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)).