Fixed-Point Theory on a Frechet Topological Vector Space

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)).