Krasnoselskii's fixed point theorem for weakly continuous maps

Abstract We consider the sum A+B : M→X , where M is a weakly compact and convex subset of a Banach space X, A : M→X is weakly continuous, and B∈ L (X) with ||Bp||⩽1, p⩾1. An alternative condition is given in order to guarantee the existence of fixed points in M for A+B. Some illustrative applications are given.