Common fixed point theorems of compatible and weakly compatible maps in Menger spaces
type: Item
instance of: scholarly article
title: Common fixed point theorems of compatible and weakly compatible maps in Menger spaces
zbMATH Open document ID: 1172.54027
author: Jin-Xuan Fang
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2009-07-01T00:00:00Z
review text: The main result of the paper is the following common fixed point theorem for compatible and weakly compatible self-mappings on Menger spaces under \(\varphi\)-contractive conditions, which improves the result of \textit{B. Singh} and \textit{S. Jain} [J. Math. Anal. Appl. 301, No.~2, 439--448 (2005; Zbl 1068.54044)]: Let \(\Phi\) be the class of all functions \(\varphi: \mathbb R_+\rightarrow \mathbb R_+\), such that \(\varphi\) is non-decreasing, upper semi-continuous from the right and \[ \sum_{n=0}^{\infty}\varphi^n(t)<\infty \] for all \(t>0.\) Let \(P, Q, L, M\) be self-maps on a complete Menger space \((X, {\mathcal F},\Delta)\) with \(\Delta\) a continuous t-norm of Hadžić-type. If (i)\ \(L(X)\subset Q(X), M(X)\subset P(X)\); (ii) either \(P\) or \(L\) is continuous; (iii)\ \((L, P)\) is compatible and \((M, Q)\) is weakly compatible; (iv) there exists \(\phi\in \Phi\) such that \[ F_{Lx, My}(\phi(t))\geq \min\{F_{Px, Lx}(t), F_{Qy, My}(t), F_{Px, qy}(t), F_{Qy, Lx}(\beta t), [F_{Px, \xi }* F_{\xi, My}]((2-\beta)t)\} \] for all \(x, y, \xi \in X\), \(\beta \in (0, 2)\) and \(t>0\), where \[ F_1 * F_2 (t):=\sup_{t_1+t_2=t} \min\{F_1 (t_1), F_2 (t_2)\} \quad (t>0), \] then \(L, M, P\) and \(Q\) have a unique common fixed point.
reviewed by: Dorel Miheţ
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 54E70
zbMATH DE Number: 5573704
zbMATH Keywords: Menger space
zbMATH Keywords: common fixed points
zbMATH Keywords: compatible maps
zbMATH Keywords: weakly compatible maps
MaRDI profile type: Publication
full work available at URL: j.na.2009.01.018
OpenAlex ID: W2052989327
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DOI: 10.1016/J.NA.2009.01.018
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description: scientific article; zbMATH DE number 5573704