On S-KKM property and-related topics
Let \(X\) be a convex subset of a linear space and \(Y\) be a topological space. If \(F,T: X\to 2^Y\) are two multifunctions such that \(T(\text{co }A)\subseteq F(A)\) for any \(A\in\langle X\rangle\) (the class of all finite subsets of \(X\)), then \(F\) is called a generalized KKM mapping with respect to \(T\). If the multifunction \(T: X\to 2^Y\) satisfies the requirement that for any generalized KKM mapping \(F\) with respect to \(T\) the family \(\{\overline{Fx}: x\in X\}\) has the finite intersection property, where \(\overline{Fx}\) denotes the closure of \(Fx\), then \(T\) is said to have the KKM property. The class \(\text{KKM}(X,Y)\) is then defined to be the set \(\{T: X\to 2^Y\): \(T\) has the KKM property\}. In Section 2 of the paper the authors introduce the class S-KKM\((X,Y,Z)\) consisting of all multifunctions \(T: Y\to 2^Z\) that have the S-KKM property. They show that S-KKM\((X,Y,Z)\) contains \(\text{KKM}(Y,Z)\) for any \(S: X\to 2^Y\). Section 3 is devoted to the establishment of some fixed point theorems for multifunctions in S-KKM\((X,Y,Z)\). In Section 4, they deduce matching theorems and KKM-type theorems, extending some results by Ky Fan and Park. For the last Section, they apply an KKM-type theorem in Section 4 to the generalization of the Fan-Browder fixed point theorem.
scientific article; zbMATH DE number 1246312