Generalized G-KKM theorems in generalized convex spaces and their applications (Q1604454)

Let \(E\) be a topological space and \(\emptyset\not=D\subset E\) and \(\Gamma\) a \(G\)-convex structure on \((E,D)\) as defined by \textit{S. Park} [J. Korean Math. Soc. 31, No.~3, 493-519 (1994; Zbl 0829.49002)]. A subset \(C\subset E\) is said to be \(G\)-convex if \(\Gamma(N)\subset C\) for each finite subset \(N\) of \(C\cap D\). Let then \(\Gamma\) be a \(G\)-structure on \((E,D)\). The author considers the following situation: \(X\) is a nonempty set, \(\emptyset\not=Y\subset E\), \(K\subset Y\) is nonempty and compact, and \(T,S\) are mappings from \(X\) to the nonempty subsets of \(Y\) such that for each \(x\in X\) the intersection of \(T(x)\) with any compact subset of \(Y\) is closed and such that the following holds: whenever \(x_1,\dotsc,x_n\in X\) there are \(y_1,\dotsc,y_n\) and a compact \(G\)-convex subset \(L\) of \(Y\) with \(y_i\in S(x_i)\cap D\cap L\) such that \(\Gamma(\{y_i|\;i\in F\})\subset\bigcup_{i\in F}T(x_i)\) for each finite subset \(F\) of \(\{1,\dotsc,n\}\) and \(L\cap\bigcap_{x\in S^{-1}(L)}T(x)\subset K\). The author proves that \(K\cap\bigcap_{x\in X}T(x)\not=\emptyset\) in this situation.