Best approximation theorems for nonexpansive and condensing mappings in hyperconvex spaces
type: Item
instance of: scholarly article
title: Best approximation theorems for nonexpansive and condensing mappings in hyperconvex spaces
zbMATH Open document ID: 1163.41005
author: Jack T. Markin
author: Nazeer Shahzad
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2009-03-11T00:00:00Z
review text: Let \((M,d)\) be a hyperconvex metric space, \(X\subset M\) and \(F:X\rightarrow 2^{M}\) a given multivalued mapping. The authors consider the following problems in this setting: 1) The best approximation problem: Find the condition on \(F\) and on a set \(A\subset M\) such that there is \(x\in A\) satisfying \(d(x,F(x))\leq d(y,F(x)),\) for all \(y\in A;\) 2) The invariant approximation problem consisting in finding conditions for the mapping \(F\) with invariant set \(A\) and with \(Fix(F)\neq\emptyset,\) implying that \(Fix(F)\cap P_{A}(p)\neq\emptyset,\) where \(P_{A}(p)\) is the metric projection of \(p\in Fix(F)\) onto A; 3) The best proximity pair problem: Find conditions on \(F:A\rightarrow2^{B},\) \(A\) and \(B\) implying that there is a point \(x\in A\) such that \(d(x,F(x))=\inf \{a,b):a\in A,\) \(b\in B\}.\) These three problems are studied when the multivalued mapping \(F\) is condensing or nonexpansive, and one obtains sufficient conditions for the existence of the solutions of these problems. For example, if the set \(X\subset M\) is admissible (respectively, bounded externally hyperconvex) and \(F:X\rightarrow2^{M}\) is condensing, continuous in the Hausdorff metric of \((M,d)\) and the values of \(F\) are nonempty bounded externally hyperconvex (respectively admissible) subset of \((M,d),\) then \(F\) has a point of best approximation (Th. 3.1).
reviewed by: Costică Mustăţa
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 5528004
zbMATH Keywords: Best approximations
zbMATH Keywords: invariant sets
zbMATH Keywords: best proximity pairs
zbMATH Keywords: nonexpansive mappings
zbMATH Keywords: condensing mappings
zbMATH Keywords: hyperconvex spaces
MaRDI profile type: Publication
full work available at URL: j.na.2008.03.045
OpenAlex ID: W2035781160
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DOI: 10.1016/J.NA.2008.03.045
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description: scientific article; zbMATH DE number 5528004