KKM mappings in metric spaces (Q1764857)

For a set \(X\), let \(\langle X\rangle\) denote the set of all nonempty finite subsets of \(X\). A bounded subset \(A\) of a metric space \((M,d)\) is said to be subadmissible if \(c_0(D)\subseteq A\) for each \(D\in\langle A\rangle\), where \(c_0(D)\equiv\bigcap\{B\subseteq M\): \(B\) is closed ball in \(M\) such that \(D\subset B\}\). For a subadmissible subset \(X\) of a metric space \((M,d)\), a multifunction \(G: X\to M\) is called a KKM mapping if \(c_0(A)\subset G(A)\) for each \(A\in\langle X\rangle\). More generally, if \(Y\) is a topological space and \(G: X\to Y\), \(F: X\to Y\) are two multifunctions such that \(F(c_0(A))\subseteq G(A)\) for any \(A\in \langle X\rangle\), then \(G\) is called a generalized KKM mapping with respect to \(F\). If the multifunction \(F: X\to Y\) satisfies that for any generalized KKM mapping \(G: X\to Y\) with respect to \(F\), the family \(\{\text{cl\,}G(x): x\in X\}\) has the finite intersection property, then \(F\) is said to have the KKM property. Define the class \(\text{KKM}(X,Y)= \{F: X\to Y\): \(F\) has the KKM property\}. In this paper, the authors obtain some fixed point theorems for this class (Theorem 2.1 and 2.2). As an application, they deduce a coincidence point result (Theorem 2.5). They also obtain a generalized Fan's matching theorem (Theorem 2.7), a generalized Fan-Browder type theorem (Corollary 2.8), and a new version of Fan's best approximation theorem (Theorem 2.13). These theorems improve the similar results of \textit{G. X.-Z. Yuan} [J. Math. Anal. Appl. 235, No. 1, 315--325 (1999; Zbl 0930.54035)], \textit{W. A. Kirk}, \textit{B. Sims} and \textit{G. X.-Z. Yuan} [Nonlinear Anal., Theory Methods Appl. 39A, No. 5, 611--627 (2000; Zbl 1068.47072)] and \textit{G. Isac} and \textit{G. X.-Z. Yuan} [Discuss. Math., Differ. Incl. 19, No. 1--2, 17--33 (1999; Zbl 0962.47022)], respectively.