Fixed point theorems with an application in generalized games
The following fixed point theorem is proved: Theorem. Let \((E, \{\Gamma_ A\})\) be an \(H\)-space, \(C\) and \(X\) be a nonempty subset and a nonempty closed \(H\)-convex subset of \(E\), respectively, and \(F\) be a topological space. Let \(h : C \to 2^ F\) be a multivalued mapping and \(S : E \to F\) be continuous; if there exists a multivalued mapping \(g : X \to 2^ C\), \(g(x) = \text{id}\) on \(X \cap C\) such that (i) for each \(x \in X\), \(s(X) \cap h(g(x)) \neq \emptyset\), \(s^{-1} (h(g(x)))\) is an \(H\)-convex set; (ii) for each \(y \in X\), the set \(\{x : s(y) \subset h(g(x))\}\) is relatively open in \(X\); (iii) there exists an \(H\)-compact set \(D \subset X\) such that for each \(x \in X \smallsetminus D\), \(s(D) \cap h(g(x)) \neq \emptyset\); (iv) for each \(x \in X \smallsetminus C\), \(s(x) \notin h(g(x))\). Then there exists a point \(x_ 0 \in C\) such that \(s(x_ 0) \in h(x_ 0)\). Using this theorem, an existence theorem for equilibrium point in a generalized game is proved.
scientific article; zbMATH DE number 703295