Some topological and geometrical properties in general Köthe-Bochner Banach spaces : Study of generalized Cauchy functional equations

Abstract

Köthe-Bocher Banach spaces of vector valued functions X(E), chich are generalizations of both the Lebesgue-Bochner and Orliez-Bochner spacesn have been studied by many authors. One of the fundamental problems here is the question of wether a geometrical property lifts from E to X(E). Although often an answer to such a question is expected, the proff of such a response is usually nontrivial. This thesis is composed with two independently parts. The first one treats some topological and geometrical properties in Köthe-Bochner Banach spaces. In the second part, we determine the solution of the generalized Cauchy fucntional equations. This thesis is organized as follows : The first chapter deals with an elementary approach to multivalued functions and Köthe-Bochner Banach spaces and presents some preliminaries results that we use in tje sequel. In the second chapter, we study strongly exposed points in general Köthe-Bochner Banach spaces X(E). We first give a characterization of strongly exposed points of the set of X-selections of a measurable multifuction Г. We then apply this result to the study of strongly exposed points of the closed unit ball of X(E). The third chapter concerns a characterization of strongly exposed points by selection methods. The main purpose of the chapter 4 is to characterize and provide some properties of the sets LXF of X-Bochner measurable selections of multifunctions in E, where E is a seperable Banach space and X is an order continous Köthe space. More precisely, we obtain a characterization of décomposable sets and a generalization of minimization theorems for integral functional in X€. In chapter 5, we give contributions to the study and resolution of the classical generalized Cauchy functional equations.