Remarks on recent fixed point theorems for compatible maps
type: Item
instance of: scholarly article
title: Remarks on recent fixed point theorems for compatible maps
zbMATH Open document ID: 0868.54036
DOI: 10.1155/S0161171296001111
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 1996-09-30T00:00:00Z
full work available at URL: https://eudml.org/doc/47681
review text: Let \(A,B,S,T\) be selfmaps of a metric space \((X,d)\). The pair \(\{A,B\}\) is said to be an \((\varepsilon,\delta)\)-contraction relative to the pair \(\{S,T\}\) if \(A(X) \subseteq T(X)\), \(B(X) \subseteq S(X)\) and there exists a function \(\delta: (0,\infty) \to (0,\infty)\) such that \(\delta (\varepsilon)> \varepsilon\) for all \(\varepsilon\), and for \(x,y\) in \(X\), we have \(\varepsilon \leq m(x,y) < \delta (\varepsilon)\) implies \(d(Ax,By)< \varepsilon\) and \(Ax=By\) whenever \(Sx=Ty\). Here \(m(x,y)= \max \{d(Sx,Ty)\), \(k\cdot d(Sx,Ax)\), \(k\cdot d(Ty,By)\), \({1\over 2} (d(Sx,By) + d(Ax,Ty))\}\), where \(k\in(0,1)\). Under this assumption, the authors prove that if one of \(A(X)\), \(B(X)\), \(S(X)\), \(T(X)\) is a complete subspace of \(X\), then \(A\) and \(S\) (resp. \(B\) and \(T)\) have a coincidence point. \(A,B,S,T\) have a unique common fixed point if \(A\) and \(S\) (resp. \(B\) and \(T)\) commute on their set of coincidence points. A similar conclusion is given if the four maps satisfy generalized contractive conditions.
reviewed by: Salvatore Sessa
Mathematics Subject Classification ID: 54H25
zbMATH DE Number: 930060
zbMATH Keywords: coincidence point
zbMATH Keywords: generalized contractive conditions
author: Shyam Lal Singh
author: Veena Chadha
author: S. N. Mishra
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 930060