinis.

ReferencesHTTPS://WWW.WHO.INT/NEWS-ROOM/FACT-SHEETS/DETAIL/CHAGAS-DISEASE-(AMERICAN-

TRYPANOSOMIASIS)NOUVELLET, P., CUCUNUBA, Z.M., GOURBIERE, S., 2015. CHAPTER FOUR - ECOL-

OGY, EVOLUTION AND CONTROL OF CHAGAS DISEASE: A CENTURY OF NEGLECTED

MODELLING AND A PROMISING FUTURE, IN: ROY, M.A., MARIA GLORIA, B. (EDS.),ADVANCES IN PARASITOLOGY. ACADEMIC PRESS, PP. 135–191.

MESK, M., MAHDJOUB, T., GOURBIERE, S., RABINOVICH, J.E., MENU, F., 2016.INVASION SPEEDS OF TRIATOMA DIMIDIATA, VECTOR OF CHAGAS DISEASE: AN APPLI-CATION OF ORTHOGONAL POLYNOMIALS METHOD. J. THEOR. BIOL. 395, 126–143.

91

CANTRELL, R.S., COSNER, C., 2004. SPATIAL ECOLOGY VIA REACTION-DIFFUSION

EQUATIONS. JOHN WILEY SONS.LABBAS, R., MEDEGHRI, A., MENAD, A., 2018. SOLVABILITY OF ELLIPTIC DIF-

FERENTIAL EQUATIONS, SET IN THREE HABITATS WITH SKEWNESS BOUNDARY CON-DITIONS AT THE INTERFACES. MEDITERR. J. MATH. 15, 128.

Groups whose proper subgroups are (locallyπ−finite)-by-(locally nilpotent)

Amel Dilmi, Nadir TrabelsiUniversity Setif

If X is a class of groups, then a group G is called a minimal non-X-group if it isnot an X-group but all its proper subgroups belong to X. Let π be a set of primesand let X be a quotient and subgroup closed class of locally nilpotent groups suchthat every infinite locally graded minimal non-X-group is a countable p-group forsome prime p. Our main result in the present paper states that G is an infinitelygenerated minimal non-(LFπ)X-group if and only if there exists a prime p /∈ πsuch that G is an infinitely generated minimal non-X p-group; where LFπ denotesthe class of locally finite π-groups.

Key words: Locally nilpotent; Locally finite; π-groups; Minimal non-nilpotent.

2010 Mathematics Subject Classification: 20F50; 20F19 .

IntroductionIf X is a class of groups, then a group G is called a minimal non-X-group if it is not anX-group but all its proper subgroups belong to X. Many results have been obtained onminimal non-X-groups for several choices of X. In particular, in [NW] a complete descrip-tion of infinitely generated minimal non-nilpotent groups having a maximal subgroup isgiven. These groups are metabelian Chernikov p-groups, where p is a prime. Later in[smith], infinitely generated minimal non-nilpotent groups without maximal subgroupshave been studied and it was proved, among many results, that they are countable p-groups. In []dilmi] it is proved that if G is a minimal non-(LF)N (respectively, non-(LF)Nc) group, then G is a finitely generated perfect group which has no proper sub-groups of finite index and G/Frat(G) is simple, where LF (respectively, N, Nc) denotesthe class of locally finite (respectively, nilpotent, nilpotent of class at most c) groups.Therefore there are no minimal non-(LF)N-groups (respectively, non-(LF)Nc-groups)which are infinitely generated (or equivalently locally graded). In the present paper,we generalize these last results by considering the classes (LFπ)N and (LFπ)Nc, whereπ is a given set of primes and LFπ denotes the class of locally finite π-groups. It turnsout that infinitely generated minimal non-(LFπ)N-groups exist. For if G is an infinitelygenerated minimal non-nilpotent group, then it is a p-group for some prime p and henceit is an infinitely generated minimal non-(LFπ)N-group for every set π not containing p.We will prove that the converse is also true. In fact our results on (LFπ)N and (LFπ)Ncwill be consequences of more general results on (LFπ)X (respectively, (LFπ)V), where

92

X (respectively, V) denotes a quotient and subgroup closed class (respectively, a variety)of locally nilpotent groups such that infinite locally graded minimal non-X-groups arecountable p-groups.

Recall that a group is locally graded if every non-trivial finitely generated subgrouphas a proper non-trivial finite homomorphic image.

Main resultsOur main results are as follows.

Let X be a quotient and subgroup closed class of locally nilpotent groups such thatinfinitely generated minimal non-X-groups are countable p-groups.

Theorem 54 Let π be a set of primes. A group G is an infinitely generated minimal non-(LFπ)X-group if and only if there exists a prime p /∈ π such that G is an infinitely generatedminimal non-X p-group.

Taking π to be the set of all primes, we deduce that an infinitely generated groupwhose proper subgroups are (LF)X-groups is itself a (LF)X-group.

Since by [NW] and [smith] a locally nilpotent minimal non-nilpotent group is a p-group for some prime p, then we can take X to be the class N; where N denotes the classof nilpotent groups.

Corollary 12 Let π be a set of primes. A group G is an infinitely generated minimal non-(LFπ)N-group if and only if there exists a prime p /∈ π such that G is an infinitely generatedminimal non-N p-group.

Let V be a variety of locally nilpotent groups.

Theorem 55 Let π be a set of primes. If G is an infinite locally graded group whose propersubgroups are in (LFπ)V, then so is G.

Clearly V can stands for the variety Nc; where Nc denotes the class of nilpotentgroups of class at most c and c is a positive integer.

Corollary 13 Let π be a set of primes. If G is an infinite locally graded group whose propersubgroups are in (LFπ)Nc, then so is G.

ConclusionLet Cπ = C∩ (LFπ), where C is the class of Chernikov groups.

In [AA-NT], it is proved that a locally graded group is a minimal non-(CN)-group ifand only if it is a minimal non-N-group without maximal subgroups. Using Theorem 54we generalise this result.

Let π be a set of primes and let G be an infinitely generated group. Then G is aminimal non-(CπN)-group if, and only if, G is a minimal non-N-group such that either Ghas no maximal subgroups or there exists a prime p /∈ π such that G is a p−group havingmaximal subgroups.

93

In [Otal-Pena, it is proved that a locally graded group whose proper subgroups are inCNc is itself a CNc-group. We generalise this result using Theorem 55.

Let π be a set of primes and let G be an infinite locally graded group whose propersubgroups are in CπNc. Then G is a CπNc−group.References

A. Arikan - N. Trabelsi, On minimal non-Baer-groups, Comm. Algebra 39 (2011), pp.2489-2497.

A. Dilmi, Groups whose proper subgroups are locally finite-by-nilpotent, Ann. Math. BlaisePascal 14 (2007), pp. 29-35.

M.F. Newman, J. Wiegold, Groups with many nilpotent subgroups, Arch. Math. 15(1964) 241-250.

J. Otal, J. M. Pena, Groups in which every proper subgroup is Cernikov-by-nilpotentor nilpotent-by-Cernikov, Arch. Math., Vol. 51 (1988), 193-197.

H. Smith, Groups with few non-nilpotent subgroups, Glasgow Math. J. 39 (1997)141-151.

N. Trabelsi, On minimal non-(torsion-by-nilpotent) and non-((locally finite)-by-nilpotent)groups, C. R. Acad. Sci. Paris, Ser. I 344 (2007) 353-356.

Recursive conditional hazard function estimator with functionalstationary ergodic data

Amina GOUTALUniversity of Djillali Liabes

In this paper, we investigate a recursive kernel estimator of the conditional haz-ard function whenever functional stationary ergodic data are considered. Underthe assumption of ergodicity, the novelty of our approach is that we do not re-quire independence of the observations. It is shown that, under some wild con-ditions, the recursive kernel estimate of the three parameters (conditional density,conditional distribution and conditional hazard) are asymptotically normally dis-tributed.Key words: Asymptotic normality, Conditional hazard function, Functional data, Recur-sive estimate, Stationary ergodic processes.


IntroductionRecently there has been an increasing interest in the study of functional data. For anoverview of the present state on nonparametric functional data (FDA), we refer to theworks of [ferraty2006nonparametric] and [ramsay2007applied], and the references therein.Conditional hazard estimation with a functional explanatory variable and a scalar re-sponse acquired considerable interest in the statistical literature. The first work was pro-posed by Ferraty et al. [ferraty2008estimation], where they introduce a kernel estimatorand prove some asymptotic properties (with rates) in various situations including cen-sored and/ or dependent variables. Quintela-del-Rıo [quintela2008hazard] extended theresults of Ferraty et al. [ferraty2008estimation] by calculating the bias and variance and

94

establishing their asymptotic normality.In recent years, the statistical modeling for functional ergodic data has been an increasinginterest and a great importance in various fields. The general framework of ergodic func-tional data has been initiated by Laıb and Louani [laib2010nonparametric],[laib2011rates]who stated consistencies with rates together with the asymptotic normality of the re-gression function estimate. This paper is organized as follows: Section 2 introduces theestimator of the conditional hazard function. Next we will define some notations andhypothesis. Finally we give the asymptotic normality of the proposed estimator of theconditional hazard function. The

Main results

The recursive estimation of the conditional hazard functionLet (Xi, Yi)i=1,...,n be a sequence of strictly stationary ergodic processes. Where Xi are val-ues in semi-metric space (F , d) and Yi are real-valued random variables. Nx will denotea fixed neighborhood of x. We assume that the regular version of the conditional prob-ability of Y given X exists. Moreover, we suppose that, for all x ∈ Nx the conditionaldistribution function of Y given X = x, the recursive kernel estimator of the conditionalhazard function h(y|x) such that

h(y|x) = f (y|x)1− F(y|x) , for y ∈ R and F(y|x) < 1

is

h(y|x) = f (y|x)1− F(y|x)

, ∀y ∈ R.

We define the recursive kernel estimator of the conditional distribution function by

F(y|x) =

n

∑i=1

K(a−1i d(x, Xi))H(b−1

i (y−Yi))

n

∑i=1

K(a−1i d(x, Xi))

(167)

where K is the kernel, H is a strictly increasing distribution function and ai, bi are asequences of positive real numbers such that lim

n→ +∞an = lim

n→ +∞bn = 0.

So, from (167) we get the conditional density f (y|x).

Notations and hypothesis(H1) (i) The function φ(x, h) := P(X ∈ B(x, h)) > 0, ∀h > 0,

where B(x, h) := x′ ∈ F/d(x′, x) < h.(ii) For all i = 1, ..., n there exists a deterministic function φi(x, .) such that almostsurely 0 < P(Xi ∈ B(x, h)|Fi−1) ≤ φi(x, h), ∀h > 0 and φi(x, h)→ 0 as h→ 0.

(iii) For all sequence (hi)i=1,...,n > 0,

n

∑i=1

P(Xi ∈ B(x, hi)|Fi−1)

n

∑i=1

φ(x, hi)

→ 1.

95

(H2) (i) Let S be a compact set of R, the conditional distribution function F(.|x) is suchthat, ∀y ∈ S , ∃β > 0, inf

y∈S(1− F(y|x)) > β, ∀ (y1,y2)∈ S × S , ∀ (x1,x2)∈ Nx × Nx

|F(y1|x1)− F(y2|x2)| ≤ C1(d(x1, x2)β1 + |y1 − y2|β2).

(ii) The density f (.|x) is such that,∀y ∈ S , ∃α > 0 f (y|x) < α, ∀(y1,y2)∈ S × S , ∀(x1,x2)∈ Nx × Nx

| f (y1|x1)− f (y2|x2)| ≤ C2(d(x1, x2)β1 + |y1 − y2|β2)

with C1 > 0, C2 > 0, β1 > 0, β2 > 0.

(H3) ∀(y1, y2) ∈ R2 |H(j)(y1)− H(j)(y2)| ≤ C|y1 − y2|, for j = 0, 1∫|t|β2 H(1)(t)dt < ∞ and

∫H′2(t)dt < ∞.

(H4) K is a function with support (0,1) such that 0 < C1 < K(t) < C2 < ∞.

(H5) The bandwidths (ai, bi) satisfied : ∀t ∈ [0, 1]

limn→+∞

∑ni=1 φ(x, tai)

∑ni=1 φ(x, ai)

= βx(t)

and

limn→+∞

√nφn(x)ϕn(x)

(n

∑i=1

aβ1i φ(x, ai) +

n

∑i=1

bβ2i φ(x, ai)

)= 0

where ϕn(x) = n−1 ∑ni=1 φ(x, ai).

Asymptotic normalityTheorem 56 Under hypotheses (H1)-(H5), we have for all x ∈ A(

nϕn(x)σ2

h (x, y)

)1/2

(h(y|x)− h(y|x)) D−→ N (0, 1) as n→ ∞ (168)

where A = x, σ2h (x, y) 6= 0 and σ2

h (x, y) =α2h(y|x)

α21(1− F(y|x))

with α1 = K(1)−∫ 1

0 K′(s)βx(s)ds and α2 = K2(1)−∫ 1

0 (K2(s))′βx(s)ds.

ConclusionThis paper provides a theoretical framework about recursive conditional hazard functionestimator with functional stationary ergodic data. The resulting recursive conditionalhazard function estimator has been shown to be consistent and asymptotically normallydistributed under appropriate conditions.References

[ferraty2006nonparametric] Ferraty, F., and Vieu, P. (2006). Nonparametric functionaldata analysis, Springer Series in Statistics, Springer New York.

[ferraty2008estimation] Ferraty, F., Rabhi, A., and Vieu, P. (2008). Estimation non-parametrique de la fonction de hasard avec variable explicative fonctionnelle,Rev. RoumaineMath. Pures Appl, 53, 1–18.

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[laib2011rates] Laib, N.,and Louani, D. (2011). Rates of strong consistencies of theregression function estimator for functional stationary ergodic data. Journal of StatisticalPlanning and Inference, 141(1), 359-372.

[laib2010nonparametric] Laib, N., and Louani, D. (2010). Nonparametric kernel re-gression estimation for functional stationary ergodic data: asymptotic properties. Journalof Multivariate analysis, 101(10), 2266-2281.

[ramsay2007applied] Ramsay, J. O., and Silverman, B. W. (2007). Applied functionaldata analysis: methods and case studies. Springer.

[quintela2008hazard] Quintela-Del-Rıo, A. Hazard function given a functional vari-able: Non-parametric estimation under strong mixing conditions. Journal of Nonparamet-ric Statistics, 20(5), 413-430. (2008).

Power Utility Maximization in Levy market modelMalika HAMMAD

Djillali Liabes University

We consider an optimal portfolio problem in continuous time for incompletemarkets model when the security prices follow a Geometric Levy process withdeterministic coefficients. Using the conditions for the existence of optimal port-folio policies to construct the state price density, we introduce and study the con-vex dual function of the power utility. Finally we formulate and solve our prob-lem.Key words: Portfolio optimization, Incomplete markets model, Levy processes, EMM,utility function.

2010 Mathematics Subject Classification: 91B28, 91B26, 60J75, 91B16.

IntroductionWe consider the problem of the investment and consumption portfolio on stock whoseprice at time t, Pt, is modelled by a geometric Levy process [T]: dPt = ρtPt−dt + σtPt−dYt,where Y is a general Levy process. Fajardo [F] proves that when the stocks returns fol-low a geometric Levy processes and using stochastic calculus for semimartingales, it ispossible to obtained conditions for the existence of optimal policies. We [H] take backthe latter result, using the martingale method of convex duality approach [K] and for aparticular utility function we formulate and study the value function for our problem.

The modelWe consider a financial market M consisting of 2 assets. The first is riskless bond andthe second is risky stock. The evolution of their respective prices are modeled by thefollowing equations:

dB(t) = r(t)B(t)dt, B(0) = 1

dP(t) = P(t−)[ρtdt + σtdYt], P(0) ∈ (0, ∞)

The sources of risk are modeled by Levy process Y(t), 0 ≤ t ≤ T.

97

Optimzation problem portfolioFor x > 0, a portfolio π(·), a consumption processes c(t), 0 ≤ t ≤ T and the ter-

minal wealth process Xx,π,C(T) defining: A0(x) ∆= (π, C) ∈ A(x)/E[

∫ T

0u−(t, c(t))dt +

g−(Xx,π,C)(T)] < ∞ where g and u(t, ·) are the utility functions. The value function isgiven by

VZ (x) = sup(π,C)∈A0(x)

E[∫ T

0u(t, c(t))dt + g(Xx,π,C)(T)] (169)

To maximizing this quantity, a random variable Ft− measurable ξ and a consumptionprocess C that satisfy

E[HZ (T)ξ +∫ T

0HZ (t−)dC(t)] = x > 0 (170)

Then if there exist a pair (π, C) ∈ A(x) and Xx,π,C(T) = ξ, If y > 0 : the optimum of lastoptimization problem is attained if

ξZ = Ig(yHZ (T)) and cZ (t) = Iu(t, yHZ (t−)), 0 ≤ t ≤ T. (171)

Theorem 57 (F) Suppose x ∈ (0, ∞) and VZ (x) < ∞, ∀x ∈ (0, ∞). For any x > 0, considerthe optimization problem with value function VZ (x) as in (169) and define ξZ and cZ (·) as in(171). Then if there is a portfolio process πZ (·) such that (πZ , CZ ) ∈ A(x) and Xx,πZ ,CZ (T) =ξZ .

Then (πZ , CZ ) are the solution of the optimal problem and the value function is given by:

VZ (x) = G(YZ (x)) where G(y) ∆=E[

∫ T

0u(t, Iu(t, yHZ )(t−))dt + g(Ig(yHZ (T)))], ∀y ∈

(0, ∞)

and the convex dual of VZ (·) is VZ (y) = G(y) − yXZ (y) = E[∫ T

0u(t, yHZ (t−))dt] +

E[g(yHZ (T))].

Main resultsTheorem 58 (H) Suppose x ∈ (0, ∞) and the constraints (170), (171) satisfied, the optimal

value of VZ (x) for Problem (169) is given by: VZ (x) =x1−p

1− p(XZ (1))p , 0 < x < ∞.

The optimal terminal wealth is is given by: ξZ =x

XZ (1)

(HZ (T)

)− 1p

.

The optimal consumption process is given as: cZ (t) =x

XZ (1)

(HZ (t−)

)− 1p

.

Where XZ (1) = E

[∫ T

0(HZ (t−))

p−1p dt + (HZ (T))

p−1p

].

ConclusionTaken the sufficient conditions for the existence of optimal consumption and investmentpolicies for any EMM, with a particular utility function and the martingale method of

98

convex duality approach. We have examined the optimal investment and consumptionproblem in incomplete Levy Market.

References[H] M. HAMMAD, F. LIMAM BELARBI, Optimal consumption and Investment with

Levy Processes for power utility functions, International Journal of Statistics and Economics.,2017, 18(03), 01–08.

[F] J. Fajardo, Optimal consumption and investment with levy processes, BrazilianReview of Econometrics., 57(4)825–848, 2003.

[K] I. Karatzas, S.E. Shreve, Methods of mathematical Finance, Springer, Verlag, NewYork, 1998.

[T] M. HAMMAD, Optimal Consumption and Investment with Bounded dounside risk forpower utility functions, Djillali Liabniversity of Sidi Bel, Algeria, 2017.

Attractors for a Nonautonomous Reaction-Diffusion Equationwith Delay

Hafidha Harraga, Mustapha YebdriUniversity Aboubekr Belkaid of Tlemcen

In this paper, we discuss the existence and uniqueness of solutions for a non-autonomous reaction-diffusion equation with delay, after we prove the existenceof a pullback D-asymptotically compact process and by a priori estimates weshow that it has a pullback D-absorbing set that allow us to prove the existenceof a pullback D-attractor for the associated process to the problem.

Key words: pullback attractors, reaction diffusion equations, delay term

2010 Mathematics Subject Classification: 35R10, 35K57.

IntroductionWe consider the following nonautonomous functional reaction-diffusion equation

∂∂t u(t, x)− ∆u(t, x) = f (u(t, x)) + b(t, ut)(x) + g(t, x) in (τ, ∞)×Ω ,u = 0 on (τ, ∞)× ∂Ω ,u(τ, x) = u0(x), x ∈ Ω ,u(τ + θ, x) = ϕ(θ, x), θ ∈ [−r, 0] and x ∈ Ω ,

(172)

where Ω ⊂ RN is a bounded domain with smooth boundary ∂Ω , τ ∈ R , u0 ∈ L2(Ω)is the initial condition in τ and ϕ ∈ L2([−r, 0]; L2(Ω)) is also the initial condition in[τ − r, τ] , r > 0 is the length of the delay effect. The other symbols verify the followingconditions :

H1) Concerning the nonlinearity, we assume that f ∈ C1(R, R) , there exist positiveconstants c, µ0, µ1, k and p > 2 , N ≤ 2p

p−2 such that

− c− µ0|u|p ≤ f (u)u ≤ c− µ1|u|p ∀u ∈ R, (173)

( f (u)− f (v)) (u− v) ≤ k(u− v)2 ∀u, v ∈ R. (174)

99

Let us denote by

F(u) :=∫ u

0f (s)ds .

From (173), there exist positive constants l, c′, µ′0, µ′1 such that

| f (u)| ≤ l(|u|p−1 + 1

)∀u ∈ R, (175)

− c′ − µ′0|u|p ≤ F(u) ≤ c′ − µ′1|u|p ∀u ∈ R. (176)

H2) The operator b : R× L2([−r, 0]; L2(Ω)) → L2(Ω) is a time-dependent externalforce with delay, such that

(I) For all φ ∈ L2([−r, 0]; L2(Ω)) , the function R 3 t 7→ b(t, φ) ∈ L2(Ω) is measurable;

(II) b(t, 0) = 0 for all t ∈ R ;

(III) ∃Lb > 0 s.t ∀t ∈ R and ∀φ1, φ2 ∈ L2([−r, 0]; L2(Ω));

‖b(t, φ1)− b(t, φ2)‖ ≤ Lb‖φ1 − φ2‖L2([−r,0];L2(Ω)) ; (177)

(IV) ∃Cb > 0 s.t ∀t ≥ τ , and ∀u, v ∈ L2([τ − r, t]; L2(Ω)) ;∫ t

τ‖b(s, us)− b(s, vs)‖2ds ≤ Cb

∫ t

τ−r‖u(s)− v(s)‖2ds . (178)

Remark 17 From (I)-(III), for T > τ and u ∈ L2([τ − r, T); L2(Ω)) the function R 3 t 7→b(t, φ) ∈ L2(Ω) is measurable and belongs to L∞([τ, T); L2(Ω)) .

H3) The function g ∈ L2loc(R; L2(Ω)) is an another nondelayed time-dependent exter-

nal force.For more details on differential equations with delay, we refer the reader to see J. Wu

[w] and J.K. Hale [h]. The purpose of this paper is to discuss the existence of pullbackD-attractor in L2(Ω)× L2([−r, 0]; L2(Ω)) by using a priori estimates of solutions to theproblem (172).

This work is motivated by the work of T. Caraballo and J. Real. [5], where they provedthe existence of pullback attractors for the following 2D-Navier-Stokes model with delays:

∂u∂t − ν∆u + ∑2

i=1 ui∂u∂xi

= f −∇p + g(t, ut) in (τ, ∞)×Ω ,div u = 0 in (τ, ∞)×Ω ,u = 0 on (τ, ∞)× ∂Ω ,u(τ, x) = u0(x), x ∈ Ω ,u(t, x) = φ(t− τ, x), t ∈ (τ − h, τ) and x ∈ Ω ,

(179)

where ν > 0 is the kinematic viscosity, u is the velocity field of the fluid, p the pressure,τ ∈ R the initial time, u0 the initial velocity field, f a nondelayed external force field, ganother external force with delay and φ the initial condition in (−h, 0) , where h is a fixedpositive number.

On the other hand, the problem (172) with critical nonlinearity was treated by J. Liand J. Huang in [3], where they the existence of uniform attractor for the following non-autonomous parabolic equation with delays :

∂u(t,x)∂t + Au(t, x) + bu(t, x) = F(ut)(x) + g(t, x) x in Ω ,

u(τ, x) = u0(x), u(τ + θ, x) = φ(θ, x), θ ∈ (−r, 0) .(180)

100

Here Ω is a bounded domain in Rn0 with smooth boundary, b ≥ 0 , A is a densely-defined self-adjoint positive linear operator with domain D(A) ⊂ L2(Ω) and with com-pact resolvent, F is the nonlinear term which is locally Lipschitz continuous for the initialcondition, g is an external force. In this work, we checked the existence of pullback D-attractor in C([−h, 0]; L2(Ω)) .

Main results

Construction of the associated processWe start with the following general existence and uniqueness of solutions result whichcan be obtained by the normal Faedo-Galerkin method.

Theorem 59 Assume that g ∈ L2loc(R; L2(Ω)) , b and f satisfy (I)-(IV) and (173)-(176) respec-

tively and if λ1 > 1+Cb/2 , Then for all T > τ and all (u0, ϕ) in L2(Ω)× L2((−r, 0); L2(Ω)) ,there exists a unique weak solution u to the problem (172).

Now, we will apply the above results in the phase space H := L2(Ω)× L2((−r, 0); L2(Ω)) ,which is a Hilbert space with the norm

‖(u0, ϕ)‖2H = ‖u0‖2 +

∫ 0

−r‖ϕ(θ)‖2dθ ,

with a pair (u0, ϕ) of H . To this aim, We consider g ∈ L2loc(R; L2(Ω)) , b : R× L2((−r, 0); L2(Ω))→

L2(Ω) with the hypotheses (I)-(IV) and f ∈ C1(R; R) verifying (173)-(176). Then the fam-ily of mappings

S(t, τ) : H → H(u0, ϕ) 7−→ S(t, τ)(u0, ϕ) = (u(t), ut) , (181)

with t ≥ τ , τ ∈ R and u is the weak solution to (172), defines a process.

Lemma 13 Let (u0, ϕ) , (v0, φ) ∈ H be two couples of initial conditions for the problem (172)and u , v be the corresponding solutions to (172). Then there exists a positive constant ν :=2( 1

2 + k + Cb2 − λ1) > 0 , such that

‖u(t)− v(t)‖2 ≤(‖u0 − v0‖2 + Cb‖ϕ− φ‖2) eν(t−τ) , ∀t ≥ τ . (182)

It also holds

‖ut − vt‖2C((−r,0);L2(Ω)) ≤

(‖u0 − v0‖2 + Cb‖ϕ− φ‖2) eν(t−r−τ) , ∀t ≥ τ + r . (183)

Theorem 60 Under the previous assumptions, the mapping S(t, τ) defined in (181) is a contin-uous process for all τ ≤ t .

Existence of pullback D-absorbing set in C((−r, 0); L2(Ω)) and HLemma 14 Assume that g ∈ L2

loc(R; L2(Ω)) , there exists a small enough α < 2λ1 − 2− Cbsuch that ∫ t

−∞eαs‖g(s)‖2ds < ∞ , (184)

101

the function f satisfies (173)-(176) and b fulfils conditions (I)-(IV) and∫ t

τeσs‖b(s, us)− b(s, vs)‖2ds ≤ Cb

∫ t

τ−reσs‖u(s)− v(s)‖2ds . (185)

Then we have

‖u(t)‖2 ≤ e−α(t−τ)‖u(τ)‖2 + Cbe−α(t−τ)∫ τ

τ−r‖u(s)‖2ds

+ 2c|Ω|α−1(

1− e−α(t−τ))+ e−αt

∫ t

−∞eαs‖g(s)‖2ds , (186)

and

ηe−αt∫ t

τeαs‖u(s)‖2ds + 2µ1e−αt

∫ t

τeαs‖u(s)‖p

Lp(Ω)ds

≤ e−α(t−τ)‖u(τ)‖2 + Cbe−α(t−τ)∫ τ

τ−r‖u(s)‖2ds + 2c|Ω|α−1

(1− e−α(t−τ)

)+ e−αt

∫ t

−∞eαs‖g(s)‖2ds , (187)

where η := 2λ1 − 2− α− Cb .

Proposition 11 Under the assumptions in lemma (14). Then the family B1(t) : t ∈ R givenby

B1(t) = BC((−r,0);L2(Ω))(0, R1(t)) ,

with

R21(t) = 2c|Ω|α−1 + e−α(t−r)

∫ t

−∞eαs‖g(s)‖2ds , ∀t ∈ R ;

is pullback D-absorbing for the mapping U(t, τ) . Moreover, the family B0(t) : t ∈ R given by

B0(t) = BL2(Ω))(0, R1(t))× BL2((−r,0);L2(Ω))

(0,√

rR1(t))⊂ H , ∀t ∈ R ,

is pullback D-absorbing for the process S defined by (181).

Conclusion

Existence of pullback D-attractorTo prove the existence of pullback D-attractor, we need to prove the following lemma.

Lemma 15 Assume that conditions of lemma (14) are satisfied. Then the process S(t, τ) cor-responding to (172) is pullback D-asymptotically compact.

By Proposition 11 and Lemma 15, we proved that the process S(t, τ) has a pullback D-absorbing set and it is pullback D-asymptotically compact, then we can deduce the fol-lowing result.

Theorem 61 The process S(t, τ) corresponding to (172) has a pullback D-attractor A =A(t) : t ∈ R in H . Furetheremore, A ⊂ L2(Ω)× C((−r, 0); L2(Ω)) .

102

References[5] T. Caraballo, J. Real. Attractors for 2D-Navier-Stokes models with delays. J. Differential

Equations 205, 271-297 (2004).[6] T. Caraballo, J. Real, G. Lukaszewicz. Pullback attractors for asymptotically compact

nonautonomous dynamical systems. Nonlinear Anal. 64, 484498 (2006).[1] J.Garcia-Luengo,P.Marin-Rubio. Reaction-diffusion equations with non-autonomous

force in H−1 and delays under measurability conditions on the driving delay term. J. Math.Anal.Appl.417, 80-95 (2014).

[2] J. K. Hale, Asymptotic Behavior of Dissipative Systems. Mathematical Surveys andMonographs, AMS, Providence, RI, 25 (1988).

[h] J. K. Hale, S. M. Verduyn-Lunel, Introduction to Functional Differential Equations.Springer-Verlag (1993).

[3] J. Li, J. Huang. Uniform attractors for non-autonomous parabolic equations with delays.Nonlinear Analysis 71, 2194-2209 (2009).

[l] J. L. Lions. Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires.Dunod, Paris, (1969)

[4] P. Marin-Rubio, J. Real. Attractors for 2D-Navier-Stokes equations with delays on someunbounded domains. J. Nonlinear Anal. 67, 2784-2799 (2007).

[w] J. Wu, Theory and Applications of Partial Functional Differential Equations. Springer-Verlag, New York (1996).

Etude de la fonction quantile pour des donnees tronqueesaleatoirement a gauche et a covariables fonctionnels

Nacera Helal , Elias OULD SAIDESI, Univ. Lille Nord de France

Let Y be a random real response which is subject to left-truncation by anotherrandom variable T. In this paper, we study the kernel conditional quantile esti-mation when the covariable X takes values in an infinite-dimensional space. Akernel conditional quantile estimator is given and under some regularity con-ditions, among which the small-ball probability, its strong uniform almost sureconvergence with rate is established. Some special cases have been studied toshow how our work extends some results given in the literature. Simulations aredrawn to lend further support to our theoretical results and assess the behaviorof the estimator for finite samples with different rates of truncation and sizes.Key words: Three to five key words.


IntroductionLet (X, Y) be a couple of random variables (r.v.’s) valued in F × R, where F is a semi-metric space, d denoting the semi-metric and Y being with distribution function (d.f.) F.Our purpose is to study the co-variation between X and Y via the quantile regressionestimation when the interest r.v. is subject to random left truncation and the regressorstake values in an infinite dimensional space. Let T be another real r.v. with unknown

103

d.f. G. We consider a sample (Y1, T1), (Y2, T2), ..., (YN , TN), N copies of (Y, T), where thesample size N is fixed but unknown. In this model (Yi, Ti) is observed only if Yi ≥ Tino data is collected otherwise. Then the observed sample size n is random (but known)with n ≤ N. In practice, such models are consodered in many applications.

Now let (Xi, Yi, Ti), 1 ≤ i ≤ N be a sequence of iid random vectors where Xi takesvalues in some normed space (S , ‖.‖), Yi and Ti are as before.

Since N is unknown and n is known (although random), our results will not be statedwith respect to the probability meseare P (related to the N-sample). Without possibleconfusion, we still denote (Yi, Ti), i = 1, 2, ..., n, (n ≤ N) the observed pairs from theoriginal N−sample. In all the remaining of this paper we suppose that T is independentof Y.

Now, for x ∈ S , we consider the conditional probability distribution of Yi given Xi =x by

F(y|x) = P(Yi ≤ y|Xi = x) (188)

where F is supposed strictly monotone. Let p ∈ (0, 1), the conditional quantile is definedby:

ζp(x) = infy : F(y|x) ≥ p. (189)

It is clear that an estimator of ζp(x) can easily be deduced from an estimator of F(·|x).We point out that ζp(x) satisfies

F(ζp(x)|x) = p. (190)

Background for truncation modelsRecall that our original sample is (Xi, Yi, Ti)1≤i≤N . Taking into account the truncationeffect we denote by (X1, Y1, T1), . . . , (Xn, Yn, Tn) the actually observed sample (i.e Yi ≥Ti, 1 ≤ i ≤ n) and suppose that α := P(Y1 ≥ T1) > 0. Note here that n is a real randomvariable itself and that from the strong law of large numbers (SLLN) we have, as N → ∞:

αn =nN−→ α P− a.s. (191)

where t ∧ u = min(t, u). Following [stu93] the distribution functions of Y and T are:

F∗(y) = α−1∫ y

−∞G(u)dF(u) and G∗(t) = α−1

∫ +∞

−∞G(t ∧ u)dF(u)

respectively and are estimed by

F∗n (y) = n−1n

∑i=1

1Yi≤y and G∗n(t) = n−1n

∑i=1

1Ti≤y

respectively, where 1A is the indicator of the set A. Note that, in what follows, the star no-tation (∗) relates to any characteristic of the actually observed data (that is, conditionallyon n). Define

C(y) = G∗(y)− F∗(y)= P(T1 ≤ y ≤ Y1|Y1 ≥ T1)

= α−1G(y)(

1− F(y))

, y ∈ [aF,+∞[

104

and consider its empirical estimate

Cn(y) = n−1n

∑i=1

1Ti≤y≤Yi

= G∗n(y)− F∗n (y−).

It is well known that the respective nonparametric maximum likelihood of F and G arethe product-limit estimators given by

Fn(y) = 1− ∏Yi≤y

[nCn(Yi)− 1nCn(Yi)

]and Gn(y) = ∏

Ti>y

[nCn(Ti)− 1nCn(Ti)

]

which were obtained by [lyn71]. provided aG ≤ aF, bG ≤ bF and∫

dF/G < ∞. Conse-

quently, α is identifiable only if aG ≤ aF and bG ≤ bF.

Quantile and distribution functions estimatorsOur estimation of the conditional distribution function is based on the choice of weights.These are obtained in [oul06. As N is unknown, we have to adapt the weights given in[LOP09 which gives the following values

Wi,n(x) =α−1

n K(‖x− Xi‖

hn,K

)n

∑i=1

G−1n (Yi)K

(‖x− Xi‖hn,K

) . (192)

Note that, in this formula and the forthcoming, the sum is taken only for i such thatGn(Yi) 6= 0. This in turn yields an estimator of conditional distribution function F(y|x)given by

Fn(y|x) = αn

n

∑i=1

Wi,n(x)1

Gn(Yi)H(y−Yi

hn,H

)

=

n

∑i=1

G−1n (Yi)K

(‖x− Xi‖hn,K

)H(y−Yi

hn,H

)n

∑i=1

G−1n (Yi)K

(‖x− Xi‖hn,K

)=

ψn(x, y)gn(x)

(193)

Here K is a real-valued kernel function, H is a d.f. and hn,K = hK (resp hn,H = hH) is asequence of positive real numbers which goes to zero as n goes to infinity.Let p ∈ (0, 1), a natural estimator of ζp(.) is given by

ζp,n(x) = infy : Fn(y|x) ≥ p (194)

which satisfiesFn(ζp,n(x)|x) = p. (195)

105

We consider partial derivative of ψn(x, y)

∂ψn(x, y)∂y

= ψ′n(x, y) =αn

nhHφ(hK)

n

∑i=1

1Gn(Yi)

K(‖x− Xi‖

hK

)H′(y−Yi

hH

)where H′ is derivative of H.Making use of (190) and (195), we getNow we are in position to state our main results:

Theorem 62 Under Assumptions A1–A4, we have

supx∈Ξ

supy∈[a,b]

∣∣∣Fn(y|x)− F(y|x)∣∣∣= O

(hβ

K + hγH

)+ O

((log n

nφ(hK)

)1/2)

a.s. as n→ ∞.

Theorem 63 Under the same assumptions as those of Theorem 62 and if f (y|x) > 0 for all y ina neighborhood of ζp(x) and x fixed, we have

supx∈Ξ

∣∣ζp,n(x)− ζp(x)∣∣ = O

(hβ

K + hγH

)+ O

((log n

nφ(hK)

)1/2)

a.s. as n→ ∞.

References[stu93] W. Stute. Almost sure representation of the product-limit estimator for truncated

data. Ann. Statist, Numero: 146–156, 1993. [lyn71] A. Lynden-Bell.A method of allowing forknown observational selection in small samples applied to 3CR quasars. Monthly Notices Roy.Astron. Soc, Numero: 95–118, 1971.

[oul06] E. Ould Saıd, M. Lemdani. Asymptotic properties of a nonparametric regressionfunction estimator with randomly truncated data.Ann. Inst. Statist. Math, Numero: 357–378,2006.

[LOP09] M. Lemdani, E. Ould Saıd, N. Poulin. Asymptotic properties of a conditionalquantile estimator with randomly truncated data.. J. Multivariate Anal., Numero: 546–559,2009.

Optimal exploitation of a fishery under conservation of fishspecies

Meryem HELLALUniversity of Tlemcen

We develop a spatially ecological-economical model to investigate fishery prof-its.Rather than imposing a marine reserve on our model, we ask ”when andwhere a marine reserve should be implemented ?”The optimal strategy dependson the selling price, conservationist pressure, size of the habita and ocean cur-rents. Analytical conditions are derived to show the impact of the pach size onoptimal strategy. The importance of ocean currents is analyzed through numer-ical simulations. We determine conditions under which marine reserves emergeas an essential tool for optimal policy.

Key words: Optimal harvesting · Fishery management. Economic-ecological op-timization · Pontryagin’s maximum principle2010 Mathematics Subject Classification: 34C60; 37N25; 92B05.

106

The ecosystem model

We assume that population is living in a patch of length T > 0. Let u(x) be thedensity of the stock at location x.We consider the system

−D d2u

dx2 = −µu− E(x)u + r− k dudx

u (0) = 0, u (T) = 0(196)

The model () describes the steady state of the corresponding time-dependent

system. The quantity−Dd2udx2 represents changes in concentration due to random

motion of the species, explained by the search of food, whereD is the diffusioncoefficient. Here E is the fishing effort which is estimated by the time of fishing,number and capacity of vessels, µ is the rate of mortality, r the rate of recruitment.Such recruits could come from local patches or from immigration of juvenilesproduced elsewhere. The constant k is the advective velocity of the species.

Optimal harvesting and ecosystem conservation

Our objective is to select a harvesting strategy that maximizes the functional

J =∫ T

0

(pE(x) + Q

)u(x)dx

subject to population equations () and control constraints 0 ≤ E ≤ Emax.The functional J consists of two terms, one expresses the income generated by

harvesting where p is the price value per unit of biomass.

Necessary conditions

Let v = dudx , then the problem () is equivalent to a coupled system of differential

equations dudx = vdvdx = µu + Eu− r + kvu (0) = 0, u (T) = 0

(197)

Firstly, we define the Hamiltonian as H = µ(x)E+ vλ1 (x)+ (µu− r + kv) λ2 (x)+Qu. where λ1, λ2 are the adjoint variables and µ(x) = (p + λ2) u is called theswitching function Clark [2].

The Hamiltonian H depends linearly on E. Consequently, the optimal controlwill be a combination of extreme and the singular controls. The optimal controlE that maximizes H must satisfy the following conditions

E = Emax, when µ(x) > 0, i.e., λ2(x) > −pE = 0, when µ(x) < 0, i.e., λ2(x) < −p

107

λ2(x) is the usual shadow price [2] and −p is the net economic revenue on aunit harvest.

According to the Pontryagin’s maximum Principle, the adjoint variables λ1and λ2 satisfy: i) the system

dλ1

dx= −∂H

∂u,

dλ2

dx= −∂H

∂vwhich implies that

dλ1dx = −pE− λ2

(µ + E

)−Q

dλ2dx = −kλ2 − λ1

(S)

and ii) the transversality conditions λ2(0) = λ2(T) = 0.Remark. Note that system S cannot be solved analytically. This fact makes op-

timal solution more difficult the corresponding analysis when k = 0, see[Leenheer].

Singular control

Lemma 16 The singular control E∗ does not occur.

Optimal harvesting

Since the optimal control is not singular, the possible states of the optimal controlare:i) No harvesting, E = 0ii) Maximal harvesting, E = Emaxiii) Switching between E = 0 and E = Emax.

Lemma 17 i) If E is constant,the system (S) admits a solution (λ1, λ2) satisfying λ2 (0) =

λ2 (T) = 0 and λ1 (0) < −s1

(pE+Qµ+E

).

ii) In addition, this solution admits a unique minimum(λmin

1 , λmin2).

iii) The minimum value is

λmin2 = es2x∗

λ1 (0) + s1

(p+EQµ+E

)s1

−( pE + Qµ + E

)

with x∗ = 1(s2−s1)

ln

(λ1(0)+s2

(p+EQµ+E

)λ1(0)+s1

(p+EQµ+E

) × s2s1

).

Optimal Harvesting

We will show that if pµ ≥ Q there is no switch of the optimal control.If Q > pµ,there exists a critical value of L such that if L > Lcri there will be exactly twoswitches of the optimal control.We distinguish two cases:

108

The case λmin2 ≥ −p

If pµ ≥ Q, the optimal harvesting is to fish at maximal rate Emax everywhere andno MPA is needed.

The case λmin2 < −p

If Q > pµ, then there exists a real value Lcri such that if L > Lcri, then thereexists an optimal control such that E = Emax on [0, L1[ ∪ ]L2, L] and E = 0 on[L1, L2] .The optimal strategy requires the installation of a single MPA in the mid-dle of [0 , L].References

[ss] S. M. BOUGUIMA, M. HELLAL, ”Marine Reserve Design with OceanCurrents and Multiple Objectives”. Environ. Model. Assess (Novembre 2016).

[2] C. W. Clark, Mathematical Bioeconomics: The Optimal Management of Renew-able Resources, Wiley, New York, 1976.

[Leenheer] P. De Leenheer, Optimal placement of Marine Protected Areas:a trade-off between fisheries goals and conservation efforts, IEEE Transactions on AutomaticControl, vol. 59, No. 6, June 2014

Estimation d’un Melange de Modeles GARCH PeriodiquesRokia Hemis, Hafida Guerbyenne, Faycal Hamdi

Universite de Setif, USTHB, USTHB

Nous nous interessons dans notre travail a une nouvelle classe de modeledes series chronologiques les modeles melange GARCH periodiques qui est uneextension des modeles melange ARCH periodiques. Dans la premiere partiede notre travail nous presentons cette nouvelle classe de modeles ainsi ses pro-prietes probabilistes. Dans la deuxieme partie nous nous interessons au problemede l’estimation des parametres de ces modeles nous proposons l’algorithme dugriddy Gibbs ou on determine le noyau de la loi a posteriori conditionnelle dechaque parametre du modele necessaire pour l’application de l’algorithme.Key words: Series chronologiques, Analyse bayesienne, Modeles GARCH,Modeles periodiques, Algorithme du Griddy Gibbs.

2010 Mathematics Subject Classification:62.

IntroductionIl est bien connu que les modeles les plus populaires et les plus utilises dans la modelisationde la volatilite instantanee dans les series chronologiques financieres sont les modelesAutoregressifs Conditionnellement Heteroscedastiques ARCH, introduit par Engle [5] etleur extention generalises de Bollerslev [3] et en suite les modeles GARCH periodiquesune classe de modeles introduite par Bollerslev et Ghysels (1996) qui ont montre unegrande capacite a capturer la periodicite dans la variance conditionnelle. Divers modeles

109

ont ete proposes afin de capturer differentes caracteristiques telles que la longue memoire,le changement de regime et periodicite dans la variance conditionnelle.

Notre but est de proposer un modele qui peut presenter des series chronologiquesavec une structure d’autocorrrelation periodique aussi bien que d’autres caracteristiques(telles que la multimodalite, le changement de regime) nous proposons la classe de melangedes modeles GARCH perioqiques notee: MPGARCH.

Melange de modeles GARCH periodiquesDefinition 1 :

Un processus stochastique yt, t ∈ Z suit un melange de K modeles autoregressifsconditionnellement heteroscedastiques periodiques, de periode S et d’ordresp1, p2, . . . , pK; q1, q2, . . . , qK, note MPGARCHS (K; p1, p2, . . . , pK; q1, q2, . . . , qK), s’il est definipar:

F (yt|Ft−1) = ∑Kk=1 λk Φ

(yt√h(k)t

), t ∈ Z,

h(k)t = ω(k)t + ∑

qki=1 α

(k)t,i y2

t−i + ∑pkj=1 β

(k)t,j ht−j k = 1, ...., K.

ht = ∑Kk=1 λkh(k)t

(1)

ou Φ (.) et F (.|Ft−1) sont respectivement, la distribution cumulative de la loi normalecentree reduite et de la distribution cumulative conditionnelle de yt, sachant les valeurspassees du processus jusqu’au temps t− 1. Les parametres ω

(k)t , α

(k)t,i et β

(k)t,j sont periodiques

en t, de periode S, i.e., ω(k)t+Sn = ω

(k)t , α

(k)t+Sn,i = α

(k)t,i , i = 1, ..., qk et β

(k)t+Sn,j = β

(k)t,j , j = 0, ..., pk

et t ∈ Z.Pour exclure la possibilite d’avoir des variances conditionnelles negatives ounulles, les parametres doivent verifies les conditions suivantes: ω

(k)t > 0, α

(k)t,i ≥ 0,

i = 0, ..., qk , k = 1, ..., K et t ∈ Z , et β(k)t,j ≥ 0, j = 0, ..., pk , k = 1, ..., K et t ∈ Z

. les constantes λk, k = 1, ..., K sont des constantes reelles strictement positives tel que∑K

k=1 λk = 1, avec λk > 0. Nous pouvons ecrire egalement (1) en fonction de sa fonctionde densite.

Proprietes ProbabilistesDans cette partie de notre travail, nous allons donner la condition de stationnarite strictedu modele MPGARCH. D’apres (Zhang et al. [9], Boshnakov [4]) nous reecrivons notremodele sous la forme markovienne, ce qui nous permettra d’etudier la stationarite stricteet au second ordre et l’existence des moments d’ordres superieurs.

Yt = At Yt−1 + Bt, t ∈ Z (2)

La stationnarite de ce modele decoule de l’existence d’une solution strictement sta-tionnaire. L’outil principal pour l’etude de la stationnarite stricte est le concept d’exposantde lyapounov, le plus grand exposant de Lyapunov

γS(A) = infn∈N

1N

E [log ‖AnS AnS−1 · · · A1‖] (3)

La condition de stationnarite periodique au second ordre est:

110

ρ

(S−1

∏s=0

A⊗2S−s

)< 1 (4)

EstimationPour proceder a l’estimation par la methode bayesienne des parametres d’un modeleMPGARCHS, via l’algorithme de Griddy-Gibbs (Ritter and Tanner [4]). Cette methodeconsiste a determiner les noyaux des lois a posteriori de chaque parametre du modeleconsidere, en utilisant l’analyse bayesienne nous avons ainsi P (θ|y) ∝ L (θ|y) · P (θ) ,ouP (θ|y), P (θ) et L (θ|y) sont respectivement les lois a posteriori, a priori et la fonction devraisemblance du parametre θ.

Dans notre travail les lois a priori des parametres sont choisies comme suit:

• pour le parametre λ (les proporions du melange) on propose la loi de DirechletD(α).

• pour les autre parametres les lois a priori sont choisies comme etant des loisuniormes sur des intervalles qui verifient les conditions de stationnarite du pro-cessus yt, t ∈ Z .

Comme il est necessaire de donner l’expression de la fonction de vraisemblance dansle cas des modeles MPGARCH.

Fonction de vraisemblance• y = (y1, y2, ..., yNS, Z1, ..., ZNS) et Z = (Z1, ..., ZNS) le vecteur contenant, re-

spectivement, les donnees completes et les donnees manquantes, avec Zt =(

z(1)t , ..., z(K)t

)′.

• Θ =(

λ′, Θ′1,1, · · · , Θ′S,1 , Θ′1,2, · · · , Θ′S,2, · · · ,Θ′1,K, · · · , Θ′S,K

)′ou Θs,k =

(ω

(k)s , α

(k)s,1 , ..., α

(k)s,q , β

(k)s,1 , ..., β

(k)s,p

)′et λ = (λ1, ..., λK)

′ .

L (Θ) ∝S

∏s=s0

K

∏k=1

[λk ·

(h(k)s+Sn0

)− 12

exp−

y2s+Sn0

2h(k)s+Sn0

]z(k)s+Sn0

×S

∏s=1

N−1

∏n=n0+1

K

∏k=1

[λk ·

(h(k)s+Sn

)− 12

exp− y2

s+Sn

2h(k)s+Sn

]z(k)s+Sn

soit τ(k)s+Sn la probabilite conditionnelle que l’observation yt soit generee par la keme

composante.References

[1] Ausin M.C., Galeano P., Bayesian estimation of the Gaussian mixture GARCHmodel. Comput. Statist.Data Anal, 2007, 51, 2636-2652.

[2] Bentarzi M., Hamdi F., Mixture Periodic Autoregressive conditional Heteroscedas-tic Models. Comput. Statist.Data Anal, 2008, 53, 1-16.

[3] Bollerslev T., Generalized autoregressive heterscedasticity. J. Econometrics,1986,51, 307-327.

111

[4] Boshnakov G., On first and Secend Order Stationarity of Random CoefficientModel. Linear Algebra Appl, 2011,434, 415-423.

[5] Engle R.F., Autoregressive Conditional Heteroskedasticity with estimates of vari-ance of the U. K inflation. Econometrica, 1982,50, 987- 1008.

[6] Ritter C., M.A Tanner M.A., Facilitating the Gibbs sampler: the Gibbs stopper andthe Griddy–Gibbs sampler J. Amer. Statist. Assoc, 1992, 87, 861–868.

[7] Shao Q., Mixture periodic autoregressive time series models. Statist. Proba. Lett,2006, 76, 609-618.

[8] Wong C.S., Li W.K., On a mixture autoregressive conditional heteroscedastic model.J. Amer. Statist.Assoc, 2001, 96, 982- 995.

[9] Zhang Z., Li W.K., Yuen K.C., On a mixture GARCH time series model. J. Timeser. Anal. 2006, 27, 577-597.

Reconstruction de la solution de l’equation de la chaleur avecdes deplacements

Hisao Fujita Yashima,Narimene AchourEcole Normale Superieure de Constantine

Nous proposons une methode de reconstruction de la solution de l’equationde la chaleur avec un terme de deplacement. L’idee initiale de cette methode est larepresentation stochastique de la solution de l’equation du type parabolique parl’equation de Kolmogorov inverse. Mais la methode que nous proposons nouspermet de nous liberer du langage de probabilites et de retourner a la versiondeterministe de la construction de la solution.

Key words: Equation de la chaleur, solution fondamentale, approximation par la discretisationdu temps.

2010 Mathematics Subject Classification: 35K58, 35K15.

Introduction - equation de Kolmogorov inverseEn 1931 Kolmogorov [Kolmo] a montre que l’esperence mathematique de la solutiond’une equation stochastique verifie l’equation de type parabolique, mais avec le tempsinverse.

THEOREME (de Kolmogorov, version [Gui-Sko]). Soit ξ un processus stochastique quiverifie cette equation

ξτ,x(s) = x +∫ s

τa(r, ξτ,x(r))dr +

m

∑k=1

∫ s

τbk(r, ξτ,x(r))dWk(r), (198)

ou a et b sont des fonctions qui verifient la condition de Lipschitz. Soit ϕ une fonction bornee,continue et possedant des derivees premieres et secondes continues et bornees. Alors la fonction

u(τ, x) = Eτ,x ϕ(ξ(s)), τ ∈ [t0, s],

112

possede des derivees premieres et secondes par rapport a xi continues, est derivable par rapport aτ, verifie l’equation

∂τu(τ, x) +m

∑i=1

ai(τ, x)∂

∂xiu(τ, x) +

12

m

∑k,i,j=1

bki(τ, x)bkj(τ, x)∂2

∂xi∂xju(τ, x) = 0 (199)

aveclimτ→s

u(τ, x) = ϕ(x) (200)

ou ai, bkj et xi sont les coordonnees des vecteurs a, bk et x respectivement.

On va essayer de transformer l’idee de l’equation de Kolmogorov inverse dans uneversion deterministe et dans la direction du temps naturel, ce qui – nous semble-t-il –donne une nouvelle maniere de construire la solution de l’equation du type parabolique.

Reconstruction de la solution de l’equation de la chaleurRappelons que la solution du probleme

∂

∂tu(t, x) =

12

∆u(t, x) dans Rm, u(0, x) = ϕ(x)

est donnee par

u(t, x) =∫

Rm

1(2πt)m/2 e−

|x−y|22t ϕ(y)dy.

Mais, si on veut, on peut l’exprimer par

u(t, x) = Eϕ(x + W(t)),

ou W(t) est le mouvement brownien canonique dans Rm. Cette relation relie la versiondans le langage de la solution fondamentale a la version probabiliste.

Pour l’equation

∂tu(t, x) + v · ∇u(t, x) =12

∆u(t, x) (201)

avec un vecteur constant v, la traduction de l’egalite de Chapman-Kolmogorov dans lelangage de “solution fondamentale” donne la reconstruction de la solution u(t, x) par:Soient

0 = t0 < t1 < · · · < tk−1 < tk < · · · .

Si on definitu(t0; x) = ϕ(x),

u(tk; x) =∫

Rn

1(2π(tk − tk−1))n/2 e

− |y|22(tk−tk−1) u(tk−1; x− (tk− tk−1)v+ y)dy, k = 1, 2, · · · ,

alors on a ∫Rn

1(2πtk)n/2 e−

|y|22tk ϕ(x− tkv + y)dy = u(tk; x) = u(tk; x).

Si v n’est pas constant, alors cette construction ne donne pas la solution de (201). Maissi on choisit une famille de temps discretises et fait tendre le pas de discretisation vers 0,on peut avoir l’eventuelle convergence vers la solution de l’equation.

113

Utilisation de l’idee de la reconstruction de la solutionde l’equation de la chaleurNous avons commence cette recherche par [achour] en utilisant le mouvement brownienpour le cas le plus simple.

On pose

δn =12n pour n ∈N

ett[n]k = kδn, k = 0, 1, 2, · · · .

Pour chaque n ∈N fixe, on definit

u[n](t0, x) = u0(x), (202)

u[n](t[n]k , x) = Eu[n](t[n]k−1, x− δnv(x) +√

2κW(δn)), k = 1, 2, · · · , (203)

u[n](t, x) =t[n]k − t

δnu[n](t[n]k−1, x) +

t− t[n]k−1

δnu[n](t[n]k , x) pour t[n]k−1 ≤ t ≤ t[n]k . (204)

Alors on a entre autres la

PROPOSITION 1. Soit τ > 0. Les fonctions u[n](t, x), n = 1, 2, · · · , definies dans (202)–(204) convergent uniformement sur [0, τ]×Rd.

L’idee de cette famille de “solutions approchees” a eu un rapide developpement du aL. Taleb et S. Selvaduray (et H.F.Y.) et illustre dans [Taleb2019].

Approximation deterministe de la solution de l’equationde transport-diffusionEn traduisant la definition (203) en langage deterministe et en y ajoutant un terme nonlineaire F(t, x, u), on pose

u[n](t[n]k , x) =∫

RdΘn(y)u[n](t[n]k−1, x− δnv(t[n]k , x)+ y)dy+ δnF(t[n]k−1, x, u[n](t[n]k−1, x)), k = 1, 2, · · · ,

(205)ou

Θn(x) =1

(4πδnκ)d/2 exp(− |x|2

4δnκ), x ∈ Rd, (206)

κ etant une constante strictement positive.Sous l’hypothese que les fonctions donnees u0(x), v(t, x) et F(t, x, u) sont suffisam-

ment regulieres, pour la suite de fonctions u[n](t, x)∞n=1 definies par (202), (205), (204),

on peut demontrer les resultats suivants ([Taleb2019]).

PROPOSITION 2. Quel que soit τ > 0, les fonctions u[n](t, x), n = 1, 2, · · · , definies par(202), (205), (204), et leurs derivees par rapport a xj, j = 1, · · · , d, jusqu’au meme ordre de laregularite des donnees, sont uniformement bornees dans [0, τ]×Rd.

Dans la suite nous utilisons le cas ou les derivees par rapport a xj jusqu’au ordre 3sont uniformement bornees.

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PROPOSITION 3. Quel que soit τ > 0, les fonctions u[n](t, x), n = 1, 2, · · · , definiespar (202), (205), (204), et leurs derivees premieres et secondes par rapport a xj, j = 1, · · · , d,convergent uniformement dans [0, t]×Rd.

Pour la demonstration de la proposition 3, l’egalite

Θn(z) =∫

RdΘn+1(z− y)Θn+1(y)dy

joue le role crucial.

PROPOSITION 4. La fonction limite u(t, x) de la suite de fonctions u[n](t, x)∞n=1 definies

par (202), (205), (204), fonction limite donnee par la proposition 3, satisfait a l’equation

∂tu(t, x) + v(t, x) · ∇u(t, x) = κ∆u(t, x) + F(t, x, u(t, x)) dans R+ ×Rd, (207)

et a la condition initialeu(0, x) = u0(x) dans Rd. (208)

Conclusion - perspectivesSans exagerer, nous pouvons croire que cette methode pourra avoir des developpementsinteressants dans le prochain futur. Par exemple on va considerer le probleme dans ledemi-espace (au lieu de l’espace entier Rm). Dans ce cas on verra le role fondamental dela solution explicite de l’equation de la chaleur, qui est tres classique, mais reste essen-tielle (voir par exemple [Budak]). Naturellement il y aura beaucoup d’applications pourdes problemes concernant les phenomenes de diffusion dans l’ecoulement d’un fluide,problemes tres communs dans la nature et dans l’environnement de notre vie.References

[achour] Achour, N.: Proprietes de l’equation de transport-diffusion. Memoire de Mas-ter ENS Constantine, 2019.

[Budak] Budak, B. M., Samarskii, A. A., Tikhonov, A. N.: Collection de problemes dephysique mathematique 4-ieme ed. (en russe). Fizmatlit, 2004.

[Gui-Sko] Guikhman, I., Skorokhod, A.: Introduction a la theorie des processus aleatoires(traduit du russe). Mir, 1980.

[Kolmo] Kolmogoroff, A. N.: Uber die analytischen Methodes in der Wahrschein-lichkeitsrechnung. Math. Ann. vol. 104 (1931), pp. 415–438.

[Taleb2019] Taleb, L., Selvaduray, S., Fujita Yashima, H: Approximation par une moyennelocale de la solution de l’equation de transport-diffusion. A paraıtre sur Ann. Math.Africaines.

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Clustering des series temporellesYamina Khemal Bencheikh, Welid Grimes

Universite Setif 1

Nous proposons dans ce papier d’utiliser des methodes du clustering pour creerdes groupes homogenes de series temporelles. Les deux approches du clus-tering, simple et croise conduisent a une reduction tres efficace des donnees.Elles permettent d’identifier puis d’isoler les groupes d’attributs fortement lies.L’utilisation du clustering croise fournit une double partition des donnees, sem-ble le plus pertinent, car il delecte des groupes ignores par une simple partitionsur les variables, ce qui peut se reveler tres utile sur des donnees de taille impor-tante en particulier sur des series temporelles.

Key words: Series temporelles, Clustering, Clustering croise.

2010 Mathematics Subject Classification: 62-07

IntroductionLa classification des donnees (ou data clustering) est une des methodes d’analyse desdonnees. Elle vise a diviser un ensemble de donnees en differents groupes (clusters)homogenes, en ce sens que les elements d’un meme cluster se ressemblent et ont descaracteristiques communes. Et les objets de clusters differents ont au contraire peu depoints en commun. Pour obtenir un bon partitionnement, il convient a la fois de : min-imiser l’inertie intra-cluster pour obtenir des clusters les plus homogenes possibles etmaximiser l’inertie inter-cluster afin d’obtenir des sous-ensembles bien differencies. Leclustering, ou groupement, de series temporelles necessite de faire face a deux grandesproblematiques. Dans un premier temps, l’utilisation de la distance euclidienne, et detoute autre metrique de Minkowski, sur les donnees brutes peut conduire a des resultatspeu intuitifs. En particulier, cette distance est tres sensible aux effets d’echelle, a lapresence de points atypiques ou manquants et ne permet pas de prendre en compted’eventuels decalages temporels donc il faut choisir une mesure de comparaison dessequences a analyser en fonction des invariances necessaires par rapport a l’application: facteur d’echelle en amplitude ou en temps, decalage en amplitude, dephasage tem-porel global ou distorsions temporelles locales, occultations de mesures ou complexitedes series temporelles. Dans un second temps, il faut choisir un algorithme de groupe-ment des series temporelles afin de retrouver les differents ensembles constituant lesdonnees etudiees. Nous proposons dans ce travail une extension du clustering des seriestemporelles au cas ou les donnees mettent en jeu les deux ensembles : c’est ce que l’onappelle le clustering croise [1]. Une etude de quelques mesures et methodes existantesdans le contexte general du groupement des series temporelles sera faite dans laquellenous presentons les differents types de mesures utilisees pour comparer des series tem-porelles, ensuite nous nous interessons a l’analyse des methodes classiquement utiliseespour le groupement des series temporelles. Ces etudes permettent de justifier les orienta-tions methodologiques, notamment l’interet de chacune d’entre elles dans le cadre d’unapprentissage non supervise. Nous proposons un nouvel algorithme permettant de fairele clustering croise des series temporelles et presentons un commentaire sur les resultatsobtenus a la suite de l’application de cet algorithme sur des donnees reelles,

116

Recherche de similarite dans l’analyse des series tem-porellesLes series temporelles sont des donnees ordonnees dans le temps et cet ordonnancementa une signification que l’on ne peut ignorer. Ainsi, on ne peut pas leurs appliquer desmethodes de fouille de donnees classiques qui supposent l’independance entre les exem-ples mais bien des methodes specialement adaptees, qui respectent la temporalite de cetype de donnee. Citons ici quelques mesures de similarite entre deux series temporelles: Mesure de similarite p-normee [2], la Dynamic Time Warping DTW [3], la methodeLongest Common Subsequence LCSS [4], la distance de Hamming et la mesure IterativeMultiscale Dynamic time warping IMs-DTW [5].

Differentes approches du clusteringLe clustering peut etre applique a chaque serie temporelle dans un ensemble. L’objectifest de regrouper des series entieres dans des clusters tels que les series temporelles danschaque cluster soient les plus similaires possibles. Un exemple d’application d’un telclustering est le clustering des entreprises dans une bourse en vue de detecter les en-treprises dont les variations des valeurs des actions se ressemblent. Le clustering sur lesseries temporelles peut etre effectue par plusieurs approches a chacune ses ingredients,on distingue les trois grandes approches suivantes : Le clustering hierarchique cree unehierarchie entre clusters, cette hierarchie peut etre representee par une structure arbores-cente appelee dendrogramme. Le clustering par partitionnement : Les algorithmes departitionnement ont pour but de determiner sans aucun ordonnancement hierarchiquedes observations, un nombre de clusters (groupes), dans notre cas de series temporelles,similaires. Le processus de partitionnement vise a optimiser une certaine fonction decout. Citons quelques methodes de partitionnement les plus utilisees : les Nuees Dy-namiques, la methode des Centres Mobiles, la methode des K-means et la methode desK-medoids. Le clustering par modelisation a pour but d’identifier des clusters de pointssur le principe que la densite des observations appartenant a un cluster augmente en serapprochant du centre ou du representant. Le centre ou representant du cluster etant parconsequent le point de densite maximale (ou mode). Pour les series temporelles, l’ideeest de les modeliser pour capturer et resumer leurs dynamiques. Deux series seront alorssimilaires si les modeles ajustes le sont.

Apres avoir expose, d’une part differents types de mesures utilisees pour comparerdes series temporelles, d’autre part les differentes methodes de regroupements pour cesmeme types de donnees, nous avons selectionne la mesure IMS-DTW, en tant que mesurede dissimilarite dans notre algorithme d’apprentissage non supervise de type (moyennemobile) . En l’occurrence nous utiliserons ici les K-medoids. L’approche combinee (IMS-DTW et K-medoids) permet un clustering des series temporelles de precipitations sanspasser par la caracterisation qui est tres sensible a ses parametres.

Clustering croiseDans le domaine du clustering, bien que la plupart des methodes utilisees cherchent aconstruire des partitions soit sur l’ensemble des objets soit sur celui des variables separement,il existe d’autres methodes du clustering croise qui considerent simultanement les deux

117

ensembles. Compare au clustering classique, en ne privilegiant pas un ensemble sur unautre le clustering croise est plus efficace pour decouvrir des blocs homogenes dans unematrice de donnees. Ces dernieres annees, cette approche a suscite un grand interet dansdifferentes communautes scientifiques et dans des domaines varies tels que la fouillede donnees textuelles, l’analyse des habitudes de consommation, les systemes de recom-mandation et la bioinformatique. Dans ce dernier, l’analyse de donnees transcriptomiquesdecrivant les expressions de genes a travers differentes conditions experimentales, leclustering croise permet d’identifier des sous-ensembles de genes co-exprimes unique-ment a travers un sous-ensemble de conditions. Le clustering croise a d’ailleurs ete l’objetd’un challenge: la societe Netflix, dont l’un des objectifs etait de reussir a classer simul-tanement les utilisateurs de cette societe de location de dvd et les films [6]. L’objectif dela methode du clustering croise des series temporelles est la recherche d’un couple departitions, l’une sur les individus (les lignes du tableau etudie), l’autre sur les variables(colonnes), tel que la perte d’information due au regroupement soit minimale ; c’est-a-dire telle que la difference entre l’information apportee par le tableau initial et celleapportee par le tableau obtenu apres regroupement soit minimale.

La methode proposee fournit une solution locale au probleme d’optimisation suivant: Il s’agit de trouver une partition P = (P1, P2, . . . , PK) de l’ensemble I des individusen K clusters, une partition Q = (Q1, Q2, . . . , QM) de l’ensemble J des variables en Mclusters et un ensemble L = λm

k : k = 1, . . . , K, m = 1, . . . , M de representants a partird’un tableau de donnees temporelles , tel que le critere d’inertie intra-cluster suivant soitminimum.

W(P×Q, L) =K

∑k=1

M

∑m=1

∑i∈Pk

∑j∈Qm

DIMS−DTW(xji , λm

k )

λmk :Representant du block de clusters.

P×Q = Pk ×Qm, k = 1,×, K, m = 1, . . . , MPk ×Qm = xj

i , i ∈ Pk, j ∈ Qm ou xji est la serie temporelle correspondant au croisement

de l’individu i avec la variable j.DIMS−DTW : la mesure de similarite Iterative Multiscale Dynamic time warping.

Partant d’un element (P×Q, L) initial, on fixe Q et on cherche a ameliorer P et L, puison fixe P et on cherche a ameliorer Q et L; on construit ainsi une suite (P(n), Q(n), L(n))qui fait decroitre le critere.

Application numerique et commentaireLes donnees utilisees sont les mesures de precipitations. Pour des raisons de confiden-tialite nous ne pouvons pas donner plus de details sur ces donnees. Le langage utilisedans ce travail est le Python : Les resultats obtenus, montrent une efficacite des algo-rithmes proposes. Un resultat important obtenu, surtout pour l’algorithme que nousavons propose : algorithme du clustering croise ; les resultats montrent que le couple departitions obtenues sur les lignes et les colonnes ne represente pas forcement la meilleurepartition pour les lignes et la meilleure partition pour les colonnes. Notre algorithme areussi a ameliorer le couple de partitions initiales qui representaient le meilleur couple departitions separement. Il serait donc interessant de tester ces algorithmes sur des donneessimulees pour confirmer une fois de plus leur efficacite.References

[1] Govaert, G., Classification croisee, These de doctorat d’etat, Universite Pierre etMarie Curie Paris 6, 1983. [2] Yi, B-K., Faloutsos, C., Fast sequence indexing for arbitrary

118

norms VLDB, 2000 [3] Yehia, H., Rubin, P., and Vatikiotis-Bateson, E., Quantitative asso-ciation of vocal-tract and facial behavior. Speech Communication, 26(1-2):23–43, 1998. [4]Vlachos et al., Indexing multi dimensional time series. International journal on very databases, vol. 15, no. 1, pp. 1–20, 2006. [5] Dilmi, M. D., Barthes, L., Mallet, C., (2018), Itera-tive Multi-Scale Dynamic Time Warping (IMs-DTW): Estimation de la dissimilarite entredes series temporelles de precipitation et application a la classification des evenementspluvieux. Presentation SAMA 13 mars 2018, Paris-Sacly. [6] Bennett, J. and Lanning, S. ,Thenetflix prize. In proceedings of KDD cup and workshop, volume 2007, page 35, 2007.

Quelque surfaces minimales r dans l’espace deLorentz-Heisenberg H1

3

Hamid KHIARCentre universitaire de Ain-temouchent

L’objet de ce travail est la classification de quelques surfaces spales dans l’espacesde Lorentz-Heisenberg. Il s’intsse recherche des surfaces minimales r dans cetespace. On commence d’abord par l’de de la mique de Lorentz-Heisenberg.On donne tous les ments qui caractsent cette mique, on calcule les symboles deChristoffel Γk

ij et les formes de connexion θij. Nous en disons ainsi les formes decourbure Ωi

j et les composantes du tenseur de courbure Rijkl et celle du tenseurde Ricci Rij. Nous irons aussi l’ation des surfaces minimales associt les ation desgques. On termine par les surfaces minimales r par des droites.On dntre que :

i) Les seules surfaces minimales de H13 r par des droites gques verticales sont

des morceaux de plans verticaux.

ii) Les seules surfaces minimales de H13 r par des droites gques de contact sont,

omie pre H13 , des morceaux de plans, des morceaux d’hcos z = γArctg(y/x)

avec γ r, ou des morceaux du parabolo hyperbolique z = ξxy. On montreaussi que la surfacez = λ]4ξ[x

√x2 − 1− log(x +

√x2 − 1)]− ξxy est minimale rar la droite Lt

de vecteur directeur (0, 1,−ξt) passant par le point (t, 0, a(t)). La droite Ltn’est pas une gque de H1

3 .

Dans ce travail on propose aussi alors une description de toutes les surfaces min-imales de H1

3 r par des droites qui ne sont pas nssairement des gques de H13 .

Cependant, dans ce dernier cas la description propos’est pas compl dans le senso les surfaces obtenues dndent de fonctions arbitraires qui sont solutions d’ationsdiffntielles qui ne sont pas rlues.References

[1] M. Bekkar : Exemples de surfaces minimales dans l’espace de Heisenberg,Rend. Del. Sem. Univ. Cagliari, Vol. 61, Fasc. 2 (1991).

[2] M. Bekkar : Miques riemanniennes qui admettent le plan comme surfaceminimale, Th de Doctorat, Universit Haute Alsace, (Sept. 1991).

119

[3] M. Bekkar, T. Sari : Surfaces minimales r dans l’espace de Heisenberg,Rend. Sem. Math Univ-Pol Torino. Vol. 50.3, (1992).

[4] T. Hangan : Sur les distributions totalement gques du groupe nilpotentriemannien H2p+1, Rend. Del. Sem. Univ. Cagliari, Vol. 55, Fasc.1 (1985).

[5] Daniel, B., The Gauss map of minimal surfaces in the Heisenberg group,arXiv:math/0606299v1.

[6] Inoguchi, J., Minimal surfaces in the 3-dimensional Heisenberg group, Dif-fer. Geom. Dyn. Syst. 10 (2008) 163169.

[7] Inoguchi, J., Flat translation invariant surfaces in the 3-dimensional Heisen-berg group, J. Geom. 82 (2005) 8390.

[8] Rahmani, S., Miques de Lorentz sur les groupes de Lie unimodulaires dedimension 3, J. Geom. Phys. 9 (1992), 295302

[9] Rahmani, N., Rahmani, S., Lorentzian geometry of the Heisenberg group,Geom. Dedicata 118 (2006), 133140

[10] Sanini, A., Gauss map of a surface of the Heisenberg group, Boll. Un.Mat. Ital. B 11 (2, supp.) (1997), 7993

[11]Wolfgang Khnel : Differential Geometry Curves-Surfaces Manifolds, 2ndedition, Vol.16, translated by Bruce Hunt, AMS (1950).

[12]I. Van De Woestijne : Minimal surfaces of the 3-dimensionnal Minkowskispace, in geometry topology of submanifolds, II, world scientific, singapore, 1999.pp.344-369.

Regular generalized Roumieu ultradistributionsFatima Zohra KORBAA , Khaled BENMERIEM

Mascara University

The aim of this work is to introduce and to study new classes of generalized func-tions measuring regularity both by the asymptotic behavior of the nets of smoothfunctions representing a Roumieu generalized ultradistributions and by their ul-tradifferentiable smoothness.

Key words: Roumieu ultradistributions, Colombeau generalized function, Oberguggen-berger regular space, microlocal analysis

2010 Mathematics Subject Classification: 46F30, 46F10, 35A18. .

IntroductionThe theory of generalized functions as a positive answer to the question of product distri-butions [17], this theory has been developed and applied in linear and nonlinear partialdifferential equations with non-smooth coefficients and distributions data by several au-thors [7] and [13].

Ultradistributions are useful in applications in quantum field theory, partial differ-ential equations, convolution equations, harmonic analysis, pseudo-differential theory,

120

time-frequency analysis, and other areas of analysis, see [15], so it is necessary to de-velop a generalized functions type theory in connection with ultradistributions, we givea contribution of generalized Roumieu ultradistribution theory. The problem of regular-ity theory in the Colombeau setting is in phase of exploration. The current research in theregularity problem in Colombeau algebra G is based, e.g. see [ober], on the Oberguggen-berger subalgebra G∞ the first measure of regularity within the Colombeau algebra. Thissubalgebra G∞ plays the same role as C∞ in D′.

We denote GM,+∞N (Ω) subalgebras of GM(Ω) representing classes of nets (uε)ε of

smooth functions having simultaneously ultradifferentiable smoothness of Denjoy-Carlemantype M = (Mp)p and regular asymptotic behavior in ε of [delc]. Elements of GM,+∞

N (Ω)are called ultraregular generalized functions.

Main resultsTo consider the algebra of generalized Roumieu ultradistributions, we first introduce thealgebra of moderate elements and its ideal of null elements. Let Ω be a non void open setof Rn and I =]0, 1].

Definition 12 The space of moderate elements, denoted EMm (Ω), is the space of ( fε)ε ∈ C∞(Ω)I

satisfying for every compact K of Ω, ∀α ∈ Zn+, ∃k > 0, ∃c > 0,∃ε0 ∈ I, ∀ε ≤ ε0,

supx∈K|∂α fε(x)| ≤ c exp(M(

kε)) (209)

The space of null elements, denoted NM(Ω), is the space of ( fε)ε ∈ C∞(Ω)I satisfying for everycompact K of Ω, ∀α ∈ Zn

+, ∀k > 0, ∃c > 0, ∃ε0 ∈ I, ∀ε ≤ ε0,

supx∈K|∂α fε(x)| ≤ c exp(−M(

kε)) (210)

Definition 13 The algebra of generalized Roumieu ultradistributions of class (Mp)p∈Z+ , de-noted GM(Ω), is the quotient algebra

GM(Ω) =EM

m (Ω)

NM(Ω).

We embed D′MN(Ω) into GM(Ω) using the sheaf properties, then we have the followingcommutative diagram

EMN p!(Ω) −→ GM(Ω)↓

D′MN(Ω)

Regular generalized Roumieu ultradistributions

Definition 14 The space of N-ultraregular moderate elements of class M, denoted EM,N,+∞m (Ω),

is the space of ( fε)ε ∈ C∞(Ω) satisfying: ∀K b Ω, ∃k > 0, ∃c > 0, ∃ε0 ∈]0, 1], ∀α ∈ Zn+

supx∈K|∂α fε(x)| ≤ c|α|+1N|α| exp(M(

kε))

The space of null elements is defined as NM,N,+∞(Ω) := NM(Ω) ∩ EM,N,+∞m (Ω).

121

The main properties of this two spaces are given in the following proposition.

Proposition 12 1) The space EM,N,+∞m (Ω) is an algebra stable by the action of N-ultradifferential

operators.

2) The space NM,N,+∞(Ω) is an ideal of EM,N,+∞m (Ω).

Proof 4 1) Let ( fε)ε, (gε)ε ∈ EM,N,+∞m (Ω) and K be a compact of Ω, then: ∃k1 > 0, ∃c1 >

0, ∃ε1 ∈]0, 1], ∀α ∈ Zn+, ∀ε ≤ ε1 :

supx∈K|∂α fε(x)| ≤ c|α|+1

1 N|α| exp(M(k1

ε))

We have also: ∃k2 > 0, ∃c2 > 0, ∃ε2 ∈]0, 1], ∀α ∈ Zn+, ∀ε ≤ ε2 :

supx∈K|∂αgε(x)| ≤ c|α|+1

2 N|α| exp(M(k2

ε))

let α ∈ Zn+, λ1, λ2 ∈ Zn

+, it’s clear that: ∃c = max(c1, c2), ∃k = (λ1 + λ2)max(k1, k2),∃ε0 = min(ε1, ε2) such that: ∀ε ≤ ε0

|∂α(λ1 fε(x) + λ2gε(x))| ≤ c|α|+1N|α| exp(M(kε))

So: (λ1 f1 + λ2 f2) ∈ EM,N,+∞m (Ω).

And we have:

|∂α( fεgε)(x)| ≤α

∑β=0

(αβ

) ∣∣∂α−β fε(x)∣∣ .∣∣∂βgε(x)

∣∣≤

α

∑β=0

(αβ

)c|α−β|+1

1 .c|β|+12 .N|α−β|.N|β| exp(M( k1

ε ) + M( k2ε ))

then: ∃A > 0, ∃H > 0, ∀t > 0

2M(t) ≤ M(Ht) + ln(A).

t = 1ε max(k1, k2) =

kε , C = max(c1, c2).

|∂α( fε.gε)(x)| ≤α

∑β=0

(αβ).A.C|α|+1N|α|. exp(M(Hk

ε ))

≤ C|α|+1.N|α|. exp(M( kε ))

Then: ( fε.gε)ε ∈ EM,N,∞m (Ω).

Let now P(D) = ΣaγDγ be an N-ultradifferential operator, then ∀h > 0, ∃b > 0, suchthat:

exp(−M( k1ε ))

1N|α||∂α(P(D) fε(x))| ≤ exp(−M( k1

ε )) ∑γ∈Zn

+

bh|γ|

N|γ|.N|α||∂α+γ fε(x)|

≤ b exp(−M( k1ε )) ∑

γ∈Zn+

A(H)|α+γ|h|γ|

N|α+γ||∂α+γ fε(x)|

≤ b ∑γ∈Zn

+

A(H)|α+γ|h|γ|

122

hence,for Hh < 12 we have:

exp(−M(k1

ε))

1N|α||∂α(P(D) fε(x))| ≤ c′H|α|

which shows that: (P(D) fε)ε ∈ EM,N,∞m (Ω)

2) The fact that NM,N,∞(Ω) = NM(Ω) ∩ EM,N,∞m (Ω) ⊂ EM,N,∞

m (Ω).And NM(Ω) is an ideal of EM

m (Ω), then NM,N,∞ is an ideal of EM,N,∞m (Ω)

Definition 15 The algebra of N−ultraregular generalized functions of class M = (Mp)p∈Z+ ,denoted GN,∞

M (Ω), is the quotient algebra

GM,∞N (Ω) =

EM,N,∞m (Ω)

NM,N,∞(Ω)

The following result is Paley-Wiener type characterization of GM,∞N (Ω).

Proposition 13 Let f = cl( fε)ε ∈ GMc (Ω), then f is N−ultraregular if and only if

∃k1 > 0, ∃k2 > 0, ∃c > 0, ∃ε1 > 0, ∀ε ≤ ε1, such that

|F ( fε)(ξ)| ≤ c exp(M(k1

ε)− N(k2 |ξ|)), ∀ξ ∈ Rn. (211)

The algebra GM,∞N (Ω) plays the same role as the Oberguggenberger subalgebra of regular

elements G∞(Ω) in the Colombeau algebra G(Ω).

Generalized Roumieu wave frontThe aim of this section is to introduce the generalized Roumieu wave front of generalizedRoumieu ultradistribution and to give its main properties.

Definition 16 A point (x0, ξ0) 6∈ WFM,Ng ( f ) ⊂ Ω × Rn\0 If ξ0 6∈ ∑M,N

g,x0( f ), i.e: there

exists φ ∈ DM(Ω), φ(x) = 1 neighborhood of x0, and conic neighborhood Γ of ξ0, ∃k1 > 0,∃k2 > 0, ∃c > 0, ∃ε0 > 0 such that: ∀ξ ∈ Γ, ∀ε ≤ ε0,

|F (φ fε)(ξ)| ≤ c exp(M(k1

ε)− N(k2 |ξ|))

The main proprieties of the generalized Roumieu wave front WFM,Ng are subsumed in the

following proposition:

Proposition 14 Let f ∈ GM(Ω), then

(1) The projection of WFM,Ng ( f ) on Ω is N − sinsuppg( f ).

(2) If f ∈ GMc (Ω), The projection of WFM,N

g ( f ) on Rn\0 is ∑M,Ng ( f ).

(3) ∀α ∈ Zn+, WFM,N

g (∂α f ) ⊂WFM,Ng ( f ).

(4) ∀g ∈ GM,∞N (Ω), WFM,N

g (g f ) ⊂WFM,Ng ( f ).

Theorem 64 Let T ∈ D′MN(Ω) ∩ GM(Ω) then WFM,MN−1 p!g (T) = WFMN−1 p!(T)

123

References[ben torino] Benmeriem, K., Bouzar, C., An Algera of generalized Roumieau Ultra-

distributions. Rend. Sem. Mat. Univ. Politec. Torino 70 (2), Numro: 101–109, 2012.[1] Benmeriem,K., Bouzar, C., Algebras of generalized Gevrey Ultradistributions. No-

viSad J. Math. 41 (1) (2011), 53–62.[ben-09] Benmeriem, K., Bouzar, C., Generalized Gevrey ultradistributions. New York

J. Math. 15, Numero: 37–72, 2009.[2] Benmeriem, K., Bouzar, C., Ultraregular generalized functions of Colombeau type.

J.Math. Sci. Univ. Tokyo 15 (4), Numero: 427–447,2008.[kor] Benmeriem, K., Korbaa F.Z, Generalzed Roumieu ultradistributions and their

microlocal analysis. NoviSad J. Math. 46 (2),Numero 181-200, 2016.[4] Colombeau, J.F., New Generalized Functions and Multiplication of Distributions.

North Holland, 1984.[colom] Colombeau, J.F., New generalized functions and multiplication of distribu-

tions. Math. Studies 84. Amsterdam: North Holland, 1984.[5] Colombeau, J.F., Elementary introduction to new generalized functions. North

Holland, 1985.[delc] Delcroix, A., Regular rapidly decreasing nonlinear generalized functions. Ap-

plication to microlocal regularity. J. Math. Anal. Appl., 327, p. 564-584, (2007).[7] Gramtchev, T., Nonlinear maps in space of distributions. Math. Z. 209 , Numero:

101–114, 1992.[8] Grosser, M., Kunzinger, M., Oberguggenberger, M., Steinbauer, R., Geometric the-

ory of generalized funcions. Kluwer, 2001.[ober] Hormann, G., Oberguggenberger, M., Pilipovic, S., Microlocal hypoellipticity

of linear differential operators with generalized functions as coefficients, Trans. Amer.Math. Soc., vol.358, N 8, p. 3363-3383, (2005).

[11] Komatsu, H., Ultradistributions I. J. Fac. Sci. Univ. Tokyo. Sect. IA 20 (1973),25–105.

[13] Nedeljkov, M., Pilipovic, S., Scarpalezos, D., The linear theory of Colombeaugeneralized functions. Longman Scientific and Technica, 1998.

[15] Rodino, L., Linear partial differential operators in Gevrey spaces. World Scientific,1993.

[17] Schwartz, L., Sur l’impossibilite de la multiplications des distributions. C. R.Acad. Sci. Paris 239, Numero:847–848, 1954.

On the analysis of unreliable Markovian multiserver queueM/M/c with retrials.

Faiza LIMAM-BELARBI, Meriem ELHADDADUniversity Djillali Liabes

In this paper, we investigate an approximate analysis of unreliable M/M/cretrial queue with c3 in which all servers are subject to breakdowns and repairs.Arriving customers that are unable to access a server due to congestion or failurecan choose to enter a retrial orbit for an exponentially distributed amount of timeand persistently attempt to gain access to a server, or abandon their request anddepart the system. Once a customer is admitted to a service station, he remainsthere for a random duration until service is complete and then depart the sys-tem. However, if the server fails during service, i.e., an active breakdown, the

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customer may choose to abandon the system or proceed directly to the retrial or-bit while the server begins repair immediately. In the unreliable model, there areno exact solutions when the number of servers exceeds one. Therefore, we seekto approximate the steady-state joint distribution of the number of customers inorbit and the status of the c servers for the case of Markovian arrival and servicetimes. Our approach to deriving the approximate steady-state probabilities em-ploys a phase-merging algorithm.Key words: Retrial queue, Multi server, Breakdown and repair of service, Phase mergingalgorithm.


IntroductionQueueing systems in which arriving customers who find all servers and waiting po-sitions (if any) occupied may retry for service after a period of time are called retrialqueues. For detailed survey of retrial queues and bibliographical information see Falin[Falin1990], Artalejo [1999a],[b], [2010], monograph by Falin and Templeton [1997] andGomez-Corral [2008]. Retrial queues with unreliable servers have been studied by Kulka-rni and Choi [1990], Aissani and Artalejo [1998] and Brian [Brian]. There are a great num-ber of numerical and approximations methods available.In this work, we investigate an approximate analysis of unreliable M/M/c retrial queuewith c ≥ 3 in which all servers are subject to breakdowns and repairs. Arriving cus-tomers that are unable to access a server due to congestion or failure can choose to entera retrial orbit for an exponentially distributed amount of time and persistently attempt togain access to a server, or abandon their request and depart the system. Once a customeris admitted to a service station, they remain there for a random duration until service iscomplete and then depart the system. However, if the server fails during service, i.e., anactive breakdown, the customer may choose to abandon the system or proceed directlyto the retrial orbit while the server begins repair immediately. In the unreliable model,there are no exact solutions when the number of servers exceeds one. Therefore, we seekto approximate the steady-state joint distribution of the number of customers in orbit andthe status of the c servers for the case of Markovian arrival and service times. Our ap-proach to deriving the approximate steady-state probabilities employs a phase-mergingalgorithm.Our work is organized as follows: First, we provide the formal model description andstate the assumptions that are needed to implement the approximation procedure. Next,We will place more emphasis on the solutions by phase merging algorithm outlined byKorolyuk and Korolyuk [23]. The algorithm is useful for the analysis of general. We givealso an illustrate example. Applying it to our model, we derive approximations for thesteady-state probabilities and several standard queueing performance measures. Finally,we assess the quality of the approximations by comparing results with those obtainedusing direct truncation method by M.G. Subramanian, Ayyappan and G. Sekar [SAS].

Model descriptionWe consider an unreliable M/M/c retrial queuing system in which customers arrive ac-cording to a Poisson process with rate λ (λ > 0). If upon arrival, the customer find one of

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the servers idle and not failed, he occupies him immediately. However, if customer doesnot find any available servers ( busy or failed) may join the retrial orbit with probabilityqa or abandons the system with probability 1− qa (0 ≤ qa ≤ 1). Customers who enterthe orbit wait for an exponentially distributed time with rate θ(θ > 0) before attemptingto access a server again. The service times are assumed to be exponentially distributedwith mean 1/µ. Failures for the c servers occur independently via a Poisson process withrate ξ(ξ > 0) and the repair times for each server are exponentially distributed with rateα(α > 0). Furthermore, interarrival times, service times, retrial times, interfailure timesand repair times are mutually independent.This model accounts for both active and idle breakdowns. For active breakdowns, thecustomer that is preempted by a server failure enters the retrial orbit with probability q for abandons the service with probability 1− q f .The state of the system can be described by a trivariate stochastic process in continoustime, (N(t), B(t),F(t)) : t ≥ 0, where N(t) is the number of customers in the orbit at time t, B(t) is thenumber of busy servers at time t and F(t) is the number of failed servers at time t.Since all the random times are exponentially distributed, the stochastic process is a continuous-time Markov chain (CTMC) on the state space S = (i, j, k) : i ≥ 0, j + k ≤ c, j, k ∈0, 1, 2, ...c. We assume that as t→ ∞ the steady-state distribution of (N(t), B(t), F(t)) :t ≥ 0 exists.

Define p(i, j, k) as the limiting probability that the system is in the state (i, j, k) where(i, j, k) ∈ S. Defined mathematically,

p(i, j, k) = limt→∞

P(N(t) = i, B(t) = j, F(t) = k).

Note that a set of onlyc−1

∑n=1

n + (2c + 1) pairs of (j, k) are needed to completely char-

acterize the status of the servers at any time. Let us introduce the following algorithmwhich gives the ordered pairs represents the number of busy servers and the number offailed servers, respectively.Var c : Integer ;beginread (c) ;for j=0 to c Dofor k=0 to c-i Dowrite (j,k) ;end .

Main results

•Mean Orbit Length

For the retrial queueing model with c servers, it was shown that the steady-state dis-tribution is Poisson with parameter λ/θ. Therefore, the long-run mean orbit length isapproximately the expected value of this Poisson random variable. Denoting N as the

126

steady-state number of customers in orbit, the mean orbit size is approximated by

E[N] ≈ λθ= λqa p0,c + ∑

j+k<cj 6=0

jξq f pj,k + ∑j+k=c

j 6=0

(λqa + jξq f )pj,kθ ∑j+k 6=c

pj,k.

•Mean Number of customer in ServiceLet Ns be defined as the random number of customers at the servers, the approximateexpression for the expected number of customers at the servers can be computed usingthe approximate steady-state joint probabilities derived in the last step of the algorithmand it is given by

E[Ns] =∞

∑i=0

(∑j 6=0

p(i, j, k))

• Steady-State System Size and Sojourn TimeLet us define L as the steady-state number of customers in the system, to calculate it, wesimply sum the expressions for E[N] and E[Ns].The steady-state mean sojourn time, W, follows directly from Little’s law.

L ≈ E[N] + E[Ns]W ≈ L

λ .

• Total Expected Time in OrbitDue to server failure and blocking when making a retrial attempt, customers may enterthe orbit more than once. Therefore, the expected time a customer spends in orbit is 1/θtimes the expected number of retrial attempts before gaining access to the server. defineY as the random number of retrials a customer performs until it gains access to a server.Then Y is a geometric random variable with parameter pu, the steady-state probabilitythat at least one server is available. The approximation for pu is given by

pu =∞

∑i=0

( ∑j+k 6=c

p(i, j, k)).

The expected number of retrials performed, E[Y], is therefore, 1/pu and letting Wr be therandom time spent in orbit once they are there we have,

E[Wr] ≈ (θpu)−1.

References[1998] Aissani, A and J.R. Artalejo, , On the single server retrial queue subject to

breakdown, Queueing Sys., 30 (1998), 309-321.[2008] Artalejo, J.R. and A. Gomez-Corral, Retrial Queueing Systems: A Computational

Approach, Springer, Spain, pp: 318. (2008)[1999a] Artalejo, J.R., A classified bibliography of research on retrial queues: Progress

in 1990-1999.,Busin. Econ.,7 (1999), 187-211.[b] Artalejo, J.R., Accessible bibliography on retrial queues, Math. Comp. Mod., 30

(1999), 1-6.

127

[2010] Artalejo, J.R., Accessible bibliography on retrial queues: Progress in 2000-2009., Math. Comp. Mod., 51 (2010), 1071-1081.

[Brian] Brian, P. Crawford, Approximate analysis of an unreliable M/M/2 retrial queue,thesis, (2012) [Falin1990] Falin, G., , A survey of retrial queues, Queueing Sys., 7, (1990),127-167.

[1997] Falin, G.I. and J.G.C. Templeton, Retrial queues, Champman and Hall, London,pp: 328. (1997)

[23] Korolyuk, V.S. and V.V. Korolyuk, Stochastic models of systems. Kluwer AcademicPublishers, Boston (1999).

[1990] Kulkarni, V.G. and B.D. Choi, Retrial queues with server subject to breakdownand repairs, Queueing Sys., 7 (1990), 191-208.

[SAS] Subramanian, M.G., G. Ayyappan and G. Sekar, M/M/c Retrial queueing sys-tem with breakdown and repair of services, Asian Journal of Mthematics and Statistics 4 (4)(2011), 214-223.

Estimation for infinite variance moving average processTawfiq Fawzi Mami

Centre universitaire de Ain-temouchent

In this paper, we propose a semi parametric estimator for a heavy tailed MA(1)process driven by positive-value stable variables innovations. We study its asymp-totic normality and finite sample performance.Key words: extreme value theory, max autoregressive processes, tail index estimation.

2010 Mathematics Subject Classification: 60G70, 62G32 .

IntroductionThe α-stable family of distributions X ∼ S(α, β, σ, µ) includes the Gaussian one as a spa-cial case. However, this class of distributions allows in addition for asymmetry and heavytails. In general, closed form density function of X are not known. The exceptions arefor α = 2 corresponding to normal distribution, α = 1 and β = 0 yielding Cauchy dis-tribution and α = 0.5, β = ±1 for the Levy distribution. The α-stable random variableis defined in terms of its characteristic function ϕX(t) (seee Samorodnitsky and Taqqu [Sa]) given by

ϕX(t) = exp

iµt− σα|t|α(

1 + iβt|t|w(t, α)

)(212)

with

w(t, α) =

tan(

απ

2) i f α 6= 1

2π

ln |t| i f α = 1,

where the characteristic exponent (index of stability, tail exponent) α ∈]0, 2], a skewnessparameter β ∈ [−1, 1], a scale parameter σ > 0, a location parameter µ ∈ R.

The family of stable laws S(α, 1, 1, µ) with 0 < α < 1, µ ≥ 0 define positive ran-dom variables with support (µ, ∞[, such distributions have become a standard tool in

128

modelling heavy tailed data in such diverse areas as finance, engineering and survivalanalysis. Quite often, one encounters insurance claims or lifetime data which displayheavy tail behaviors which make positive stable laws good candidates for fitting thistype of data.

In many applications, the desire to model the phenomena under study by non-negativedependent processes has increased. Recently, advancements in such models have shiftedfocus to some specialized features of the model, e.g. heavy tail innovations of the model.Let the MA(1) process

Xt = λ Zt−1 + Zt, (213)where 0 < λ < 1 and ∑∞j=0 λjω < ∞ for 0 < ω < α and

Zt ∼ i.i.d. which, for simplicity, we take to be positive stable S(α, 1, 1, µ), 0 < α <1, Sa]theserandomvariableshavethe f ollowingapproximateo f thetaildistribution f orx→ ∞

1− FZ(x) ∼ 2π

Γ(α) sin(απ

2)x−α, (214)

and Xt ∼ S(α, 1, (λα + 1)1/α, µ(1 + λ)).

Main resultsWe consider the MA(1) process in (), we have

limx→∞

P(Xt > x)P(Zt > x)

= 1 + λα,

thus we have the following approximation

1− FX(x) ∼ 2π

Γ(α) sin(απ

2)(1 + λα)x−α.

Hence we can estimate 2π Γ(α) sin( απ

2 )(1 + λα) by kn XαX

n−k,n, where

k = k(n)→ ∞, k/n→ 0

and

αX =

(1k

k

∑i=1

log Xn−i+1,n − log Xn−k,n

)−1

.

is the Hill estimator [], with Xi,n denoting the i-th ascending order statistics 1 ≤ i ≤ n,associated to the random sample (X1, X2, . . . , Xn). It follows that

λn =

πkXαXn−k,n

2nΓ(αX) sin(

παX

2

) − 1

1/αX

(215)

The asymptotic normality of λn is established in the following Theorem.

Theorem 65 Suppose the MA(1) process in () and k = kn be such that k→ ∞, k/n→ 0, then

√k

log (n/k)(λn − λ)

D−→ N

0,α2(1 + 3λα)λ2−2α

(1 + λα)

(2π

Γ(α) sin(απ

2)

)2

.

129

We can estimate the extreme index α by the t-Hill estimator given by :

αt−HX =

(1k

k

∑j=1

Xn−k,n

Xn−j+1,n

)−1

− 1

−1

, (216)

introduced in Fabian and Stehlık [Fa2009].

ConclusionNow, we generate 100 replicates of sizes 10000 from the MA(1) in (), we compare the biasand the root mean squared error (RMSE) of the estimators of λ (our estimator λn and λt−H

nin which we use the t-Hill estimator in (216) for estimating the tail index α). We remarkthat our estimator λn has the smallest bias and the λt−H

n estimator has the smallest RMSE.

References[DH] Drees, H., Extreme quantile estimation for dependent data, with applications to

finance, Bernoulli, 2003, 4, 617–657.[Fa2009] Fabian ,Z., and Stehlık, M., On robust and distribution sensitive Hill like

method. Tech. rep. IFAS Reasearch Paper Series, 2009, 43.[Sa] Samorodnitsky, G., and Taqqu, M., Non-Gaussian Stable Processes: Stochastic

Models with Infinite Variance. Chapman ft Hall, London, 1994.

Asymptotic result of the conditional estimator of cumulativedistribution in functional spatial data case

Hamza Daoudi, Boubaker MechabIbn Khaldoun University, Djillali Liabes University

The purpose of the present paper is to investigate by the local linear methoda nonparametric estimate of the conditional quantile of scalar response variablegiven a functional variable when the observations are spatially dependent. Themain goal is to establish the almost complete convergence with rate of this es-timate under some general conditions. Key words: Spatial functional data, Local

linear estimation, Conditional function, Strongly mixing process.

2010 Mathematics Subject Classification: 62G05, 62G20.

IntroductionThe conditional quantile arise in a variety of fields including econometrics, epidemiology,environmental science and many others. There is a large literature on the estimation ofthis function for independent as well as for dependent mixing data. In recent years,there has been a considerable interest in functional data analysis. We refer to Ferraty andVieu [?], Ramsay and Silverman [?]. In view of the importance of the estimate by thelinear local method, the local polynomial sitting has been recognized to have superior

130

bias properties than the kernel method (see Fan and Gijbels [?] for more details). Worksin the context of the functional non spatial data as well as the functional spatial data arevery limited and key references of this topic can be found in Barrientos-Marin et al. [?],Demongeot et al. [?], Chouaf and Laksaci[?].

Presentation of the spatial dataIn order to define the spatial functional version of the local linear estimator of the con-ditional distribution function, let Zi = (Xi, Yi) for i ∈ ZN and N ≥ 1, be (F × R)-valued measurable and strictly stationary spatial process, defined on a probability space(Ω,A, P), where (F , d) is a semimetric space. Moreover, a point i = (i1, ..., iN) ∈ ZN willbe referred to as a site.On the other hand, we assume that the process Zi = (Xi, Yi), under study, is observedover a rectangular domain:

In =

i = (i1, ..., iN) ∈ ZN , 1 ≤ ik ≤ nk, k = 1, ..., N

where n = (n1, ..., nN) ∈ ZN .All along the paper, when no confusion is possible, we denote by C and C’ any genericpositive constants and we will write:

n→ ∞ if mink=1,...,N

nk → ∞ and |nj/nk| < C

for all j, k ∈ 1, ..., N. This kind of design is known as an asymptotically increasing do-main, which allows the area of observations to become larger but keeping a minimumdistance between observation sites. Throughout this paper, we denote by ℵx a fixedneighborhood of a fixed point x ∈ F , and by § a fixed compact subset of R. Assumethat the Zi ’s are identically distributed to Z = (X, Y) and that there exists a regularversion of the conditional probability of Y given X. Then, let Fx be the conditional dis-tribution of the variable Y given X = x. Moreover, we suppose that Fx has a continuousprobability density f x with respect to (w.r.t.) the Lebesgue’s measure over R.Then, in what follows, we mainly study the local linear estimation of the functions Fx

and f x when the functional random field (Zi = (Xi, Yi), i ∈ ZN) satisfies the followingmixing condition:

There exists a function ϕ (t) ↓ 0 as t→ ∞, such that∀ E, E

′subsets of NN with finite cardinals

α(B (E) , B

(E′))

= supB∈B(E), C∈B(E′)

|P (B ∩ C)−P (B)P (C)|

≤ ψ(

Card (E) , Card(

E′))

ϕ(

dist(

E, E′))

,

(217)

where B (E) (respectively,B(E′))denotes the Borel σ-field generated by (Zi,i ∈ E) (re-

spectively, (Zi, i ∈ E′)), Card(E) (respectively, Card(E′)) is the cardinality number of E(respectively, E′),dist(E, E′) is the Euclidean distance between E and E′ and ψ : Z2 → R+

is a symmetric positive nondecreasing function in each variable such that:

∀n, m ∈ Z, ψ(n, m) ≤ C min(n, m). (218)

131

Remark that, when (217) holds with ψ ≡ 1 or N = 1, then the random field Zi = (Xi, Yi)is called strongly mixing. We also assume that the process Z satisfies the following mixingcondition:

∞

∑i=1

ik ϕ(i) < ∞, k > 0. (219)

Model and hypothesesHere, we adopt the fast functional local modeling, that is, the conditional cumulativedistribution function Fx is estimated by a where the couple (a, b) is obtained by theoptimization rule:

(a, b) = arg min(a,b)∈R2

∑i∈In

(H(h−1

H (y−Yi))− a− bβ(Xi, x))2 K(h−1

H δ(x, Xi)) (220)

where β(., .) and δ(., .) are locating functions defined from F 2 into R, such that:

∀ξ ∈ F , β(ξ, ξ) = 0 and d(., .) = |δ(., .)|

and where the function K is a kernel, H is a distribution function and hK = hK,n(respectively,hH = hH,n) is a sequence of positive real numbers which converges to 0 when n → ∞.Clearly, the estimator a, given by (291), can be explicitly written as follows:

Fx(y) =

∑i,j∈In

i 6=j

Wij(x)H(hH−1(y−Yj))

∑i,j∈In

i 6=j

Wij(x), ∀y ∈ R (221)

whereWij(x) = β(Xi, x)(β(Xi, x)− β(Xj, x))K(h−1

K δ(x, Xi))K(h−1K δ(x, Xj))

Main resultsWe first give the hypotheses that are necessary in deriving the almost-complete conver-gence (a.co.) of the functional locally modeled estimator of Fx(y).

Theorem 66 Under some assumptions, and (217)-(219), we have:

supy∈S|Fx(y)− Fx(y)| = O

(hb1

K + hb2H

)+ O

((log n

n φx(hK)

) 12)

a.co. (222)

ConclusionThis work presents the study of the almost complete convergence of the estimator of thedistribution function by the linear local method in the spatial context in the presence of afunctional explanatory variable.References

132

Barrientos-Marin J., Ferraty F., Vieu P., Locally Modelled Regression and FunctionalData, Journal of Nonparametric Statistics., 2010, 22(5), 617–632.

Chouaf A., Laksaci A., On the functional local linear estimate for spatial regression.Stat. Risk. Model., 2012, 29(3), 189–214.

Demongeot J., Laksaci A., Madani F., RachdiM., Local linear estimation of the condi-tional density for functional data, C. R., Math., Acad. Sci. Paris, 2010, 348, 931–934.

Fan J., Gijbels I., Local Polynomial Modelling and its Applications. London, Chapman& Hall, 1996.

Ferraty F., Vieu, P., Nonparametric Functional Data Analysis: Theory and Practice,Springer Series in Statistics, 2006.

Ramsay J. O., Silverman B. W., Applied functional data analysis; Methods and casestudies. Springer-Verlag, New York, 2002.

A New Error Estimate on Uniform Norm of Schwarz Algorithmfor Elliptic Quasi-Variational Inequalities with Nonlinear Source

TermsAllaoua Mehri , Samira Saadi

University May 8th 1945 Guelma, University Badji Mokhtar Annaba

The Schwarz algorithm for a class of elliptic quasi-variational inequalitieswith nonlinear source terms was studied in this work. The author proved a newerror estimate in uniform norm, making use of a stability property of the dis-crete solution. The domain was splitted in two sub-domains with overlappingnon-matching grids. This approach combined the geometrical convergence of so-lutions and the uniform convergence of variational inequalities.Key words: Schwarz Algorithm, overlapping grids, Quasi-Variational Inequalities, ErrorEstimates.

2010 Mathematics Subject Classification: 65N30, 65K15, 05C38, 65N15.

IntroductionIn the present paper, we consider the numerical solution of elliptic quasi-variational in-equalities with nonlinear right-hand side. This kind of problems have many applicationsin impulse control (see [Boulbrachene2,Cortey-Dumont2]). To estimate a new error ofthe solution we apply Schwarz algorithm, so we split the domain in two overlappingsub-domains such that each sub-domain has its own generated triangulations [Saadi2,Mehri2]. In this approach we transform the nonlinear problem into a sequence of linearproblems in each sub-domain.To prove the main result of this paper, we construct two discrete auxiliary sequencesof Schwarz, and we estimate the error between continuous and discrete Schwarz iter-ates. The proof stands on a discrete L∞-stability property with respect to both the bound-ary condition and the source term for variational inequality, while in [Saadi1] the proofstands on a stability property with respect to the boundary condition for variational in-equality.

133

An overlapping Schwarz method for elliptic quasi-variationalinequalities with nonlinear source terms

Formulation of the problem

Let Ω be an open bounded polygon in R2 with sufficiently smooth boundary ∂Ω. Wedefine the bilinear form, for any u, v ∈ H1(Ω)

a(u, v) =∫

Ω

(∑

1≤i,j≤2aij

∂u∂xi

∂v∂xj

+ ∑1≤j≤2

aj∂u∂xj

v + a0uv

)dx, (223)

the coefficients aij(x), aj(x), a0(x) are supposed to be sufficiently smooth and satisfy thefollowing conditions

∑1≤i,j≤2

aij(x)ξiξ j ≥ α|ξ|2, ξ ∈ R2, α > 0, x ∈ Ω, (224)

a0(x) ≥ β > 0, ∀x ∈ Ω. (225)

We also suppose that the bilinear form is continuous and strongly coercive

∃α > 0 : a(v, v) ≥ α‖v‖2H1(Ω). (226)

Let the obstacle Mu of impulse control defined by

Mu(x) = k + inf u(x + ξ), x ∈ Ω, ξ ≥ 0, x + ξ ∈ Ω, k > 0, (227)

the operator M maps L∞(Ω) into itself and possesses the following properties

Mu ≤ Mu, whenever u ≤ u, (228)

M(u + c) ≤ Mu + c, with c a positive constant (229)

and a closed convex set

Kg(u) = v ∈ H1(Ω) : v = g on ∂Ω, v ≤ Mu in Ω, (230)

where g is a regular function. Let f (.) be the right-hand side supposed nondecreasingand Lipschitz continuous of constant σ, such that

σ/β < 1. (231)

We consider the following elliptic quasi-variational inequality (Q.V.I)find u ∈ Kg(u) solution of

a(u, v− u) ≥ ( f (u), v− u), ∀v ∈ Kg(u),(232)

(., .) denotes the usual inner product in L2(Ω).

134

The continuous Schwarz algorithmWe consider the problem: find u ∈ K0(u) such that

a(u, v− u) ≥ ( f (u), v− u), ∀v ∈ K0(u), (233)

where K0(u) defined in (230) with g = 0.We split Ω into two overlapping polygonal sub-domains Ω1 and Ω2 such that

Ω1 ∩Ω2 6= ∅, Ω = Ω1 ∪Ω2,

and u satisfies the local regularity condition

u|Ωi∈W2,p(Ωi), 2 ≤ p < ∞.

We set Γi = ∂Ωi ∩Ωj where ∂Ωi denotes the boundary of Ωi. The intersection of Γ1 andΓ2 is assumed to be empty. We will always assume to simplify that Γ1, Γ2 are smooth.For w ∈ C0(Γi), we define

V(w)i =

v ∈ H1(Ωi)/v = 0 on ∂Ω ∩ ∂Ωi, v = w on Γi

, i = 1, 2.

We associate with problem (233) the couple (u1, u2) ∈ V(u2)1 ×V(u1)

2 such that a1(u1, v− u1) ≥ ( f (u1), v− u1), ∀v ∈ V(u2)1 ,

u1 ≤ Mu1, v ≤ Mu1 in Ω1,(234)

a2(u2, v− u2) ≥ ( f (u2), v− u2), ∀v ∈ V(u1)2 ,

u2 ≤ Mu2, v ≤ Mu2 in Ω2,(235)

where

ai(u, v) =∫

Ωi

(∑

1≤l,j≤2al j

∂u∂xl

∂v∂xj

+ ∑1≤j≤2

aj∂u∂xj

v + a0uv

)dx, i = 1, 2

ui = u|Ωi, i = 1, 2.

Let u0 ∈ C0(Ω) be the initial value such that

a(u0, v) = ( f (u0), v), ∀v ∈ H10(Ω), (236)

we define the Schwarz sequence (un+11 ) on Ω1 such that un+1

1 ∈ V(un2 )

1 solvesa1(un+1

1 , v− un+11 ) ≥ ( f (un

1), v− un+11 ), ∀v ∈ V(un

2 )1 ,

un+11 ≤ Mun

1 , v ≤ Mun1 in Ω1,

(237)

and respectively (un+12 ) on Ω2 such that un+1

2 ∈ V(un1 )

2 solvesa2(un+1

2 , v− un+12 ) ≥ ( f (un

2), v− un+12 ), ∀v ∈ V(un

1 )2 ,

un+12 ≤ Mun

2 , v ≤ Mun2 in Ω2,

(238)

where

u01 = u0 in Ω1, u0

2 = u0 in Ω2,un+1

1 = 0 in Ω \Ω1, un+12 = 0 in Ω \Ω2.

135

The discretizationLet τhi be a standard regular and quasi-uniform finite element triangulation in Ωi; i =1, 2, hi being the mesh size. We assume that τh1 and τh2 are mutually independent onΩ1∩Ω2, in the sense that a triangle belonging to τhi does not necessarily belong to τhj , i 6=j. Let Vhi = Vhi(Ωi) be the space of continuous piecewise linear functions on τhi whichvanish on ∂Ω ∩ ∂Ωi. For given w ∈ C0(Γi) we set

V(w)hi

=

vhi ∈ Vhi / vhi = πhi(w) on Γi

, i = 1, 2

where πhi denotes a suitable interpolation operator on Γi. We give the discrete counter-part of Schwarz algorithm defined in (237) and (238) as follows.Let u0

hi= rhi u

0 be given, we define the discrete Schwarz sequence (un+11h1

) on Ω1 such that

un+11h1∈ V

(un2h2

)

h1solves

a1(un+11h1

, vh1 − un+11h1

) ≥ ( f (un1h1

), vh1 − un+11h1

), ∀vh1 ∈ V(un

2h2)

h1,

un+11h1≤ rh1 Mun

1h1vh1 ≤ rh1 Mun

1h1in τh1 ,

(239)

and on Ω2 the sequence un+12h2∈ V

(un1h1

)

h2solves

a2(un+12h2

, vh2 − un+12h2

) ≥ ( f (un2h2

), vh2 − un+12h2

), ∀vh2 ∈ V(un

1h1)

h2,

un+12h2≤ rh2 Mun

2h2vh2 ≤ rh2 Mun

2h2in τh2 ,

(240)

withu0

1h1= u0

h1in Ω1, u0

2h2= u0

h2in Ω2.

We will also assume that the respective matrices produced by problems (239) and (240)are M-matrices

Main resultsTheorem 67 (Main result) . Let (un+1

i ), (un+1ih ), i = 1, 2 be the respective solutions of (237),

(238), (239), (240). Then, for n large enough, there exists a constant c independent of h and nsuch that ∥∥∥ui − un+1

ih

∥∥∥i≤ ch2 |log h|3 ,

∥∥∥ui − un+1ih

∥∥∥W1,∞(Ωi)

≤ ch |log h|3 .

ConclusionIn this work, we have established a new approach of an overlapping Schwarz algorithmon non-matching grids for a class of elliptic quasi-variational inequalities with nonlinearsource terms. We have obtained a new error estimate in uniform norm which is opti-mal for these problems. The error estimate obtained contains a logarithmic factor with

136

an extra power of |log h| than expected. We will see that this result plays an importantrole in the study of error estimate for evolutionary problems with nonlinear source terms.

References[Boulbrachene2] M. Boulbrachene, Pointwise error estimates for a class of elliptic

quasi-variational inequalities with nonlinear source terms, J. Applied Mathematics Com-puter, 2005, 161, 129 - 138.

[Cortey-Dumont2] P. Cortey-Dumont, On finite element approximation in the L∞-norm of variational inequalities, J. Numerische Mathematik, 1985, 47, 45-57.

[Mehri2] A. Mehri and S. Saadi, A new error estimate on uniform norm of Schwarz al-gorithm for elliptic quasi-variational inequalities with nonlinear source terms, Journal ofInequalities and Applications, SpringerOpen, March 2018, https://doi.org/10.1186/s13660-018-1649-3.

[Saadi1] S. Saadi and A. Mehri, L∞-Error estimate of Schwarz algorithm for ellipticquasi-variational inequalities related to impulse control problem, The Australian Journalof Mathematical Analysis and Applications (AJMAA), 2014, 11(1), 1-13.

[Saadi2] S. Saadi and A. Mehri, L∞-Error estimate of Schwarz algorithm for noncoer-cive variational inequalities, J. Applied Mathematics, 2014, 5(3), 572-580.

Numerical analysis of the Bloch-Torrey equation in deformingmedia: Application to cardiac diffusion MRI

Imen MekkaouiDr. Tahar Moulay University

We investigate in this work the diffusion magnetic resonance imaging (MRI)in deformable organs such as the living heart. The difficulty comes from thehight sensitivity of diffusion measurement to tissue motion. Commonly in lit-erature, the diffusion MRI signal is given by the complex magnetization of wa-ter molecules described by the Bloch-Torrey equation. When dealing with de-formable organs, the Bloch-Torrey equation is no longer valid. Our aim is thento introduce a new mathematical description of the Bloch-Torrey equation in de-forming media. Moreover, some numerical simulations are presented to quantifythe influence of cardiac motion on the computation of diffusion coefficient.

Key words: Bloch-Torrey equation, cardiac diffusion magnetic resonance imaging, finiteelement method, apparent diffusion coefficient.

2010 Mathematics Subject Classification: 35K57, 35E10, 65M60.

IntroductionDiffusion magnetic resonance imaging (diffusion MRI) is an imaging technique that iscapable of providing MRI images with a contrast sensitive to the diffusion motion ofwater molecules [b02]. Diffusion MRI can study the microscopic structure of biologicaltissues through the measure of the diffusion coefficient of water molecules in the tissues.

137

This technique was successfully applied to static organs such as in brain imaging [b3].However, its implementation on moving organs like the beating heart is very difficult be-cause of the cardiac motion during acquisition. The tissue motion induces an attenuationof the signal measured in diffusion MRI which can considerably degrade the quality ofestimation of the diffusion coefficient.

The signal measured in diffusion MRI is described by the magnetization density ofthe water molecules which can be modeled by the complex valued Bloch-Torrey partialdifferential equation (PDE) [b9]. This equation provides a mathematical description ofthe diffusion phenomenon of water molecules in the field of magnetic resonance imaging.In static organs, such as the human brain, several approaches have been proposed in theliterature to model the diffusion MRI signal from the Bloch-Torrey equation. For example,a multi-compartment model for the Bloch-Torrey equation has been developed in [b18]within a more complex geometry and under more realistic experimental conditions.

In this abstract, the Bloch-Torrey equation in static medium is presented and diffusionmeasurement technique used in practice is described. Then, The Bloch-Torrey equationestablished in the presence of physiological motion is introduced. Finally, we show somenumerical simulations of the diffusion MRI images during the cardiac cycle. These im-ages are reconstructed by means of the complex magnetization obtained by solving theBloch-Torrey equation with motion by a finite element method on Matlab.

Mathematical ModelLet Ω ⊂ Rd(d = 2, or 3) be a biological cell and Γ its boundary. For t ∈ [0, T], theBloch-Torrey equation is [b9]:

∂t M(x, t)− div (σ∇M(x, t)) + iq · x f (t)M(x, t) = 0 in Ω× (0, T),σ∇M · nx = 0 on Γ× (0, T)M(x, 0) = ρ(x) on Ω× 0

(241)

M is the water proton magnetization, σ intrinsic diffusion coefficient, q = γg, γ the gyro-magnetic ratio of the hydrogen atom, g the diffusion-encoding magnetic field gradient, fthe normalized time profile of g:

f (t) =

1 if 0 < t ≤ δ,−1 if ∆ < t ≤ ∆ + δelsewhere.

The Bloch-Torrey equation with a deforming domain Ω is introduced in [articlePMB] as:∂t M− div(J’

−1σJ’−t∇M) + iq · ϕ(t, x) f (t)M = 0 in Ω× (0, T),

J’−1σJ’

−T∇M · nx = 0 on Γ× (0, T),M(x, 0) = ρ(x) on Ω× 0.

(242)

ϕ(t, x) is a geometric time-space deformation field of the cell, J’ the Jacobian matrix of ϕ,J’−t = (J’

−1)t.

The dMRI signal measured at TE = ∆ + δ:

S =∫

ΩRealM(x, TE)dx. (243)

138

The apparent diffusion coefficient (ADC) is:

ADC = −1b

ln(SS0

) (244)

b-value is b := ‖q‖2∫ TE

0 F(t)2dt, and F(t) =∫ t

0 f (s)ds. S0 is the dMRI signal with q = 0.

Numerical results: Application to cardiac diffusion MRI

Analytical 2D short-axis cardiac motion modelTo study the effect of cardiac motion on diffusion MRI images, we implement a 2D car-diac image simulator introduced in [b28]. This simulator is based on a spatial-temporalanalytical cardiac motion model mimicking a realistic deformation of the heart. It canbe applied to a real short-axis image of the left ventricle to generate from it a syntheticsequence of MRI images of the beating heart (Fig. 5). The initial form of a short-axis sliceof the left ventricle can be modeled by a ring delimited by two circles of internal Rint andexternal Rext radii, respectively (Fig. 5).

Figure 5: (Left) Cardiac MRI images generated by the simulator introduced in [b28]. (Right) Adomain Ω in the form of a ring is chosen for representing the left ventricle zone.

We denote by (R0, θ0) the polar coordinates at an initial time t0 of a material point c0in the ring Ω(0) = [Rint, Rext]× [0, 2π], and by (R(t), θ(t)) its coordinates during defor-mation. The heart deformation ϕ is defined in polar coordinates by [b28]:

ϕ : [0, T]×Ω(0) → Ω(t)(t, c0) 7→ ϕ(t, c0) = ct

with

ct = ϕ(t, (R0, θ0)) =

R(t) =(

R20−R2

intλ(t) g(θ0) + Rint(θ0, t)2

)1/2

θ(t) = θ0 + ψ(t, R0) + χ(t)

λ(t) = 1− 0.2S(t), and Rint(θ0, t) = Rint − (Rint − Rmin)g(θ0)S(t), withg(θ0) = 1 + 0.1(cos(θ0 + 3π/4) + 1), Rmin = Rint − 5(Rext − Rint)/6, and

ψ(t, R0) = −(0.2π/180)S(t)R0 ; χ(t) = (10π/180)S(t).

The function S is:

S(t) =

12 (1− cos(πt

Ts)) if 0 ≤ t ≤ Ts

12 (1− cos(π(t−Ts−Td)

Td)) if Ts < t < Ts + Td,

139

Ts is the cardiac contraction duration (systolic phase), Td is the cardiac expansion duration(diastolic phase). The duration of one cardiac cycle in human heart is about 1000ms. Thesystolic duration covers one third of the cardiac cycle and the diastolic duration coverstwo thirds.

Then, the heart deformation field in Cartesian coordinates is:

ϕ(t, (x, y)) =(

R(t) cos θ(t), R(t) sin θ(t)).

Numerical simulationsThe Bloch-Torrey equation Eq. (242) is solved by finite elements method with cardiacdeformation field ϕ introduced in Section . The diffusion MRI signal is then calculatedaccording to Eq. (243). The diffusion MRI images are reconstructed according to the ADCformula (244) at different times in the cardiac cycle between 0 and 950ms. The resultingimages are shown on Figure 6 as well as the exact diffusion coefficient.

Figure 6: (Top) Diffusion MRI images at different moments of cardiac cycle. (Bottom) Exactdiffusion coefficient.

ConclusionIn this abstract, we presented the Bloch-Torrey equation that can be used to simulate thediffusion MRI images in medical imaging. We introduced a generalization form of thisequation which is derived when the imaged organ is moving like in the living heat. Thenumerical results presented in this work confirm that the diffusion is unaffected by thecardiac deformation at two points in the cardiac cycle called sweet spots. These points cor-respond to times when the cardiac deformation approximates its time average during thecardiac cycle (see [articlePMB]) and are identified at mid-systole and mid-diastole. These

140

results are in agreement with the previous experimental studies ([Dou2002], [b424]), inwhich it has been shown that there exist two time points in the cardiac cycle, sweet spots,relative to which the effect of myocardial deformation on the diffusion measurement isapproximately zero.References

[b28] P. Clarysse, C. Basset, L. Khouas, et al., Two-dimensional spatial and temporaldisplacement and deformation field fitting from cardiac magnetic resonance tagging, Clarysse, P.et al. Medical Image Analysis, 4 (2000), 253–268.

[Dou2002] J. Dou, T. G. Reese, W. Y. Tseng and V. J. Wedeen, Cardiac diffusion MRIwithout motion effects, Magn Reson Med, 48 (2002), 105–114.

[b3] M. Lazar, Mapping brain anatomical connectivity using white matter tractogra-phy, NMR Biomed., 23 (2010), 821–835.

[b02] D. Le Bihan and E. Breton, Imagerie de diffusion in vivo par resonance magnetiquenucleaire, CR Academie des Sciences, 301 (1985), 1109–1112.

[articlePMB] I. Mekkaoui, K. Moulin, et al. Quantifying the Effect of Tissue Defor-mation on Diffusion-Weighted MRI: A Mathematical Model and an Efficient SimulationFramework applied to Cardiac Diffusion Imaging, Physics in Medicine and Biology, 61(2016), 5662–5686.

[b18] D. V. Nguyen, J. R. Li, D. Grebenkov and D. Le Bihan, A finite element meth-ods to solve the Boch-Torrey equation applied to diffusion magnetic resonance imaging,Journal of Computational Physics, 263 (2014), 283–302.

[b424] C. T. Stoeck, A. Kalinowska, C. V. Deuster, J. Harmer, R. W. Chan and M.Niemann et al., Dual-phase cardiac diffusion tensor imaging with strain correction, PloSOne, 9 (2014), e107159.

[b9] H. C. Torrey, Bloch equation with diffusion terms, Physical Review, 104 (1956),563–565.

Study of Generalized finite operators and orthogonalityHadia Messaoudene , Nadia! Mesbah

Larbi Tebessi University -Tebessa, Algeria.

Let H be a separable infinite dimensional complex Hilbert space, and L(H)the algebra of all bounded linear operators onH.

The inner derivation induced by A ∈ L(H) is the map defined by:δA : L (H)→ L (H) : δA(X) = AX− XA.It is known that the identity operator I is not a commutator i.e. for all A ∈

L(H); I /∈ R (δA), where R (δA) is the range of the derivation δA.Nevertheless J.H Anderson [1] proved the existence of an operator B ∈ L(H)

such that: I ∈ R(δB), where R(δB) is the closure of R (δB) in the norm topology.

This permitted him to define a new class of operators noted, JA(H).An operator A ∈ L(H) is said to be in the class of Joel Anderson if the distance

between the rang of the inner derivation and the identity operator is minimal .i.e.∃(Xn) ∈ L(H) : AXn − Xn A→ I ⇐⇒ I ∈ R(δB), so:

JA(H) = A ∈ L(H) : I ∈ R(δA)= A ∈ L(H); ∃(Xn) ∈ L(H) : AXn − Xn A→ I .

141

The concept of finite operators which is the class of operators where the dis-tance between of the identity operator and the derivation range is maximal wasintroduced by J. P. Williams [22], he noted this class by:

F (H) = A ∈ L(H) : ‖AX− XA− I‖ ≥ 1, ∀X ∈ L(H).Williams proved that F (H) contain the normal, hyponormal, dominant and

operators satisfies the quadratic equation.

It is clear that F (H) ∩ JA(H) = ∅, and F (H) is not the complementary ofJA(H), as there are elements of L(H) which are neither operators in JA(H)nor operators in F (H). [10] proved the existence of such operators (There existsan operator A ∈ L(H), such that:‖AX− XA− I‖2 ≥ 0.97; ∀X ∈ L(H)).

In the nineties and beyond, several class of operators were introduced withoutstudy if they belong or not to the classes JA(H) and F (H).

In this paper we are aiming to investigate these classes and compare the classesof Joel Anderson and finite operator .

And prove that the classe JA(H) has no algebraic structure and to give a nec-essary and sufficient condition for a bounded linear operator A to be in JA(H)and to obtain some results concerning the form of operators in JA(H).

We present some properties ofF (H) and give some classes of operators whichare in F (H).

Key words: finite operator, class of Joel Anderson, approximate reduced spectrum, classR1.

2010 Mathematics Subject Classification: 47B47secondary 47A12.References

[A1] J. H. Anderson, Derivation range and identity, Bull of Amer. math. soc.vol 79 N4(1973)705-708.http://arxiv.org/abs/0805.4100.

[D1] B. P. Duggal and In Ho Jeon ,On p-quasihyponormal operators, Linear.Algeb. and its appl 422 (2007) p 331-340.

[F1] C. K. Fong and V. I. Istratescu, Some characterizations of Hermitien op-erators and related classes of operators, Proc of Amer. Math. Soc. V76 (1979) p107-112.

[F2] T. Furuta, Invitation to linear operators, from matrices to bounded linear opra-tors on Hilbert space, Taylor and Francis Ltd (2001) .

[F3] T. Furuta, On the class ofT. Furuta, Invitation to linear operators, from ma-trices to bounded linear oprators on Hilbert space, Taylor and Francis Ltd (2001) .

[H1] D. Herrero and A. Fialkow, Finite operators and similarity orbits, Proc ofAmer. Math. Soc.V93 N 4 (1985) p 601-609.

[Ha 1] P. R. Halmos, A Hilbert space probleme book, second edition. Springer-Verlag (1982).

[Ha 2] P. R. Halmos,Commutators of Operators II, American Journal of Mathe-maticsVol. 76, No. 1 (Jan., 1954), pp. 191-198.

[I1] M. Ito, Some classes of operators associated with generalized Aluthgetransformation, SUT. Jour. of Mathematics V 35 N1 (1999) p 149-165.

[M1] S. Mecheri, Non normal derivation and orthogonality, Proc of Amer. Math.Soc.V 133 N 3 (2004) p 759-762.

142

http://arxiv.org/abs/0805.4100

[M2] S. Mecheri, Finite operators, Demonstratio Mathematica Vol XXXV N 2(2002) p 357-366.

[ME1] H. Messaoudene, On derivation range and identity operator, Int. Jour-nal of Math. Analysis, Vol. 4, 2010, no. 24, 1189-1196.

Optimal Control Problems for a Semilinear Evolution Systemwith Infinite Delay

Fatima Zahra MOKKEDEMUniversite de TLEMCEN

In the first part of this paper, we study the existence of optimal controls fora class of infinite-delayed semilinear evolution systems. Without assuming theintegral cost function to be convexe, the results hold for bounded and unboundedadmissible control sets.

In the second part of this work, We prove the existence of time optimal con-trols to a target set. In addition, we show the convergence of time optimal controlsto a point target set.Key words: Evolution equations; infinite delay; fundamental solution; optimal control;time optimal control.

2010 Mathematics Subject Classification: 34K30, 34K35, 35R10, 93B05, 93C10.

IntroductionIn this paper we study the standard optimal control and time optimal control problemsfor the following class of infinite-delayed semilinear evolution systems:

ddt

x(t) = Ax(t) + L (xt) + F(t, xt) + Bu(t), 0 ≤ t ≤ T,

x0 = ϕ ∈ B,(245)

where the state function x(·) ∈ X and the control function u(·) ∈ U with X and U are twoHilbert spaces. The histories xt : (−∞, 0]→ X, given in the usual way by xt(θ) = x(t+ θ)for θ ≤ 0, belong to some abstract phase space B defined axiomatically, see [HK]. Theoperator A : D(A) ⊆ X → X is the infinitesimal generator of an analytic semigroup(S(t))t≥0 on X, see [EN]. The operators L : B → X and B : L2(0, T; U) → L2(0, T; X)are bounded linear operators and F : [0, T]× B → X is a nonlinear Lipschitz continuousfunction.

System (329) is an abstract form of partial functional differential equations (PFDEs)with infinite delay. A variety of mathematical models for physical, chemical or biolog-ical processes are most appreciatively formed as PFDEs with infinite delay [Wu1], forexample the reaction-diffusion logistic equation with infinite delay and equations of heatconduction in materials with fading memory.

In the existing works on control problems, the desired results are obtained by usingthe semigroup theory and by assuming that L(xt) = 0. Here, because of the additionaloperator L(xt), the mild solutions expressed by the semigroup (S(t))t≥0 take an implicit

143

form which makes the convergence arguments very complicated and difficult to be veri-fied. Hence, an other operator, named the fundamental solution, is constructed to replacethe semigroup one. The mild solutions expressed by the fundamental solution are of ex-plicit form which makes the proofs shorter and easier.

Main results

Fundamental solution and basic lemmas

The mild solutions of System (329) are expressed by the C0−semigroup (S(t))t≥0 as fol-lows

x(t, ϕ) =

S(t)ϕ(0) +∫ t

0S(t− s)

(L (xs(·, ϕ)) + F(s, xs) + Bu(s)

)ds, t ≥ 0,

ϕ(t), t ≤ 0.(246)

This implicit form makes the convergence arguments very difficult and complicated. Toovercome this difficulty, we need to present the mild solutions of System (329) in a moreexplicit way. To this end, we first construct the fundamental solution G(t) to be the uniquesolution of the operator equation

G(t) =

S(t) +∫ t

0S(t− s)L (Gs) ds, t ≥ 0,

0, t < 0,

where Gt(θ) := G(t + θ), θ ≤ 0. Then, by using Laplace transform techniques, we canprove that the mild solutions of System (329) are given equivalently by

x(t, ϕ) =

G(t)ϕ(0) +∫ t

0G(t− s)

(L (ϕs) + F(s, xs) + Bu(s)

)ds, t ∈ (0, T],

ϕ(t), t ≤ 0,

where the function ϕ(·) is defined as

ϕ(t) =

ϕ(t), t ≤ 0,0, t > 0.

It is worthy to mention that the fundamental solution G(t) does not satisfy the semigroupproperty which makes our discussion greatly different and more complicated than theones in the existing works.

Let the mild solution x(·) of System (329) be denoted by x(·; u) (to show its depen-dence on the control function u) and define the solution operator W : L2(0, T; U) →C([0, T]; X) by

u(·) 7→ (Wu)(·) = x(·; u).

Lemma 18 If the semigroup (S(t))t≥0 is compact, then the solution operator W is compact fromL2(0, T; U) to C([0, T]; X).

144

Next define the so-called Nemitsky operator F : L2(0, T; U) → L2(0, T; X) corre-sponding to the nonlinear function F(·, ·), i. e.,

(Fu)(·) = F(·, (Wu)·).

Then for this operator we have

Lemma 19 Under the assumptions of Lemma 20, the operator F is compact from L2(0, T; U) toL2(0, T; X).

Existence of Optimal controlsLet the admissible set Uad be a closed convex subset of L2(0, T; U) and define the integralcost function

J = J(u, x(·; u))φ0(x(T; u)) +∫ T

0

(φ1(x(t; u), xt(θ; u), t) + φ2(u(t), t)

)dt, (247)

where θ ≤ 0 and φi, i = 0, 1, 2, satisfy that

(H5) (i) φ0 : X → R is continuous.(ii) φ1 : X× B× [0, T]→ R is positive, i. e., φ1 ≥ 0 and measurable in t ∈ [0, T] foreach (x, ψ) ∈ X× B and continuous in (x, ψ) ∈ X× B for a. e. t ∈ [0, T].(iii) φ2 : U × [0, T] → R is integrable on [0, T] for each u ∈ Uad and the functionalΓ : Uad → R given by

(Γu) =∫ T

0φ2(u(t), t)dt

is continuous and convex.

We need to find a control u ∈ Uad which minimizes the cost J subject to the constraint(329) in both cases of bounded and unbounded admissible control sets.

Theorem 68 Let Uad be bounded in L2(0, T; U) and suppose that (H1)− (H5) are all satisfied.If the semigroup (S(t))t≥0 is compact, then there exists at least one control u ∈ Uad whichminimizes the cost function J subject to (329).

Next we consider the case that Uad is unbounded in L2(0, T; U). To this end, besidesthe previous assumptions we assume that the followings also hold true.

(H6) (i) There exists a constant c0 > 0 such that φ0(·) ≥ −c0 on X.(ii) There exists a constant c1 > 0 such that φ1(·, ·, ·) ≥ −c1 on X× B× [0, T].(iii) There exists a monotone increasing function θ0 ∈ C(R+; R) such that lim

r→∞θ0(r) =

+∞ and

(Γu) =∫ T

0φ2(u(t), t)dt ≥ θ0

(‖u‖L2(0,T;U)

), for u ∈ Uad.

Theorem 69 Let Uad be unbounded in L2(0, T; U) and suppose that (H1)− (H6) are all satis-fied. If the semigroup (S(t))t≥0 is compact, then there exists at least one optimal control u ∈ Uadthat minimizes the cost function J subject to (329).

145

Time optimal controlNow we study some time optimal control problems for System (329). Let the admissibleset Uad be a bounded, closed and convex subset in L2(0, T; U) and W be a bounded, closedand convex subset in X. Define

U0

u ∈ Uad | x(t; u) ∈W for some t ∈ [0, T]

,

and suppose that U0 6= ∅. For each u ∈ U0 we can define the transition time that is thefirst time t(u) such that x(t; u) ∈ W. This transition time t(u) is well defined for eachu ∈ U0. The set W is called a target set and the time optimal control problem remains tofind a control u ∈ U0 such that

t(u) ≤ t(u), for all u ∈ U0,

subject to the constraint (329).

Theorem 70 Let ϕ ∈ B. Assume that U0 6= ∅ and (H1)− (H4) are satisfied. If the semigroup(S(t))t≥0 is compact, then there exists a time optimal control u ∈ U0.

Next, we consider the case in which the target set W is singleton. Put W = w0 suchthat ϕ(0) 6= w0 and ϕ(θ) 6= w0 for some θ < 0. Since X is reflexive, we can choose adecreasing sequence of non-empty, bounded, closed and convex sets Wnn≥1 in X suchthat

w0 =+∞⋂n=1

Wn and dist(w0, Wn) = supx∈Wn

|x− w0| → 0 as n→ +∞. (248)

DefineUn

0

u ∈ Uad∣∣ x(t; u) ∈Wn, for some t ∈ [0, T]

.

Then the time optimal control problem with the target set w0 can be solved as follows.

Theorem 71 Let Wnn≥1 be a decreasing sequence of non-empty, bounded, closed and convexsets in X satisfying the condition (248) and Un

0 6= ∅, for n ≥ 1. Let unn≥1 be a sequence suchthat, for any n ≥ 1, un is the time optimal control with the optimal time tn to the target set Wn.Then there exists a time optimal control u0 with the optimal time t0 = sup

n≥1tn to the point

target set w0 which is given by the weak limit of some subsequence of unn≥1 in L2(0, t0; U).

ConclusionThe purpose of this work is to extend the existence results on optimal and time optimalcontrols for semilinear evolution systems with finite delay achieved in Papers [JYC, JS,Naka] to the systems with infinite delay of the form (329). Note that many practical mod-els with fading memory can be represented by this kind of equations. The main tool usedin this work is the fundamental solution theory with infinite delays established in ourprevious work [MFE]. We first proved the existence and uniqueness of mild solution ofthe control system (329). Then, we showed the compactness of the solution operator Wand hence of the Nemitsky operator F defined above. Based on these results, the mildsolution of the discussed System becomes implicit and the discussion becomes more con-venient. In this work, we have proved the existence of optimal controls for certain integralcost function in bounded and unbounded admissible control sets respectively. Since the

146

solution operator W is completely continuous, the functions φ0 and φ1 in (247) are notsupposed to be convex which improves somehow the previous results on this topic in-cluding the ones in [JS, Naka]. On the other hand, we have investigated the existence oftime optimal controls to a target set. In addition, a convergence theorem of time optimalcontrols to a point target set has also been proved. Clearly, the obtained results of thispaper extend and develop directly those in papers [JYC, JS, Naka]to semilinear infinite-delayed control systems.References

[EN] Engel K. J. and Nagel R., One-Parameter Semigroups for Linear Evolution Equa-tions, Springer, New York, 2000. [HK] Hale J. and Kato J., Phase space for retarded equa-tions with infinite delay, Funk. Ekvac., 1978, 21, 11–41.

[JYC] Jeong J. M., Ju E. Y. and Cheon S. J., Optimal control problems for evolutionequations of parabolic type with nonlinear perturbations, J. Optim. Theory Appl., 2011,151, 573–588.

[JS] Jeong J. M. and Son S. J., Time optimal control of semilinear control systems in-volving time delays, J. Optim. Theory Appl., 2015, 165, 793–811.

[Liu] Liu K., The fundamental solution and its role in the optimal control of infinitedimensional neutral systems, Appl. Math. Optim., 2009, 609, 1–38.

[MFE] Mokkedem F. Z. and Fu X., Approximate controllability for a semilinear evo-lution system with infinite delay, J. Dyn. Control Sys., 2016, 22, 71–89.

[Naka] Nakagiri S., Optimal control of linear retarded systems in Banach spaces, J.Math. Anal. Appl., 1986, 120, 169–210.

[Wu1] Wu J., Theory and Applications of Partial Functional Differential Equations,Springer Verlag, New York, 1996.

Estimation of extremal index for heavy tailed ARMAX processHakim Ouadjed


Based on the theory of extreme values, we propose an asymptotically normalsemiparametric estimator of the extremal index for heavy tailed ARMAX(1) pro-cess.Key words: extreme value theory, max autoregressive processes, tail index estimation.

2010 Mathematics Subject Classification: 60G70, 62G32 .

IntroductionThe extremal index parameter characterizes the degree of local dependence in the ex-tremes of a stationary time series and has important applications in a number of areas,such as hydrology, telecommunications, finance and environmental studies. This pa-rameter is the key for extending extreme value theory results from i.i.d. to stationarysequences.

Many applications as in insurance and finance, telecommunication and other areasof technical risk, usually exhibit a dependence structure. Leadbetter et al. [Led] puta mixing condition D(un) based on the probability of exceedances of a high thresholdun, it limits the degree of long-term dependence of the sequence, providing asymptoticindependence between far apart extreme observations.

147

Definition 17 (D(un) condition) A strictly stationary sequence Xi, whose marginal distri-bution F has upper support point xF = supx : F(x) < 1, is said to satisfy D(un) if, for anyintegers i1 < . . . < ip < j1 < . . . < jq with j1 − ip > ln,∣∣∣PXi1 ≤ un, . . . , Xip ≤ un, Xj1 ≤ un, . . . , Xjq ≤ un

− P

Xi1 ≤ un, . . . , Xip ≤ un

P

Xj1 ≤ un, . . . , Xjq ≤ un

∣∣∣ ≤ δ(n, ln),

where δ(n, ln)→ 0 for some sequences ln = o(n) and un → xF as n→ ∞.

Let X1, . . . , Xn be a strictly stationary sequence with marginal distribution F , andX1, . . . , Xn an i.i.d. sequence of random variables with the same distribution F, definethe following quantities Mn = max(X1, . . . , Xn) and Mn = max(X1, . . . , Xn). Under theD(un) condition, with un = anx + bn, if

P[a−1n (Mn − bn) ≤ x]→ G(x), as n→ ∞, (249)

for normalizing sequences an > 0 and bn ∈ R, Leadbetter et. al. [Led] showed that

P[a−1n (Mn − bn) ≤ x]→ [G(x)]θ , as n→ ∞, (250)

where G is one of the three extreme value types distributions

Type I (Gumbel):G(x) = exp(−e−x), x ∈ R,

Type II (Fret):

G(x) =

0, x ≤ 0exp(−x−α), x > 0, α > 0,

Type III (Weibull):

G(x) =

exp(−(−x)α), x ≤ 0, α > 01, x > 0,

and θ ∈ (0, 1] is the extremal index, this parameter characterizes the short-range depen-dence of the maxima. In particular, θ−1 gives a measure of the degree of clustering oflarge values of the sequence.

Main resultsThe ARMAX(1) process with infinite variance is defined by

Xi = max (λ Xi−1, Zi) , (251)

where 0 < λ < 1 and Z1, . . . , Zn are independent and identically distributed, withdistribution function FZ(x) = exp(−x−α), 0 < α < 2. For this process θ = 1− λα

As a result we get

θn =nk

X−αXn−k,n (252)

where k = k(n)→ ∞, k/n→ 0 and

αX =

[1k

k

∑i=1

log Xn−i+1,n − log Xn−k,n

]−1

,

148

is the Hill estimator [Hi], with Xi,n denotes the i-th ascending order statistics 1 ≤ i ≤ n,associated to the random sample (X1, X2, . . . , Xn).

The asymptotic normality of θn is established in the following theorem.

Theorem 72 Suppose (251) and k = kn be such that k→ ∞, k/n→ 0, then√

klog (n/k)

(θn − θ)D−→ N

(0, σ2

2)

,

whereσ2

2 = α4θ3(2− θ). (253)

ConclusionWe compare, in terms of bias and RMSE, our estimator in (252) with that of Ferro andSegers [FS] where we established the performance of our estimator for α ∈ [0, 1] then,with that of Olmo [4] where we have shown its efficiency always for α ∈ [0, 1].References

[FS] Ferro, C. A. T., and Segers, J., Inference for Clusters of Extreme Values, Journal ofthe Royal Statistical Society, Ser. B., 2003, 65, 545–556.

[Hi] Hill, B. M., A simple approach to inference about the tail of a distribution, Ann.Statist., 1975, 3, 1136–1174.

[Led] Leadbetter, M. R., Lindgren, G. and Rootzen, H., Extremes and Related Proper-ties of Random Sequences and Processes, Springer, New York, 1983.

[ol] Olmo, J., A New Family of Consistent and Asymptotically-Normal Estimators forthe Extremal Index, Econometrics., 2015, 3, 633–653.

Special values of generalized multiple Hurwitz zeta function atnon-positive integers

Boualem SADAOUIKhemis Miliana University

In this talk, we provide an alternative method to calculate the values of gen-eralized multiple Hurwitz zeta function at non-positive integers by means ofRaabe’s formula and the Bernoulli numbers.Key words: Generalized multiple Hurwitz zeta function; integral representation; specialvalues; Bernoulli numbers; Raabe’s formula.

2010 Mathematics Subject Classification: 11B68; 11M32; 11M35; 11M41.

IntroductionThe multiple Hurwitz zeta function is defined by

ζn(α; s1, . . . , sn) = ∑m=(m1,...,mn)∈Nn

1(m1 + α)s1 . . . (m1 + · · ·+ mn + α)sn

(254)

149

where α 6= 0,−1,−2, ..., and (s1, . . . , sn) ∈ Cn, which introduced by Akiyama and Ishikawaand proved its analytic continuation to Cn [akiyama02]. Matsumoto and Tanigawa provedthe analytic continuation of wide class of multiple Dirichlet series and multiple Hurwitzzeta functions in [matsumoto03] and [matsumoto2003], and the analytic continuation ofthe series (254) is a special case of [Theorem 1][matsumoto03].

Our main result in this work is the values at non positive integers of the followingseries

ζn(α; s) = ∑m=(m1,...,mn)∈Nn

n

∏i=1

1(m1 + · · ·+ mi + αi)si

(255)

where, α = (α1, . . . , αn) ∈ Rn verified some conditions, this series is called the general-ized multiple Hurwitz zeta function.

The key of this study is the use of the Raabe formula [friedman04] which expressesthe integral in terms of the sum.

In what follows, for any elements x = (x1, . . . , xn) and y = (y1, . . . , yn) of Cn ands = (s1, ..., sn) denote a vector in Cn .

Main resultsFor real numbers α = (α1, . . . , αn) ∈ Rn, such that, for all 1 ≤ i ≤ n:

αi 6= 0,−1,−2, ...,

So, for a complex n−tuples s = (s1, . . . , sn) ∈ Cn, we define the generalized multipleHurwitz zeta function by

ζn(α; s) := ζ(α1, . . . , αn; s1, . . . , sn) (256)

= ∑m=(m1,...,mn)∈Nn

n

∏i=1

1(m1 + · · ·+ mi + αi)si

(257)

= ∑m1>···>mn≥0

n

∏i=1

1(mi + αi)si

(258)

and the corresponding integral function associated to the generalized multiple Hurwitzzeta function by

Yn(α; s) =∫[0,+∞[n

n

∏i=1

1(x1 + · · ·+ xi + αi)si

dx. (259)

We give now the main result in this work

Theorem 73Let N = (N1, . . . , Nn) a point of Nn, if the point (s = −N) is not a polar divisor for

the integral function Yn(α; s), then the value of the generalized multiple Hurwitz zeta functionζn(α; s) at the point (s = −N) exists and is given by

ζn(α;−N) = (−1)n

∑k=(k2,...,kn)∈Nn−1 ∑ v=(v1,...,vn)∈Nn

vj≤kj ∀ 2≤j≤n;v1≤(∑ni=1 Ni+n−∑n

i=2 ki)A(−N) Bv ∏n

j=11

(∑ni=j Ni+n−j+1−∑n

i=j+1 ki)

(260)

150

with

A(−N) =

(∑n

i=1 Ni + n−∑ni=2 ki

v1

)α(∑n

i=1 Ni+n−∑ni=2 vi)

n

∏j=2

(∑n

i=j Ni + n− j + 1−∑ni=j+1 ki

k j

)(k j

vj

).

(261)and

T(N) :=

k = (k2, ..., kn) ∈Nn−1 : 0 ≤ k j ≤

n

∑i=j

Ni + n− j + 1−n

∑i=j+1

ki, ∀ 2 ≤ j ≤ n

.

and

Bv =n

∏j=1

Bvj

where Bvj is the vj−th Bernoulli number.

References[akiyama02] S. Akiyama and H. Ishikawa, On analytic continuation of multiple L−functions

and related zeta-functions , ’Analytic Number Theory’,edited by C. JIA and K. MAT-SUMOTO,Kluwer 1–16,2002.

[apostol76] T.M. Apostol, Introduction to Analytic Number Theory, Springer 1976.[friedman] E. Friedman and A. Pereira, Special Values of Dirichlet Series and Zeta

Integrals, Int. J. Number Theory,08,3, 697–714, 2012.[friedman04] E. Friedman and S. Ruijsenaars, Shintani-Barnes zeta and gamma func-

tions, Advances Math. 187, 362–395, 2004.[gun2018] S. Gun and B. Saha, Multiple Lerch Zeta Functions and an Idea of Ra-

manujan, Michigan Math. J., 67, no. 2, 267–287, 2018.[hoffman92] M. Hoffman, Multiple harmonic series, Pacific J. Math., 152, 257–290,

1992.[kamano06] K. Kamano, The Multiple Hurwitz Zeta Function and a Generalization

of Lerch’s Formula,TOKYO J. MATH., 29, 1, 62–73, 2006.[matsumoto06] K. Matsumoto, Analytic proprieties of multiple zeta-functions of Barnes,

of Shintani, and Eisenstein series, Nagoya Math.J., 172,59–102,2003.[matsumoto03] K. Matsumoto, The analytic continuation and the asymptonic be-

haviour of certain multiple zeta-functions I, J. Number Theory, 101, 223–243, 2003[matsumoto2003] K. Matsumoto and Y. Tanigawa, The analytic continuation and the

order estimate of multiple Dirichlet series, J. Theorie des Nombres de Bordeaux, 15, 267–274, 2003

[sadaoui14] B. Sadaoui, Multiple zeta values at the non-positive integers, ComptesRendus Mathematique,352, 12, 977-984, 2014.

[sadaoui16] B. Sadaoui and A. Derbal, Behaviour at the non-positive integers of Dirich-let series associated to polynomials of several variables, Manuscripta Mathematica,151,2, 183-207, 2016.

151

Stability and Bifurcations in 2D Spatiotemporal DiscreteSystems

M. L. Sahari, A. K. Taha, L. RandriamihamisonBadji Mokhtar-Annaba University, Federal University of Toulouse, Institut National

Polytechnique de Toulouse

This talk deals with stability and local bifurcations of two-dimensional (2D)spatiotemporal discrete systems. Necessary and sufficient conditions for asymp-totic stability of the systems are obtained. Some definitions for the bifurcationsof 2D spatiotemporal discrete systems are also given, and an illustrative exampleare provided to explain our results.

Key words: Bifurcation, 2D spatiotemporal discrete system, shift operator, spectrum, sta-bility.

2010 Mathematics Subject Classification:37B20, 37L15, 37L60, 37K50.

IntroductionIn the present work, we are interested in the following 2D spatiotemporal discrete sys-tems:

xm+1,n+1 = f (xm,n , xm+1,n), (262)

where (m , n) ∈ (Z , N) and f : R2 −→ R is a nonlinear function. We introduce inthe first place, the notion of dynamics in the weighted space of two-sided real sequences,then we give some definitions concerning the singularities in a more general frameworkand formulate the necessary and sufficient conditions for the stability which are more ac-curate than those in [Chen,Lin,Singh,Tian,Wang,Zhang,Randriamihamison]. Some casesof local bifurcations are outlined based on the stability results that we have already es-tablished, before ending with the study of a 2D quadratic recurrence of the form

xm+1,n+1 = x2m,n + bxm+1,n + a. (263)

Let X :=

[x] = (xi)∞i=−∞ ∈ RZ : ‖[x]‖ :=

√√√√ i=∞

∑i=−∞

αix2i < ∞

, the weighted space

of two-sided sequences, we can define an 1-D recurrence on X in the following way :For an initial condition [x]0 = (xm,0)

∞m=−∞ ∈ X, called also “boundary condition”, we

recursively construct a solutions sequence[x]n = (xm,n)

∞m=−∞ , n = 0, 1, 2, ...

⊂ X, by

[x]n+1 = ( f (xm−1,n , xm,n))∞m=−∞ . (264)

Otherwise, if F : X −→ X is the map defined by

F ([x]) = ( f (xi−1 , xi))∞i=−∞ , (265)

for all [x] = (xi)i=∞i=−∞ ∈ X, then the system (262) is equivalent to the following infinite-

dimensional discrete (dynamical) system:

[x]n+1 = F ([x]n) , (266)

The map F defined by (265)-(266) is said to be induced by system (262). Obviously, asolution sequence [x]n, n = 0, 1, 2, ... is a solution of system (262) if and only if it is asolution of the system (266).

152

SingularitiesDefinition 18 (Fixed points) The point [x]∗ = (x∗i )

∞i=−∞ ∈ X is called a fixed point or 1-

period cycle of F, ifF ([x]∗) = [x]∗, (267)

i.e.f (x∗i−1, x∗i ) = x∗i , i ∈ Z. (268)

Definition 19 [x]∗ = (x∗i )∞i=−∞ ∈ X is a periodic point of period two or 2-periodic point for F,

if there exists [y]∗ = (y∗i )∞i=−∞ ∈ X, such as [y]∗ 6= [x]∗ and

F2 ([x]∗) = [x]∗,F ([x]∗) = [y]∗.

(269)

Particular cases of 2-periodic pointsIn [Randriamihamison], a 2-periodic point is given by: Let (c11, c21, c12, c22) ∈ R4 be suchthat

(i) f (c11, c21) = c22 and f (c21, c11) = c12,(ii) f (c12, c22) = c21 and f (c22, c12) = c11,(iii) c11 6= c21 or c11 6= c12.Now, if (x∗, y∗, v∗, z∗) ∈ R4 , is such that f (x∗, y∗) = z∗, f (y∗, x∗) = v∗, f (v∗, z∗) = y∗,

and f (z∗, v∗) = x∗, with x∗ 6= v∗ or y∗ 6= z∗, then the point [x]∗ = (x∗i )∞i=−∞ ∈ X, where

x∗2i = x∗,

x∗2i+1 = y∗,, i ∈ Z, (270)

verifies (269). Therefore [x]∗ is a 2-periodic point for F , moreover, the quadruplet (x∗, y∗, v∗, z∗)verifies (i), (ii) and (iii).

Among the 2-periodic points verifying (270), we can distinguish two types, that is tosay:

2-periodic points of Horizontal type (H): If (x∗, y∗) ∈ R2 is such thatf (x∗, x∗) = y∗ 6= x∗,f (y∗, y∗) = x∗.

(271)

Then[x]∗ =

(x∗i = x*)∞

i=−∞ , [y]∗ = (y∗i = y∗)∞i=−∞

⊂ X is a 2-periodic orbit for F .

2-periodic points of Diagonal type (D): If [x]∗ = (x∗i )∞i=−∞ ∈ X is such that

x∗2i = x∗ 6= y∗,

x∗2i+1 = y∗,i ∈ Z,

where (x∗, y∗) ∈ R2 satisfy f (x∗, y∗) = x∗,f (y∗, x∗) = y∗.

Then [x]∗ is a 2-periodic point for F .

153

StabilityLet T ∈ L(X) be a perturbations of the identity by a weighted shift operator given by

T = W + D, (272)

where D is a diagonal operator with diagonals dii∈Z (see [Sahari]). Through the follow-ing result, we provides a complete description of the spectrum of diagonal perturbationof weighted shift operator W.

Let the matrix representation of the operator DF([x]∗) denoted by J∗F = (J∗i,j)i,j∈Z

(often called ”Jacobian” matrix [Berezansky]), is given by

DF([x]∗)ei = ∑j∈Z

J∗i,jej; i ∈ Z,

where

J∗i,j =

f ′y(x∗i−1 , x∗i ) :=

∂ f (x, y)∂y

∣∣∣∣(x∗i−1 , x∗i )

if j = i,

f ′x(x∗i−1 , x∗i ) :=∂ f (x, y)

∂x

∣∣∣∣(x∗i−1 , x∗i )

if j = i + 1,

0 otherwise.

Proposition 15 Let [x]∗ ∈ X be a fixed point and J∗F ∈ L(X) is the Jacobian matrix at [x]∗ of F.

(i) Ifr(J∗F) < 1,

then [x]∗ is asymptotically stable.(ii) If

r(J∗F) > 1,

then [x]∗ is unstable.(iii) If

σ(J∗F) ⊂ C\D(0, 1),

then [x]∗ is repulsive.

Corollary 14 If [x]∗ = (· · · , x∗, x∗, (x∗), x∗, x∗, · · · ) ∈ X is a fixed point of type I of F, then1) [x]∗ is asymptotically stable if | f ′x(x∗, x∗)|+

∣∣∣ f ′y(x∗, x∗)∣∣∣ < 1.

2) [x]∗ is unstable if | f ′x(x∗, x∗)|+∣∣∣ f ′y(x∗, x∗)

∣∣∣ > 1.

3) [x]∗ is repulsive if∣∣∣| f ′x(x∗, x∗)| −

∣∣∣ f ′y(x∗, x∗)∣∣∣∣∣∣ > 1 .

BifurcationsAssume that the dynamic (266) is parameterised by the real parameters a and b, i.e.Fa,b(x) ≡ F(x; a, b) and fa,b(x) ≡ f (x, y; a, b). The bifurcations are studied in the (a, b) ∈R×R parameters plane. According to the Proposition 15, a fixed point [x]∗ ∈ X of thesystem (266) is asymptotically stable or repulsive if the spectrum σ(J∗Fa,b

) of the JacobianJ∗Fa,b

at [x]∗ is completely included or completely excluded from the unit disk. So, it is obvi-ous that a bifurcation of a fixed point [x]∗ of F occurs when the spectrum σ(J∗Fa,b

) crosses

154

the unit circle S(0, 1) for certain values (called bifurcations values) of the parameters aand b. The pair (a, b) thus obtained belongs to a set Λ ⊂ R×R called the bifurcationcurve.

Definition 20 Let [x]∗ ∈ X be a fixed point of F, then the pair (a, b) ∈ R×R belongs to thebifurcation curve Λ if σ(J∗Fa,b

) ∩ S(0, 1) 6= ∅.

References[Berezansky] Yu M Berezansky and Yuri Grigorevich Kondratiev. Spectral methods

in infinite-dimensional analysis, volume 12. Springer Science & Business Media, 2013.[Chen] Guanrong Chen, Chuanjun Tian, and Yuming Shi. Stability and chaos in 2-d

discrete systems. Chaos, Solitons & Fractals, 25(3):637–647, 2005.[Lin] Yi-Zhong Lin and Sui Sun Cheng. Stability criteria for two partial difference

equations. Computers & Mathematics with Applications, 32(7):87–103, 1996.[Randriamihamison] L. Randriamihamison, and A. K. Taha. ”About the singularities

and bifurcations of double indices recursion sequences.” Nonlinear Dynamics 66.4 (2011):795-808.

[Sahari] M. L. Sahari, A. K. Taha, and L. Randriamihamison. ”A note on the spec-trum of diagonal perturbation of weighted shift operator.” Le Matematiche 74.1 (2019):35-47. [Singh] Vimal Singh. Stability analysis of 2-d discrete systems described by theFornasini–Marchesini second model with state saturation. IEEE Transactions on Circuitsand Systems II: Express Briefs, 55(8):793–796, 2008. [Tian] Chuanjun Tian and GuanrongChen. Stability and chaos in a class of 2-dimensional spatiotemporal discrete systems.Journal of Mathematical Analysis and Applications, 356(2):800–815, 2009. [Wang] ZidongWang and Xiaohui Liu. Robust stability of two-dimensional uncertain discrete systems.IEEE Signal processing letters, 10(5):133–136, 2003. [Zhang] Bing Gen Zhang and XingHua Deng. The stability of certain partial difference equations. Computers & Mathemat-ics with Applications, 42(3):419–425, 2001.

TH-surfaces dans les espaces 3-dimensionnel euclidien E3 etlorentzien E3

1

Bendehiba SenoussiEcole Normale Superieure de Mostaganem

L’objet de ce travail est la recherche des TH-surfaces minimales et developpablesdans les espaces 3-dimensionnel Euclidien E3 et Lorentzien E3

1.Key words: Surfaces de translation, surfaces factorables, surface minimale, surface developpable.

2010 Mathematics Subject Classification: 51B20, 53A10, 53C45

IntroductionLa theorie des surfaces minimales de E3 a debute au 18eme siecle avec notamment lestravaux de Lagrange, d’Euler et de Meusnier. Par la suite, de nombreux mathematiciensse sont penches sur cette question, on peut par exemple citer Scherk. H.F et Weierstrass.K au cours du 18eme siecle. Un des interets des surfaces minimales locales est de resoudre

155

le probleme de Plateau: Trouver une surfaces d’aire minimum absolu de frontiere une courbedonnee de l’espace.

Parmi toutes les surfaces contenant une courbe fermee donnee γ, les surfaces mini-males sont celles qui realisent le minimum de l’aire limitee par γ.

Une immersion r : M2 → E3 de la surface M2 dans E3 est dite minimale si sa courburemoyenne est partout nulle.

Lagrange a etabli l’equation satisfaite par les surfaces de la forme z = f (u, v) quiminimisent l’aire pour des variations fixant un contour.

fuu(1 + f 2v )− 2 fu fv fuv + fvv(1 + f 2

u) = 0.

Lagrange a remarque que les fonctions f (u, v) = a1u+ a2v+ a3, ai ∈ R, sont solutionsde cette equation.

Scherk a determine les solutions de l’equation de Lagrange qui sont de la formef (u, v) = g(u) + h(v).

Les fonctions solutions sont de la forme

f (u, v) =1λ(ln cos(λu)− ln cos(λv)), λ ∈ R∗.

Soit M2 une surface reguliere de E3. La courbure de Gauss est definie par

KG =LN −M2

EG− F2 .

Les surfaces dont la courbure de Gauss KG = 0 sont dites developpables.

La courbure moyenne d’une surface reguliere M2 de E3, notee H, est definie par

H =EN + LG− 2FM

2(EG− F2).

Main resultsSoit r : M2 → E3 une immersion isometrique d’une TH- surface dans l’espace euclidiende dimension 3 muni de la metrique induite.

M2 peut etre parametree par

r(u, v) = (u, v, z = A( f (u) + g(v)) + B f (u)g(v)); A, B ∈ R, (273)

ou f et g des fonctions regulieres des variables u et v, respectivement.

TH- surfaces minimales dans E3

Par des calculs immediats. Il vient alors

H =12

W−3H1, (274)

(275)

ouH1 = α f ′′(1 + g′2γ2)− 2Bαγ f ′2g′2 + γg′′(1 + f ′2α2).

La condition de minimalite H1 = 0 conduit a l’equation

αγ′′(B2 + α′2γ2)− 2αγα′2γ′2 + γα′′(B2 + γ′2α2) = 0. (276)

On a le theoreme suivant:

156

Theorem 74 Soit M2 une surface dans E3 donnee comme graphe d’une fonction z = z(u, v),representee par (273).

Si M2 est une surface minimale, alors elle est congruente a l’une des surfaces suivanti) f (u) = λ1u + λ2 et g(v) = λ3 tan(λ4v + λ5)− λ6, λi ∈ R.ii) f (u) = δ1 tan(δ2u + δ3)− δ4 et g(v) = δ5v + δ6, δi ∈ R.iii) f (u) = c, c ∈ R et g(v) = av + b; a, b ∈ R.iv) f (u) = au + b; a, b ∈ R et g(v) = c; c ∈ R.

TH- surfaces developpables dans E3

La courbure de Gauss est

K =αγ f ′′g′′ − B2 f ′2g′2

EG− F2 .

Si K = 0, alors

αγα′′γ′′ − γ′2α′2 = 0. (277)

Theorem 75 Soit M2 une surface dans E3 donnee comme graphe d’une fonction z = z(u, v),representee par (273).

Si M2 est une surface developpables, alors elle est congruente a l’une des surfaces suivanti) z(u, v) = δ1g(v) + δ2; δ2, δ1 ∈ R.ii) z(u, v) = λu + δ1, (λ, δ) ∈ R2.iii) z(u, v) = A( f (u) + g(v)) + B f (u)g(v),ou f (u) = λ3ek1u + λ4, g(v) = λ5ek2v + λ6; λ3, λ4, λ5, λ6 ∈ R.iv) z(u, v) = A( f (u) + g(v)) + B f (u)g(v),ou f (u) = c3((1 − λ)k1u + c1)

11−λ + c4, g(v) = c5((

λ−1λ )k2v + c2)

λλ−1 + c6; c3, c4, c5,

c6 ∈ R.

TH- surfaces minimales dans E31

On appelle espace de Minkowski E31 l’espace vectoriel R3 constitue des vecteurs (x1, x2, x3)

muni du produit scalaire

gL(X, Y) = −x1y1 + x2y2 + x3y3,

ou X = (x1, x2, x3), Y = (y1, y2, y3).

Theorem 76 Soit M2 une surface dans E31 donnee comme graphe d’une fonction z = z(u, v),

representee par (273).Si M2 est une surface minimale, alors elle est congruente a l’une des surfaces suivanti) f (u) = λ1u + λ2 et g(v) = λ3 coth(λ4v + λ5)− λ6, λi ∈ R.ii) f (u) = δ1 coth(δ2u + δ3)− δ4 et g(v) = δ5v + δ6, δi ∈ R.iii) f (u) = c, c ∈ R et g(v) = av + b; a, b ∈ R.iv) f (u) = au + b; a, b ∈ R et g(v) = c; c ∈ R.

v) f (u) = λ2e√

ku − AB et g(v) = λ1e

√kv − A

B .

157

TH- surfaces developpables dans E31

Theorem 77 Soit M2 une surface dans E31 donnee comme graphe d’une fonction z = z(u, v),

representee par (273).Si M2 est une surface developpables, alors elle est congruente a l’une des surfaces suivanti) z(u, v) = δ1g(v) + δ2; δ2, δ1 ∈ R.ii) z(u, v) = λu + δ1, (λ, δ) ∈ R2.iii) z(u, v) = A( f (u) + g(v)) + B f (u)g(v),ou f (u) = λ3ek1u + λ4, g(v) = λ5ek2v + λ6; λ3, λ4, λ5, λ6 ∈ R.iv) z(u, v) = A( f (u) + g(v)) + B f (u)g(v), ou f (u) = c3((1 − λ)k1u + c1)

11−λ + c4,

g(v) = c5((λ−1

λ )k2v + c2)λ

λ−1 + c6; c3, c4, c5, c6 ∈ R.

ConclusionTH-surfaces sont une generalisation naturelle des surfaces de Translation et homothetiques.L’objet de ce travail est la recherche des TH-surfaces (Translation et homothetique sur-face) minimales et developpables dans les espaces 3-dimensionnel Euclidien E3 et LorentzienE3

1.References

[BeSe] Bekkar M., Senoussi B., Translation surfaces in the 3-dimensional space satis-fying ∆I I Iri = µiri, J. Geom., 2012, (103), 367-374.

[LLI] Liu H., Translation surfaces with dependent Gaussian and mean curvature in3-dimensional spaces. (Chinese) J. Northeast Univ. Tech., 1993, 14, 88-93.

[Liu] Liu H., Translation surfaces with constant mean curvature in 3-dimensionalspaces, J. Geom., 1999, 64, 141-149.

[Lpez3] Lopez R., Moruz M., Translation and homothetical surfaces in Euclideanspace with constant curvature, arXiv:1410.2512v1 [math.DG]., 2014.

[BeSe] Senoussi B., Bekkar M., Translation and homothetical TH-surfaces in the 3-dimensional Euclidean space E3 and Lorentzian-Minkowski space E3

1, Open J. Math. Sci.,2019, 3, 234-244.

[YuLi] Yu Y., Liu H., The factorable minimal surfaces, Proceedings of the Eleventh Inter-national Workshop on Differential Geometry, 33-39, Kyungpook Nat. Univ., Taegu, 2007.

Compatibilite des structures riemanniennes et des structures deJacobi

Yacine Aıt Amrane, Zeglaoui Ahmed

Universite des Sciences et de la Technologie Houari Boumediene, University Dr MoulayTahar

On definit une notion de compatibilite entre une structure riemannienne et une struc-ture de Jacobi. On montre que dans le cas des structures de Poisson, des structures decontact et des structures localement conformement symplectiques, des exemples fonda-mentaux de structures de Jacobi, on obtient respectivement des structures de Poissonriemanniennes au sens de M. Boucetta, des structures (1/2)-Kenmotsu et des structureslocalement conformement kahleriennes.

158

Mots cles: Variete de Jacobi, Variete de Poisson pseudo-riemannienne, Variete de Ken-motsu, Variete localement conformement kahlerienne, algebroıde de Lie.2010 Mathematics Subject Classification: 53C15.

Introduction

Les varietes de Jacobi ont ete introduites separement par A. Lichnerowicz et A. Kirillov.Elles generalisent a la fois les varietes de Poisson, les varietes de contact et les varieteslocalement conformement symplectiques. On se pose la question naturelle de l’existenced’une notion de compatibilite entre une structure de Jacobi et une structure pseudo-riemannienne, qui pour des structures de Jacobi particulieres, donne lieu a des structuresgeometriques remarquables. Dans ce travail, on introduit une telle notion qui dans lecas d’une variete de Poisson donne une structure de Poisson pseudo-riemannienne ausens de M. Boucetta. On montre que pour une structure de contact riemannienne, aveccette notion de compatibilite on obtient une structure (1/2)-Kenmotsu, et que dans le casd’une structure localement conformement symplectique et d’une metrique riemannienne”associee”, on retrouve une structure localement conformement kahlerienne.

Prealgebroıdes de Lie associes a une variete de Jacobi

Tout au long de ce travail M designe une variete differentiable, π un champ de bivecteurset ξ un champ de vecteurs sur M.

Le couple (π, ξ) est une structure de Jacobi sur M si [π, π] = 2ξ ∧ π et [ξ, π] :=Lξπ = 0, ou [., .] designe le crochet de Schouten-Nijenhuis. Le cas ξ = 0 correspond aune structure de Poisson (M, π).

Soit ]π : T∗M→ TM le morphisme de fibres vectoriels defini par β (]π (α)) = π (α, β)et soit l’application [., .]π : Ω1(M)×Ω1(M) → Ω1(M) definie par [α, β]π := L]π(α)β−L]π(β)α− d (π(α, β)), appelee le crochet de Koszul. Considerons le morphisme de fibresvectoriels ]π,ξ : T∗M −→ TM defini par

]π,ξ(α) = ]π(α) + α(ξ)ξ

et, pour une 1-forme λ ∈ Ω1(M), l’application [., .]λπ,ξ : Ω1(M) × Ω1(M) −→ Ω1(M)definie par

[α, β]λπ,ξ := [α, β]π + α(ξ)(Lξ β− β

)− β(ξ)

(Lξα− α

)− π(α, β)λ.

Le triplet (T∗M, ]π,ξ , [., .]λπ,ξ), associe a (π, ξ, λ), est un algebroıde alterne sur M.

Dans le cas ou ξ = λ = 0, le triplet (T∗M, ]π,ξ , [., .]λπ,ξ) n’est rien d’autre que l’algebroıdealterne (T∗M, ]π, [., .]π) associe au champ de bivecteurs π. Rappelons que quelles quesoient les formes differentielles α, β, γ ∈ Ω1(M) on a

γ (]π ([α, β]π)− []π(α), ]π(β)]) =12[π, π] (α, β, γ) . (278)

Ainsi, (T∗M, ]π, [., .]π) est un prealgebroıde de Lie si et seulement si π est un tenseur dePoisson. Dans ce cas, le triplet (T∗M, ]π, [., .]π) est meme un algebroıde de Lie, appelel’algebroıde cotangent de la variete de Poisson (M, π). Dans le cas d’une structure deJacobi on a le resultat suivant

159

Supposons que (π, ξ) est une structure de Jacobi sur M et soit λ ∈ Ω1(M). On a

]π,ξ([α, β]λπ,ξ)−[]π,ξ(α), ]π,ξ(β)

]= π(α, β)

(ξ − ]π,ξ(λ)

),

quelles que soient les formes α, β ∈ Ω1(M).Supposons que (π, ξ) est une structure de Jacobi sur M et soit λ ∈ Ω1(M). Si

]π,ξ(λ) = ξ, alors l’algebroıde alterne (T∗M, ]π,ξ , [., .]λπ,ξ) est un prealgebroıde de Lie,c’est-a-dire

]π,ξ([α, β]λπ,ξ) =[]π,ξ(α), ]π,ξ(β)

],

quelles que soient les formes α, β ∈ Ω1(M). La reciproque aussi est vraie si π 6= 0.

Algebroıde cotangent a une variete de contact. Supposons M de dimension impaire2n + 1, n ∈ N∗. Supposons egalement que (π, ξ) est la structure de Jacobi associee aune 1-forme de contact η sur M, c’est-a-dire que π(α, β) = dη

(]η(α), ]η(β)

), ou ]η est

l’isomorphisme inverse de [η : TM → T∗M, [η(X) = −iXdη + η(X)η, et que ξ = ]η(η).Le champ ξ est appele le champ de Reeb associe a la structure de contact (M, η), il estcaracterise par iξdη := dη(ξ, .) = 0 et iξη := η(ξ) = 1.

Il est facile de montrer que ]π,ξ = ]η , d’ou en particulier ]π,ξ(η) = ]η(η) = ξ. Donc sion pose [., .]η = [., .]ηπ,ξ , d’apres le corollaire et du fait que ]π,ξ = ]η est un isomorphisme,l’algebroıde alterne (T∗M, ]η , [., .]η), associe naturellement a la variete de contact (M, η),est un algebroıde de Lie isomorphe a l’algebroıde tangent de M. On pourra l’appelerl’algebroıde cotangent de la variete de contact (M, η).

Algebroıde cotangent a une variete localement conformement symplectique. Une struc-ture localement conformement symplectique sur M est un couple (ω, θ) compose d’une1-forme differentielle fermee θ et d’une 2-forme differentielle non degeneree ω sur Mtelles que dω + θ ∧ω = 0. Dans le cas particulier ou la forme θ est exacte, i. e. θ = d f , ondit que (ω, d f ) est conformement symplectique, ce qui est equivalent a e f ω est symplec-tique, d’ou la terminologie.A tout couple (ω, θ) compose d’une 1-forme differentielle θ et d’une 2-forme differentiellenon degeneree ω on associe le couple (π, ξ) defini par : i

ξω = θ et i

]π (α)ω = −α, pour

tout α ∈ Ω1(M). On a le resultat suivant.

Proposition 16 Le couple (ω, θ) est une structure localement conformement symplectique si etseulement si le couple (π, ξ) est une structure de Jacobi.

Supposons maintenant que (ω, θ) est une structure localement conformement sym-plectique sur M et que (π, ξ) est la structure de Jacobi associee. Comme on a ]π,ξ(θ) = ξet qu’on montre que ]π,ξ est un isomorphisme, si on pose ]ω,θ := ]π,ξ et [., .]ω,θ :=[., .]θπ,ξ , d’apres le corollaire et du fait que ]π,ξ est un isomorphisme, l’algebroıde alterne(T∗M, ]ω,θ , [., .]ω,θ), associe naturellement a la variete localement conformement symplec-tique (M, ω, θ), est un algebroıde de Lie isomorphe a l’algebroıde tangent de M. Onpourra l’appeler l’algebroıde cotangent de la variete localement conformement symplec-tique (M, ω, θ).

160

Derivee de Levi-Civita contravariante associee au triplet (π, ξ, g)

Dans toute la suite, on designe par g une metrique pseudo-riemannienne sur M, par[g : TM → T∗M l’isomorphisme de fibres vectoriels tel que [g(X)(Y) = g(X, Y), par]g l’isomorphisme inverse de [g, et par g∗ la cometrique de g, c’est-a-dire le champ detenseurs defini par g∗(α, β) := g

(]g(α), ]g(β)

).

Au couple (π, g) on associe les endomorphismes J de TM et J∗ de T∗M definis re-spectivement par

g(J]g(α), ]g(β)) = π(α, β) et g∗(J∗α, β) = π(α, β). (279)

On a J = ]g J∗ [g. Au triplet (π, ξ, g) on associe la 1-forme λ definie par λ = g(ξ, ξ)[g(ξ)−[g(Jξ), et on note [., .]gπ,ξ au lieu de [., .]λπ,ξ . On appelle la derivee de Levi-Civita contravari-ante associee au triplet (π, ξ, g) la derivee D : Ω1(M)×Ω1(M) → Ω1(M) definie par laformule de Koszul :

2g∗ (Dαβ, γ) = ]π,ξ(α) · g∗(β, γ) + ]π,ξ(β) · g∗(α, γ)− ]π,ξ(γ) · g∗(α, β)−g∗([β, γ]

gπ,ξ , α)− g∗([α, γ]

gπ,ξ , β) + g∗([α, β]

gπ,ξ , γ). (280)

Dans le cas ou l’algebroıde alterne (T∗M, ]π,ξ , [., .]gπ,ξ) est un prealgebroıde de Lie et que]π,ξ est une isometrie, on deduit de (280) la formule suivante qui relie D a la connexionde Levi-Civita ∇ de g :

]π,ξ (Dαβ) = ∇]π,ξ (α)]π,ξ(β). (281)

Algebroıde alterne associe a une variete riemannienne presque de contact. Supposonsque (Φ, ξ, η, g) est une structure pseudo-riemannienne presque de contact sur M. L’applicationπ definie par π(α, β) = g(]g(α), Φ(]g(β))) est un champ de bivecteurs sur M et ]π,ξ estune isometrie. Si l’algebroıde alterne (T∗M, ]π,ξ , [., .]gπ,ξ) est un prealgebroıde de Lie, alorsla formule (281) est verifiee.

Supposons en outre que η est une forme de contact. On dit que la variete (M, η, g) est(pseudo-)riemannienne de contact, ou que g est associee a la forme de contact η, s’il existeun endomorphisme Φ de TM tel que (Φ, ξ, η, g) est une structure (pseudo-)riemanniennepresque de contact et que g(X, Φ(Y)) = dη(X, Y).

Supposons que (M, η, g) est une variete pseudo-riemannienne de contact. On a

]η (Dαβ) = ∇]η(α)]η(β).

Metrique riemannienne associee a une structure localement conformement symplec-tique. Supposons que ω ∈ Ω2(M) est une 2-forme non degeneree et soit θ ∈ Ω1(M).Supposons que le couple (π, ξ) est associe au couple (ω, θ). On dit que la metriquepseudo-riemmannienne g est associee au couple (ω, θ) si ]ω,θ := ]π,ξ est une isometrie,c’est-a-dire si

g (]ω,θ(α), ]ω,θ(β)) = g∗(α, β),

pour tous α, β ∈ Ω1(M). Si θ = 0, alors ξ = 0 et ]ω,θ = ]ω, et si J est l’endomorphismedefini par (279), alors g(JX, JY) = g(X, Y) et ω(X, Y) = g(JX, Y), pour tous X, Y ∈X(M). Donc, si de plus g est definie positive, le couple (ω, g) est une structure presquehermitienne sur M et J est la structure presque complexe associee.

161

Supposons que (ω, θ) est une structure localement conformement symplectique etque g est une metrique associee. On a

]ω,θ (Dαβ) = ∇]ω,θ(α)]ω,θ(β).

Remarquons que comme ω (ξ, ]π(α)) = −i]π(α)ω(ξ) = α(ξ) et de meme ω(ξ, ]π(β)) =β(ξ), alors

ω(]π,ξ(α), ]π,ξ(β)) = π(α, β). (282)

Ainsi, un calcul direct sous les memes hypotheses que le theoreme ci-dessus, donne

Dπ(α, β, γ) = ∇ω(]ω,θ(α), ]ω,θ(β), ]ω,θ(γ)). (283)

Compatibilite du triplet (π, ξ, g)

On dit que la metrique g est compatible avec le couple (π, ξ) ou que le triplet (π, ξ, g) estcompatible si

Dπ(α, β, γ) =12(γ(ξ)π(α, β)− β(ξ)π(α, γ)− J∗γ(ξ)g∗(α, β) + J∗β(ξ)g∗(α, γ)) , (284)

pour tous α, β, γ ∈ Ω1(M). La formule (284) peut aussi s’ecrire sous la forme

(Dα J∗) β =12(π(α, β)[g(ξ)− β(ξ)J∗α + g∗(α, β)J∗[g(ξ) + J∗β(ξ)α

), (285)

pour tous α, β ∈ Ω1(M). La compatibilte dans le cas ou ξ est nul signifie que (M, π, g)est une variete de Poisson pseudo-riemannienne, et de Poisson riemannienne si de plusla metrique g est definie positive.

Supposons que (η, g) est une structure riemannienne de contact sur M et soit (Φ, ξ, η, g)la structure riemannienne presque de contact associee. Supposons que (π, ξ) est la struc-ture de Jacobi associee a η. Alors le triplet (π, ξ, g) est compatible si et seulement si(M, Φ, ξ, η, g) est (1/2)-Kenmotsu.

Supposons que (ω, d f ) est une structure conformement symplectique sur M et que(π, ξ) est la structure de Jacobi associee. Si g est une metrique riemannienne associee a ωet au couple (ω, d f ). Alors, le triplet (π, ξ, g) est compatible si et seulement si le triplet(ω, d f , g) est une structure conformement kahlerienne.References

[amrzeg] Y. Aıt Amrane, A. Zeglaoui, Compatibility of Riemannian structures and Jacobistructures, Journal of Geometry and Physics 133 (2018) 71-80.

[blair] D. E. Blair, ”Riemannian geometry of contact and symplectic manifolds”, Pogressin mathematics, vol. 203, 2nd ed., Birkhauser, 2010.

[boucetta1] M. Boucetta, Compatibilite des structures pseudo-riemanniennes et des struc-tures de Poisson, C. R. Acad. Sci. Paris, Ser. I 333 (2001) 763-768.

[marle] C.-M. Marle, On Jacobi manifolds and Jacobi bundles, in ”Symplectic geometry,groupoids, and integrable systems”, Seminaire Sud Rhodanien de Geometrie a Berkeley(1989), Math. Sci. Res. Inst. Publ. 20, Springer-Verlag, New York, 1991, pp 227-246.

162

On the regular vortex patch topic for the planar Boussinesqsystem with fractional dissipation

Mohamed Zerguine

Batna University

The present communication treats essenstially the global persistence of a smooth ge-ometric structure for the planar Boussinesq system partially viscous with fractional dis-sipation (−∆)

α2 , α ∈ 1, 2 for the density. Roughly speaking, we show that if the initial

vorticity has a smooth regularity, that is ω0 = 1Ω0 , with Ω0 is a bounded domain whoseboundary ∂Ω0 is a Jordan curve with Holder’s regularity C1+ε, 0 < ε < 1. Then the ve-locity vector field is a Lipschitz function globally in time and the transported vorticityΩt = Ψ(t, Ω0) keeps its initial regularity, with Ψ refers to the flow. Even though, the vor-ticity can be decomposed into a singular part which is a vortex patch term and a regularpart, which is deeply related to the smoothing effect for density, i.e., ω(t, x) = 1Ωt + ρ(t).Key words: Boussinesq system, Asymptotique behavior, Logarithmic estimate, Paradif-ferential calculus.

2010 Mathematics Subject Classification: 76D03, 35B30.

Introduction

The incompressible Boussinesq equations with fractional dissipation are given as follows,∂tv + v · ∇v +∇p = ρ~e2 if (t, x) ∈ R+ ×R2,∂tρ + v · ∇ρ + (−∆)

α2 ρ = 0 if (t, x) ∈ R+ ×R2,

div v = 0,(v, ρ)|t=0 = (v0, ρ0).

(Bα)

Here, v refers to the velocity vector field in R2, the pressure p and the density ρ are twoscalar functions. The buoyancy force ρ~e2 models the effects of the gravity in the direction~e2 = (0, 1) and (−∆)

α2 designates the fractional laplacian with α ∈]0, 2] which is given for

f ∈ S(2) by

(−∆)α2 f (x) =

12π

∫R2

eix·ξ |ξ|αF f (ξ)dξ, (−∆)α2 f (x) = CαP.V

∫2

( f (x)− f (y))|x− y|2+α

dy.

where P.V. designate the principle value and Cα > 0.In R2, the vorticity ω associated to the velocity field v may be identified by ω , ∂1v2 −∂2v1. Consequently, the equivalent vorticity-density formulation of (Bα) is read as fol-lows,

∂tω + v · ∇ω = ∂1ρ if (t, x) ∈ R+ ×R2,∂tρ + v · ∇ρ + (−∆)

α2 ρ = 0 if (t, x) ∈ R+ ×R2,

(ω, ρ)|t=0 = (ω0, ρ0),(Vα)

where the velocity v is closely determined by the well-known Biot-Savart law, v = N2 ? ωwith

N2(x) =1

2π

x⊥

|x|2 , x⊥ = (−x2, x1).

163

If ρ ≡ ρ0, the system (Vα) is reduced to the classical 2d− incompressible Euler equationsexpressed by the following vorticity formulation

∂tω + v · ∇ω = 0 if (t, x) ∈ R+ ×R2,ω|t=0 = ω0. (E)

We point out that the regularity well-posedness topic for the system (E) is in satisfactoryway. In particular, Kato showed in [Kato] that whenever v0 ∈ Hs(N), with s > N

2 +1. Then the system (E) is locally well-posed, i.e., v ∈ C([0, T?[, Hs) and satisfying thefollowing blow-up criterion.

T? < +∞⇒∫ T?

0‖∇v(τ)‖L∞ dτ = +∞, (286)

with T? is the maximal lifespan of the solution. Later, in [B-K-M] Beale-Kato-Majda es-tablished another alternative that (286) which relies on the singularities accumulation ofthe vorticity in finite time,

T? < +∞⇒∫ T?

0‖ω(τ)‖L∞ dτ = +∞, s >

N2+ 1. (287)

A worthwhile remark is (286) and (287) are equivalent. In the case N = 2, we observefrom (E) that ω evolves a nonlinear transport equation, thus by means of the characteristicmethod we conclude that

ω(t, x) = ω0(Ψ−1(t, x)) (288)

That means ω is constant along the mappings trajectories Ψ. Consequently, one obtainsan infinite Lp conservation laws ‖ω(t)‖Lp = ‖ω0‖Lp , with p ∈ [1, ∞]. In accordance with(287), the Kato’s solution is globally in time, that is to say, T? = ∞. Under this pattern,Yudovich succeed to formulate a new weak solution: If ω0 ∈ L1 ∩ L∞, then (E) admitsa unique global solution v ∈ LL, with LL designates the set of log−Lipschitz functions.Even though, the associated flow Ψ is only homeomorphic and degenerating in time,

Ψ− I ∈ Ce−αt, t ≥ 0. (289)

An important sub-class of Yudovich’s class is the set of so-called vortex patch, meaningthat the initial vorticity is a characteristic function of bounded domain Ω0, that is ω0 =1Ω0 . This structure is preserved during the time, meaning that ω(t) = 1Ωt , with Ωt =Ψ(t, Ω0) is the patch that moves with the flow. The serious problem that arises is theregularity of the time evolution domain Ωt. At a first time, We can’t draw anything fromYudovich’s theory because the flow Ψ loses its regularity. The first successful attempt inthis way is due to Chemin [Chemin] where he has measured the Lipschitz norm of thevelocity with respect to the tangential (co-normal) regularity of the vorticity by means ofthe following logarithmic estimate.

‖∇v‖L∞ ≤ C‖ω‖L2∩L∞ log(

e +‖ω‖Cε(Xt)

‖ω‖L∞

).

where Cε(Xt) stands the anisotropic Holder spaces and the family Xλ = (Xt,λ)λ∈Λ is se-lected is the sense that commutes with the advective operator ∂t + v · ∇ and satisfying thefollowing inhomogeneous transport equation ∂tXt + v ·∇Xt = Xt ·∇v. As a consequence

‖∇v‖L∞ ≤ CeCt, t > 0. (290)

164

Furthermore, Chemin proved that if ω0 = 1Ω0 , where ∂Ω0 is a Jordan curve of C1+ε, with0 < ε < 1. Then the boundary ∂Ωt persists in time and preserves its initial regularity.Later, the Chemin’s result was extended for several systems and differents regularities byvarious authors, see for instance [Bertozi-Constantin,Constantin-Wu,Danchin,Depauw,Fanelli,Gamblin-Raymond,Hmidi,Hmidi-Zerguine].

Main resultsThe first study of the smooth vortex patch problem for (Bα) has been started recently in

[Hmidi-Zerguine] where Hmidi and the author studied the case α = 2 and formulatedthe following theorem.

Theorem 78 Assume that ∂Ω0 ∈ C1+ε, with 0 < ε < 1, ω0 = 1Ω0 and ρ ∈ L1 ∩ L∞. Then thesystem (Bα), with α = 2 is globally well-posed.

(v, ρ) ∈ L∞loc(+; )× L∞(+; L1 ∩ L∞).

More precisely:

(1.i) ‖∇v(t)‖L∞ ≤ C0eC0t log2(2+t),

(1.ii) ω(t, x) = 1Ωt(x) + ρ(t, x), ρ ∈ Cη for all η < 1,

(1.iii) ‖ω(t)− 1Ωt‖Lp ≤ C0 log2− 2p (2 + t).

Let us give a bunch of comments about Theorem 78 in the following few remarks.

Remark 18 Compared to the incompressible Euler system, we see that a Lipschitz norm of thevelocity has a logarithmic growth for large time. This is due to the logarithmic factor in the growthof the vorticity,

‖ω(t)‖L∞ ≤ C0log2(1 + t).

Remark 19 It is interesting to know whether or not the logarithmic growth for the differencebetween the vorticity and its singular part is optimal.

In the second result, we intend to lead the same result for the critical case α = 1.Roughly speaking, we shall prove the following result.

Theorem 79 Assume that ∂Ω0 ∈ C1+ε, with 0 < ε < 1 ω0 = 1Ω0 and ρ ∈ L1 ∩ L∞ ∩ Bεp,∞,

with p > max(

21−ε , 2

ε

). Then (Bα), with α = 1 admits a unique global solution.

(v, ρ) ∈ L∞loc(+; )× L∞(+; L1 ∩ L∞).

To be precise:

(1.iv) ‖∇v(t)‖L∞ ≤ C0eC0t,

(1.v) ω(t, x) = 1Ωt(x) + ρ(t, x), ρ ∈ Cη for all η < ε− 2p < 1,

(1.vi) ‖ω(t)− 1Ωt‖Lp ≤ C0.

165

Remark 20 In the recent paper [Hmidi-Zerguine], Hmidi and the author proved the followingstatement: If we replace the critical dissipation (−∆)

12 by the usual Laplacian (−∆) for the den-

sity equation, one obtains a logarithmic growth for the vorticity given by (1.i) -Theorem 78. How-ever, in our case we obtain the uniform estimate.

‖ω(t)‖L∞ ≤ C0,

which is sharp compared to the incompressible Euler equations. The logarithmic losing in the fullLaplacian is a low frequency problem. Indeed, for higher frequency the solutions of the trans-portdiffusion model perform an exponential decay for large time. However for low frequency thedissipation of the fractional Laplacian is much stronger than the Laplacian one and this will leadto a better time decay.

References[B-K-M] Beale J T, Kato T, Majda A, Remarks on breakdown of smooth solutions for

3-D Euler equations, Comm. Math. Phys., (1984), 94, 61–66. [Bertozi-Constantin BertozziA L, Constantin P, Global regularity for vortex patches, Comm. Math. Phys., (1993) 152(1),19–28.

[Chemin] Chemin J Y, Perfect incompressible Fluids., Oxford University Press., 1998.[Constantin-Wu] Constantin P, Wu J, Inviscid limit for vortex patches, Nonlinearity 8.,

1995, 735–742.[Danchin] Danchin R, Poches de tourbillon visqueuses, J. Math. Pures Appl., 1997, 9,

76(7), 609–647.[Depauw] Depauw N, Poche de tourbillon pour Euler 2D dans un ouvert a bord, J.

Math. Pures Appl., 9, 78 (3) 1999, 313–351.[Fanelli] Fanelli F, Conservation of geometric structures for non-homogeneous invis-

cid incompressible fluids, Comm. Partial Differential Equations., 2012, 37(9), 1553–1595.[Gamblin-Raymond] Gamblin P, Saint Raymond X, On three-dimensional vortex patches,

Bull. Soc. Math. France., 1995, 123(3) , 375–424.[Hmidi] Hmidi T, Regularite holderienne des poches de tourbillon visqueuses, J.

Math. Pures Appl., 2005, (9)84, 11,1455–1495.[Hmidi-Zerguine] Hmidi T, Zerguine M, Vortex patch for stratified Euler equations,

Commun. Math. Sci., 2014, 12( 8), 1541–1563.[Kato] Kato T, On classical solutions of the two dimensional non stationnary Euler

equations, Arch. Mech. Anal., 1968 27, 188–200.[Meddour-Zerguine] Meddour H, Zerguine M, Optimal rate of convergence in strati-

fied Boussinesq system, Dynamics of PDE., 2018, 15(4), 235–263.[Zerguine] M. Zerguine, The regular vortex patch for stratified Euler equations with

critical fractional dissipation, J. Evol. Equ., 2015, 15, 667–698.

166

Table 1: Model parameters and their meaning

bB constant source rate of B-CLL produced by bone marrowr replication rate of the leukemic B cells

dB natural mortality rate of the leukemic B cellsdBN killing rate of B-CLL cells by N cellsdBT killing rate of B-CLL cells by T cellsbN constant source rate of N cellsdN mortality rate of N cellsdNB deactivation rate of N cells by contact with B-CLL cellsbT constant production rate of T cellsdT natural mortality rate of T cellsdTB T cells activity suppression rate by contact with B-CLL cells

k fraction of TH cell activation that results in T cells recruitmentaTH maximal TH cells activation rate by contact with B-CLL cellsbTH constant production rate of TH cellsdTH natural mortality rate of TH cells

s half saturation constantL parameter shaping the saturating sigmoid response

n PZ(n) B(n)−4 0.001318 0.0013−3 0.0214 0.0214−2 0.1359 0.1357−1 0.3413 0.34010 0.3413 0.34011 0.1359 0.13572 0.0214 0.02143 0.0013 0.0013

Table 2: Comparison of PZ(n) and B(n) for some values of n

x Φ(x) FZ+U(x) Φ(x)−FZ+U(x)Φ(x)

0, 1 0, 5398 0, 5341 1%0, 2 0, 5792 0, 5682 2%0, 3 0, 6179 0, 6024 3%0, 4 0, 6554 0, 6365 3%0, 8 0, 7881 0, 7730 2%0, 9 0, 8159 0, 8072 1%

Table 3: Relative error: Comparison of Φ(x) and FZ+U for some values of x

167

n x ∈ [n, n + 1[ Φ(x) ∼= xn ψn(xn)

0 [0; 1[ 0, 5 + 0, 3413 · x 0, 5584 0, 02111 [1; 2[ 0, 8413 + 0, 1359 · (x− 1) 1, 4675 0, 02112 [2; 3[ 0, 9772 + 0, 0214 · (x− 2) 2, 4188 0, 0060

Table 4: ψn(xn) = Max(Φ(x)− FZ+U) on the interval [n; n + 1]

168

Posters

Functional local linear estimate of the conditional cumulativedistribution function

Khadidja Abdelhak

University of Naama

In this paper, the nonparametric estimation is investigated by the local linear method ofthe conditional cumulative distribution function of univariate response variable Yi giventhe functional variable Xi. The aim is to show almost complete convergence (with rates)of the constructed estimator under some general conditions.

Key words: Nonparametric local linear estimation, Conditional cumulative distributionfunction, Functional variable, Missing at random.

2010 Mathematics Subject Classification: 62G05,62G20.

IntroductionIn recent years, the considerable progress in computing power makes it possible To collectand analyze more and more cumbersome data. Many multivariate statistical techniques,concerning parametric models, have been extended to functional data and a good reviewon this topic can be found in [Am2] or [Am6]. Recently, new studies have been carried outin order to propose nonparametric methods taking into account functional data. We referto [Am5] For a more comprehensive review on this subject. The kernel density estimationhas been an important topic in statistics. A large number of works have dealt with thekernel density estimation. However, it is well known that a local polynomial smoothingprocedure has many advantages over the kernel method (see [Am3] and [Am4] for moredetails). In particular, the former method has better properties, in terms of bias estima-tion. Missing data often arise in various settings, including surveys, clinical trials, andlongitudinal studies. Responses may be missing, and methods for handling the missingdata often depend on the mechanism that generates the missing values, see [Am1]. Inmany practical works including for instance sampling survey, pharmaceutical tracing orreliability, data are often incompletely observed and part of the responses are missing atrandom (MAR). In this paper, the nonparametric estimation is investigated by the locallinear method of the conditional cumulative distribution function with data missing atrandom, of univariate response variable Yi given the functional variable Xi. The aim ofthis work is to show the almost convergence of the estimator under some general condi-tions.

169

Model, Notations and Assumptions

Description of the model and estimator

Let us consider a sequence (Xi, Yi)i≥1 of independent and identically random pair ac-cording to the distribution of the pair (X, Y), all of them defined on the same probabilityspace (Ω,A, P) and taking their values in a space F ×R, where (F , d) is a semi-metricspace.We suppose that F ×R is endowed with the product σ-algebra of the Borel σ-algebrasB(F ) and B(R) on F and on R respectively. For a fixed x ∈ F , we denote by Fx theconditional cumulative distribution function (cdF) of Y given (X = x).However, in the case of missing at random for the response variable, an available incom-plete sample of size n from (X, Y, δ) is (Xi, Yi, δi), 1 ≤ i ≤ n, where Xi is observedcompletely, δi = 1 if Yi is observed, and δi = 0 otherwise. Meanwhile the Bernoullirandom variable δ is satisfied with

P(δ = 1|X, Y) = P(δ = 1|X) = P(X),

where P(X) is a functional operator, which is called the conditional probability of theobserving response given the predictor and is often unknown. This mechanism showsthat δ and Y are conditionally independent given X. Missing at random is a commonassumption for statistical analysis with missing data and is reasonable in many practicalsituations, the reader can be referred to [FSV13, LLV15]. As indicated by [Am3] the func-tion Fx(·) can be viewed as a nonparametric regression model with response variableH(h−1

H (y− Yi)) where H is some cumulative distribution function and hH is a sequenceof positive real numbers. This consideration is motivated by the fact that

E[H(h−1H (y−Yi))|Xi = x]→ Fx(y) as hH → 0.

We use technique extended the local linear ideas to the infinite dimensional framework(see [BFV2010] and [DLMR2010]). This idea is combined with the consideration that thedata are missing at random. Here, we adopt the fast functional local modeling, that is,the conditional cumulative distribution function Fx is estimated by a where the couple(a, b) is obtained by the optimization rule:

min(a,b)∈R2

n

∑i=1

(H(h−1

H (y−Yi))− a− bβ(Xi, x))2

δiK(h−1K $(x, Xi)) (291)

where β(., .) and $(., .) are locating functions defined from F × F into R. K is a kernelappropriately chosen, H is a distribution function and hK = hK,n(respectively, hH = hH,n)is a sequence of positive real numbers which converges to 0 when n → ∞. Clearly, afterdirect computations, we get

Fx(y) =∑

1≤i,j≤nWij(x)H(hH

−1(y−Yj))

∑1≤i,j≤n

Wij(x), ∀y ∈ R (292)

with Wij(x) = βi(βi − β j)δiδjK(h−1K $(x, Xi))K(h−1

K $(x, Xj)) and βi = β(Xi, x).

170

Notations and AssumptionsThe hypotheses that are necessary in deriving the almost-complete convergence 1 (a.co.)of the functional locally modeled estimator of Fx are given in this section.In what follows x (resp. y) will denote a fixed point in (F (resp. R), Nx (resp. Ny) will de-note a fixed neighborhood of a fixed point x (resp. of y) and φx(r1, r2) = P(r2 < $(X, x) <r1). Then, we assume that our nonparametric model satisfies the following conditions:

(H1) For any r > 0, φx(r) := φx(−r, r) > 0.

(H2) The conditional distribution function Fx is such that: there exist some positive con-stants b1 and b2, ∀(y1, y2) ∈ Ny × Ny and ∀(x1, x2) ∈ Nx × Nx:

|Fx1(y1)− Fx2(y2)| ≤ C(|$(x1, x2)|b1 + |y1 − y2|b2

)where C is a positive constant depending on x.

(H3) The functions $(·, ·) and β(·, ·) are such that:

∀z ∈ F , β(z, z) = 0, |$(x, z)| = d(x, z) and C1|$(x, z)| ≤ |β(x, z)| ≤ C1|$(x, z)|where C1 > 0, C2 > 0.

(H4) The kernel K is a positive, differentiable function which is supported within [−1, 1].

(H5) The kernel H is a differentiable function.

(H6) The bandwidth hK satisfies: that there exists a positive integer n0, such that, ∀n >n0:

− 1φx(hK)

∫ 1

−1φ(zhK, hK)

ddz

(z2K(z))dz > C3 > 0

and

hK

∫B(x,hK)

β(u, x)dP(u) = o( ∫

B(x,hK)β2(u, x)dP(u)

)where B(x, r) = z ∈ F/|$(z, x)| ≤ r and dP(x) is the cumulative distribution ofX.Also that

limn→∞

nγhH = ∞ for some γ > 0 and limn→∞

log n

nhjHφx(hK)

= 0 for j ∈ 0, 1.

(H7) The operator P(.) is continuous on Nx and such that P(X) > 0.

Main Result: almost-complete convergenceTheorem 80 Under assumptions (H1)-(H7), we have

|Fx(y)− Fx(y)| = O(

hb1K + hb2

H

)+ O

((log n

n φx(hK)

) 12)

, a.co.

1Let (zn)n∈N be a sequence of real r.v.’s; we say that zn converges almost completely (a.co.)to zero if, and only if, ∀ε > 0, ∑∞

n=1 P(|zn| > ε) < ∞. Moreover, we say that the rate of almostcomplete convergence of zn to zero is of order un (with un → 0) and we write zn = Oa.co.(un) if,and only if, ∃ε > 0, ∑∞

n=1 P(|zn| > εun) < ∞.

171

Conclusion

In this paper, we have estimated the conditional cumulative distribution function by thelocal linear method and used missing at random. We also presented the result of almostcomplete convergence of the estimator.

References[BFV2010] J. Barrientos-Marin, F. Ferraty, P. Vieu, Locally modelled regression and func-tional data, Journal of Nonparametric Statistics 22(5): 617–632, 2010.[Am2] D. Bosq, Linear Processes in Function Spaces: Theory and applications, LectureNotes in Statistics, Springer. 2000.[DLMR2010] J. Demongeot, A. Laksaci, F. Madani, M. Rachdi, Local linear estimation ofthe conditional density for functional data, C. R., Math., Acad. Sci. Paris 348: 931–934,2010.[Am1] S. Efromovich, Nonparametric regression with responses missing at random, Jour-nal of Statistical Planning and Inference, 141 (2011), 3744-3752.[Am3] J. Fan, I. Gijbels, Local Polynomial Modelling and its Applications, Monographson Statistics and Applied Probability, Chapman & Hall, 1996.[Am4] J. Fan, Q. Yao, Non linear time series. Nonparametric and parametric methods,Springer Series in Statistics, Springer-Verlag, New York, 2003.[FSV13] F. Ferraty, F. Sued, P. Vieu, Mean estimation with data missing at random forfunctional covariables, Statistics, 47(2013), 688-706.[Am5] F. Ferraty, P. Vieu, Nonparametric Functional Data Analysis, Springer Series inStatistics, New York, USA, 2006.[LLV15] N. Ling, Y. Liu, P. Vieu, Conditional mode estimation for functional stationary er-godic data with responses missing at random, Statistics, (2016) doi: 10.1080 02331888.2015.1122012.[Am6] J.O. Ramsay, B.W. Silverman, Applied functional data analysis; Methods and casestudies, Springer-Verlag, New York, 2002.

On the conditional distribution function estimate in the singlefunctional index model under censored data

Fatima Akkal

University of Sidi Bel Abbes

In this presentation, we consider a nonparametric conditional distribution estimationgiven a functional explanatory variable in the case of censored response, when the dataare sampled from a stationary ergodic process and based on the single index structure.A kernel-type estimator of the conditional distribution function is introduced. Then, astrong consistency rate of the estimator is established. Key words: Nonparametric esti-

mation, conditional distribution function, single functional index model, censored data,ergodic data.

2010 Mathematics Subject Classification: 62G05, 62G07, 62G20, 62N01.

172

Introduction

Functional data analysis (FDA) is an important field of statistics that has received an in-creasing interest in recent years. This great consideration is due to the interaction withother applied fields. Typically, this area concerns the modelization of variables taking val-ues in infinite dimensional spaces, which appear in applied sciences such as economics,soil science, environmetal science, biometrics, chemometrics, meteorology, medical sci-ences.... Some key references on this topic are the monographs of Ramsay and Silverman[ramsay2005], Bosq [bosq2000], in addition to Ferraty and Vieu [ferraty2006].On the other hand, the functional index model plays a major role in statistics. The interestof this approach comes from its use to reduce the dimension of the data by projection infractal space.This trend has attracted the attention of many researchers as Aıt Saidi et al.[ait2008] in the case where the functional single-index is unknown. Attaoui et al. [Atta2011] who investigated the functional single-index model via its conditional density Ker-nel estimator. They established its pointwise and uniform almost complete convergencerates. Recently, Mahiddine et al. [Mahi2014] studied the nonparametric estimation ofsome characteristics of the conditional distribution in single functional index model. Theproblem of the conditional hazard function estimate with functional explicatory variablein single functional index was treated by Bouchentouf et al. [Bouch2014] in various situa-tions, including censored and non-censored data. More recently and concerning the sametopic, we can cite Attaoui and Ling [Atta2016] for the conditional cumulative distributionestimation.

Main results

The model and the estimates

Let (Xi, Ti)i≥1 be a sequence of stationary ergodic samples with identically distributionas (X, Y), where T is valued in R and X is valued in some infinite dimensional Hilbertianspace H with scalar product < ·, · >. The semi-metric dθ , associated to the single indexθ ∈ H defined by ∀x1, x2 ∈ H : dθ(x1, x2) = | < x1 − x2, θ > |. For x ∈ H, we dfine theconditional cumulative distribution function F(θ, t, x) of T given < X, θ >=< x, θ > by:∀t ∈ R, F(θ, t, x) = P(T ≤ t| < X, θ >=< x, θ >).

In the censorship situations, the lifetime Ti may not be directly observable. Instead,we observe only censored lifetimes of items under study. That is, assuming that (Ti)i≥1 isa sequence of independent and identically distributed (i.i.d.) random variables, and thatthey form a strictly stationary sequence of lifetimes. We suppose that there exists a sam-ple of i.i.d. censoring random variable (r.v) (Ci)i≥1 with common unknown continuousdistribution function (df) G.In the censored setup, the observed random variables are not couples (Ti, Xi), but ratherthe triplets (Yi, δi, Xi) whith

Yi = minTi, Ci and δi = 1Ti≤Ci .1 ≤ i ≤ n,

Where 1A denotes the indicator function of the set A.

173

• Conditional cumulative distribution estimation:

The kernel type estimator of the conditional distribution F(θ, t, x) adapted for censorshipmodel is given by :

F(θ, t, x) =

n

∑i=1

δi

G(Yi)K(

h−1K (< x− Xi, θ >)

)H(

h−1H (t−Yi)

)n

∑i=1

K(

h−1K (< x− Xi, θ >)

) ,

where, the functions K and H are kernels and hK := hK,n(resp.hH := hH,n) is a sequenceof positive real numbers which goes to zero as n tends to infinity.

In practice G(·) = 1−G(·) is unknown, then we replace G(·) by its Kaplan and Meier(1958) estimate Gn(·) given by

Gn(t) = 1− Gn(t) =

n

∏i=1

(1−

1− δ(i)

n− i + 1

)1Y(i)≤t

, if t < Y(n);

0, if t ≥ Y(n).

where Y(1) < Y(2) < . . . < Y(n) are the order statistics of Yi and δ(i) is the non-censoringindicator corresponding to Y(i).

Therefore feasible estimator of F(θ, t, x) is given by

F(θ, t, x) =

n

∑i=1

δi

Gn(Yi)K(

h−1K (< x− Xi, θ >)

)H(

h−1H (t−Yi)

)n

∑i=1

K(

h−1K (< x− Xi, θ >)

) .

Almost complete rate of convergenceTheorem 81 Under some hypotheses, we have as n goes to infinity

supt∈SR

∣∣∣F(θ, t, x)− F(θ, t, x)∣∣∣ = O

(hβ1

K + hβ2H

)+Oa.co.

(√log n

nφθ,x(hK)

)

ConclusionIn this presentation, we examine conditional distribution estimation for functional sta-tionary ergodic data based on the single index model under random censorship. Thepointwise almost complete convergence of the kernel estimator with rate are presentedunder some mild conditions.

References[ait2008] Aıt Saidi A., Ferraty F., Kassa R., and Vieu P., Cross-validated estimation in thesingle functionalindex model, Journal of theoretical and applied statistics., 2008, 42, 475–494.[Atta 2011] Attaoui S., Laksaci A., Ould Saıd E., A note on the conditional density esti-mate in the single functional index model, Statist. Probab. Lett., 2011, 81(1), 45–53.[Atta2016] Attaoui S., Ling N., Asymptotic results of a nonparametric conditional cumu-lative distribution estimator in the single functional index modeling for time series datawith applications, Metrika., 2016, 79, 485–511.

174

[Bouch2014] Bouchentouf A. A., Hamel E. H., Rabhi A., and Soltani S., NonparametricEstimation of Hazard Function with Functional Explicatory Variable in Single FunctionalIndex, Journal of applied mathematics and statistics., 2014, 1(1), 20–41.[bosq2000] Bosq D., Linear Processes in function spaces: Theory and applications. Leturenotes in ctatistics, Berlin: Springer Verlag, 2000.[chaouch2015] Chaouch M., Khardani S., Randomly censored quantile regression estima-tion using functional stationary ergodic data, Journal of nonparametric statistics., 2015, 27,65–87.[ferraty2006] Ferraty F., and Vieu P., Nonparametric functional data analysis: theory andpractice, New York: Springer, 2006.[Mahi2014] Mahiddine A., Bouchentouf A. A., and Rabhi A., Nonparametric estimationof some characteristics of the conditional distribution in single functional index model,Malaya J. Mat., 2014, 2(4), 392–410.[ramsay2005] Ramsay J. O., and Silverman B. W., Functional data analysis (2nd ed), NewYork: Springer Verlag, 2005.

Numerical solution for time-fractional partial differentialequation

N. Attia, D. Seba, A. NourUniversity of Boumerdes

In this work an efficient algorithm is used to find numerical approach of a time-fractionalpartial differential equation. The numerical solutions converge uniformly. The resultsdemonstrate the efficiency and the simplicity of the given method. Key words: Numeri-

cal solution, Partial differential equation, Convergence.

2010 Mathematics Subject Classification: 34K28, 35-XX.

IntroductionA great deal of important and complex phenomena in physics and applied mathematicscan be achively modeled by using nonlinear partial differential equations with fractionalorder. There are many numerical methods to find approximate solution for these equa-tions, among them, we mention Adomian decomposition method (ADM, for short) [A1].In this research, we present an iterative method to find numerical approach of the time-fractional KdV-Burger equation (KdVBE, for short) of the form

Dαt υ(x, t) + συ(x, t)υx(x, t) + ευxx(x, t) + $υxxx(x, t) = 0, 0 < x, t < 1, 0 < α ≤ 1,

(293)Here σ, ε and $ are constants, α is a parameter describing the time- Caputo fractionalderivative, x stands for the space coordinate, t is the time. υ(x, t) represents a functionof two variables x and time t which is vanishing for t < 0. The initial and boundaryconditions of this equation are given by

υ(x, 0) = υ0(x),υ(0, t) = f1(t) and υ(1, t) = f2(t),

(294)

175

where υ0(x), f1(t) and f2(t) are analytic functions.This study is organized as follows : in Section 2. some important definitions, mathemat-ical preliminaries and concepts of fractional calculus theory are presented. We constructour approach solutions of model problem by extend the application of the given methodin Section. 3. In Section 4. Two numerical applications of time-fractional KdVBE areachievely solved to show the simplicity and efficiency of this implementation.

Basic definitionsIn this part, we present some necessary definitions, properties and theorems shall be usedin this study.

Definition 21 The Caputo fractional derivative of υ(x, t) of order α > 0 is

(Dαt υ)(x, t) :=

1Γ(m− α)

∫ t

0(t− r)m−α−1∂rυ(x, r)dr, (295)

for m− 1 < α ≤ m, m ∈N, t ≥ 0, x ≥ 0.

Definition 22 (see[?]) H is a Hilbert space of function g : X → F on a set X. A functionK : X×X→ C is a reproducing kernel of H if the following are satisied

1. Kx(·) ∈ H for all x ∈ X.

2. 〈g,Kx(·)〉 = g(x) for all g ∈ H and for all x ∈ H.

The Space W42 [0, 1]

We define the space W42 [0, 1] as; W4

2 [0, 1] = υ(x)|υ(j) are absolutely continuous real-valued functions on [0, 1], υ(4) ∈ L2[0, 1], j = 0, 1, 2, 3,



2 [0, 1] = υ(t)|υ(j) are absolutely continuous real-valued functions on [0, 1], υ′′ ∈ L2[0, 1], j = 0, 1, and υ(0) = 0,



2 [0, 1] = υ(t)|υ are absolutely continuous real-valuedfunctions on [0, 1], υ′ ∈ L2[0, 1],

The Space W(4,2)2 (D)

Let D = [0, 1]× [0, 1].We define the space W(4,2)

2 (D) as; W(4,2)2 (D) = υ(x, t)| ∂4υ

∂x3∂t is completely continuous in

D, ∂6υ∂x4∂t2 ∈ L2(D), υ(x, 0) = 0, where the inner product and the norm in W(3,2)

2 (D) are

176

defined, respectively, by

〈υ(x, t),κ(x, t)〉W(4,2)

2 (D)=

3

∑i=0

∫ 1

0

[∂2

∂t2∂i

∂xi υ(0, t)∂2

∂t2∂i

∂xiκ(0, t)]

dt +1

∑j=0

⟨∂j

∂tj υ(x, 0),∂j

∂tjκ(x, 0)⟩

W42

+∫ 1

0

∫ 1

0

∂4

∂x4∂2

∂t2 υ(x, t)∂4

∂x4∂2

∂t2κ(x, t)dxdt

and ‖υ‖W(4,2)

2 (D)=√〈υ, υ〉

W(4,2)2 (D)

in which υ,κ ∈W(4,2)2 (D).

The Space W(2,1)2 (D)

We define the space W(2,1)2 (D) as; W(2,1)

2 (D) = υ(x, t)| ∂υ∂x is completely continuous in

D, ∂3υ∂x2∂t ∈ L2(D),

〈υ(x, t),κ(x, t)〉W(2,1)

2 (D)=

1

∑i=0

∫ 1

0

[∂

∂t∂i

∂xi υ(0, t)∂

∂t∂i

∂xiκ(0, t)]

dt + 〈υ(x, 0),κ(x, 0)〉W22

+∫ 1

0

∫ 1

0

∂2

∂x2∂

∂υ(x, t)

∂2

∂x2∂

∂tκ(x, t)dxdt

and ‖υ‖W(2,1)

2 (D)=√〈υ, υ〉

W(2,1)2 (D)

in which υ,κ ∈W(2,1)2 (D).

Representation solution of Eqs. 293 - 294After homogenizing the initial and boundary conditions, we need to describe a linearoperator L as,

L : W(4,2)2 (D)→W(2,1)

2 (D) (296)

such thatL℘(x, t) = (Dα

t (℘(x, t) + U(x, t))) + G(x, t,℘xx,℘xxx) (297)

The problem 293 - 294 can be written as follow:L℘(x, t) = F (x, t,℘,℘x), 0 ≤ x, t ≤ 1℘(x, 0) = 0,℘(0, t) = ℘(1, t) = 0,

(298)

Next, we construct an orthogonal system ψi∞i=1 which is can be derived from the Gram-

Schmidt orthogonalization process of ψi∞i=1 :

ψi(x, t) =i

∑k=1

βikψk(x, t), βii > 0, i = 1, 2, . . . , (299)

where βik are orthogonalization coefficients and are given by the following subroutine :βij =

1‖ψ1‖ for i = j = 1,

βij =1di

for i = j 6= 1,βij = − 1

di∑i−1

k=j Cikβkj for i > j,

such that di =√‖ψi‖2 −∑i−1

k=1 C2ik, Cik =

⟨ψi, ψk

⟩W3

2, and ψi∞

i=1 are the orthonormal

system in W(4,2)2 (D).

177

Theorem 82 if (xi, ti)∞i=1 is dense on D and the solution is unique on W(4,2)

2 (D), then thesolution of 293 - 294 is given by

υ(x, t) =∞

∑i=1

Aiψi(x, t). (300)

Numerical applicationsIn this section, we present two numerical examples to show the accuracy of the givenmethod.

Example 3 We consider the following time-fractional KdVBE as follows:

Dαt υ(x, t) + συ(x, t)υx(x, t) + ευxx(x, t) + $υxxx(x, t) = 0, 0 < x, t < 1, 0 < α ≤ 1,

subject the initial condition

υ(x, 0) =−1

50σε$

[250µ$2 + 6ε3sech2

(εx

10$

)(−1 + sinh

(εx5$

))].

The exact solution of problem is given as:

υ(x, t) =−1

50σε$

[250µ$2 + 6ε3sech2

(12

(εx5$

+µtα

Γ(α + 1)

))(−1 + sinh

(εx5$

+µtα

Γ(α + 1)

))].

Taking xi = ip , i = 1, 2, . . . , p, ti = i

q , i = 1, 2, . . . , q. Here, we take σ = 6, ε = −1, $ = 2and µ = 1. The absolute error of Example 3 is given in Table ?? and the graphics of approximatesolution are given in Figure 9 and Figure with α = 0.75 and α = 0.9 respectively.

x/t 0.2 0.3 0.4 0.5 0.6 0.7 0.80.1 1.46833E-3 2.79680E-3 4.03999E-3 5.21834E-3 6.34158E-3 7.41499E-3 8.44173E-30.2 1.46833E-3 2.80387E-3 4.04863E-3 5.22763E-3 6.35075E-3 7.42339E-3 8.44882E-30.3 1.47692E-3 2.81075E-3 4.0570E-3 5.23656E-3 6.35949E-3 7.43129E-3 8.45536E-30.4 1.48107E-3 2.81745E-3 4.06509E-3 5.24514E-3 6.36780E-3 7.43871E-3 8.46143E-30.5 1.48512E-3 2.82396E-3 4.07291E-3 5.25337E-3 6.37568E-3 7.44562E-3 8.46676E-30.6 1.48908E-3 2.83028E-3 4.08045E-3 5.26124E-3 6.38314E-3 7.45204E-3 8.47164E-30.7 1.49293E-3 2.83640E-3 4.08772E-3 5.26875E-3 6.39017E-3 7.45797E-3 8.47595E-30.8 1.49669E-3 2.84234E-3 4.09472E-3 5.27591E-3 6.39676E-3 7.46341E-3 8.47972E-3

178

ConclusionIn this article, an efficient method has been achievely implemented to find an approx-imate solution of the time-fractional KdVBE. The convergence of iterative process isdemonstrated. Numerical results displayed that the present iterative approach is power-ful for solving time-fractional KdVBE.

References[A1] Adomian G.,Solving Frontier Problems of Physics: The Decomposition Method,Nova Science Publishers, Inc, New York., 2009. [B2] Cui M., Lin Y., Nonlinear NumericalAnalysis in the Reproducing Kernel Space, Edition E., 2016. [T1] Sahoo S., Saha Ray S.,A new method for exact solutions of variant types of time fractional Korteweg-de-Vriesequations in shallow water waves, Math Meth- ods Appl Sci., 2017, 40, 106-14.

Estimation of discrete semi-Markov processChafiaa Ayhar

University of Saida

We consider a discrete-time semi-Markov process, with a finite state space. Takinga censored history, we study the empirical estimators for the semi-Markov kernel, semi-Markov transition function, its strong consistency and the asymptotic normality for eachestimator. Key words: Semi-Markov chain; Discrete-time semi-Markov kernel; Nonpara-

metric estimation; Asymptotic properties.

2010 Mathematics Subject Classification: 60K15, 62F12, 62G30.

IntroductionSemi-Markov processes are a class of stochastic processes which generalize Markov chains.For a discrete (resp. continuous) time Markov process, the sojourn time in each state isgeometrically (resp. exponentially) distributed. In the semi-Markov, the sojourn timedistribution can be any distribution on N∗ (resp. R+) This is the reason why the semi-Markov approach is much more suitable for applications than the Markov one [3,4].

In this work we introduce an empirical estimator of the semi-Markov kernel, whichproves to be also an approached maximum likelihood estimator. The kernel estimatorprovides us with empirical estimators for the Markov renewal function and transitionprobability function. After having obtained these general results, we can use them inany inference problem which involves functionals of the semi-Markov kernel (or Markovrenewal function).

PreliminariesLet us consider:

• E, the state space. We suppose E to be finite, with |E| = s.

179

• The stochastic process J = (Jn)n∈N with state space E for the system state at the nthjump, N = 0, 1, ...

• The stochastic process S = (Sn)n∈N with state space N for the nth jump. We sup-pose S0 = 0 and 0 < S1 < S2 < · · · < Sn < Sn+1 < · · ·

• The stochastic process X = (Xn)n∈N∗ with state space N∗ = N− 0 for the so-journ time Xn in state Jn−1 before the nth jump. Thus, Xn = Sn − Sn−1.

Definition 23 The stochastic process (J, S) = (Jn, Sn)n∈N is said to be a discrete-time Markovrenewal process (DTMRP) if for all n ∈ N, for all i, j ∈ E and for all k ∈ N it almost surelysatisfies

P(Jn+1 = j, Sn+1 − Sn = k | J0, ..., Jn; S0, ..., Sn) = P(Jn+1 = j, Sn+1 − Sn = k|Jn). (301)

Moreover, if equation (1) is independent of n, (J, S) is said to be homogeneous, with discretesemi-Markov kernel q(k) = (qij(k))i,j∈E ∈ ME defined by qij(k) := P(Jn+1 = j, Xn+1 =k|Jn = i), k > 0, and qij(0) = 0.

Definition 24 The transition matrix p = (pij)i,j∈E of the embedded Markov chain (Jn)n∈N isdefined by

pij := P(Jn+1 = j|Jn = i), i, j ∈ E, n ∈N.

We do not allow virtual transitions, i.e. we take pii = 0 for all i ∈ E.

Definition 25 For all i, j ∈ E such that pij 6= 0, let us denote by(i) fij(·), the conditional distribution of sojourn time in state i before going to state j :

fij(k) = P(Xn+1 = k|Jn = i, Jn+1 = j), k ∈N∗, (302)

(ii) hi(·), the sojourn time distribution in state i:

hi(k) = P(Xn+1 = k|Jn = i) = ∑l∈E

qil(k), k ∈N∗,

(iii) Hi(·), the sojourn time cumulative distribution function in state i:

Hi(k) = P(Xn+1 ≤ k|Jn = i) =k

∑l≥1

hi(l), k ∈N∗,

(iv) Hi(·), the survival function of sojourn time in state i:

Hi(k) = P(Xn+1 > k|Jn = i).

Obviously, for all i, j ∈ E and k ∈N, we have qij(k) = pij fij(k).

Main resultsWe suppose the following assumptions.A1 The Markov chain (Jn)n∈N is irreducible.A2 The mean sojourn times are finite, i.e. ∑k≥0 khi(k) < ∞ for any state i ∈ E.A3 The Markov renewal process (Jn, Sn)n∈N is aperiodic.

180

Definition 26 Let us consider a sample path of the DTMRP (Jn, Sn)n∈N, censored at time M ∈N

H(M) := (J0, X1, ..., JN(M)−1, XN(M), JN(M), uM),

where N(M):= maxn|Sn ≤ M, is the discrete-time counting process of the number of jumps in[1, M] ∈N and uM = M - SN(M),

For all i,j ∈ E and k ≤M, we define:

(i) Ni(M) := ∑N(M)−1n=0 1Jn=i: the number of visits to state i, up to time M;

(ii) Nij(M) := ∑N(M)n=1 1Jn−1=i,Jn=j: the number of transitions from i to j, up to time M;

(iii) Nij(K, M) := ∑N(M)n=1 1Jn−1=i,Jn=j,Xn=k: the number of transitions from i to j, up to time

M, with sojourn time in state i equal to k, 1 ≤ k ≤ M.

Taking a sample path H(M) of a DTMRP, for all ∀ i,j ∈ E and k ∈ N, k ≤ M, wedefine the empirical estimators of the probability transition function pij, of the conditionalsojourn time fij(k) and of the discrete semi-Markov kernel qij(k) by

pij(M) :=Nij(M)

Ni(M), fij(k, M) :=

Nij(k, M)

Nij(M), qij(k, M) :=

Nij(k, M)

Ni(M). (303)

Proposition 17 [2] Let µii denote the mean recurrence time of the state i for the DTMRP (Jn, Sn)n∈N.We have

Ni(M)M

a.s.−−−→M→∞

1µii

, Nij(M)M

a.s.−−−→M→∞

pijµii

The asymptotic properties of the proposed estimators.

Proposition 18 [2] The empirical estimator fij(k, M) proposed in equation (3) is uniformlystrongly consistent i.e.

maxi,j∈E max0≤k≤M| fij(k, M) − fij(k)|a.s.−−−→

M→∞0.

Theorem 83 [2] (1) The empirical estimator of the semi-Markov kernel proposed in equation (3)is uniformly strongly consistent, i.e.

maxi,j∈E max0≤k≤M|qij(k, M) − qij(k)|a.s.−−−→

M→∞0.

(2) For i, j ∈ E and k ∈ N,√

M[qij(k, M) − qij(k)] converges in distribution, as M → ∞, to azero mean normal random variable with variance µiiqij(k)[1− qij(k)].

Let us consider the matrix functions Q = (Q(k); k ∈N) ∈ ME(N), defined by

Qij(k) := P(Jn+1 = j, Xn+1 ≤ k|Jn = i) =k

∑l=1

qij(l), i, j ∈ E, k ∈N.

Let us denote by Q(k, M), the estimators of Q(k), defined by:

Q(k, M) :=k

∑l=1

q(l). (304)

We can easily see tha Q are uniformly strongly consistent estimators.

181

ConclusionAn important advantage of the discrete-time semi Markov process is its suitability forapplications, when the time scale is discrete. In this work we have considered empiri-cal estimators for transition probability, the conditional sojourn time and of the discretesemi-Markov kernel for this model. other problems can be considered, like empiricalestimators for the reliability and the availability of this process.

References[A1] Barbu, V., Boussemart, M. and Limnios, N., 2004, Discrete time semi-Markov modelfor reliability and survival analysis. Communications in Statistics-Theory and Methods,33(11), 2833-2868.[B2] Barbu, V.,LIMNIOS, N, Empirical estimation for discrete-time semi-Markov pro-cesses with applications in reliability, 2006, 483–498. [T1] Cinlar, E., 1975, Introduc-tion to Stochastic Processes (NY: Prentice Hall). [C1] Limnios, N. and Opri¸ san, G.,2001, Semi-Markov Processes and Reliability (Boston: Birkhauser). [D1] M. Dumitrescua,M.L. Gamizb and N. Limniosc, Minimum divergence estimators for the Radon-Nikodymderivatives of the Semi-Markov kernel, April 30, 2014.

Mean-field reflected delayed Backward Stochastic DifferentialEquations with jumps

Khalida Bachir CherifUniversity A

We establish sufficient conditions for the existence and uniqueness of Mean-field reflectedbackward stochastic differential equations with time delayed generator in the sense thatat time t, the generator may depend on previous values up to a delay constant δ. For suf-ficiently small delay constant δ and for any finite time horizon, we get a unique solution.

Keywords: Reflected backward stochastic differential equations; Time delayed gener-ator; Mean-field; Levy process; Poisson random measure.

IntroductionBackward stochastic differential equations (BSDEs) appear in their linear form as an ad-joint equation when Bismut [b] is the first how dealt with stochastic optimal control.Later, this theory has been developed for the nonlinear case, similar works have beengiven by Pardoux and Peng [PP], and the first work in finance has been presented by ElKaroui et al. [EPQ]. After that, El Karoui et al. [EL] introduced the notion of a reflectedBSDE on one lower barrier in one dimension, the solution is constrained to remain abovea continuous lower-boundary process.

Given a driven Brownian Motion B, a generator f : Ω × [0, T] × R2 −→ R and aterminal condition ξ. Solving a reflected BSDE is to find a triple process (Yt, Zt, Rt)0≤t≤Tadapted to the considered filtration (the Brownian one) without continuous reflectinglower barrier L(t), such that, at time t, (Y(t), Z(t), R(t))t≥0 satisfies the equation

Y(t) = ξ +∫ T

tf (s, Ys, Zs)ds + R(T)− R(t)−

∫ T

tZ(s)dB(s); (305)

182

Where, Yt ≥ Lt, 0 ≤ t ≤ T, almost surely. Moreover, the process (Rt)0≤t≤T is nonde-creasing, continuous, and the role of Rt is to push the state process upward with minimal

energy, in order to keep it above L: in this sense, it satisfies∫ T

t(Ys − Ls)dRs = 0.

The crucial question now is which conditions should be satisfied by the generator f andthe terminal value ξ in order to get the existence and the uniqueness of such a solution to(305).In this paper we are interested in a generalization of Reflected BSDE, where at time s thecoefficient f depends on past information and the law of the solution process and weconsider also the discontinuous case. More precisely, we are interested in the Mean-FieldReflected Delayed BSDE (MF-RDBSDE) with jumps of the form

Y(t) = ξ +∫ T

tf (s, Ys, Zs, Ks,L(Ys,Zs,Ks))ds−

∫ Tt Z(s)dB(s)

−∫ T

t

∫R0

K(s, ζ)N(ds, dζ) + R(T)− R(t); t ∈ [0, T] ,

Y(t) = Y(0), Z(t) = 0, K(t, ·) = 0; t < 0,

(306)

Here, for a given delay constant δ > 0

Ys = (Y(s + r))r∈[−δ,0] ,Zs = (Z(s + r))r∈[−δ,0] ,Ks(·) = (K(s + r, ·))r∈[−δ,0] ,

Theorem 84 Suppose that f is a lipschitz driver with lipschitz constant C. Then MF-RDBSDE(306) admits a unique solution (Y, Z, K(·), R) ∈ S2 ×H2 ×H2

v ×A2.

References[AO3] Agram, N., & Øksendal, B. (2016). Model uncertainty stochastic mean-field con-trol. arXiv preprint arXiv:1611.01385.[AO2] Agram, N., & Øksendal, B. (2017). Stochastic Control of Memory Mean-Field Pro-cesses. Applied Mathematics & Optimization, 1-24.[A] Agram, N. (2016). Stochastic optimal control of McKean-Vlasov equations with an-ticipating law. arXiv preprint arXiv:1604.03582.[AR] Agram, N., & Røse, E.E. (2017). Optimal control of forward-backward mean-fieldstochastic delay systems. Afr. Mat. https://doi.org/10.1007/s13370-017-0532-6.[AR2] Agram, N., & Røse, E.E. (2017). Mean-field delayed BSDEs in finite and infinitehorizon. arXiv preprint arXiv:1509.08777.[BBP] Barles, G., Buckdahn, R., & Pardoux, E. (1997). Backward stochastic differentialequations and integral-partial differential equations. Stochastics: An International Jour-nal of Probability and Stochastic Processes, 60(1-2), 57-83.[b] Bismut, J. M. (1978). An introductory approach to duality in optimal stochastic con-trol. SIAM review, 20(1), 62-78.[EL] EL KAROUI, N. et al. (1997a). Reflected solutions of backward SDE’s, and relatedobstacle problems for PDE’s. Ann. Prob. 25, 702-737.

183

[EPQ] El Karoui, N., Peng, S., & Quenez, M. C. (1997). Backward stochastic differentialequations in finance. Mathematical finance, 7(1), 1-71.[jm] J.-P. Lepeltier, A. Matoussi and M. Xu. (2005). Reflected Backward Stochastic Differ-ential Equations under Monotonicity and General Increasing Growth Conditions.Advancesin Applied Probability Vol. 37, No. 1 (Mar., 2005), pp. 134-159.[iy] Imade Fakhouri,& Youcef Ouknine (2018). L2-solution for reflected BSDEs withjumps under monotonicity and general growth conditions: a penalization method.[ml] Ma, H., & Liu, B. (2017). Infinite horizon optimal control problem of mean-field back-ward stochastic delay differential equation under partial information. European Journalof Control.[SEM] Mohammed, S. E. A.: Stochastic differential equations with memory: Theory, ex-amples and applications. Stochastic analysis and related topics VI. The Geilo Workshop,1996, Progress in Probability, Birkhauser, 1998.[OS] Oksendal, B., & Sulem, A. (2015). Risk minimization in financial markets modeledby Ito-Levy processes. Afrika Matematika, 26(5-6), 939-979.[SQ] Quenez, M. C., & Sulem, A. (2013). BSDEs with jumps, optimization and appli-cations to dynamic risk measures. Stochastic Processes and their Applications, 123(8),3328-3357.[PP] Pardoux, E., & Peng, S. (1990). Adapted solution of a backward stochastic differ-ential equation. Systems & Control Letters, 14(1), 55-61. [P] Pardoux, Etienne. (1999).BSDEs, weak convergence and homogenization of semilinear PDEs. NATO ASI Series CMathematical and Physical Sciences-Advanced Study Institute, 528, 503-550.[R] Royer, M. (2006). Backward stochastic differential equations with jumps and relatednon-linear expectations. Stochastic processes and their applications, 116(10), 1358-1376.

[TL] Tang, S., & Li, X. (1994). Necessary conditions for optimal control of stochastic sys-tems with random jumps. SIAM Journal on Control and Optimization, 32(5), 1447-1475.

A model of competition between plasmid-bearing andplasmid-free in a chemostat with external lethal inhibitor

Bar Bachir, Mohamed Dellal

University of Tlemcen, University of Tiaret

A model of competition between plasmid-bearing and plasmid-free organisms in a chemo-stat with a lethal external inhibitor was proposed in a survey of Hsu and Waltman, Suchmodels are relevant to commercial production by genetically altered organisms in con-tinuous culture. This paper provides a complete and rigorous analysis of the asymptoticbehavior of the governing equations. By means of operating diagrams, we describe theasymptotic behavior of the model with respect to those operating parameters. Someexamples are given to illustrate the mathematical results. Key words: Chemostat, com-

petition, plasmid-bearing, plasmid-free, inhibitor, stability, operating diagram.

2010 Mathematics Subject Classification: 34C80, 34D20, 92D25.

184

IntroductionThe chemostat is an important laboratory apparatus used for the continuous culture ofmicro-organisms. Genetically altered organisms are used in biotechnology to manufac-ture a product (e.g., the production of insulin or protein). The alteration is accomplishedby the introduction of DNA into the cell in the form of a plasmid. In chemostat Modelswhere the competitors are plasmid-bearing (genetically altered) and plasmid-free organ-isms, the plasmid can be lost in reproduction, creating a better competitor (one whichdoes not carry the metabolic load imposed by the plasmid). To deal with this, an addi-tional piece of genetic material is added to the plasmid, one that codes for resistance to aninhibitor, and the inhibitor is added to the reactor, competition between plasmid-bearingand plasmid-free organisms is extensively studied in the literature

The model introduced in [S.B. Hsu1] is the followingS′ = (S0 − S)D− f1(S) x

β − f2(S)yβ

x′ = [ f1(S)− D− γp]x + q m2SK2+S y

y′ = [(1− q) f2(S)− D]yp′ = (p0 − p)D− δp

K+p y

(307)

where f1(S) = m1SK1+S and f2(S) = m2S

K2+S .With the initial condition S(0) ≥ 0, x(0) > 0, y(0) > 0 and p(0) ≥ 0. The biological

parameters of the model are m1, m2, δ, K1, K2, K, β, γ, and q. The constant q represents Theprobability that a plasmid is lost in reproduction. These are called biological parameterssince they depend on the organisms, substrate and inhibitor considered. These param-eters are measurable in the laboratory. In contrast, the operating parameters are the inputconcentration of the nutrient S0, the input concentration of the inhibitor p0 and the di-lution rate D of the chemostat. These parameters are called operating parameters sincethey are under the control of the experimenter.

Main resultsWe suppose only that fi, i = 1, 2, and g in system (307) are C1-functions satisfying thefollowing conditions:

(H1) For i = 1, 2, fi(0) = 0 and f ′i (S) > 0 for all S ≥ 0.

(H2) g(0) = 0 and g′(p) > 0 for all p ≥ 0.

We define the break-even concentrations:

λ2(D) = f−12

( D1− q

), λ+(D, p0) = f−1

1 (D + γp0) (308)

Note that λ2 is defined for D ∈ Ic = (1− q)I2, and λ+ is defined for (D, p0) such thatD + γp0 ∈ I1;

Let p∗(D) = f1(λ2)−Dγ , and p = maxp∗, 0. We define the function K by

K(p, D, p0) = W1(p, D, p0)W2(p, D), for p ∈ ( p, p0]

Where, W1(p, D, p0) =D(p0 − p)

g(p), and W2(p, D) =

11− q

[q f1(λ2)

γ(p− p∗)+ 1]

185

Let pc be the unique solution of the equation

K(p, D, p0) = β(S0 − λ2) (309)

Proposition 19 Assume that (H1)-(H2) are satisfied. System (307) has the following equilibria:

• The washout equilibrium E0 = (S0, 0, 0, p0), that always exists.

• The equilibrium E1 = (λ+, x, 0, p0) of extinction of plasmid bearing organism y, wherex = Dβ(S0−λ+)

D+γp0 and λ+ is given by (308). This equilibrium exists if and only if λ+ < S0.

• The coexistence equilibrium Ec = (λ2, xc, yc, pc), where λ2 (respectively pc) given by (308)(respectively by (309) ) and yc, xc are given by

yc = W(pc, D, p0), xc =qDyc

γ(1− q)(pc − p∗)(310)

This equilibrium exists if and only if λ2 < minλ+, S0.

Theorem 85

1. If S0 < minλ2, λ+, then the washout equilibrium E0 of system (307) exists and isglobally asymptotically stable.

2. If S0 < λ2 and S0 > λ+, then the boundary equilibrium E1 of system (307) exists and isglobally asymptotically stable with respect to solutions with x(0) > 0.

Proposition 20 Assume that (H1) and (H2)are satisfied. The stability of equilibria of (307) is asfollows:

• The equilibrium E0 is LES if and only if S0 < minλ2, λ+.

• The equilibrium E1, if it exists, has at least three dimensional stable manifolds and is LESif and only if λ+ < λ2.

• The equilibrium Ec, if it exists, is LES if and only if it satisfies the following Routh-Hurwitzcondition

A3(A1A2 − A3) > A21A4 (311)

Where Ai, i = 1..4 depend on S0, D, and p0.

We use the following values to run some simulations.

Case m1 m2 K1 K2 δ K γ q1 5.0 6.0 0.5 3.5 5.0 0.3 6.0 0.3

Table 5: Biological parameters values used in the numerical simulations.

186

(a)

t t

(b)

Figure 7: The biological parameters values are given in Table 5, where p0 = D =1, x red, y blue. (a): Time course for S0 = S1 = 1.7 (Ec is stable). (b): Time coursefor S0 = 2 (Ec is unstable).

ConclusionIn this work we have studied the model (307) of competition between plasmid-bearingand plasmid-free in the chemostat with an external lethal inhibitor introduced in [S.B.Hsu1] by considering the model (307) with general growth rate functions of competitorsand absorption rate of external inhibitor. Our mathematical analysis of the model hasrevealed several possible behaviors.

References[S.B. Hsu1] S. B. Hsu and P. Waltman, A survey of mathematical models of competitionwith an inhibitor, Mathematical Biosciences. 2004, 187 , 53–91.

Least Squares Estimation in Periodic Restricted Expar(p) ModelsSabah Becila, Mouna Merzougui

University of Annaba

A generalization of the periodic restricted exponential autoregressive model (PEXPAR(1))to order p is introduced. The least squares method is used for estimating the parameters.The asymptotic properties of estimates for strictly stationary restricted PEXPAR are de-rived. A small simulation study is carried out to check the asymptotic properties. Key

words: Nonlinear time series, Periodic restricted exponential autoregressive model, Leastsquares estimation, Asymptotic properties.

2010 Mathematics Subject Classification: 62F12, 62M10.

IntroductionPeriodic time series models have been extensively used in the recent decades to describemany series with periodic dynamics. The inability of SARIMA models to adequately

187

represent many seasonal time series exhibiting a periodic autocovariance structure hasmotivated the research in the periodically correlated processes. This notion, introducedby Gladyshev (1960), was exploited in a variety of new classes of time series models,among them, the periodic GARCH (Bollerslev and Ghysels (1996)), the periodic bilinear(Bibi and Gautier (2005)) and the mixture periodic autoregressive model (Shao (2006)).

In this paper, we extend the class of periodic restricted exponential autoregressivemodel (PEXPAR(1)) discussed in Merzougui et al. (2016) to order p. PEXPAR seriessatisfy a nonlinear difference equation similar to that for EXPAR models with parametersand white noise variances which change periodically with season.

The class of exponential autoregressive (EXPAR) models introduced by Ozaki (1980)and Haggan and Ozaki (1981) has shown their appropriateness in capturing certain well-known features of nonlinear vibration theory, such as amplitude dependent frequency,jump phenomena and limit cycle behavior, these models are autoregressive in form withamplitude dependent exponential coefficients.

This paper deals with the least squares estimation of the periodic restricted EXPAR(p)model.

Main results

Periodic restricted EXPAR modelThe proces Yt; t ∈ Z is said to follow a Periodic Restricted Exponential AutoregressivePEXPARS(pt), with period S(S ≥ 2) , if it is a solution of a nonlinear periodic stochasticdifference equation of the form:

Yt =pt

∑j=1

(ϕt,j + πt,jexp(−γY2t−1))Yt−j + εt, t ∈ Z (312)

Where εt; t ∈ Z is i.i.d. process with continuous density fσt(.), not necessarily Gaus-sian, with mean 0 and finite variance σ2

t . The autoregressive parameters ϕt,j, πt,j, ∀t ∈ Z

and j = 1, · · · , p, the order pt and the innovation variance σ2t are periodic, in time, with

period S, i.e.,ϕt+kS,j = ϕt,j, πt+kS,j = πt,j, pt+kS = pt, and σ2

t+kS = σ2t , ∀k, t ∈ Z and j = 1, · · · , pt.

The nonlinear parameter,γ > 0, is known.Putting t = i + Sτ, i = 1, 2, . . . , S and τ ∈ Z and taking p = maxpi, i ∈ 1, 2, · · · , S.where ϕi,j = 0, πi,j = 0, for each j > pi, one can rewrite equation (1) in the equivalent

form:

Yi+Sτ =p

∑j=1

(ϕi,j + πi,jexp(−γY2i+Sτ−1))Yi+Sτ−j + ε i+Sτ, i = 1, . . . , S, τ ∈ Z (313)

Let ϕi= (ϕi,1, πi,1, . . . , ϕi,p, πi,p)

′, i = 1, . . . , S and ϕ = (ϕ′1, . . . , ϕ′

S)′ ∈ R2pS. We make

the following assumptions: A1: The periodic exponential autoregressive parameters ϕsatisfy the strict stationarity periodically condition of (2). A sufficient condition is: Allthe roots of associated characteristic equation

zp − ci,1zp−1 · · · − ci,p = 0

are inside the unit circle, where ci,j = max|ϕi,j|, |ϕi,j + πi,j|, j = 1, · · · , p; i = 1, · · · , S.A2: The periodically ergodic process Yt; t ∈ Z is such that E(Y4

t ) < ∞, for any t ∈ Z.

188

Parameter estimationThe estimation of the parameters ϕ of the model(2) is a linear optimisation problem,we can solve it using the least squares procedure. Suppose that we have observationsY1, · · · , YN from (2), N = mS, and define the conditional sum of squares

LN(ϕ) =S∑

i=1Li,m(ϕ) LN(ϕ) =

S∑

i=1(

m−1∑

τ=r+1(YSτ+i−

p∑

j=1(ϕi+j +πi+jexp(−γY2

Sτ+i−1))YSτ+i−j)2).

The estimate ϕi= (ϕi,1, πi,1, · · · , ϕi,p, πi,p)

′, for a fixed season i, is a solution to the esti-mating equations

∂Li,m(ϕ)

∂ϕi,j= 0 and

∂Li,m(ϕ)

∂πi,j= 0, j = 1, . . . , p.

The solution for a fixed season i is

ϕi=

Mi,1,1 · · · Mi,1,p...

. . ....

Mi,p,1 · · · Mi,p,p

−1

×

m−1∑

τ=r+1YSτ+i−1YSτ+i

m−1∑

τ=r+1YSτ+i−1YSτ+iexp(−γY2

Sτ+i−1)

...m−1∑

τ=r+1YSτ+i−pYSτ+i

m−1∑

τ=r+1YSτ+i−pYSτ+iexp(−γY2

Sτ+i−1)

, (314)

σ2i =

1m− r− 1

m−1

∑τ=r+1

(YSτ+i −p

∑j=2

(ϕi,j + πi,jexp(−γY2Sτ+i−1))YSτ+i−j)

2, (315)

Where for j, k = 1, · · · , p,

Mi, j, k =m−1∑

τ=r+1YSτ+i−jYSτ+i−k

m−1∑

τ=r+1YSτ+i−jYSτ+i−kexp(−γY2

Sτ+i−1)

m−1∑

τ=r+1YSτ+i−jYSτ+i−kexp(−γY2

Sτ+i−1)m−1∑

τ=r+1YSτ+i−jYSτ+i−kexp(−2γY2

Sτ+i−1)

.

Theorem 86 Suppose that Yt, satisfying (2), is strictly stationary, then the least squares esti-mators (3) and (4) are strongly consistent as m −→ ∞. That is

ϕi

a.s−→ ϕi

and σ2i

a.s−→ σ2i ,

and we have

√m(ϕ

i)− ϕ

i) D−→m −→ ∞

N(02p, σ2i Γ−1

i ),

where

Γi =

Γi,1,1 · · · Γi,1,p...

. . ....

Γi,p,1 · · · Γi,p,p

,

and

Γi,j,k =

(E(Yi−jYi−k) E(Yi−jYi−kexp(−γY2

i−1))E(Yi−jYi−kexp(−γY2

i−1)) E(Yi−jYi−kexp(−2γY2i−1))

), j, k = 1, · · · , p.

189

Simulation resultsThe performance of the estimation is shown via small simulation. The restricted PEXPAR2(2)model is used to generate time series for sizes n=200,400,800. We consider 1000 MonteCarlo replications and report the LS estimations, their bias and their standard deviations.The table 1 gives the estimation with the parameters ϕ = (0.6,−1, 0.3,−0.5;−0.5, 1,−0.4, 0.8)′, γ =

1 and normal white noise σ2 = (0.6, 1)′. The choice of the values of the parameters wastaken such that the model fulfill the condition A1.

ConclusionIn this study, we have used the linear least squares method for the estimation of theperiodic restricted EXPAR(p) model, consistency and asymptotic normality are derivedand simulated series checked the asymptotic properties. This LS estimator can be usedas an initial estimator in adaptive estimation.

As a part of future research, the authors study the Nonlinear LS and Quasi ML esti-mation of the periodic (unrestricted) EXPAR(p) model.

We have considered, here, a sufficient condition of strict stationarity but this subjectmerit further research.Haggan, V. and Ozaki,T. (1981), Modeling nonlinear random vibrations using an amplitude-dependent autoregressive time series model. Biometrik. 68(1), 96-189. Chan, K. S. andTong, H. (1985). On the use of the deterministic Lyapunov function for the ergodicity ofstochastic difference equations. Advances in applied probability, 17(3), 666-678.Koul, H. L. and Schick, A.(1997). Efficient estimation in nonlinear autoregressive timeseries models. Bernoulli, 3, 247-277. Merzougui, M. (2017) : Estimation in periodic re-stricted EXPAR(1) models. Communications in Statistics - Simulation and Computation,D.O.I. 10.1080/03610918.2017.1361975. Merzougui, M., Dridi, H. and Chadli, A. (2016) :Test for Periodicity in restrictive EXPAR Models. Communications in Statistics, Theoryand Methods, 45(9), 2770-2783.

Growth of solutions of a class of linear differential equationsnear a singular point

Samir Cherief

University of Mostaganem

In this paper, we investigate the growth of solutions of the differential equation

f ′′ + A (z) exp

a(z0 − z)n

f ′ + B (z) exp

b

(z0 − z)n

f = 0,

where A (z) , B (z) are analytic functions in the closed complex plane except at z0 anda, b are complex constants such that ab 6= 0 and a = cb (c > 1). Another case has beenstudied for higher order linear differential equations with analytic coefficients having thesame order near a finite singular point.

190

Key words: Linear differential equations, growth of solutions, near a finite singular

point.

2010 Mathematics Subject Classification: Primary 34M10; Secondary 30D35.

IntroductionFor the fundamental results and the standard notations of the Nevanlinna value dis-tribution theory of meromorphic function on the complex plane C and in the unit discD = z ∈ C : |z| < 1 , see [haym, yang, lain]. The importance of this theory has inspiredmany authors to find modifications and generalizations to different domains. Extensionsof Nevanlinna Theory to annuli have been made by [bieb, khri, kond, korh, mark]. Re-cently in [fet,ham3], Fettouch and Hamouda investigated the growth of solutions of cer-tain linear differential equations near a finite singular point. In this paper, we continuethis investigation near a finite singular point to study other types of linear differentialequations.

First, we recall the appropriate definitions. Set C = C ∪ ∞ and suppose that f (z)is meromorphic in C− z0 where z0 ∈ C. Define the counting function near z0 by

Nz0 (r, f ) = −r∫

∞

n (t, f )− n (∞, f )t

dt− n (∞, f ) log r, (316)

where n (t, f ) counts the number of poles of f (z) in the region z ∈ C : t ≤ |z− z0| ∪∞ each pole according to its multiplicity; and the proximity function by

mz0 (r, f ) =1

2π

2π∫0

ln+∣∣∣ f (z0 − reiϕ

)∣∣∣ dϕ. (317)

The characteristic function of f is defined in the usual manner by

Tz0 (r, f ) = mz0 (r, f ) + Nz0 (r, f ) . (318)

In addition, the order of meromorphic function f (z) near z0 is defined by

σT ( f , z0) = lim supr→0

log+ Tz0 (r, f )− log r

. (319)

For an analytic function f (z) in C− z0 , we have also the definition

σM ( f , z0) = lim supr→0

log+ log+ Mz0 (r, f )− log r

, (320)

where Mz0 (r, f ) = max | f (z)| : |z− z0| = r . If f (z) is meromorphic in C − z0 offinite order 0 < σT ( f , z0) = σ < ∞, then we can define the type of f as the following:

τT ( f , z0) = lim supr→0

rσTz0 (r, f ) .

191

If f (z) is analytic in C − z0 of finite order 0 < σM ( f , z0) = σ < ∞, we have alsoanother definition of the type of f as the following:

τM ( f , z0) = lim supr→0

rσ log+ Mz0 (r, f ) .

By the usual manner, we define the hyper order near z0 as follows:

σ2,T ( f , z0) = lim supr→0

log+ log+ Tz0 (r, f )− log r

, (321)

σ2,M ( f , z0) = lim supr→0

log+ log+ log+ Mz0 (r, f )− log r

. (322)

Remark 21 It is shown in [fet] that if f is a non constant meromorphic function in C− z0and g (w) = f

(z0 − 1

w

)then g (w) is meromorphic in C and we have

T (R, g) = Tz0

(1R

, f)

;

and so σ ( f , z0) = σ (g) . Also, if f (z) is analytic in C− z0 , then g (w) is entire and thusσT ( f , z0) = σM ( f , z0) and σ2,T ( f , z0) = σ2,M ( f , z0) .

So, we can use the notation σ ( f , z0) without any ambiguity. But concerning the type,as in the complex plane, τT ( f , z0) does not equal to τM ( f , z0) .

Main resultsRecently, Fettouch and Hamouda proved the following two results.

Theorem 87 [fet] Let z0, a, b be complex constants such that arg a 6= arg b or a = cb (0 < c < 1)and n be a positive integer. Let A (z) , B (z) 6≡ 0 be analytic functions in C − z0 withmax σ (A, z0) , σ (B, z0) < n. Then, every solution f (z) 6≡ 0 of the differential equation

f ′′ + A (z) exp

a(z0 − z)n

f ′ + B (z) exp

b

(z0 − z)n

f = 0,

satisfies σ ( f , z0) = ∞ with σ2 ( f , z0) = n.

Theorem 88 [fet] Let A0 (z) 6≡ 0, A1 (z) , ..., Ak−1 (z) be analytic functions in C− z0 satis-fyingmax

σ(

Aj, z0)

: j 6= 0< σ (A0, z0) . Then, every solution f (z) 6≡ 0 of the differential equa-

tionf (k) + Ak−1 (z) f (k−1) + ... + A1 (z) f ′ + A0 (z) f = 0. (323)

satisfies σ ( f , z0) = ∞ with σ2 ( f , z0) = σ (A0, z0) .

In this paper, we will investigate the case c > 1 to complete the remaining case inTheorem 87, in the following two results.

192

Theorem 89 Let n be a positive integer and A (z) 6≡ 0, B (z) 6≡ 0 be analytic functions inC − z0 such that max σ (A, z0) , σ (B, z0) < n; let a, b be complex constants such thatab 6= 0 and a = cb (c > 1). Then, every solution f (z) 6≡ 0 of the differential equation

f ′′ + A (z) exp

a(z0 − z)n

f ′ + B (z) exp

b

(z0 − z)n

f = 0, (324)

that is analytic in C− z0 , satisfies σ ( f , z0) = ∞.

Theorem 90 Let n be a positive integer and A (z) 6≡ 0, B (z) 6≡ 0 be polynomials; let a, b becomplex constants such that a = cb (c > 1). Then, every solution f (z) 6≡ 0 of the differentialequation

f ′′ + A(

1z0 − z

)exp

a

(z0 − z)n

f ′ + B

(1

z0 − z

)exp

b

(z0 − z)n

f = 0, (325)

that is analytic in C− z0 , satisfies σ ( f , z0) = ∞ with σ2 ( f , z0) = n.

In the following result, we will improve Theorem 88 by studying the case when

max

σ(

Aj, z0)

: j 6= 0≤ σ (A0, z0) .

Theorem 91 Let A0 (z) 6≡ 0, A1 (z) , ..., Ak−1 (z) be analytic functions in C− z0 satisfyingthe following conditionsi) 0 < σ

(Aj, z0

)≤ σ (A0, z0) < ∞ (j = 1, ..., k− 1) ,

ii) max

τM(

Aj, z0)

: σ(

Aj, z0)= σ (A0, z0)

< τM (A0, z0).

Then, every solution f (z) 6≡ 0 of (323), that is analytic in C− z0 , satisfies σ ( f , z0) = ∞with σ2 ( f , z0) = σ (A0, z0) .

We can find the analogous of Theorem 96 in the complex plane and in the unit disc.References

[bieb] L. Bieberbach, Theorie der gewohnlichen Differentialgleichungen, Springer-Verlag,Berlin, Heidelberg, New York, 1965.[chen3] Chen Z. X. and Yang C. C., Some further results on zeros and growths of entiresolutions of second order linear differential equations, Kodai Math. J., 22 (1999), 273-285.[fet] Fettouch H. and Hamouda S., Growth of local solutions to linear differential equa-tions around an isolated essential singularity, Electron. J. Differential Equations, Vol 2016(2016), No. 226, pp. 1-10.[gund1] Gundersen G. G., Estimates for the logarithmic derivative of a meromorphicfunction, plus similar estimates, J. Lond. Math. Soc. (2), 37 (1988), 88-104.[ham12] Hamouda S., Properties of solutions to linear differential equations with analyticcoefficients in the unit disc, Electron. J. Differential Equations, Vol 2012 (2012), No. 177, pp.1-9.[ham] Hamouda S., Iterated order of solutions of linear differential equations in the unitdisc, Comput. Methods Funct. Theory, 13 (2013) No. 4, 545-555.[ham3] Hamouda S., The possible orders of growth of solutions to certain linear differen-tial equations near a singular point, J. Math. Anal. Appl. 458 (2018) 992–1008.[haym] Hayman W. K., Meromorphic functions, Clarendon Press, Oxford, 1964.[khri] Khrystiyanyn A.Ya. and Kondratyuk A.A., On the Nevanlinna theory for mero-morphic functions on annuli, Matematychni Studii 23 (1) (2005) 19–30.

193

[kond] Kondratyuk A.A. and Laine I., Meromorphic functions in multiply connected do-mains, in: Fourier Series Methods in Complex Analysis, in: Univ. Joensuu Dept. Math.Rep. Ser., vol. 10, Univ. Joensuu, Joensuu, 2006, pp. 9-111.[korh] Korhonen R., Nevanlinna theory in an annulus, in: Value Distribution Theory andRelated Topics, in: Adv. Complex Anal. Appl., vol. 3, Kluwer Acad. Publ., Boston, MA,2004, pp. 167-179.[ain] Laine I., Nevanlinna theory and complex differential equations, W. de Gruyter, Berlin,1993.[mark] Mark E. L. and Zhuan Y., Logarithmic derivatives in annulus, J. Math. Anal. Appl.356 (2009) 441-452.[tsu] Tsuji M., Potential theory in modern function theory, Chelsea, New York, 1975, reprintof the 1959 edition.[tu] Tu J. and Yi C-F. On the growth of solutions of a class of higher order linear differ-ential equations with coefficients having the same order, J. Math. Anal. Appl. 340 (2008)487–497.[wa] Wittaker J. M., The order of the derivative of a meromorphic function, J. LondonMath. Soc. 11 1936, 82-87, Jbuch 62, 357.[yang] Yang L., Value distribution theory, Springer-Verlag Science Press, Berlin-Beijing. 1993.

On the local linear estimation of the nonparametric robustregression for dependent functional data

Souheyla Chemikh


This work deals with the prediction problem via the M-regression when the regres-sors are functional random variables based on the local linear technique. The main pur-pose of this paper is to study the almost complete convergence (with rate) of the ro-bust local linear estimator of the regression function when the observations are α-mixing.Moreover, a simulation study is given in order to evaluate, on a finite sample, the perfor-mance of our estimator.Key words: Functional data analysis; Local linear method; Robust estimation; Almost

complete convergence; Strong mixing.

2010 Mathematics Subject Classification: 37A25; 62G05; 62G08; 62G20;62G35.

IntroductionThe functional nonparametric statistics is a new branch of statistics which is devoted tothe study of functional data, i.e., random variables which are observed in several pointsof an interval and take values in infinite dimensional space called functional space, werefer to Ferraty and Vieu (2006) for more discussions. Recall that, in functional non-parametric statistics, there are several models which are devoted to the study of the linkbetween two or more random variables, in this paper, we are interested to the model ofrobust regression by using the linear local method in the case where the observations arestrongly mixing.

194

It is well known that the classical regression is very affected by the presence of out-liers or heteroscedasticity phenomena that motivated the interest for the study of the M-regression. Historically, the first consequent result in this area dates back to Huber(1965)and it has been widely studied for real data (see for instance Robinson(1984) and Boenteet al.(2009)). For the robust local linear estimation, we refer to Cai etal.(2003) for theα-mixing case. In this paper, we propose the local linear estimator of the M-regression.

It is important to note that the local linear method is an alternative statistical approachto the kernel method, and which has many advantages over the latter. In particular, thegreatest advantage is the reduction of the bias term of the estimator in a various situ-ations. We refer to Fan(1992) for more discussions on the importance of this method.The generalization of the local linear technique in the case of functional explanatory vari-ables has been discussed very recently, the first interesting article, on this topic, datesback to Baıllo and Grane (2009), they proposed the first version of the functional locallinear estimate by establishing the L2 convergence rate of the local linear estimator of theregression function when the explanatory variable is Hilbertian. Barrientos et al.( 2010)proposed another easy-to-use functional version. The authors in this article studied thealmost complete convergence of the local linear estimator of the regression function. Re-cently, Al-Awadhi et al.(2018) studied the almost complete convergence and asymptoticnormality of the conditional quantiles estimator.The nonparametric robust regression es-timation has been studied by many authors, in the nonparametric functional framework,we cite for example Azzedine et al.(2008) for for independent and identically distributedobservations and Attouch et al.(20012) for functional time series data.

In an earlier paper (see Belarbial et al.(2018), we studied the almost complete con-vergence (with rate) and the asymptotic normality of a robust local linear estimator ofregression function for independent and identically distributed observations. Our in-terest in this work is to generalize these results to dependent case and we carry out asimulation study in order to show the advantages of this estimate.

The robust local linear estimator

Consider n independent pairs of random variables (Xi, Yi) for i = 1, ..., n that we assumedrawn from the pair (X, Y). The latter is valued in F × IR where F is a semi-metric spaceand d denotes a semi- metric. The object of this paper is to study the co-variation betweenXi and Yi by the nonparametric robust regression function. For x ∈ F the nonparametricrobust regression, denoted by θx, is defined as the unique minimizer of

θx = arg mint∈IR

IE [ρ(Y− t)|X = x] (326)

where ρ(.) is a real-valued Borel function satisfying some regularity conditions to bestated below. In this paper, we adopt the fast version proposed by Barrientos-Marinet al. (2010) for which the function θx is approximated by

∀z in neighborhood of x θz = a + bβ(z, x)

where a and b are estimated by a and b are solution of

min(a,b)∈IR2

n

∑i=1

ρ(Yi − a− bβ(Xi, x))K(

h−1δ(x, Xi))

(327)

195

where β(., .) is a known function from F ×F into IR such that, ∀ξ ∈ F , β(ξ, ξ) = 0, withK is kernel and h = hn is a sequence of positive real numbers and δ(., .) is a function ofF ×F such that d(., .) = |δ(., .)|. It is clear that, under this consideration, we can write

θx = a and θx = a.

Main resultsTheorem 92 Under hypotheses

|θx − θx| = O(hmin(k1,k2)) + O

( √ξ1/2

x log(n)nφ2

x(h)

)a.co.

Simulation studyRecall first that, our model has several advantages, it is considered as a generalizationof several models of the functional regression. More precisely, we can easily obtain thedefinition the classical functional robust regression (C.F.R.R.) proposed by Attouch et al.(2010) if we take b = 0 in 327, its estimator is given by

mina∈IR

n

∑i=1

ρ(Yi − a)K(h−1d(x, Xi)).

We obtain the local linear functional regression (L.L.F.R.) constructed by Barrientos etal.(2010) if we take ρ(y) = y2 in the model 327, its estimator is defined by

min(a,b)∈IR2

n

∑i=1

(Yi − a− bβ(Xi, x))2K(h−1δ(x, Xi)).

And if we replace b by 0 and we take ρ(y) = y2 in the same model, we obtain theclassical functional regression (C.F.R) which was introduced in NDFA by Ferraty andVieu(2006), its estimator is defined by

mina∈IR

n

∑i=1

(Yi − a)2K(h−1d(x, Xi)).

The objective of this section is to see the behavior of our proposed estimator (thelocal linear functional robust regression (L.L.F.R.R)) for a finite sample. In particular, ourfirst purpose is to show how can implement easily our estimator in the practice and thesecond purpose is to compare the sensitivity to outliers of the four estimators of ((C.F.R),(C.F.R.R.),(L.L.F.R.),(L.L.F.R.R)). For this we consider the explanatory curves generated inthe following way:

Xi(t) = cos(4(Wi − t)π) + ait2 f or i = 1, 2, . . . , n and ∀t ∈ [0, 1],

All the curves Xi’s are discretized on the same grid which is composed of 100 equidis-tant values in [0, 1], and are represented in Figure 8

Then, we define the scalars responses variables Yi’s by:

Yi = R(Xi) + εi

196

Time

0 20 40 60 80 100−4

−20

24

68

10

Figure 8: The curves Xi=1,...,100(t), tj=1,...,100 ∈ [0, 1]

where the errors εi’s are i.i.d normally distributed asN (0, 0.2) and R(Xi) = exp−∫ 1

0

dt1 + Xi(t)2

.

In the first illustration which is given in Table 6, we compare between the four re-gressions models in the case of the absence of outliers where it appears clearly that themodels ((C.F.R), (C.F.R.R.),(L.L.F.R.),(L.L.F.R.R)) have a good behavior but the MSE for((L.L.F.R.),(L.L.F.R.R)) is smaller than for (C.F.R.) and (C.F.R.R.).

Table 6: Comparison between the four regressions models in the absence of outliersSize of sample C.F.R. C.F.R.R L.L.F.R. L.L.F.R.R

n=100 0.960 0.954 0.860 0.758n=200 0.664 0.668 0.591 0.442

The second illustration is given in Table 7 where we introduced the outliers by multi-plying some values of the response variable Y, in the learning sample by 50. In thiscase, we observe that (C.F.R.R.) and (L.L.F.R.R) models give better results than (C.F.R)and (L.L.F.R.). More precisely, the MSE of the (C.F.R)and (L.L.F.R.) increases substan-tially relatively to the number of the outliers, but it remains is very low for (C.F.R.R) and(L.L.F.R.). we observe also that the error of (L.L.F.R.) and (L.L.F.R.R) is smaller than thatof (C.F.R) and (C.F.R.R.) respectively. Another remarkable point, the error of (L.L.F.R.R)remain very low with respect to the other regressions models studied, in the all cases.

Table 7: Comparison between the four regressions models in the presence of outliersSize of sample number of artificial outliers C.F.R. C.F.R.R L.L.F.R. L.L.F.R.R

One outlier 1.981 1.051 1.516 0.971n=100 10 outliers 80.189 8.280 75.830 7.811

25 outliers 359.33 32.468 321.970 28.409One outlier 1.280 0.981 1.115 0.795

n=200 10 outliers 68.871 6.921 59.124 5.14225 outliers 311.652 26.422 270.354 21.789

197

ConclusionIn this work, we have positioned our contribution in the extensive literature of nonpara-metric analysis of functional data by giving the asymptotic results in the case α-mixing.The results obtained, in the simulation part, confirm that the robust local linear estimatorhas an insensitive and effective behavior in the presence of the outliers observations witha reduced bias term.

References[A1] Attouch, M., Gheriballah, A., Laksaci, A. Robust regression for functional time seriesdata, Journal of Japan Statist.Soc,, 2012, 42,125–143.[B2] Barrientos-Marin, J., Ferraty, F. and Vieu, P., Locally modelled regression and func-tional data, J. of Nonparametric Statistics, 2010,22, 617–632.[T1] Belarbi,F.,Chemikh,S., Laksaci, A. Local linear estimate of nonparametric robust re-gression in function data, Statistic and Probability letters,,(2018),134, 128–133.

Well-posedness of sea-ice modelCHATTA. SofianeUSTHB, Algeria

In this paper we have two models in one-dimension to which we examine the well-posedness. These models of equations describe the balance of linear momentum com-bined with simplified thermodynamics represented by two continuity equations for ef-fective ice thickness and ice concentration. The first model is a constitutive model for seaice [hibler1979dynamic] in which we changed the viscosity-coefficient ξ and the secondmodel is the pressure replacement model with a changed ξ similarly to the first model.The aim is to examine the well-posedness of these new models.Key words: Well-posedness, pressure replacement, nonlinear problem, depth-averged

stress

IntroductionThe sea-ice plays a major role in climate variability and change. Most sea-ice dynamicmodels currently used in the community are based on a viscous-plastic (VP) formulation[hibler1979dynamic]. This model treats ice as an isotropic, continuum, and viscous fluidat low strain rates and as a plastic solid at higher strain rates. The result of combiningthese behaviors is a nonlinear, viscous, and compressible fluid model for ice. The ques-tion that is raised from the past, is the Hibler’s original VP model [hibler1979dynamic]well-posed or ill-posed. As with any mathematical model, it is of interest to understand ifsolutions to the equations have physically realistic features. [hadamard1902problemes]argued on physical grounds that the equations making up a mathematical model thatdescribe a time-dependent physical process should have three features:

1. existence a solution for arbitrary initial data.

2. uniqueness a solution determined by the initial data.

198

3. the solution should depend continuously on the initial data.

In this paper, We concentrate our attention on the third property of a well-posedproblem, the continuity of solutions with respect to initial data.

The difficulty in analyzing nonlinear systems of equations has led to confusion inthe literature about the difference between unstable solutions and the ill-posedness ofthe initial-value problem. [gray1995stability] were apparently the first to notice a shortwavelength linear instability in the VP model, but stated that this instability could be con-trolled by nonlinearities in the problem. [schulkes1996asymptotic] showed that the totalenergy of the system is bounded and claimed this fact made the VP model asymptoticallystable. [dukowicz1997comments] argued similarly that the dissipative nature of the VPmodel controlled the instabilities found in [gray1995stability]. [gray1997reply] further re-sponded to [dukowicz1997comments] and [schulkes1996asymptotic] by correcting someflaws in the analysis and clarifying that [gray1995stability] showed ill-posedness becausethe growth rate of the instability increased with decreasing wavelength. But they statederroneously that the nonlinear analysis means that the initial-value problem was well-posed for certain boundary conditions. In this paper, we take the constitutive model of[hibler1979dynamic] and changed the viscosity-cofficient ξ in σ and in the second partwe do the same with the pressure replacement model of [guba2013] and in the future wewill try to see if these models well-posedness or no.

Main resultsWe present here the balance equations for the prognostic variables: velocity v, ice con-centration or area fraction A, and effective ice thickness h. The governing equations are:

ρhdvdt−∇.σ = Fext. (328)

dAdt

+ A∇.v = SA, and (329)

dhdt

+ h∇.v = Sh. (330)

where ρ > 0 is a constant ice density and σ is depth-averaged stress. Fext, SA and Share respectively the external forces and the two thermodynamic source terms. These willbe amitted because they do not affect the analysis.

The goal is to investigate the well-posedness of the initial-value problem in one spacedimension.

Model 1The constitutive model for σ is given by

σ = P[KEux tanh(1

2√

EK|ux|)− 1

2] (331)

where K, E are a constants K = 2.5 ∗ 108, E = 1 + 1e2 and the isotropic compressive

strength P is assumed to depend on the mean ice thickness and area fraction throughthe relation

199

P = P∗he−C(1−A) (332)

with P∗ and C being positive constants. Usually, e = 2, C = 20, and values of P∗ from2.0× 103 to 5.0× 104Nm−2 have been used.

And the viscosity coefficients used here are:

ξ = KP tanh(1

2K√

E|ux|) η =

ξ

e2 . (333)

The linearized system of equations for this model is:

ut = −uu0x − u0ux −hh0

(u0t + u0u0x)

+KEρh0

tanh(1

2√

EKu0x)(P0uxx + Pxu0x + Pu0xx + P0xux)−

Px

2ρh0,

At = −uA0x − u0Ax − A0ux − Au0x, andht = −uh0x − u0hx − h0ux − hu0x,

Note that there are two cases here (ux > 0, ux < 0).

Model 2

The pressure replacement model for σ is given by

σ = KP tanh(1

2√

EK|ux|)[Eux −

√E|ux|]] (334)

In the same way we find the linearized system of equations for this model:

ut = −uu0x − u0ux −hh0

(u0t + u0u0x)

+K(E−

√E)

ρh0tanh(

12√

EKu0x)(P0uxx + Pxu0x + Pu0xx + P0xux),

At = −uA0x − u0Ax − A0ux − Au0x, andht = −uh0x − u0hx − h0ux − hu0x,

200

On the resolution of nonlinear fractional elliptic problemSara Dob

University of Skikda

In this work, we study the existence of weak solutions for the non-linear fractionalelliptic problem

−divs(a(x, u(x))Dsu(x)) = f (x, u(x)), in Ω,u = 0. on n \Ω,

with s ∈]0, 1[ and Ω is an open bounded subset of n. To prove the existence of solutionsunder suitable assumptions on the non linearities a and f , we use Schauder’s fixed point.Key words: Nonlinear elliptic equations; fractional gradient; weak solution.

2010 Mathematics Subject Classification: 35J60, 35D30.

IntroductionFractional derivative equations are today one of the important themes of scientific under-standing and are of great help in modeling many problems of mathematical physics.

The theory of fractional calculus is a subject almost as old as differential calculus andit dates back to when Leibniz, Gauss, Newton developed the foundations of this typeof calculation (See references [int], [into], but it is only in the last three decades that thefractional computing has enjoyed wider interest; see the works [e] and [c]. Fractional cal-culation has a very wide field of application (see [into], [d]), for example: viscoelasticity,control theory, diffusion equation, electricity, electromagnetic, biology...Etc

In this work, we investigate the existence of weak solutions to the nonlinear fractionalelliptic equations of the type

−divs(a(x, u(x))Dsu(x)) = f (x, u(x)), in Ω,u = 0. on n \Ω,

(335)

where s ∈]0, 1[ and Ω is an open bounded of Rn such that n > 2s.We recall that the geometric interpretation of the fractional gradient for all u ∈ C∞

c (n) is

Dsu = I1−s ∗ Du.

where the Riesz potentials for 0 < α < N defined by the formula

Iαu := Iα ∗ u,

Iα(x) :=γ(N, α)

|x|N−α,

and the constant

γ(N, α) = Πn2−α Γ( α

2 )

Γ( n−α2 )

.

201

The purpose of the present paper is to extend the results of [?] to the problem (359) underthe following conditions on the functions a and f :

a : Ω× −→ Caratheodory function,there exists α > 0 and β > 0 such that α ≤ a(., s) ≤ β a.e. and for all s ∈,f ∈ L∞(Ω×).

(336)

Preliminaries and the main resultLet

Hs0(Ω) = u : u ∈ Hs(n), u = 0 in Ωc.

with the norm, that we will denote by ‖.‖Hs0

‖u‖Hs0= ‖Dsu‖L2 .

Under the assumptions (336), we tries to show the existence of u solution of the followingproblem:

u ∈ Hs0(Ω),

∫Ω a(x, u(x))Dsu(x)Dsv(x)dx =

∫Ω f (x, u(x))v(x)dx. for all v ∈ Hs

0(Ω)

(337)

Theorem 93 Assume that (336) are fulfilled. Then (359) has a solution u ∈ Ds,2(Ω).

Proof of the main resultFor u ∈ L2(Ω), the existence and uniqueness of u solution of the linear elliptic problem

u ∈ Hs0(Ω),

∫Ω a(x, u(x))Dsu(x)Dsv(x)dx =

∫Ω f (x, u(x))v(x)dx. for all v ∈ Hs

0(Ω),

(338)

it is given in other works.

We put T(u) = u. The application T is therefore an application of E in E with E =L2(Ω). A fixed point of T is a solution of (359). To proof the existence of such a fixedpoint, we will use the Schauder fixed point theorem.

References[n]S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrodingertype problem involving the fractional Laplacian, Le Matematiche, Vol. 68, No. 1, pp. 201-216,(2013).[hhh] Mouhamed M. Fall, Fractional Elliptic Equations, African Institute for MathematicalSciences (AIMS) Senegal, Submitted in Partial Fulfillment of a Masters II at AIMS, (15June 2014).[int] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore,(2000).

202

[into] A. A. Kilbas and S. A. Marzan, Nonlinear differential equations with the Caputo frac-tional derivative in the space of continuously differentiable functions, Differential Equations,Vol. 41, pp. 84-89, (2005).[11] Thierry Gallouet and Raphaele Herbin, Equations aux derivees partielles, Master 2 demathematiques, Universite Aix Marseille, (4 decembre 2018).[e]A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Dif-ferential Equations, Elsevier Science B.V, Amsterdam, (2006).[f] V. Lakshmikantham and A.S. Vatsala, Basic theory of fractional differential equations, Non-linear Anal, Vol. 69, 2677-2682, (2008).[nn]Alexander Quaas and Aliang Xia, Existence results of positive solutions for nonlinearcooperative elliptic systems involving fractional Laplacian, Communications in ContemporaryMathematicsVol, Vol. 20, No. 3, (2018).[d] M. M. Meerschaert and A. Sikorskii, Stochastic Models for Fractional Calculus, Walter deGruyter GmbH & Co. KG., Berlin/Boston, (2012).[introo] I. Podlubny, Fractional Differential Equation, Academic Press, San Diego, (1999).[c]Wei Qi, LIN Zhao and Xingjie Yan, Multiple Solutions of Nonlinear Fractional EllipticEquation via Morse Theory, Electronic Journal of Differential Equations. Vol. 2017, No.220, 1-12.[k]Tien-Tsan Shiel and Daniel E. Spector, On a new class of fractional partial differentialequations, (pripent).

Spectral element methods a priori and a posteriori errorestimates for a penalized unilateral obstacle problem

Bochra DjeridiUniversity of Badji-Mokhtar, Annaba

The purpose of this paper is the determination of the numerical solution of a classicalunilateral elliptic obstacle problem. The numerical thecnique combines Moreau-Yoshidapenalty, spectral nite element and Newton method.The penalized method transforms theobstacle problem into a family of semilinear equations.

Body Math The discretisation, uses a non-overlapping spectral nite element methodwith Legendre-Gauss-Lobatto nodal basis using a conforming mesh. The strategy isbased on approximating the solution using a spectral nite element method. Then thediscrete system obtained will be solved by an iterative Newton iterative method.

In addition, by coupling the penalty and the discretization parameters, we prove apriori and a posteriori error estimates where reliability and e ciency of the estimators areshown for Legendre spectral nite element method.

Body Math Moreover, the numerical results given by the Newton iterative method areused to corroborate our error estimates and the e ciency of the algorithms. Key words:

Obstacle problem, penalty approximation, spectral method, finite element method, a pri-ori error estimate, a posteriori error estimate.

2010 Mathematics Subject Classification: 49J20, 65M60, 35B45, 35R35.Let σ : H1

0 × H10 → R is a coercive bilinear form such that

∃ c > 0, for all y in H10 (Ω) , σ (y, y) > c ‖y‖2

H10 (Ω) ,

203

and it is continuous in H10 (Ω)× H1

0 (Ω) , such that

∃ C > 0, for all (y, v) in H10 (Ω)× H1

0 (Ω) , σ (y, v) ≤ C ‖y‖H10 (Ω) ‖v‖H1

0 (Ω) ,

moreover, for all ϕ in H10 (Ω) , we define

K (ϕ) =

y ∈ H10 (Ω) | | | |ϕ ≤ y a.e. in Ω(1)

And we consider the problem : find y in K (ϕ) such that

σ (y, v− y) ≥ ( f , v− y) for all v in K (ϕ) . (P0)

the problem (P0) has a unique solution y dans H10 (Ω) [2].

We consider the technique used in [3] to transform the variational inequality P0 intoan elliptic equation.

For this, we define

βδ (r) =1δ

0 if r ≥ 0,

r if r < 0,(2)

where β (·) be a maximal monotone graph in R×R.We note that β ∈ C1 (R) with

β′δ (r) =

1δ

0 if r ≥ 0,

1 if r < 0(3)

Where, we obtain the following nonlinear elliptic equation for the problem of theunilateral obstacle:

−∆y + βδ (y− ϕ) = f dans Ω,

y = 0 sur ∂Ω.(4)

Thus, δ > 0 is a constant, ϕ is the obstacle where y ≥ ϕ, and it is known that theproblem (4) has a unique solution.

Body Math

References[1] C.Bernardi, Y.Maday et F.Rapetti. Discretisation variationnelles de problemes auxlimites elliptiques, Springer-Verlag,Berlin Heidelberg 2004.[2] D.Kinderlehrer and G.Stampacchia. An introduction to variational inequalities andtheir applications, SIAMS, Philadelphia, 2000.[3] V.Barbu.Analysis and Control of non linear infinite Dimoentional Systemd, Math.Sci.Engrg. 190,Academic Press,San Siego,1993[4] P.G.Ciarlet. The Finite Element Method for Elliptic Problems, North-Holland1978.

204

Solvability for fractional Sturm-Liouville boundary valueproblems with p-Laplacian operator in Banach spaces

Choukri DerbaziUniversity of Ghardaia

In this paper, we study the existence of solutions for fractional Sturm-Liouville bound-ary value problems with p-Laplacian operator in Banach spaces. The main results areproved by applying Monch’s fixed point theorem combined with the technique of mea-sures noncompactness. In addition, an example is given to demonstrate the applicationof our results.Key words: Fractional differential equations, Sturm-Liouville boundary value problem,

measure of noncompactness, Monch’s fixed point theorem,p-Laplacian operator, Banachspaces.

2010 Mathematics Subject Classification: 26A33, 34B15, 34A08, 34G20.

IntroductionFractional calculus generalizes the integer order integration and differentiation conceptsto an arbitrary(real or complex) order. Fractional calculus is the most well known andvaluable branch of mathematics which gives a good framework for biological and phys-ical phenomena, mathematical modeling of engineering etc. To get a couple of develop-ments about the theory of fractional differential equations, one can allude to the mono-graphs of Kilbas et al. [b0], Lakshmikanthem et al. [Lakshmikantham], Miller and Ross.[b1], Pudlubny [b2], Tarasov [Tarasov] and the references therein.

By the use of techniques of nonlinear analysis, many authors have studies the exis-tence and uniqueness of solutions of nonlinear fractional differential equations with avariety boundary conditions as special cases because they can accurately describe the ac-tual phenomena. They include two-point, three-point, multi-point and nonlocal bound-ary value problems with integral boundary conditions as special cases, see [b6,Shoaib]and references therein.

In recent years, fractional diferential equations involving the nonlinear p-Laplacianoperator have achieved great deal of interest and attention of several researchers. Forsome developments on the existence results of the fractional differential equation withthe p-Laplacian operator we can refer to [Bai,X.Liu,Lu,Luo,mah,T. Shen,Jing-jing, Tan,Xue,Yang,Zhai0].

At the present day, there are many results on the existence of solutions for fractionalorder differential equations, but here, we focus on which that use techniques of measuresof noncompactness and Darbo’s fixed point theorem. Such measures have proved to bean efficient tool in the study for the existence of fixed points, and hence, in the study ofsome classes of functional Riemann-Liouville or Caputo fractional differential equationsin Banach spaces see, for instance, [Abas,DS, ben,ben1,hamza,Jing-jing,Tan] and the re-cent book [Banas3], where several applications of the measure of noncompactness can befound.

Inspired by the above findings, this paper studies the BVP subjected to Sturm-Liouvilleboundary conditions for fractional differential equations with p-Laplacian operator:

cDβ0+

(φp

[cDα

0+(u(t)])

+ f (t, u(t)) = θ, t ∈ J := [0, 1], (340)

205

ξx(0)− ηu′(0) = θ, γu(1) + δu′(1) = θ, cDα0+u(0) = θ, (341)

where 1 < α ≤ 2, 0 < β ≤ 1, cDα0+ , cDβ

0+ are the standard Caputo fractional derivatives,φp(u) is a p-Laplacian operator, i.e., φp(s) = |s|p−2s for p > 1, φ−1

p = φq where 1p + 1

q =

1, f : [0, 1]× E −→ E is given function satisfying some assumptions that will be specifiedlater, E is a Banach space where θ is the zero element of E and ξ, η, γ, δ are constants withsatisfying ρ = ξγ + ξδ + ηγ 6= 0.

The paper is organized as follows. The second section provides the definitions andpreliminary results that we will need to prove our main results. Existence results are pre-sented in Section 3. An example illustrating the obtained results is presented in Section4, we conclude our exposition with some final remarks and observations.

References[Abas] S. Abbas, M. Benchohra and J. Henderson Weak Solutions for Implicit FractionalDifferential Equations of Hadamard Type. Advances in Dynamical Systems and ApplicationsVolume 11, Number 1, pp. 1-13 (2016).[DS] Ravi P. Agarwal, M. Benchohra and D. Seba On the Application of Measure of non-compactness to the existence of solutions for fractional differential equations Results. Math. 55(2009), 221-230.[Ahmadb] B. Ahmad, S.K. Ntouyas, A note on fractional differential equations with fractionalseparated boundary conditions, Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID818703, 11 pages.[Bai] C. Bai Existence and uniqueness of solutions for fractional boundary value problems withp-Laplacian operator. Advances in Difference Equations 2018:4.[bnas] J. Banas and K. Goebel, Measures of Noncompactness in Banach Spaces. Marcel Dekker,New York, 1980.[Banas3] J. Banas, Mohamed Jleli, Mohammad Mursaleen. Bessem Samet, Calogero,Vetro (Editors): Advances in Nonlinear Analysis via the Concept of Measure of Noncompact-ness. Springer, singpagor 2017.[ben] M. Benchohra, J. Henderson and D. Seba, Measure of noncompactness and fractionaldifferential equations in Banach spaces. Commun. Appl. Anal. 12 (4) (2008), 419-428.[b6] M. Benchohra, S. Hamani, S.K. Ntouyas , Boundary value problems for differential equa-tions with fractional order and nonlocal conditions, Nonlinear Anal.71, 2391-2396 (2009).[ben1] M. Benchohra and Fatima-Zohra Mostefai Weak solutions for nonlinear fractional dif-ferential equationswith integral boundary conditions in Banach spaces. Opuscula MathematicaVol. 32. No. 1. 2012.[b0 A. A. Kilbas,H. M. Srivastava, and J. J. Trujillo, Theory and Applications of FractionalDifferential Equations, vol. 204 of North-Holland Mathematics Sudies Elsevier ScienceB.V. Amsterdam the Netherlands, 2006.[Lakshmikantham] V. Lakshmikantham, and J. Vasundhara Devi, Theory of fractional dif-ferential equations in a Banach space. Eur. J. Pure Appl. Math. 1 (2008), 38-45.[X.Liu] X. Liu, M. Jia, X. Xiang, On the solvability of a fractional differential equation model in-volving the p-Laplacian operator. Computers and Mathematics with Applications 64 (2012)3267-3275.[Lu] H. Lu, Z. Han, S. Sun, Multiplicity of positive solutions for Sturm-Liouville boundaryvalue problems of fractional differential equations with p-Laplacian. Boundary Value Problems,2014:26[Luo] Z. Luo and J. Liang, Existence of solutions for a multi-point boundary value problem witha p(r)-Laplacian. Advances in Difference Equations 2018:453[mah] N. I. Mahmudov and S. Unul, Existence of solutions of fractional boundary value prob-lems with p-Laplacian operator.Boundary Value Problems (2015) .

206

[b1] K.S. Miller, B. Ross: An Introduction to Fractional Calculus and Fractional Differen-tial Equations, wiley, New YorK, 1993.[DO] H. Monch, Boundary value problems for nonlinear ordinary differential equations of secondorder in Banach spaces. Nonlinear Anal. 4 (1980), 985-999.[b2] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego (1993).[hamza] H. Rebaia, Djamila Sebab Weak Solutions for Nonlinear Fractional Differential Equa-tion with Fractional Separated Boundary Conditions in Banach Spaces.Filomat 32:3 (2018),1117-1125.[H.A.H] H.A.H. Salem, A.M.A. El-Sayed, O.L. Moustafa A note on the fractional calculus inBanach spaces. Studia Sci. Math. Hungar. 42 (2005) 115-130.[b3] G. Samko, A.A Kilbas and O.I Marichev, Fractional integral and derivative; theory AndApplications. Gordon and Breach, Yverdon.(1993).[T. Shen] T. Shen,W. Liu, and X. Shen, Existence and uniqueness of solutions for several BVPsof fractional diferential equations with p-Laplacian operator. Mediterranean Journal of Mathe-matics, vol. 13, no. 6, pp. 4623-4637, 2016.[Shoaib] M. Shoaib, K. Shah, R. Ali Khan, Existence and uniqueness of solutions for coupledsystem of fractional differential equation by means of topological degree method.Journal Nonlin-ear Analysis and Application 2018 No.2 (2018) 124-135[Jing-jing] J. Tan, C. Cheng, Existence of Solutions of Boundary Value Problems for FractionalDifferential Equations with p-Laplacian Operator in Banach Spaces, Numerical FunctionalAnalysis and Optimization 2017.1293091.[Tan] J. Tan and M. Li, Solutions of fractional differential equations with p-Laplacian operatorin Banach spaces. Boundary Value Problems (2018) 2018:15.[Tarasov] V. E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamicsof Particles, Fields and Media. Springer, Heidelberg & Higher Education Press, Beijing,2010.[Xue] T. Xue, W. Liu and T. Shen Existence of solutions for fractional Sturm-Liouville boundaryvalue problems with p(t)-Laplacian operator. Boundary Value Problems (2017) 2017:169.[Yang] C. Yang, J. Yan, Positive solutions for third-order Sturm-Liouville boundary value prob-lems with p-Laplacian. Comput. Math. Appl. 59(6), 2059-2066 (2010).[Zhai0] C. Zhai, C. Guo, Positive solutions for third-order Sturm-Liouville boundary-valueproblems with p-Laplacian. Electron. J. Differ. Equ. 2009(154), 1-9 (2009).

EXISTENCE, UNIQUENESS, COMPACTNESS OF THESOLUTION SET, AND DEPENDENCE ON A PARAMETER

FOR AN IMPULSIVE PERIODIC BOUNDARY VALUEPROBLEMKhelifa Daoudi


The authors consider a second order impulsive differential equation with a parameterand periodic boundary conditions and prove results on the existence, uniqueness, com-pactness of the solution set, and dependence of the solutions on the parameter. Variousfixed point techniques are used.Key words: existence, uniqueness, impulsive differential equations, periodic boundary

value problem, compactness of the solution set, dependence on a parameter.

2010 Mathematics Subject Classification: 34A37, 34B15, 34B37, 34K45, 39A12.

207

IntroductionIn this paper, we consider the impulsive periodic boundary value problem with a param-eter

y′′ − ρ2y = − f (t, y, λ), t ∈ J := [0, 2π], t 6= tk, k = 1, ..., m, (342)

y(t+k )− y(t−k ) = Ik(y(t−k )), t = 1, ..., m, (343)

y′(t+k )− y′(t−k ) = Ik(y(t−k )), t = 1, ..., m, (344)

y(0) = y(2π), y′(0) = y′(2π). (345)

where ρ ∈∗, λ is a real parameter, f : J ×× → is a given function,Ik, Ik ∈ C(, ), tk ∈ [0, 2π], 0 = t0 < t1 < ... < tm < tm+1 = 2π,y(t+k ) =h→0+ y(tk + h) and y(t−k ) =h→0+ y(tk − h) represent the right and left limits ofy(t) at t = tk.

One aim of this paper is to give an impulsive version of the results obtained on theproblem (342)-(345) considered by Graef et al. [Graef]. Our approach here is based onfixed point theory.

Main results

Uniqueness of solutionsTheorem 94 Assume that the following hypotheses hold:

(H1) There exists a constant d ≥ 0 such that

| f (t, y, λ)− f (t, y, λ)| ≤ d|y− y|,

for each t ∈ J, ∀ y, y, λ ∈.

(H2) There exist constants ck ≥ 0 such that

|Ik(y)− Ik(y)| ≤ ck|y− y|, for each k = 1, ..., m, ∀ y, y ∈.

(H3) There exist constants ck ≥ 0 such that

| Ik(y)− Ik(y)| ≤ ck|y− y|, for each k = 1, ..., m, ∀ y, y ∈.

Existence of solutionsTheorem 95 Suppose that:

(H4) f : J ×× → is an Caratheodory function and Ik, Ik ∈ C(, ).

(H5) There exists P ∈ L1(J,+ ) and a constants α ∈ [0, 1) such that

| f (t, y, λ)| ≤ P(t)|y|α, for each y, λ ∈ and t ∈ J.

(H6) There exist constants dk > 0 and αk ∈ [0, 1) such that

|Ik(y)| ≤ dk|y|αk , for each y ∈, k = 1, ..., m.

(H7) There exist constants dk > 0 and αk ∈ [0, 1) such that

| Ik(y)| ≤ dk|y|αk , for each y ∈, k = 1, ..., m.

208

Positive solutionsTheorem 96 Assume (H1)− (H3) and the following conditions are satisfied:

(H8) g(t)h(y) ≥ 0 for each y ∈, and t ∈ J.

(H9) Ik(y) ≤ 0, for each y ∈, t ≥ tk, and k = 1, ..., m.

(H10) L(t, tk) Ik(y) ≤ 0, for each y ∈, t ∈ J, and k = 1, ..., m.

Then the problem (342)-(345) has a unique positive solution on J.

Theorem 97 Assume (H4)-(H10) and the following conditions are satisfied:

(H11) there exist R > 0 and r > 0, with r < R such that(i) λ(t,s)∈J×J |G(t, s)|‖g‖∞h∗(r) +m

k=1 t∈J |G(t, tk)|dkrαk +mk=1 t∈J |L(t, tk)|dkrαk < r;

whereh∗(r) =u∈(0,r] |h(u)|;

(ii)t∈[0,2π]

(λ2π

0 G(t, s)g(s)h(w(s))ds−mk=1

[G(t, tk)Ik(w(tk)) + L(t, tk) Ik(w(tk))

)> R,

if r < w.

Then the problem (342)-(345) has at least has at least two positive solutions y1, y2 such that‖y1‖ < r, r < ‖y2‖ ≤ R.

ConclusionIn this work, we have proved the existence, uniqueness, compactness of the solution setthe dependence of the solutions on the parameter. Various fixed point theorems are used.

References[AiEz] E. Ait Dads and K. Ezzinbi, Boundedness and almost periodicity for some state-dependent delay differential equations, Electron. J. Differential Equations 2002 No. 67(2002), pp. 1-13.[AgMeOr] R.P. Agarwal, M. Meehan, D. O’Regan, Fixed Point Theory and Applications,Cambridge Tracts in Mathematics, 141. Cambridge University Press, Cambridge, 2001.[K.De] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985.[Graef] J.R. Graef, L. Kong, H. Wang, Existence, multiplicity, and dependence on a param-eter for a periodic boundary value problem, J. Differential Equations 245 (2008) 1185–1197.

A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.[Gufe] D.J. Guo, The fixed point and eigenvalue of a class of concave and convex operator,Chinese Sci. Bull. 15 (1985) 1132-1135 (in Chinese).[Gyori] I. Gyori, On approximation of the solutions of delay differential equations byusing piecewise constant arguments, Int. J. Math. Math. Sci. 18 (1) (1991), 111-126.[Har2] F. Hartung, Linearized stability in periodic functional differential equations withstate-dependent delays. J. Comput. Appl. Math. 174 (2) (2005), 201–211.[Har3] F. Hartung, On differentiability of solutions with respect to parameters in neutraldifferential equations with state-dependent delays. Annali di Matematica. (2013) 192:17-47.[Heik] S. Heikkila and V. Lakshmikantham, Monotone Iterative Techniques for Discontin-uous Nonlinear Differential Equations, Monographs and Textbooks in Pure and AppliedMathematics, vol. 181, Marcel Dekker, New York, 1994.

209

[KoMy] V. Kolmanovskii, and A. Myshkis, Introduction to the Theory and Applications ofFunctional-Differential Equations, Kluwer Academic Publishers, Dordrecht, 1999.[H.Mo] H. Monch, Boundary value problems for nonlinear ordinary differential equa-tions of second order in Banach spaces, Nonlinear Anal. 4 (5) (1980), 985-999.[ReWu] A.V. Rezounenko and J. Wu, A non-local PDE model for population dynamicswith state-selective delay: Local theory and global attractors, J. Comput. Appl. Math. 190(1-2) (2006), 99-113.[Se] N. P. Semenchuk, On one class of differential equations of noninteger order, Differ-ents. Uravn., 10 (1982), 1831-1833.[Stwrt] D. E. Stewart, Existence of solutions to rigid body dynamics and the Painleveparadoxes, Comptes Rendus de l’Academie des Sciences. Serie I. Mathematique 325 (1997),no. 6, 689-693.[Wu] J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996.[Yujun] D. Yujun, Periodic boundary value problems for functional-differential equationswith impulses, J. Math. Anal. Appl. 210 (1997), 170-181.

Estimation of the jump-diffusion parameter models using realdata

Frihi Zahrate El Oula

University of Badji-Mokhtar, Annaba

The study and understanding of option pricing is an increasingly important issue infinancial markets. Nowadays, mathematicians seek to develop new, more accurate mod-els for representing the random dynamics of the underlying asset. That’s why in thiswork in order to see which of the two jumps diffusion models (Merton [3] and Kou [2])is the best fit for the S&P 500. We will analyze the index real data distribution. After thatwe use the maximum likelihood estimation to determined the parameters of the previousmodels. Finally, we use Matlab to compare the densities of the S&P 500 log-returns to thedensities of the simulated data from both models.Key words: Financial mathematics, Jump diffusion model, Parameters estimation, Skew-

ness and Kurtosis, Data analysis.


IntroductionI The probability density of the double exponential jump (Kou) model is given by

f∆t(x) =1− λ∆tσ√

∆tϕ

(x− µ∆tσ√

∆t

)+ λ∆t

p.η1e(σ

2η21 ∆t)/2e−(x−µ∆t)η1 Φ

(x− µ∆t− σ2η1∆t

σ√

∆t

)+q.η2e(σ

2η22 ∆t)/2e(x−µ∆t)η2 Φ

(− x− µ∆t− σ2η2∆t

σ√

∆t

)(346)

210

Where the constants µ and σ > 0 are drift and volatility of the diffusion part, λ is the rateof the jumps in the time unit, p, q ≥ 0, p + q = 1 represent the probabilities of upwardsand downward jumps, ϕ(.) is the standard normal density function.

I The probability density of the Merton jump diffusion model is given by

f∆t(x) =∞

∑n=0

e−λ∆t(λ∆t)n

n!√

2π(∆tσ2 + nσ2j )

exp

−(x− ∆t(µ− σ2

2− λk)− nµj)

2

2(∆tσ2 + nσ2j )

(347)

Analyze the S&P 500 historical data

Testing the normality of the historical index from the period 18th of May 2004 -13th ofNovember 2018 using Kolmogorov-Smirnov test and histogram plotting.

Maximum-likelihood estimation of jump-diffusion mod-els parameters

The likelihood function is given by the following product:

L(X|θ) =n

∏i=1

f (Xi∆, θ) (348)

The log likelihood function is given by :

ln L(X|θ) =n

∑i=1

ln f (Xi∆, θ) (349)

Using the same database in section 2, the MATLAB ’fmincon’, and parameter estimatesfrom the cumulant matching method as initial parameter, negative of the log-likelihoodspecified in (4) can be minimized which is equivalent to maximizing the log-likelihoodfunction and the parameters that maximize the log-likelihood function also maximizethe likelihood function and are the MLE estimates for Merton model. In this work MAT-LAB built-in function ’mle’ is used to estimate the parameters for both Merton and Koumodels.

Comparison between the theoretical and the empiricaldistribution of the S&P 500 log returns

In order to assess the goodness of fit of the Merton and Kou distributions to the log-returns S&P 500 historical data we replace the parameters estimated and simulate thetwo theoretical distribution using monte carlo algorithm firstly, for the sample paths ofKou model :

X = (µ− 0.5σ2) + σ√

∆Z + Jump (350)

211

"""kou".jpg

A sample path for simulated Kou’s Model using estimated parameters

And next for the Merton model :

X = (µ− 0.5σ2 − λk) + σ∆Z + Jump (351)

"""merton".jpg

A sample path for simulated Merton Model using estimated parameters

Finally, we plot different figures to compare between the density of the simulateddata and the empirical one.

ConclusionIn this study, we have concluded that the double exponential jumps distribution explainsbetter the higher peak, asymmetric and heavier tails of the index than the normal distri-bution (Merton model).

References[T1] Black, F., and Scholes, M. The pricing of options and corporate liabilities. The Journal ofPolitical Economy, 81, No. 3, (1973), 637-654, 10.1086/260062.

212

[L1] Kou, S, G. A Jump-Diffusion Model for Option Pricing. Management Science Vol. 48,No. 8, pp. 1086?1101, (8 August 2002).[A1] Merton, R.C. Option pricing when underlying stock returns are discontinuous. Journalof Financial Economics, 3, 125-144 (1976).

[oneside,11pt]article rama11 amsthm, amscd, amsfonts, amssymb, graphicx,tikz, color,environ latexsym

Relative error regression

under random censorship data Omar FETITAH12

1 Laboratory of Statistics and Stocastic Processes University ofDjillali Liabes,BP 89,Sidi Bel Abbes 2200, [email protected]

hyperref In this work, we investigate the asymptotic properties of a nonparametricestimator of the relative error regression, in the case of a scalar censored response, weuse the mean squared relative error as a loss function to construct a nonparametric esti-mator of the regression function of these censored data. We establish the strong almostcomplete convergence rate of these estimators. Key words: nonparametric estimation,

relative error regression, censored data, almost complete convergence.

2010 Mathematics Subject Classification: 62G08.

Introduction

Let (Yi)1≤i≤n be a sequence of independent and identically distributed (iid) random vari-ables (rv), with a common unknown absolutely continuous distribution function (df) H.

In survival analysis, the random variables can be lifetimes of patients under study.In reality it is not possible to observe the survival time of all patients, and often some ofthem are still alive at the end of the study, or withdraw, or die from other causes thanthose addressed by the study. In those cases, we observe another rv C called censoring.Then the observable rv is the minimum of the survival time and the censoring time.

Assuming that (Ci)1≤i≤n is a sequence of iid censoring random variables with a com-mon unknown continuous distribution function G.Then in the right censorship model, we only observe the n pairs (Ti, δi) with Ti = Yi ∧Ci and δi = IYi≤Ci, 1 ≤ i ≤ n, where IA denotes the indicator function of the set A.To follow the convention in biomedical studies and as indicated before, we assume that(Ci)1≤i≤n and (Yi, Xi)1≤i≤n are independent.

2Underline the speaker

213

mailto:

The Model And The EstimatesThe ordinary way to study the relationship between X and Y is to suppose that Y =r(X) + ε, where ε is a random error variable independent to X and r is the regressionfunction.[?] propose Nonparametric relative regression to estimate the function r by an alternativeloss function, then we obtain

r(x) =E[Y−1|X = x]E[Y−2|X = x]

=:g1(x)g2(x)

.

In censorship model only the (Xi, Ti, δi)1≤i≤n are observed, we define r(x) as an estimateof r(x) by

r(x) =∑n

i=1 δiT−1i G(Ti)

−1K( x−Xih )

∑ni=1 δiT−2

i G(Ti)−1K( x−Xi

h )=:

g1(x)g2(x)

, (352)

In practice G is unknown, we use the Kaplan and Meier (1958) estimator of G givenby:

Gn(t) =

∏n

i=1

(1− 1−δ(i)

n−i+1

)IT(i)≤tif t ≤ T(n)

0 otherwise,(353)

where T(1) ≤ T(2) ≤ ... ≤ T(n) are the order statistics of (Ti)1≤i≤n and δ(i) is the concomi-tant of T(i).

Therefore, the estimator of r(x) is given by:

rn(x) =∑n

i=1 δiT−1i Gn(Ti)

−1K( x−Xih )

∑ni=1 δiT−2

i Gn(Ti)−1K( x−Xi

h )=:

g1,n(x)g2,n(x)

, (354)

AssumptionsLet C be a compact set of R. And assume that, (Ci)i≥1 are independent and let’s considerthe following hypotheses:

(A1) The bandwidth hn satisfies: limn→+∞

hn = 0 , limn→+∞

nhn

log n= +∞.

(A2) The kernel K is bounded, symmetric and has compact support. It is also Holderianof order γ > 0.Furthermore ∫

Rt|K(t)|dt < +∞ and

∫R

t2K(t)dt < +∞ hold .

(A3) The function g1(x) is twice differentiable and

supx∈C

∣∣g′′1 (x)∣∣ < +∞.

(A4) The marginal density f of X and g2(x) are twice differentiable and satisfies a Lips-chitz condition.Furthermore

g2(x) > Γ for all x ∈ C and Γ > 0.

214

(A5) The functions r2(x) and r1(x) defined by r2(x) =∫

Ry−2l fX,Y(x, y)dy and r1(x) =∫

Ry−l fX,Y(x, y)dy are twice differentiable and

supx∈C

∣∣r′′2 (x)∣∣ < +∞.

ResultUnder (A1)-(A5), we have as n goes to infinity

supx∈C|rn(x)− r(x)| = Oa.s.

(h2

n +

√log nnhn

).

SimulationIn this part we will simulate 3 functions in the case of censored data, where the percentageof censure of each of the following models is 30%.

Linear modelYi = −5Xi + 2.5 + εi

Figure 9: Linear model with n = 100, 500 and 1000.

Nonlinear model (Exponential model)Yi = exp (−4Xi − 2) + εi

Figure 10: Exponential model with n = 100, 500 and 1000.

ConclusionThe results given in this work when explanatory variable and response variable are realdeal with relative regression. This can open several perspectives. For example, one canalso consider extending to the case of a vector or functional explanatory variable. Thechoice of the smoothing parameter is still an open problem when the sample has a func-tional value. functional nonparametric regression models are of great interest in func-tional statistics.

References

215

[Demongeot2016] J. DEMONGEOT, A. HAMIE, A. LAKSACI, M. RACHDI, Relative-ErrorPrediction in Nonparametric Functional Statistics: Theory and Practice;Statistica Neerlandica,147 (2016), 261-268.

[Guessoum2008] Z. GUESSOUM, E. OULD SAID, On nonparametric estimation of the regres-sion function under random censorship model; Statistics and Decisions, 26 (2008), 159-177.

[Ferraty2010] F. FERRATY, A. LAKSACI, A. TADJ, P. VIEU ,Rate of uniform consistency fornonparametric estimates with functional variables; J. Statist. Plann. and Inf, 140 (2010), 335-352.

Some p-summing operators and their conjugatesFERRADI Athmane

University of M’sila

The space of mid p summing operators was introduced by Karn and Sinha in [ks14]and was later developed by Botelho, Compos and Santos in 2017 in their article [mid].In this conference we give a representation of the dual and bidual of the space Πp,midof mid p-summing operators. We give a new factorization theorem for the well-knownconcept of strongly p-summing operators and we terminate with some relationships be-tween these classes.Key words: p-summing operators, strongly p-summing operators, Cohen p-nuclear op-

erators, Domination theorem, factorization theorem.

2010 Mathematics Subject Classification: 46A20, 46B45, 47B10, 47L20.

IntroductionLet X be a Banach space and X∗ its dual. BX denotes the closed unit ball of X. For1 < p < ∞, let p∗ be its conjugate, i.e., 1/p + 1/p∗ = 1.Let us define the new sequences spaces. We write the space of mid p-summable sequences(xi)i in X with the norm

‖(xi)i‖p,mid : = sup‖(x∗n)n‖p,w=1

(∑i

∑n|〈x∗n, xi〉|p)

1p .

The concept of absolutely summing linear operators was mainly introduced by Crothendieckin the 1950’s and was generalized by Pietsch in the 1967’s [p67]. A linear operatorT : X −→ Y is absolutely p−summing if sends weakly p -summable sequences to ab-solutely p -summable sequences. The concept of strongly p-summing linear operatorswas introduced by J. S. Cohen in 1973 [c73] as a characterization of the conjugates of p∗-summing linear operators. An operator T between two Banach spaces X, Y is stronglyp-summing if there is a positive constant C such that for any (xi)i in lp(X) and (y∗i )i inlwp (Y∗) we have

‖(〈T (xi) , y∗i 〉)i‖1 ≤ C ‖(xi)i‖p ‖(y∗i )i‖p∗,ω . (355)

216

We denote by Dp(X, Y) the class of all strongly p-summing operators from X into Y anddp(T) the smallest constant C such that the inequality (355) .The notion of Cohen p-nuclear operators was initiated by Cohen in 1973 [c73]. An opera-tor T between two Banach spaces X, Y is Cohen p-nuclear if there is a positive constantCsuch that for any (xi)i in lw

p (X) and (y∗i )i in lwp (Y∗) we have∣∣∣∣∑

i〈T (xi) , y∗i 〉

∣∣∣∣ ≤ C ‖(xi)i‖p,ω ‖(y∗i )i‖p∗,ω . (356)

We denote by Np(X, Y) the class of all Cohen p-nuclear operators from X into Y andnp(T) the smallest constant C such that the inequality.

Main resultsIn this section we give a representation of the dual and bidual of the space Πp,mid of midp-summing operators. We give a new factorization theorem for the well-known conceptof strongly p-summing operators, and we study some properties concerning this classes.A linear operator T : X −→ Y is absolutely mid p−summing if sends mid p-summablesequences to absolutely p-summable sequences.

Definition 27 An operator T : X → Y is strongly mid p-summing, if there is a positive constantC such that for any (xi)i∈ lp(X), (y∗i )i ∈ lmid

p (Y∗) we have

∑i|〈T (xi) , y∗i 〉| ≤ C ‖(xi)i‖p ‖(xi)i‖p∗,mid . (357)

We denote by Dp,mid(X, Y) the class of all strongly mid p-summing operators from X into Y anddp,mid(T) the smallest constant C such that the inequality.

Theorem 98 (Dp,mid(X, Y), dp,mid(.)) is Banach operator ideal.

Proposition 21 Consider 1 < p ≤ q < ∞.and T linear operator If T ∈ Dq,mid(X, Y) thenT ∈ Dp,mid(X, Y) and dp,mid(T) ≤ dq,mid(T).

Definition 28 (mid) Let X and Y be a Banach spaces. An operator T : X → Y is weakly midp-summing (1 < p < ∞), if there is a positive constant C such that for any (xi)i ∈ lp,ω (X)

‖(Txi)i‖p,mid ≤ C ‖(xi)i‖p,ω . (358)

We denote by Wp,mid(X, Y) the class of all weakly mid p-summing operators from X into Yand wp,mid(T) the smallest constant C such that the inequality (358) . we have Wp(X, Y) ⊂Wp,mid(X, Y) and wp,mid(T) ≤ wp(T).

Proposition 22 Let X and Y be a Banach spaces We have(1) Dp(X, Y) ⊆ Dp,mid(X, Y) and dp,mid(T) ≤ dp(T).(2) ∏p(X, Y) ⊆ ∏p,mid(X, Y) and πp,mid(T) ≤ πp(T).(3) Np(X, Y) ⊆ Wp,mid(X, Y)and np,mid(T) ≤ wp(T).

Theorem 99 Let X and Y be a Banach spaces. An operator T : X → Y is weakly mid p-summing(1 < p < ∞) if and only if T∗ is weakly mid p-summing i.e[

Wp,mid(X, Y)]∗

=Wp,mid(Y∗, X∗)

217

Theorem 100 For 1 < p < ∞, the following statements are equivalent1) The operator T is absolutely mid p-summing2) The operator T∗ is strongly mid p-summing3) The operator T∗∗ is absolutely mid p-summing

Theorem 101 For 1 < p < ∞, the following statements are equivalent1) The operator T is strongly mid p-summing2) The operator T∗ is absolutely mid p-summing3) The operator T∗∗ is strongly mid p-summing

Theorem 102 Every strongly p-summing linear operator factors throgh weakly mid p-summingand strongly mid p-summing linear operators i.e

Dp =Wp∗,mid Dp,mid

Theorem 103 Let (1 < p < ∞), X and Y be a Banach spaces. the following statements areequivalent:(1) Dp,mid(X, Y) or ∏p,mid(X, Y) is reflexive(2) X and Y are reflexive(3) Dp(X, Y) or ∏p(X, Y) is reflexive

218

ConclusionLet X and Y be a Banach spaces, Let (1 < p < ∞). We have

Πp = Πp,mid Wp,mid

Dp =Wp∗,mid Dp,mid

[Wp,mid(X, Y)

]∗=Wp,mid(Y∗, X∗)[

Wp,mid(X, Y)]∗∗

=Wp,mid(X∗∗, Y∗∗)

[Πp,mid(X, Y)

]∗= Dp∗,mid(Y∗, X∗)[

Πp,mid(X, Y)]∗∗

= Πp,mid(X∗∗, Y∗∗)

[Dp,mid(X, Y)

]∗= Πp∗,mid(Y∗, X∗)[

Dp,mid(X, Y)]∗∗

= Dp,mid(X∗∗, Y∗∗)

References[c73] J. S. Cohen, Absolutely p-summing, p-nuclear operators and their conjugates, Math.Ann., 1973, 201, 177-200.[p67] A. Pietsch, Absolut p-summierende Abbildungen in normierten Raumen, StudiaMath., 1967, 28, 333-353.[mid] G. Botelho, J.R. Compos, and J. Santos, Operator ideals related to absolutely sum-ming and Cohen strongly summing operatorsJ Math., 2017, 287, 1-17 .[ks14] A. Karn and D. Sinha, An operator summability of sequences in Banach spaces.Glasg.Math. J., 2014, 2, 427-437.

The perfect mixing and the logistic equationELBETCH Bilel

University of Moulay Tahar, Saida

On considere le probleme de n sites avec un modele logistique sur chaque site, etnous supposons qu’ils sont couples par des termes de migrations. Nous allons etudierle comportement du modele quand le terme de migration tend vers l’infini, dans le casou on a trois sites avec dispersion non symetrique, et dans le cas de n sites mais avecdispersion symetrique. Pour cela, nous utilisons la theorie des perturbations singuliereset le theoreme de Tikhonov [12,13,14,15].Mots cles: dynamique de population, champ lente-rapide, variete lente

219

Le modele mathematique

Le probleme de n sites avec une variable d’etat Ni sur chacun des sites peut etre modelisepar un systeme d’equations differentielles autonome et non lineaire de la forme

dN(t)dt

= f (N(t)) + AN(t). (359)

ou

• N(t) = (N1(t), . . . , Nn(t)) est un vecteur de Rn.

• f (N(t)) = ( f1(N1(t)), ..., fn(Nn(t))) le champ de vecteurs representant la dynamiquenon lineaire sur l’ensemble des sites.

• A ∈ Mn(R): la matrice representant les relations de migration entre les sites.

On suppose que la population suit une loi d’accroissement logistique sur chaque site,c’est-a-dire

fi(Ni) = riNi

(1− Ni

Ki

), i = 1, ..., n. (360)

avec:

• Ni: Le nombre de population dans le site i.

• ri: Le taux de croissance du site i.

• Ki: La capacite de site i.

On note par βγij le terme de migration du site i vers le site j. On a donc un systemed’equations differentielles de dimension n qui modelise le probleme sous la forme suiv-ante

Ni = riNi

(1− Ni

Ki

)+ β

n

∑j=1,j 6=i

[γjiNj − γijNi

]:= Ψi(N), i = 1, . . . , n. (361)

Le systeme (361) est de la forme (359) avec une matrice An = (aij)1≤i,j≤n de la forme:

aij =

βγji, i 6= j,

−β ∑nj=1,j 6=i γij, j = i.

La somme des elements d’une colonne vaut zero, car ce qui sort d’un site se distribueentre les (n− 1) autres sites. Pour β = 0, il est connu que le systeme admet (K1, . . . , Kn)comme unique equilibre non trivial et globalement asymptotiquement stable.

Resultats obtenus

Dans cette section nous donnons deux theoremes dans les cas suivants:

220

Cas de trois sites avec dispersion non symetriqueTheorem 104 Soit (N1(t, β), N2(t, β), N3(t, β)) la solution du systeme (361) avec la conditioninitiale (N0

1 , N02 , N0

3 ) qui satisfait N0i ≥ 0, i = 1, 2, 3.

Soit N(t) la solution du systeme

dNdt

=δ1r1 + δ2r2 + δ3r3

δ1 + δ2 + δ3N

1− N

(δ1 + δ2 + δ3)δ1r1 + δ2r2 + δ3r3

δ21α1 + δ2

2α2 + δ23α3

. (362)

ou

δ1 = γ21γ31 +γ21γ32 +γ23γ31, δ2 = γ12γ31 +γ12γ32 +γ13γ32 et δ3 = γ12γ23 +γ13γ21 +γ13γ23.(363)

et avec la condition initiale N(0) = N01 + N0

2 + N03 .

Quand β→ ∞ alors les solutions du systemes (361) sont approchees par les solutions du systeme(362), c’est a dire

Ni(t, β) = N(t) + o(1), i = 1, 2, 3, pour tout t ∈ [0, ∞]. (364)

Cas de n sites avec dispersion symetriqueOn suppose dans l’equation (361) que pour tout i, j ∈ 1, 2, 3 i 6= j les γij sont egaux a 1,c’est a dire le systeme s’ecrit sous la forme

Ni = riNi

(1− Ni

Ki

)+ β

(−(n− 1)Ni +

n

∑j=1,j 6=i

Nj

), i = 1, . . . , n. (365)

Le systeme (365) est de la forme (359) avec une matrice An de la forme

An = β

−(n− 1) 1 . . . 1

1 −(n− 1) 1...

... . . .. . . 1

1 . . . 1 −(n− 1)

.

Theorem 105 Soit (N1(t, β), . . . , Nn(t, β)) la solution du systeme (365) avec la condition ini-tiale (N0

1 , . . . , N0n) qui satisfaite N0

i ≥ 0 pour tout i ∈ 1, . . . , n.Soit N(t) la solution du systeme

dNdt

=

[1n

n

∑i=1

ri

]N

1− N

n ∑ni=1 ri

∑ni=1 αi

. (366)

avec la condition initiale N(0) =1n(

N01 + . . . + N0

n).

Quand β→ ∞ alors

Ni(t, β) = N(t) + o(1), i = 1, . . . , n, pour tout t ∈ [0, ∞]. (367)

221

References[1]Roger Arditi, Claude Lobry, Tewfik Sari, In dispersal always beneficial to carrying capac-ity? New insights from the multi-patch logistic equation, Theoretical population biology 106,45-59, 2015.[2] Roger Arditi, Claude Lobry, Tewfik Sari, Asymmetric dispersal in the multi-patch logisticequation, Theoretical population biology 120, 11-15, 2018.[3] DeAngelis, D.L, Travis, C.C, Post, W.M, Persistence and stability of seed-dispersel speciesin a patchy environment, Theoretical population biology 16, 107-125, 1979.[4] Y. Takeuchi, Cooperative Systems Theory and Global Stability of Diffusion Models, ActaApplicandae Mathematicae 14(1989), 49− 57.[7] H. I. Freedman, Bindhyachal Rai and Paul waltman, Mathematical Models of PopulationInteractions with Dispersal II: Differential Survival in a Change of Habitat, journal of mathe-matical analysis and applications 115, 140-154 (1986).[8]H. I. Freedman and Paul Waltman, Mathematical Models of Population Interactions withDispersal I: Stabilty of two habitats with and without a predator, SIAM J; Appl Math 32,631-648(1977).[9]Freedman H.I. and Takeuchi,Y., Global stability and predator dynamics in a model of preydispersal in a patchy environment, Nonl. Anal. TMA. 13,993- 1002(1989).[10] HOLT,R.D, Population dynamics in two patch environments: some anomalous consequencesof an optimal habitat distribution, Thero. popul. biol. 28,181-2018 (1985).[11] DeAnglais, zhang,Effects of dispersal in a non-uniform environment on population dy-namics and competition: a patch model approach. Discrete Contin. Dyn.Syst. Ser. B 19,3087–3104(2014).[12] Tikhonov, A.N., Systems of differential equations containing small parameters in the deriva-tives. Mat. Sb. (N.S.) 31 (73), 575–586. (in Russian). 1952.[13] Wasow, W.R., Asymptotic Expansions for Ordinary Differential Equations. Robert E.Krieger Publishing Company, Huntington, NY,1976.[14] Lobry, C., Sari, T., Touhami, S., On Tykhonov’s theorem for convergence of solutions ofslow and fast systems. Electron. J. Differential Equations 1998 (19).[15] Nicolas Petit, Pierre Rouchon. Automatique: Dynamique et controle des systemes. Engi-neering school. MINES ParisTech, 2009, pp.236. cel-00439061.[16] Y. Lu and Y. Takeuchi, Global asymptotic behavior in single-species discrete diffusion sys-tems, J. Math. Biol. 32 (1993) 67-77.[17] A. Berman and R. J. Plemmons, Nonnegative Matrices in the Applied Mathematical, Sci-ences (Academic Press, New York, 1979).[18] Hadeler, K.P. and Glas, D. (1983), Quasimonotone systems and convergence to equilibriumin a population genetic model, J. Math. Anal. Apph, 95,297-303.

222

Stability of The Lame System With A Time Delay Condition ofFractional Type

GAOUAR Soumia


We consider, in a bounded and smooth domain, the Lame system with a delayed ve-locity term and mixed Dirichlet-Newmann boundary condition. We prove well-posednessby semigroup theory through the Hille-Yosida theorem. Moreover, we prove the expo-nential stability of the solution exploring the dissipative properties of the linear operatorassociated to the damped model. Key words: Lame system, Fractional feedback, Expo-

nential stability, Semigroup theory.

2010 Mathematics Subject Classification: 35B40, 47D03, 74D05.

Introduction

We consider the initial boundary value problem for the Lame system given by:

(P)utt − µ∆u− (µ + λ)∇(div u) + a1∂

α,ηt u(x, t− τ) + a2ut(x, t) = 0 in Ω× (0,+∞)

u = 0 in Γ× (0,+∞)u(x, 0) = u0(x), ut(x, 0) = u1(x), in Ωut(x, t− τ) = f0(x, t− τ), in Ω× (0, τ)

where µ, λ are Lame constants, u = (u1, u2, ..., un)T. Here Ω is a bounded domainin Rn with a smooth boundary ∂Ω. Moreover, τ > 0 is a time varying delay, a1, a2 arepositive real numbers, and the initial data (u0, u1, f0) belong to a suitable function space.The notation ∂

α,ηt stands for the generalized Caputo’s fractional derivative of order α with

respect to the time variable. It is defined as follows

∂α,ηt w(t) =

1Γ(1− α)

∫ t

0(t− s)−αe−η(t−s) dw

ds(s) ds 0 < α < 1, η ≥ 0.

Our aim in this work is to prove that the stability of our system holds with fractionaltime delay condition and to obtain an exponential decay.

Main results

We give well-posedness results for problem (P) using semigroup theory. Let U = (u, v, φ)T,the problem (P) is equivalent to: Let us denote U = (u, v, φ, z)T, where v = ut. The prob-lem (P) can be rewrite

Ut = AU,U(0) = (u0, u1, φ0, f0)

223

where the operator A is defined by

A

uvφ

=

v

µ∆u + (µ + λ)∇(div u)− ζ∫ +∞−∞ µ(ξ)φ(x, ξ) dξ − a2v

−(ξ2 + η)φ + z(x, 1)µ(ξ)−τ−1zρ(x, ρ)

(368)

with domainD(A) = U ∈ H/AU ∈ H (369)

where the energy spaceH is defined as:

H =(

H10(Ω)

)n×(

L2(Ω))n ×

(L2(Ω× (−∞,+∞)× (0, 1))

)n

We have the following existence and uniqueness result.

Theorem 106 (Existence and uniqueness)

(1) If U0 ∈ D(A), then system (P) has a unique strong solution

U ∈ C0(R+, D(A)) ∩ C1(R+,H).

(2) If U0 ∈ H, then system (P) has a unique weak solution

U ∈ C0(R+,H).

ConclusionOur main result is the following theorem to prove the strong stability of the system

Theorem 107 The C0-semigroup etA is strongly stable in H; i.e, for all U0 ∈ H, the solution of(P) satisfies

limt→∞‖etAU0‖H = 0.

References[T1] A. Benaissa and S. Benazzouz. Well-posedness and asymptotic behavior of Timoshenkobeam system with dynamic boundary dissipative feedback of fractional derivative type. SpringerInternational Publishing AG, 2016; 10.1007.[A1] Mbodje B. Wave energy decay under fractionel derivative controls. IMA Journal ofMathematical Control And information 2006; 23.237-257.

224

On a Markovian multiserver vacation queueing system withwaiting servers and impatient customers

Abdelhak GuendouziTaher Moulay University of Saida

This work deals with an infinite-buffer multiserver Markovian queueing model withbatch arrival, waiting servers, and impatient customers, under single and multiple vaca-tion policies. Using probability generating functions (PGFs), we obtain explicit expres-sions of the steady-state probabilities of the queueing model and derive useful perfor-mance measures.Key words: Multi-server queueing systems. Single/Multiple vacation. Waiting servers.

Impatient customers. Probability generating function.

2010 Mathematics Subject Classification: 60K25, 68M20, 90B22.

IntroductionVacation queueing models with impatient customers have a great advantageous in pro-viding basic framework for efficient design and study of many practical situations includ-ing telephone switchboard, inventory problems with perishable goods, computer/ com-munication network telecommunication, data/voice transmission, manufacturing sys-tem, and several other engineering systems. For more details on the subject, on mayrefer to the research papers of Altman and Yechiali [Altman], Padmavathy et al. [Pad-mavathy], Yue et al. [Yue2], Bouchentouf and Yahiaoui [Bouchentouf2017], Laxmi andRajesh [Laxmi2017], Ammar [Ammarnew] and the references therein.

In this investigation, we consider an MX/M/c queueing system under single andmultiple vacations wherein customers may leave the system due to impatience duringthe absence of the servers. The model considered in this work is based on followingassumptions:− Customers arrive in batches according to a Poisson process with rate λ. The sizes

of successive arriving batches are i.i.d. random variables X1, X2,...distributed with prob-ability mass function P(X = l) = bl ; l = 1, 2, 3, ....− The customers are served on a First-Come First-Served (FCFS) queue discipline.

The service times are assumed to follow exponential distribution with mean 1/µ.−When the busy period is finished the servers wait a random duration of time before

beginning on a vacation. This waiting duration is exponentially distributed with mean1/η.− The queueing model consists of c servers. In synchronous vacation policy, all the

servers leave for a vacation simultaneously, once the system becomes empty and theyalso comeback to the system as one at the same time.In multiple vacation policy (MVP), the servers continue to take vacations until they findthe system nonempty at a vacation completion instant. While, in single vacation policy(SVP), when the vacation ends and servers find the system empty, they remains idle untilthe first arrival occurs. Vacation periods are assumed to be exponentially distributedwith mean 1/φ.− Customers in batches are supposed to enter the queueing system, join the queue,

if the servers are unavailable due to vacation, a batch of customers activates an inde-

225

pendent impatience timer T, with exponentially distributed duration, with mean 1/ξ. IfT expires while the servers are still on vacation, the customer abandons the queue andnever returns.−We assume that the inter-arrival times, batch sizes, server waiting times, vacation

times, service times and impatience times are independent of each other.

Main resultsLet L(t); t ≥ 0 be the number of customers in the system at time t, and S(t) be the stateof servers at time t, where S(t) is defined as follows:

S(t) =

1, when the servers are in busy period at time t;0, when the servers are in vacation period at time t.

Then, let (S(t), L(t)); t ≥ 0 be a two-dimensional continuous Markov process withstate space

Ω = (s, n) : s = 0, 1, n = 0, 1, ....

Let Ps,n = limt→∞

PS(t) = s, L(t) = n, s = 0, 1, n = 0, 1, ..., denote the system steady-

state probabilities.

Theorem 108 If λE(X) < cµ, then the steady-state probabilities of the queueing system undermultiple vacation policy (MVP) are given as

P0,. =ξ

φK(1)P0,0,

and

P1,. =φG′0(1) + µR(1)P0,0

cµ− λB′(1),

where

P0,0 =

µ

cµ− λB′(1)R(1) +

ξ

φK(1)

(φλB′(1)

(cµ− λB′(1))(ξ + φ)+ 1)−1

,

and

G′0(1) =ξλB′(1)

(ξ + φ)φK(1)P0,0,

with

R(1) =c−1

∑n=0

(c− n)θn, K(1) =∫ 1

0e

λξ H(x)(1− x)

φξ −1dx, H(x) =

∫ x

0

B(z)− 11− z

dz,

and

θn =

θ0, if n = 0,λ+η

µ θ0, if n = 1,

ρn−1θn−1 −φ

nµωn−1 −

λ

nµ

n−1

∑i=1

biθn−1−i, if 2 ≤ n ≤ c− 1,

226

ωn =

1, if n = 0;λ−ηθ0

ξ , if n = 1.

ψn−1ωn−1 −λ

nξ

n−1

∑i=1

biωn−1−i if 2 ≤ n ≤ c− 1.

ρn−1 =λ + (n− 1)µ

nµ, ψn−1 =

λ + φ + (n− 1)ξnξ

, θ0 =ξ − φK(1)

ηK(1),

where B(z) is the probability generating function of the batch arrival size X, and B′(1) =E(X) is the first moment of random variable X.

Theorem 109 If λE(X) < cµ, then the steady-state probabilities of the queueing system undersingle vacation policy (SVP) are given as

P0,. =ξ

φK(1)P0,0,

and

P1,. =

ξλB′(1)

K(1)(ξ + φ)(cµ− λB′(1))+

µ

cµ− λB′(1)Q(1)

P0,0,

with

P0,0 =

ξ

φK(1)

(1 +

φλB′(1)(cµ− λB′(1))(ξ + φ)

)+

µ

cµ− λB′(1)Q(1)

−1

,

where

Q(1) =c−1

∑n=0

(c− n)Mn, M0 =ξ

ηK(1), and M1 =

ξ(λ + η)

ηµK(1)− φ

µ,

and

Mn =

M0, if n = 0,M1, if n = 1,

ρn−1Mn−1 −φ

nµγn−1 −

λ

nµ

n−1

∑i=1

bi Mn−1−i, if 2 ≤ n ≤ c− 1.

γn =

1, if n = 0;λ+φ−ηM0

αξ , if n = 1.

ψn−1γn−1 −λ

nξ

n−1

∑i=1

biγn−1−i if 2 ≤ n ≤ c− 1.

Corollary 15 If the stability condition is fulfilled, thenThe mean system size E[L] under (MVP) is

E[L] = E[L0] + E[L1],

where

E[L0] =ξλB′(1)

(ξ + φ)φK(1)P0,0,

227

and

E[L1] =

(φ(2cµ + λB′′(1))2(cµ− λB′(1))2 +

λφB′(1)(cµ− λB′(1))(2ξ + φ)

)E[L0]

+λφB′′(1)

2(cµ− λB′(1))(2ξ + φ)P0,. +

(µλ(2B′(1) + B′′(1))

2(cµ− λB′(1))2 R(1) +µ

cµ− λB′(1)R′(1)

)P0,0,

with B′′(1) is the second moment of random variable X, and

R′(1) =c−1

∑n=1

n(c− n)θn.

Corollary 16 If the stability condition is fulfilled, thenThe mean system size E[L] under (SVP) is

E[L] = E[L0] + E[L1],

where

E[L0] =ξλB′(1)

φK(1)(ξ + φ)P0,0,

and

E[L1] =

(φ(2cµ + λB′′(1))2(cµ− λB′(1))2 +

λφB′(1)(cµ− λB′(1))(2ξ + φ)

)E[L0]

+λφB′′(1)

2(cµ− λB′(1))(2ξ + φ)P0,. +

(µλ(2B′(1) + B′′(1))

2(cµ− λB′(1))2 Q(1) +µ

cµ− λB′(1)Q′(1)

)P0,0,

with

Q′(1) =c−1

∑n=1

n(c− n)Mn.

[Altman] Altman E., Yechiali U., Analysis of customers’ impatience in queues with servervacation, Queueing Syst., 2006, 52(4), 261–279.[Ammarnew] Ammar S. I., Transient solution of an M/M/1 vacation queue with waitingserver and impatient customers, Journal of the Egyptian Mathematical Society., 2017, 27(2),337–342.[Bouchentouf1017] Bouchentouf A. A., Yahiaoui L., On feedback queueing system withreneging and retention of reneged customers, multiple working vacations and Bernoullischedule vacation interruption, Arab. J. Math., 2017, 6(1), 1–11.[Laxmi2017] Laxmi V. P., Rajesh P., Performance measures of variant working vacationon batch arrival queue with reneging, International Journal of Mathematical Archive., 2017,8(8), 85–96.[Padmavathy] Padmavathy R., Kalidass K., Ramanath K., Vacation queues with impa-tient customers and a waiting server, Int. J. Latest Trends Softw. Eng., 2011, 1(1), 10–19.[Yue2] Yue D., Yue W., Xu G., Analysis of customers impatience in an M/M/1 queuewith working vacations, Journal of Industrial and Management Optimization., 2012, 8(4),895-908.

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RAMA 11 Recueil des Resum´ es´ RAMA11.pdfLa Direction Gen´ ´erale de la Recherche Scientiﬁque et du Developpement Technologique´ L’Universite Djillali Liabes, Sidi´ Bel Abbes