Remark on Krasnoselskii’s fixed point theorem

A well known result due to Krasnoselskii ensures the existence of a fixed point for an operator K=A+BK=A+B which is defined on a non-empty bounded closed convex subset SS of a Banach space XX, where (i) AA is a contraction, (ii) BB is a compact operator, and (iii) A(S)+B(S)⊂SA(S)+B(S)⊂S. In the present note, an easy sufficient condition for fulfilling (iii) in the case of a locally convex space is given. An application to the existence of solutions of a nonlinear integral equation illustrates this result.