Fixed point theorems for the class S-KKM mappings in abstract convex spaces
type: Item
instance of: scholarly article
title: Fixed point theorems for the class S-KKM mappings in abstract convex spaces
zbMATH Open document ID: 1116.47046
author: N. E. Zubov
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2007-01-09T00:00:00Z
review text: An abstract convex space \((X,{\mathcal C})\) consists of a nonempty set \(X\) and a family \({\mathcal C}\) of subsets of \(X\) such that \(X\) and \(\varnothing\) belong to \({\mathcal C}\) and \({\mathcal C}\) is closed under arbitrary intersection. An abstract convex space \((X,{\mathcal C})\) is called a \(\Phi\)-space if \(X\) is a uniform space and for each neighbourhood \(V\) there is a \(\Phi\)-mapping \(T:X\multimap X\) such that \(\text{Graph}T\subseteq V\). For a \(\Phi\)-space \((X,{\mathcal C})\), we say that a mapping \(F:X\multimap X\) has the approximate fixed point property if for any \(U\in{\mathcal U}\), where \({\mathcal U}\) is a basis of the uniform structure of \(X\), there exists \(x\in X\) such that \(U[x]\cap F(x)\neq\varnothing\). In this paper, the authors define KKM mappings and S-KKM mappings for abstract convex spaces and establish some approximate fixed point theorems for multifunctions with the S-KKM property on \(\Phi\)-spaces. They also establish a new version of Sadovskij's fixed point theorem, similar to that of \textit{M.\,A.\thinspace Khamsi} [Nonlinear Anal., Theory Methods Appl.\ 53, No.\,6(A), 829--837 (2003; Zbl 1028.47042)] in abstract convex spaces.
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 54H25
zbMATH DE Number: 5083392
zbMATH Keywords: KKM property
zbMATH Keywords: abstract convex space
zbMATH Keywords: \(\Phi\)-space
zbMATH Keywords: approximate fixed point property
zbMATH Keywords: condensing maps
reviewed by: T. D. Narang
MaRDI profile type: Publication
full work available at URL: j.na.2005.11.005
OpenAlex ID: W2012316295
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DOI: 10.1016/J.NA.2005.11.005
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description: scientific article; zbMATH DE number 5083392