Best proximity results in regular cone metric spaces
type: Item
instance of: scholarly article
title: Best proximity results in regular cone metric spaces
zbMATH Open document ID: 1235.54036
publication date: 2012-02-09T00:00:00Z
review text: In 1980, \textit{B. Rzepecki} introduced a generalized metric \(d_{E}\) on a set \(X\) in a way that \(d_{E} : X \times{X} \longrightarrow K\), replacing the set of real numbers with a Banach space \(E\) in the metric function where \(K\) is a normal cone in \(E\) with a partial order \(\leq\) [Publ. Inst. Math., Nouv. Sér. 28(42), 179--186 (1980; Zbl 0482.47029)]. Seven years later, \textit{S.-D. Lin} considered the notion of \(K\)-metric spaces by replacing the set of non-negative real numbers with a cone \(K\) in the metric function [Indian J. Pure Appl. Math. 18, 685--690 (1987; Zbl 0622.47057)]. In 1997, \textit{P. P. Zabrejko} gave a survey of the theory of \(K\)-metric spaces [Collect. Math. 48, No.4-6, 825--859 (1997; Zbl 0892.46002)]. Ten years after Zabrejko's work, \textit{L.-G. Huang} and \textit{X. Zhang} [J. Math. Anal. Appl. 332, No. 2, 1468--1476 (2007; Zbl 1118.54022)] announced the notion of a cone metric space by replacing real numbers with an ordered Banach space, which is the same as the definition of Rzepecki, or of Lin, or of Zabreiko. \textit{A. Sönmez} [Appl. Math. Lett. 23, No. 4, 494--497 (2010; Zbl 1187.54022)] defined the notion of distance between two subsets \(A\) and \(B\) of a cone metric space \(X\). In the paper under review, the authors establish some conditions which guarantee the existence of best proximity points for cyclic contraction mappings on regular cone metric spaces.
reviewed by: Hüseyin Çakallı
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 54E35
zbMATH DE Number: 6004387
zbMATH Keywords: lower bounds
zbMATH Keywords: contraction-type mappings
zbMATH Keywords: fixed point theorems
zbMATH Keywords: cone L-function
zbMATH Keywords: regular cone metric space
author: R. H. Haghi
author: Vladimir Rakočević
author: Sh. Rezapour
author: Nazeer Shahzad
MaRDI profile type: Publication
full work available at URL: s12215-011-0050-6
OpenAlex ID: W2142367722
cites work: Convergence and existence results for best proximity points
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cites work: Q4470093
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cites work: Some notes on the paper ``Cone metric spaces and fixed point theorems of contractive mappings''
cites work: Best proximity points: convergence and existence theorems for \(p\)-cyclic mappings
cites work: Common fixed points in cone metric spaces
DOI: 10.1007/S12215-011-0050-6
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 6004387