Fixed-Point Theory on a Frechet Topological Vector Space

Afif Ben Amar,Mohamed Amine Cherif,Maher Mnif

doi:10.1155/2011/390720

Abstract

We establish some versions of fixed-point theorem in a Frechet topological vector spaceE. The main result is that every mapA=BC(whereBis a continuous map andCis a continuous linear weakly compact operator) from a closed convex subset of a Frechet topological vector space having the Dunford-Pettis property into itself has fixed-point. Based on this result, we present two versions of the Krasnoselskii fixed-point theorem. Our first result extend the well-known Krasnoselskii's fixed-point theorem forU-contractions and weakly compact mappings, while the second one, by assuming that the family{T(⋅,y):y∈C(M)whereM⊂EandC:M→Ea compactoperator}is nonlinearφequicontractive, we give a fixed-point theorem for the operator of the formEx:=T(x,C(x)).

Highlights

Fixed-point theorems are very important in mathematical analysis
We present two versions of the Krasnoselskii fixed-point theorem
We recall for instance the Banach fixed-point theorem, which asserts that a strict contraction on a complete metric space into

Summary

Introduction

Fixed-point theorems are very important in mathematical analysis. They are an interesting way to show that something exists without setting it out, which sometimes is very hard, or even impossible to do. The main result is that every map A BC where B is a continuous map and C is a continuous linear weakly compact operator from a closed convex subset of a Frechet topological vector space having the Dunford-Pettis property into itself has fixed-point.

Results

Conclusion