Théorèmes du Point Fixe de Kransnoselskii sous la Topologie Faible et Applications

Abstract

The aim of this memoir is first to study new version of Krasnoselskii fixed point theorems under weak topology of Banach spaces then to provide as applications some existence results for some nonlinear integral equation. This work is divided into fours chapters. The first chapter provides some definitions and auxiliary results that will be used later on. In Chapter 2, we give an exposition of Krasnoselskii fixed point theorems for contraction mappings involving the weak topology. Our main result is applied to solve the following nonlinear integral equation x(t) = f(x) + Z T 0 g(s, x(s))ds, t 2 [0,T ]. In chapter 3, we provide some expansive Krasnoselskii-type fixed point theorems under weak topology and apply our result to prove the existence of solution of above equation. In the last chapter we present an existence theories for the two operator equations AxBx = x and AxBx+Cx = x, x 2 M, in Banach algebra endowed with weak topology. Where M is bounded, closed and convex subset of Banach algebra. A, B and C three operators defined on M. Then we apply this result to the following integral equations. x(t) = a(t) + T(x)(t) 2 64 Ö q(t) + Z(t) 0 p(t,s,x(s),x( s))ds è .u 3 75 , 0 < < 1.