A fixed point-equilibrium theorem with applications
In the present paper the author, using the Kakutani-Fan-Glicksberg fixed point theorem, proves the following theorem: Theorem 1. Let \(X\) be a nonempty compact convex subset of a locally convex Hausdorff topological vector space, \(Z\) be a nonempty set, \(\rho\) be a relation on \(2^X\) and \(T: X\multimap X\), \(F: X\times X\multimap Z\) and \(C: X\multimap Z\) be three mappings satisfying the following conditions: {\parindent8mm \begin{itemize}\item[(i)] \(T\) is upper semicontinuous with nonempty compact convex values, \item[(ii)] for each \(x\in X\), the set \(\{y\in X: F(x, y)\rho^c C(x)\}\) is convex, where \(\rho^c\) denotes the complementary relation of \(\rho\), \item[(iii)] for each \(y\in X\), the set \(\{x\in X: F(x,y)\rho C(x)\}\) is closed in \(X\), \item[(iv)] for each \(x\in X\) and \(y\in(\bigcup_{\lambda\geq 1}(\lambda x+(1- \lambda)T(x)))\cap X\), \(F(x,y)\rho C(x)\). \end{itemize}} Then there exists \(\widetilde x\in X\) such that \(\widetilde x\in T(\widetilde x)\) and \(F(\widetilde x,y)\rho C(\widetilde x)\) for all \(y\in X\).
scientific article; zbMATH DE number 5835852