Fixed-Point Theory on a Frechet Topological Vector Space

We establish some versions of fixed-point theorem in a Frechet topological vector space E. 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. 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 for U-contractions and weakly compact mappings, while the second one, by assuming that the family {T ·, y : y ∈ C M where M ⊂ E and C : M → E a compact operator} is nonlinear φ equicontractive, we give a fixed-point theorem for the operator of the form Ex : T x,C x .