Some results for the family 2-gKKM\((X,Y)\) and the \(\Phi\)-mapping in hyperconvex metric spaces
Let \(X\) be a bounded set of a metric space \((M,d)\). The admissible hull of \(X\) is defined by \(\text{ad}(X)\cap\{B\subset M: B\) is a closed ball in \(M\) such that \(X\subset B\}\). Let \(Z\) be a nonempty set and \(Y\) a metric space. A set-valued mapping \(F: Z\to 2^Y\) is called a generalized 2-gKKM map if for each \(z_1,z_2\in Z\), there exist \(y_1,y_2\in Y\) such that \(y_1\in F(z_1)\), \(y_2\ni F(z_2)\) and \(\text{ad}(\{y_1,y_2\})\subset F(z_1)\cup F(z_2)\). Let \(X\) be a metric space, \(Z\) a nonempty set and \(Y\) a topological space. If \(T: X\to 2^Y\), \(F: Z\to 2^Y\) are two set-valued mappings satisfying that for each \(z_1,z_2\in Z\), there exist \(x_1,x_2\in X\) such that \(T(x_1)\subset F(z_1), T(x_2)\subset F(z_2)\) and \(T(\text{ad}(\{x_1,x_2\})\subset F(z_1)\cup F(z_2)\), then \(F\) is called a generalized 2-gKKM mapping with respect to \(T\). If the set-valued mapping \(T: X\to 2^Y\) satisfies the requirement that for any generalized 2-gKKM mapping \(F: Z\to 2^Y\) with respect to \(T\), the family \(\{\overline{F(z)}: z\in Z\}\) has the two-intersection property, then \(T\) is said to have the 2-gKKM property. In this paper, a fixed point theorem for this family without compact assumption in hyperconvex space is proved. Let \(X\) be a topological space, and \(Y\) a hyperconvex metric space. A set-valued mapping \(T: X\to 2^Y\) is called a \(\Phi\)-mapping if there exists a set-valued mapping \(F: X\to 2^Y\) such that: {\parindent=6mm \begin{itemize}\item[(i)] for each \(x\in X\), \(N\in \langle F(x)\rangle\) implies \(\text{ad}(N)\subset T(x)\), and \item[(ii)] \(X= \bigcup_{y\in Y} \operatorname{int}F^{-1}(y)\). \end{itemize}} In the last part of this paper, some variational inequalities and minimax inequalities for the family 2-gKKM\((X,Y)\), where 2-gKKM\((X,Y)\) denotes the set \(\{T: X\to 2^Y\mid T\) has the 2-gKKM property\}, and the \(\Phi\)-mapping are proved.
scientific article; zbMATH DE number 5355001