Fixed-Point Theory on a Frechet Topological Vector Space

We establish some versions of fixed-point theorem in a Frechet topological vector space . The main result is that every map = (where is a continuous map and 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 {(？,)∶∈() where ？ and ∶→ a compact operator} is nonlinear equicontractive, we give a fixed-point theorem for the operator of the form ∶=(,()).