Equilibria for set-valued maps on nonsmooth domains
We say that a set-valued map \(\varphi: X\to 2^E\) defined on a closed subset \(X\) of a normed space \(E\) admits an equilibrium (or zero) if there exists a point \(u\in X\) such that \(0\in \varphi(u)\). In the paper under review, the following result is proved. Assume that \(X\) is a compact Lipschitzian retract of a normed space \(E\) having nontrivial Euler characteristic. Assume further that \(\varphi: X\to 2^E\) is upper semicontinuous map and belongs to the class of strongly approchable maps. If, moreover, \(\varphi\) satisfies so-called tangency condition, then \(\varphi\) admits an equilibrium. More than half of the paper consists of necessary definitions and preliminary results; only a short part of the paper is devoted to present this extension of an older result of the author for the class of nonconvex set-valued maps.
approximative absolute neighborhood retracts
scientific article; zbMATH DE number 5604836