A generalization of Krasnoselskii's eigenvalue theorem (Q2488723)

The purpose of this work is to extend Krasnoselskii's eigenvalue theorem as follows: Let \(V\) be a bounded closed neighborhood of the origin in a Banach space \(E\) and \(K\) be a closed cone in \(E\). Suppose that \(F:\partial V\cap K\to\kappa c(K)\) is a countably \(k\)-condensing map with respect to a given measure of noncompactness on \(K\) where \(k>0\). If there is a real number \(a>k\) such that \(F(\partial V\cap K)\cap a(\text{int}\, V)=\varnothing\), then there exist a positive real number \(\lambda_0\) and an element \(x_0\in\partial V\cap K\) such that \(\lambda_0x_0\in F(x_0)\). The author uses a fixed point theorem for compact set-valued maps due to \textit{C.~J.\ Himmelberg} [J.\ Math.\ Anal.\ Appl.\ 38, 205--207 (1972; Zbl 0225.54049)] and \textit{T.--W.\ Ma}'s extension theorem [``Topological degrees of set-valued compact fields in locally convex spaces'' (Diss.\ Math.~92) (1972; Zbl 0211.25903)]. Many known results can be deduced from the author's result.