Remarks on weakly KKM maps in abstract convex spaces
type: Item
instance of: scholarly article
title: Remarks on weakly KKM maps in abstract convex spaces
zbMATH Open document ID: 1186.47045
DOI: 10.1155/2008/423596
author: Sehie Park
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2008-08-19T00:00:00Z
full work available at URL: https://eudml.org/doc/45275
review text: 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 \(z\in D\).
reviewed by: Valeriu Popa
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 46A55
Mathematics Subject Classification ID: 47N10
zbMATH DE Number: 5313277
zbMATH Keywords: abstract convex space
zbMATH Keywords: KKM principle
zbMATH Keywords: KKM space
zbMATH Keywords: weakly KKM map
MaRDI profile type: Publication
OpenAlex ID: W2043491427
Wikidata QID: Q58644757
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description: scientific article; zbMATH DE number 5313277