Krasnoselskii's fixed point theorem for weakly continuous maps. (Q1413250)

The author proves the following Krasnosel'skiĭ-type theorem: Let \(X\) be a Banach space, let \(M\subset X\) be weakly compact and convex and \(A:M\to X\) weakly continuous. Assume that there is a continuous linear operator \(B:X\to X\) such that \(\| B^p\| \leq1\) for some \(p\geq1\) and such that \(x=\lambda Bx+Ay\) with \(y\in M\) and \(0<\lambda<1\) implies \(x\in M\). The conclusion is that the fixed point sets of \(A\) and of \(A+B\) are nonempty subsets of \(M\). There is an application to a Dirichlet problem in \(\mathbb{R}^n\).