Generalized G-KKM theorems in generalized convex spaces and their applications
type: Item
instance of: scholarly article
title: Generalized G-KKM theorems in generalized convex spaces and their applications
zbMATH Open document ID: 1006.47041
author: Xieping Ding
published in: Journal of Mathematical Analysis and Applications
publication date: 2002-07-04T00:00:00Z
review text: Let \(E\) be a topological space and \(\emptyset\not=D\subset E\) and \(\Gamma\) a \(G\)-convex structure on \((E,D)\) as defined by \textit{S. Park} [J. Korean Math. Soc. 31, No.~3, 493-519 (1994; Zbl 0829.49002)]. A subset \(C\subset E\) is said to be \(G\)-convex if \(\Gamma(N)\subset C\) for each finite subset \(N\) of \(C\cap D\). Let then \(\Gamma\) be a \(G\)-structure on \((E,D)\). The author considers the following situation: \(X\) is a nonempty set, \(\emptyset\not=Y\subset E\), \(K\subset Y\) is nonempty and compact, and \(T,S\) are mappings from \(X\) to the nonempty subsets of \(Y\) such that for each \(x\in X\) the intersection of \(T(x)\) with any compact subset of \(Y\) is closed and such that the following holds: whenever \(x_1,\dotsc,x_n\in X\) there are \(y_1,\dotsc,y_n\) and a compact \(G\)-convex subset \(L\) of \(Y\) with \(y_i\in S(x_i)\cap D\cap L\) such that \(\Gamma(\{y_i|\;i\in F\})\subset\bigcup_{i\in F}T(x_i)\) for each finite subset \(F\) of \(\{1,\dotsc,n\}\) and \(L\cap\bigcap_{x\in S^{-1}(L)}T(x)\subset K\). The author proves that \(K\cap\bigcap_{x\in X}T(x)\not=\emptyset\) in this situation.
Mathematics Subject Classification ID: 47H04
Mathematics Subject Classification ID: 49J35
Mathematics Subject Classification ID: 47J05
zbMATH DE Number: 1763693
zbMATH Keywords: KKM mapping
zbMATH Keywords: minimax inequality
zbMATH Keywords: saddle point
zbMATH Keywords: coincidence
zbMATH Keywords: \(G\)-convex structure
reviewed by: Christian Fenske
MaRDI profile type: Publication
full work available at URL: jmaa.2000.7207
OpenAlex ID: W1972852620
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DOI: 10.1006/JMAA.2000.7207
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description: scientific article; zbMATH DE number 1763693