Remarks on weakly KKM maps in abstract convex spaces
Let \(\langle D\rangle\) denote the set of all nonempty finite subsets of a set \(D\). An abstract convex space \((E,D;\Gamma)\) consists of a set \(E\), a nonempty set \(D\) and a multimap \(\Gamma:\langle D\rangle\multimap E\) with nonempty values. A KKM space is an abstract convex topological space satisfying the KKM principle. Let \((E,D;\Gamma)\) be an abstract convex space and \(Z\) be a set. For a multimap \(F: E\multimap Z\) with nonempty values, if a multimap \(G: D\multimap Z\) satisfies \(F(x)\cap G(A)\neq\emptyset\) for all \(A\in \langle D\rangle\) and all \(x\in \Gamma(A)\), then \(G\) is called a weakly KKM map with respect to \(F\). In this paper, the author obtains variants of the KKM principle for KKM spaces related to weakly KKM maps and some applications of them. For example, the following theorem generalizes the results by \textit{F.\,C.\thinspace Liu} [J.~Math.\ Anal.\ Appl.\ 155, No.\,2, 420--436 (1991; Zbl 0734.47030)], \textit{M.\,Balaj} [ibid.\ 294, No.\,1, 237--245 (2004; Zbl 1053.54028)] and \textit{G.\,S.\thinspace Tang}, \textit{Q.\,B.\thinspace Zhang} and \textit{C.\,Z.\thinspace Cheng} [ibid.\ 334, No.\,2, 1481--1491 (2007; Zbl 1123.49002)]. Theorem 4.3. Let \((X,D;\Gamma)\) be a compact KKM space, \(Y\) a nonempty set, and \(F: X\multimap Y\) and \(G: D\to Y\) maps such that (1) \(G\) is weakly KKM map with respect to \(F\), (2) for each \(z\in d\), the set \(\{x\in X: F(x)\cap G(z)\neq\emptyset\}\) is closed. Then there exists an \(x_0\in X\) such that \(F(x_0)\cap G(z)\neq \emptyset\) for each \(z\in D\).
scientific article; zbMATH DE number 5313277
scientific article