Perturbation theorems for positive eigenvalues of countably condensing maps (Q596842)

Drawing extensively upon Jerofsky's PhD thesis [Zur Fixpunkttheorie mengenwertiger Abbildungen, TU Dresden (1983)], the author derives a perturbation result for solutions of nonlinear parametrized equations with countable condensing right hand side. This applies, in particular, to perturbations of nonlinear eigenvalue problems: Let \(E\) be a metrizable quasi-complete locally convex vector space and \(\psi\) a measure of noncompactness on \(E\). Let \(Q\) be a topological space, \(Y\subset X\times Q\), and \(k\geq0\). An upper semicontinuous map \(F:Y\multimap E\) is said to be countably condensing if \(\psi(F(Y))\) is defined and if for any countable set \(X\subset Y\) we can have \(\psi(\text{pr}_E(X))\leq\psi(F(X))\) only if \(F(X)\) is relatively compact. The author's main result then reads as follows: Let \(E\) be as above, \(K\subset E\) be closed convex and \(Q\) be a compact metric. Let \(Y\subset K\times Q\) be closed and \(X\subset Y\) closed. Let \(F:Y\multimap K\) be a countably condensing map with compact convex values and assume that \(x\notin F(x,q)\) whenever \((x,q)\in X\) and that \(F| \,X\) is essential with respect to the set of countably condensing maps \(G:Y\multimap K\) such that \(G\) has compact values and \(x\notin G(x,q)\) for all \((x,q)\in Y\). If \(U\subset Y\) is a neighbourhood of \(\{(x,q)\in Y| \;x\in F(x,q)\}\), then there is a neighbourhood \(V\ni0\) in \(E\) such that for each countably condensing map \(G:\bar{U}\multimap K\) with compact values and \(G(x,q)\subset F(x,q)+V\), whenever \((x,q)\in\bar{U}\), then there is a solution \((x,q)\in\bar{U}\) to \(x\in G(x,q)\).