Generalized KKM theorems and common fixed point theorems
type: Item
instance of: scholarly article
title: Generalized KKM theorems and common fixed point theorems
zbMATH Open document ID: 1229.47066
author: Lai-Jiu Lin
author: Chih-Sheng Chuang
author: Zenn-Tsuen Yu
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2011-07-22T00:00:00Z
review text: Let \(Y\) be an arbitrary nonempty set and \(X\) be a nonempty convex set in a vector space. A multivalued mapping \(T:Y\multimap X\) is said to be a generalized \(KKM\)-mapping if for any nonempty finite subset \(\{y_1,\dots,y_n\}\) of \(Y\), there exists a subset \(\{x_1,\dots,x_n\}\) in \(X\) such that \(\text{co}\; \{x_i:i\in I\} \subseteq \bigcup\limits_{i\in I}T(y_i)\) for each nonempty subset \(I\) of \(\{1,\dots,n\}\). If \(\mathcal{T}\) is a family of multivalued mappings from \(Y\) into \(X\), then \(\mathcal{T}\) is said to be equi-KKM if for any nonempty finite subset \(\{y_1,\dots,y_n\}\) of \(Y\), there exists a subset \(\{x_1,\dots,x_n\}\) in \(X\) such that \(\text{co}\; \{x_i:i\in I\} \subseteq \bigcup\limits_{i\in I}T(y_i)\) for each nonempty subset \(I\) of \(\{1,\dots,n\}\) and each \(T\in \mathcal{T}\). The authors show first that, if \(T\) is a generalized \(KKM\) multivalued mapping defined on an arbitrary set and with closed values into a closed subset of a topological vector space, then the family of its values has the finite intersection property. Relying on this result, the authors establish several intersection theorems for the family of values of a generalized \(KKM\) mapping, and common fixed point theorems for families of equi-\(KKM\) mappings.
Mathematics Subject Classification ID: 47H04
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 5930952
zbMATH Keywords: generalized \(KKM\) mapping
zbMATH Keywords: equi-\(KKM\) family of mappings
zbMATH Keywords: common fixed point theorem
reviewed by: Mircea Balaj
MaRDI profile type: Publication
full work available at URL: j.na.2011.05.044
OpenAlex ID: W1989508615
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DOI: 10.1016/J.NA.2011.05.044
description: scientific article; zbMATH DE number 5930952