Nonlinear Analysis 55 (2003) 25–31 www.elsevier.com/locate/na Krasnoselskii’s xed point theorem for weakly continuous maps Cleon S. Barroso ∗ Departamento de Matem atica, Universidade Federal do Cear a, Campus do Pici, B. 914, 60455-760 Fortaleza, CE, Brazil Received 23 December 2002; accepted 12 June 2003 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 ‖B p ‖ 6 1, p ¿ 1. An alternative condition is given in order to guarantee the existence of xed points in M for A + B. Some illustrative applications are given. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Fixed points; Weakly continuous maps; Krasnoselskii; Dirichlet problem; Eigenvalue problem; Integral equations 1. Introduction The existence of xed points for the sum of two operators has been focus of interest for several years and their applications often are in nonlinear analysis. For instance, in [7] O’Regan proved a variety of such results with applications to boundary value problems of second order with nonlinearities. Krasnoselskii’s theorem [5] asserts that the sum A + B has a xed point in a closed, convex and nonempty subset M of a Banach space (X; ‖·‖), where A and B satises: (i) Ax + By ∈ M for all x;y ∈ M , (ii) A is continuous on M and A(M ) is contained in a compact subset of X , (iii) B is a -contraction on X with ¡ 1. * Tel.: +55-852889885; fax: +55-852889889. E-mail address: cleonbar@mat.ufc.br (C.S. Barroso). 0362-546X/03/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0362-546X(03)00208-6