Mediterr. J. Math. (2018) 15:121 https://doi.org/10.1007/s00009-018-1141-9 c  Springer International Publishing AG, part of Springer Nature 2018 The Schauder and Krasnoselskii Fixed-Point Theorems on a Frechet Space Toufic El Arwadi and Mohamed Amine Cherif Abstract. In this manuscript, we study some fixed-point theorems of the Schauder and Krasnoselskii type in a Frechet topological vector space E. We prove a fixed-point theorem which is for every weakly compact map from a closed bounded convex subset of a Frechet topological vector space having the Dunford–Pettis property into itself has a fixed point. Using our results, we will establish a new version of the Krasnoselskii fixed-point theorem. Mathematics Subject Classification. 65M12, 65J10. Keywords. Fixed-point theory, Frechet topological vector space, Kras- noselskii fixed-point theorems, Schauder fixed-point theorems, Dunford– Pettis property. 1. Introduction Several algebraic and topological settings in the theory and applications of nonlinear operator equations lead naturally to the investigation of fixed points of a sum of two nonlinear operators, or more generally, fixed points of a map- ping on the Cartesian product X × X into X, where X is some appropriate space. Fixed-point theorems in topology and nonlinear functional analysis are usually based on certain properties (such as complete continuity, monotonic- ity, and contractiveness) that the operator considered as a single entity must satisfy. We recall for instance the Banach fixed-point theorem, which asserts that a strict contraction on a complete metric space into itself has unique fixed point, and the Schauder principle, which asserts that a continuous mapping A on a closed convex set M in Hausdorff locally convex topological vector space E into M such that A(M ) is contained in a compact set, has a fixed point. In many problems of analysis, one encounters operators which may be split into the form T = A + B, where A is a contraction in some sense, and B is completely continuous, and T itself has neither of these properties (see [1, 3, 4]). Thus, neither the Schauder fixed-point theorem nor the Banach 0123456789().: V,-vol