Generalized KKM-type theorems in GFC-spaces and applications
A triplet \((X, Y, \Phi )\) is called a generalized finitely continuous topological space (\(GFC\)-space in short) if \(X\) is a topological space, \(Y\) is a nonempty set and \(\Phi\) is a family of continuous function \(\phi:\Delta_n \rightarrow X\), \(n\in \mathbb{N}\), such that to each finite subset \(N=\{ y_0,\dots, y_n\}\) of \(Y\) is associated a function \(\phi_N:\Delta_n \rightarrow X\) from the family \(\Phi\). Let \((X, Y, \Phi )\) be a \(GFC\)-space, \(Z\) be a topological space, \(F: Y\rightarrow 2^Z\) and \(T:X \rightarrow 2^Z\). \(F\) is said to be a \(T-KKM\) mapping if, for each finite subset \(N=\{ y_0,\dots, y_n\}\) of \(Y\) and each \(\{y_{i_0},\dots, y_{i_k}\}\subseteq N\), one has \(T(\phi_N(\Delta_k))\subseteq \bigcup_{j=0}^k F(y_{i_j})\). A set-valued mapping \(T:X \rightarrow 2^Z\) has the generalized \(KKM\) property if, for each \(T-KKM\) mapping \(F: Y\rightarrow 2^Z\), the family \(\{ \overline{F(y)}: y\in Y\}\) has the finite intersection property. The main result of the present paper is the following \(KKM\)-type theorem: If \(T: X \rightarrow 2^Z\), \(F:Y\rightarrow 2^Z\) and \(S:Y\rightarrow 2^X\), then \(\overline{T(S(Y))}\cap \bigcap_{\in Y} F(y)\neq \emptyset\) whenever the following conditions are satisfied: (i) for each finite subset \(N=\{ y_0,\dots, y_n\}\) of \(Y\), \(\phi_N(\Delta_k)\subseteq S(Y)\). (ii) \(\overline{T(S(Y))}\) is compact. (iii) \(T\) has the generalized \(KKM\) property. (iv) \(F\) is \(T-KKM\) and transfer compactly closed-valued.
transfer compactly closed-valued (open-valued) mappings
scientific article; zbMATH DE number 5569417
scientific article