A unified approach to generalized KKM maps in generalized convex spaces (Q2764700)

The classical KKM (Knaster, Kuratowski, Mazurkiewicz) maps are multimaps \(F: D \multimap \Delta_n\), where \(D\) is the set of vertices of an \(n\)-simplex \(\Delta_n\), satisfying \(co A \subset F(A)\) for all \(A \subset D\). The KKM principle says that \(\bigcap_{z \in D} F(z) \neq \emptyset\) for any such map with closed (or open) values. Various generalizations of the KKM theory were given by a series of authors. In the present paper, generalized KKM maps in generalized convex spaces are considered. A generalized convex (\(G\)-convex) space \((X,D;\Gamma)\) consists of a topological space \(X\) and a nonempty set \(D\) such that for each \(N=\{z_0,z_1,\dots,z_n\} \subset D\) there is a subset \(\Gamma (N)\) of \(X\) and a continuous function \(\Phi_N: \Delta_n \to \Gamma(N)\) such that \(J\subset \{0,1,\dots,n\}\) implies \(\Phi_N (\Delta_J) \subset \Gamma(\{z_j;j \in J\})\), where \(\Delta_n = co\{e_0,e_1,\dots,e_n\}\) is an \(n\)-simplex and \(\Delta_J = \text{co} \{e_j;j\in J\}\). A multimap \(F:D \multimap X\) is a KKM map if \(\Gamma(A) \subset F(A)\) for all nonempty finite subsets \(A\) of \(D\). For a nonempty set \(I\), a map \(F:I \multimap X\) is a generalized KKM map if for each nonempty finite subset \(N\) of \(I\) there exists a function \(\sigma:N \to D\) such that \(\Gamma(\sigma(M)) \subset F(M)\) for all nonempty finite subsets \(M\) of \(N\). The authors give a characterization of generalized KKM maps and prove a new type of KKM theorems for them. Furthermore, it is shown that generalized KKM maps are closely related to certain general convexity of corresponding extended real-valued functions.