Éditeur(s) : HAL CCSDYokohama Résumé : International audience We introduce a convergent method for approximating a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a monotone equilibrium problem. Using the main convergence theorem we obtain results which improve, develop and unify several results in fixed point problems, variational inequalities and equilibrium problems. Journal of Nonlinear and Convex Analysis
Éditeur(s) : HAL CCSDSpringerOpen Résumé : International audience We introduce methods which seem to be a new and promising tool in hierarchical fixed-point problems. The goal of this note is to analyze the convergence properties of these new types of approximating methods for fixed-point problems. The limit attained by these curves is the solution of the general variational inequality,, where denotes the normal cone to the set of fixed point of the original nonexpansive mapping and a suitable nonexpansive mapping criterion. The link with other approximation schemes in this field is also made. ISSN: 1687-1820
Éditeur(s) : HAL CCSD Résumé : International audience The main purpose of this note is to propose viscosity approximation methods which amount to selecting a particular fixed-point of a given nonexpansive self mapping in a general Hilbert space. The connection with the selection principles of Attouch, in the context of convex minimization and monotone inclusion problems, is made and an application to a semi-coercive elliptic problem is then stated. Key words: nonexpansive mappings, fixed-points, monotone operators, iterative methods, convex optimization, selection, penalty ... Lecture notes in economics and mathematical systems
Éditeur(s) : HAL CCSDIOP Publishing Résumé : International audience Based on the very recent work by Censor and Segal (2009 J. Convex Anal. 16 587-600) and inspired by Xu (2006 Inverse Problems 22 2021-34) and Yang (2004 Inverse Problems 20 1261-6), we investigate an algorithm for solving the split common fixed-point problem for the class of demicontractive operators in a Hilbert space. Our results improve and/or develop previously discussed feasibility problems and related algorithms. It is worth mentioning that the convex feasibility formalism is at the core of the modeling of many inverse problems and has been used to model significant real-world problems, for instance, in sensor networks, in radiation therapy treatment planning, in computerized tomography and data compression, see Censor et al (2006 Phys. Med. Biol. 51 2353-65) and Combettes (1996 Adv. Imaging Electron. Phys. 95 155-270) and references therein. ISSN: 0266-5611
Éditeur(s) : HAL CCSDYokohama Résumé : International audience We make use of the viscosity technique to develop a convergent method for approximating a common element of the set of fixed points of a demicontractive operator and the set of solutions of a monotone equilibrium problem. The proposed algorithm is obtained by coupling a modified hybrid steepest descent method with the extragradient algorithm. Under mild conditions, the strong convergence of the sequences generated by the algorithm is obtained. Using this result we obtain two corollaries which improve or develop several corresponding results in this field. Journal of Nonlinear and Convex Analysis
Éditeur(s) : HAL CCSD Résumé : I propose a self contained research paper. I hope it adds some news ideas and results to the …xed point theory in the framework of generalized functions algebras, with application to the Cauchy-Lipschitz problem in a generalized formulation including strongly irregular cases. This leads to the transport equation with distributions as coe¢ cients we wish to treat later. https://hal.univ-antilles.fr/hal-01231272
Éditeur(s) : HAL CCSDTaylor & Francis Résumé : International audience We introduce the notions of conditioning and well-posedness for equilibrium problems. Using these concepts, we obtain finite and strong convergence results for the proximal method that improve, develop, and unify several theorems in optimization and nonlinear analysis. ISSN: 0163-0563
Éditeur(s) : HAL CCSD Résumé : International audience It is established that the multivalued quasi variational inequalities in uniformly smooth Banach spaces are equivalent to the fixed-point problem. We use this equivalence to suggest and analyze some iterative algorithms for quasi variational inequalities with noncompact sets in Banach spaces. Our results are new and represent a significant improvement of the previously known results.; JIPAM
Éditeur(s) : HAL CCSD Résumé : International audience We consider in this paper a cascade system of two parabolic equations, governed by a quasi-linear nonlocal operator. We study the null controllability of this problem with a control subject to constraints. We begin by proving the existence of a unique control of minimal norm for the linear problem. We combine these results with a fixed-point theorem to conclude. ECC15 Linz, Austria
Éditeur(s) : HAL CCSDElsevier Résumé : International audience Based on the very recent work by Censor and Segal (2009), and inspired by Xu (2006), Zhao and Yang (2005) , and Bauschke and Combettes (2001), we introduce and analyze an algorithm for solving the split common fixed-point problem for the wide class of quasi-nonexpansive operators in Hilbert spaces. Our results improve and develop previously discussed feasibility problems and related algorithms. ISSN: 0362-546X
Éditeur(s) : HAL CCSDElsevier Résumé : International audience Two splitting procedures for solving equilibrium problems involving the sum of two bifunctions are proposed and their convergence is established under mild assumptions. Journal of Mathematical Analysis and applications
Éditeur(s) : HAL CCSDSpringer Verlag Résumé : International audience We study the convergence of the Mann Iteration applied to the partial complement of a firmly nonexpansive operator with respect to a linear subspace of a Hilbert space. A new concept considered here. A regularized version is also proposed. Furthermore, to motivate this concept, some applications to robust regression procedures and location problems are proposed. ISSN: 1862-4472
Éditeur(s) : HAL CCSDElsevier Résumé : International audience Relying on the Yosida approximate, we investigate the problem of finding zeroes for a difference of two maximal monotone operators in Hilbert spaces. These zeroes are compared to those of the corresponding regularized and dual problems. The behavior of the regularized operator is also studied and a splitting algorithm involving the resolvents of both operators is suggested via a fixed-point formulation of the regularized problem. A particular attention is given to the DC programming case. ISSN: 0096-3003
Éditeur(s) : HAL CCSDElsevier Résumé : International audience The main purpose of this paper is to propose a deterministic mathematical model with control variable which describes the spatiotemporal spread of emph{Salmonella} within a laying flock. This model assume an indirect transmission of the disease within the flock through the bacteria density in the environment. We provide a mathematical analyze of the null controllability of the model: we begin to construct a family of solutions to the linear null-controllability problem using the the Carleman inequality. Then, by a generalized Leray-Shauder fixed point theorem, we show that one can bring the system to rest at final time T. ISSN: 0307-904X
Éditeur(s) : HAL CCSDUniversity of Szeged, Bolyai Institute Résumé : International audience We consider the semilinear heat equation in a boundeddomain of R^m. We prove the null controllability of the systemwith a finite number of constraints on the normal derivative, whenthe control acts on a bounded subset of the domain. First, weshow that the problem can be transformed into a null controllabilityproblem with constraint on the control, for a linear system.Then, we use an appropriate observability inequality to solve thelinearized problem. Finally, we prove the main result by means ofa fixed-point method. ISSN: 1417-3875
Éditeur(s) : HAL CCSDTaylor & Francis Résumé : International audience This article is concerned with a generalization of the hybrid steepest descent method from variational inequalities to the multivalued case. This will be reached by replacing the multivalued operator by its Yosida approximate, which is always Lipschitz continuous. It is worth mentioning that the hybrid steepest descent method is an algorithmic solution to variational inequality problems over the fixed point set of certain nonexpansive mappings and has remarkable applicability to the constrained nonlinear inverse problems like image recovery and MIMO communication systems ISSN: 0163-0563
Éditeur(s) : HAL CCSDTaylor & Francis Résumé : International audience We study the convergence of a variable version of the Krasnoselski-Mann algorithm applied to a primal dual fixed point problem. The link with Spingarn's partial inverse method is made, and an application to feasibility problems and mathematical programming is also proposed. ISSN: 0163-0563
Éditeur(s) : HAL CCSDHindawi Publishing Corporation Résumé : International audience This paper is concerned with the study of a penalization-gradient algorithmfor solving variational inequalities, namely, find x ∈ C such that Ax, y − x ≥ 0 for all y ∈ C, where A : H → H is a single-valued operator, C is a closed convex set of a real Hilbert space H. Given Ψ : H → ∪ { ∞} which acts as a penalization function with respect to the constraint x ∈ C, and a penalization parameter βk, we consider an algorithm which alternates a proximal step with respect to ∂Ψ and a gradient step with respect to A and reads as xk I λkβk∂Ψ −1 xk−1 − λkAxk−1 . Under mild hypotheses, we obtain weak convergence for an inverse strongly monotone operator and strong convergence for a Lipschitz continuous and strongly monotone operator. Applications to hierarchical minimization and fixed-point problems are also given and the multivalued case is reached by replacing themultivalued operator by its Yosida approximatewhich is always Lipschitz continuous. ISSN: 0161-1712
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