Hichem Ben-El-Mechaiekh

Approachability and Fixed Points for Non-convex Set-Valued Maps*

This paper studies the approachability of upper semicontinuous nonconvex set-valued maps by single-valued continuous functions and its applications to fixed point theory. The use of “approximate selections” was initiated by J. von Neumann [41] in the proof of his well-known minimax principle. This remarkable property was extended to the normed space setting by Cellina [S] and Cellina and Lasota [9] in the context of a degree theory for upper semicontinuous convex compact-valued maps. It also holds for maps with compact contractible values defined on finite polyhedra of Iw” (Ma+Cole11 [40]) and more generally defined on compact ANRs (McLennan [39]). It was recently taken up by Gbrniewicz, Granas, and Kryszewski [22-241 in the context of an index theory for non-convex maps defined on compact ANRs. The content of this paper is divided into three parts. The first part is devoted to background material concerning the types of spaces and set-valued maps studied herein. In the second part we define the abstract class d of approachable set-valued maps; that is, the class of maps A: X+ Y

The coincidence problem for compositions of set-valued maps

The main purpose of this work is to give a general and elementary treatment of the fixed point and the coincidence problems for compositions of set-valued maps with not necessarily locally convex domains and to display, once more, the central role played by the selection property.

Spaces and maps approximation and fixed points

Abstract A general approximation property for topological spaces is studied in relation with fixed point theory for set-valued maps. A particular instance of this property is the admissibility in the sense of Klee. Examples of “convex” sets of topological spaces equipped with a local topological convexity structure as well as general classes of approximative neighborhood retracts are shown to have this approximation property. A general topological principle on the preservation of the fixed point property under this space approximation is proved. It allows the passage from basic classes of spaces to more elaborate ones for general classes of nonconvex set-valued maps.

A Leray-Schauder type theorem for approximable maps: A simple proof

We present a simple and direct proof for a Leray-Schauder type alternative for a large class of condensing or compact set-valued maps containing convex as well as nonconvex maps. The aim of this note is to extend the Leray-Schauder type nonlinear alternative presented in [BI] to a condensing upper semicontinuous approximable set-valued map F: X -> E when X is a closed subset with nonempty interior of a locally convex topological vector space E. The proof presented here is even shorter and simpler than the one given in [BI]. In what follows E stands for a Hausdorff locally convex topological vector space with a fundamental basis JK. of convex, symmetric neighborhoods of the origin; if X, Y are nonempty subsets of E, then F: X -Y is a set-valued map with nonempty values (simply called map). The boundary, the interior, the closure, and the convex hull of a subset A in E are denoted by AA, int A, A, and co A respectively. Definition 1. F is said to be upper semicontinuous (u.s.c.) on X if and only if for any open subset V of Y, the set {x E X: F(x) C V} is open in X. Definitions 2 ([BD], [BI], see also [GGK] for metric spaces). (1) Given U, V E X, a function s: X -+ Y is said to be a (U, V)-approximative selection of F if for any xE X, S(x) E (F[(x+ U) nX] +V) n Y. (2) F: X -* Y is said to be approachable if it has a continuous (U, V)approximative selection for any (U, V) E KV x JK. A(X, Y) denotes the class of such maps. We write .A(X) for A(X, X). (3) F is said to be approximable if its restriction FIK to any compact subset K of X is approachable. Note that an approachable map is approximable (cf. [B]). Examples. It is well-known that if F is u.s.c. with nonempty convex values, then F is approachable provided X is paracompact and Y is convex (cf. [DG]). Obviously, F is approximable without conditions on X (see [C]). Received by the editors August 12, 1996 and, in revised form, January 16, 1997. 1991 Mathematics Subject Classification. Primary 47H04, 47H10, 54C60.

The Ran–Reurings fixed point theorem without partial order: A simple proof

The purpose of this note is to generalize the celebrated Ran–Reurings fixed point theorem to the setting of a space with a binary relation that is only transitive (and not necessarily a partial order) and a relation-complete metric. The arguments presented here are simple and straightforward. It is also shown that extensions by Rakotch and by Hu and Kirk of Edelstein’s generalization of the Banach contraction principle to local contractions on chainable complete metric spaces are derived from the Ran–Reurings theorem.

A Simpler Proof of the Von Neumann Minimax Theorem

Abstract This note provides an elementary and simpler proof of the Nikaidô-Sion version of the von Neumann minimax theorem accessible to undergraduate students. The key ingredient is an alternative for quasiconvex/concave functions based on the separation of closed convex sets in finite dimension, a result discussed in a first course in optimization or game theory.

Equilibrium for perturbations of multifunctions by convex processes

We present a general equilibrium theorem for the sum of an upper hemicontinuous convex-valued multifunction and a closed convex process defined on a noncompact subset of a normed space. The lack of compactness is compensated by inwardness conditions related to the existence of viable solutions of some differential inclusion.

Some Fundamental Topological Fixed Point Theorems for Set-Valued Maps

This chapter introduces the reader to two of the most fundamental topological fixed point theorems for set-valued maps: the Browder–Ky Fan and the Kakutani–Ky Fan theorems. It provides a concise discussion including motivations, techniques, as well as some most important applications. The exposition is driven by clarity and simplicity. Generality of statements is deliberately sacrificed to the benefit of conceptual significance. Generalizations based on technicalities or artificial definitions which, with little effort, can be reduced to classical settings are set aside, unless they are motivated by convincing applications. Rather, the treatment here is reduced to the classical convex case, which is—we firmly believe—where the essence belongs. The arguments are kept elementary, as to allow the use of this chapter in a first course in topological fixed point theory and its applications.

On the Closedness of the Convex Hull in a Locally Convex Space

The question of the closedness of the convex hull of the union of a closed convex set and a compact convex set in a locally convex space does not appear to be widely known. We show here that the answer is affirmative if and only if the closed convex set is bounded. The result is first proven for convex compact sets ”of finite type” (polytopes) using an induction argument. It is then extended to arbitrary convex compact sets using the fact that such subsets in locally convex spaces admit arbitrarily small continuous displacements into polytopes.

Recent Contributions to Fixed Point Theory and Its Applications

The flourishing field of fixed point theory started in the early days of topology with seminal contributions by Poincare, Lefschetz-Hopf, and Leray-Schauder at the turn of the 19th and early 20th centuries. The theory vigorously developed into a dense and multifaceted body of principles, results, and methods from topology and analysis to algebra and geometry as well as discrete and computational mathematics. This interdisciplinary theory par excellence provides insight and powerful tools for the solvability aspects of central problems in many areas of current interest in mathematics where topological considerations play a crucial role. Indeed, existence for linear and nonlinear problems is commonly translated into fixed point problems; for example, the existence of solutions to elliptic partial differential equations, the existence of closed periodic orbits in dynamical systems, and more recently the existence of answer sets in logic programming. The classical fixed point theorems of Banach and Brouwer marked the development of the two most prominent and complementary facets of the theory, namely, the metric fixed point theory and the topological fixed point theory.Themetric theory encompasses results and methods that involve properties of an essentially isometric nature. It originates with the concept of Picard successive approximations for establishing existence and uniqueness of solutions to nonlinear initial value problems of the 1st order and goes back as far as Cauchy, Liouville, Lipschitz, Peano, Fredholm, and most particularly, Emile Picard. However, the Polish mathematician Stefan Banach is credited with placing the underlying ideas into an abstract framework suitable for broad applications well beyond the scope of elementary differential and integral equations. Metric fixed point theory for important classes of mapping gained respectability and prominence to become a vast field of specialization partly and not only because many results have constructive proofs, but also because it sheds a revealing light on the geometry of normed spaces, not to mention its many applications in industrial fields such as image processing engineering, physics, computer science, economics, and telecommunications. A particular interest in fixed points for set-valued operators developed towards the mid-20th century with the celebrated extensions of the Brouwer and Lefschetz theorems by Kakutani and Eilenberg-Montgomery, respectively. The Banach contraction principle was later on extended to multivalued contractions by Nadler. The fixed point theory for multivalued maps found numerous applications in control theory, convex and nonsmooth optimization, differential inclusions, and economics. The theory is also used prominently in denotational semantics (e.g., to give meaning to recursive programs). In fact, it is still too early to truly estimate the importance and impact of set-valued fixed point theorems in mathematics in general as the theory is still growing and finding renewed outlets. This special issue adds to the development of fixed point theory by focusing on most recent contributions. It includes works on nonexpansive mappings in Banach and metric spaces, multivalued mappings in Banach and metric spaces, monotone mappings in ordered spaces, multivalued mappings in ordered spaces, and applications to such nonmetric spaces as modular spaces, as well as applications to logic programming and directed graphs.