Common best proximity points: global minimal solutions
type: Item
instance of: scholarly article
title: Common best proximity points: global minimal solutions
zbMATH Open document ID: 1276.47077
author: S. Sadiq Basha
published in: Top
publication date: 2013-04-23T00:00:00Z
review text: Given non-empty subsets \(A\) and \(B\) of a metric space, let \(S: A \to B\) and \(T: A \to B\) be non-self mappings. Taking into account the fact that, given any element \(x\) in \(A\), the distance between \(x\) and \(Sx,\) and the distance between \(x\) and \(Tx\) are at least \(d(A, B),\) a common best proximity point theorem affirms the existence of a global minimum of both functions \(x \mapsto d(x, Sx)\) and \(x \mapsto d(x, Tx)\) by imposing a common approximate solution of the equations \(Sx = x\) and \(T x = x\) to satisfy the constraint that \(d(x, Sx) = d(x, T x) = d( A, B )\). In this paper, the author discusses a common best proximity point theorem for a pair of nonself-mappings, one of which dominates the other proximally, thereby yielding common optimal approximate solutions of some fixed point equations when there is no common solution. A suitable example to demonstrate the validity of the hypotheses is given. The main result includes the common fixed point theorem due to \textit{G. Jungck} [Am. Math. Mon. 83, 261--263 (1976; Zbl 0321.54025)] for commuting continuous self-mappings, which in turn is an extension of Banach's contraction principle.
reviewed by: Hemant Kumar Nashine
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 54E40
zbMATH DE Number: 6156822
zbMATH Keywords: global optimal approximate solution
zbMATH Keywords: common best proximity point
zbMATH Keywords: common fixed point
zbMATH Keywords: proximally dominating mappings
MaRDI profile type: Publication
full work available at URL: s11750-011-0171-2
OpenAlex ID: W2058996947
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DOI: 10.1007/S11750-011-0171-2
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description: scientific article; zbMATH DE number 6156822