A characterization for Meir-Keeler contractions
type: Item
instance of: scholarly article
title: A characterization for Meir-Keeler contractions
zbMATH Open document ID: 1402.54041
author: Hassan Khandani
publication date: 2018-09-21T00:00:00Z
review text: Let \((X,d)\) be a complete metric space. A mapping \(T:X\to X\) is called a Meir-Keeler contraction if for each \(\epsilon>0\) there exists a \(\delta>0\) such that \(d(Tx,Ty)<\epsilon\) for all \(x,y\in X\) with \(\epsilon\leq d(x,y)\leq\epsilon+\delta\). A sequence \((x_n)\) in \(X\) is said to be asymptotically regular if \(\lim_{n\to\infty}d(x_n,x_{n+1})=0\). The author provides the following characterization of Meir-Keeler contractions: A mapping \(T:X\to X\) is a Meir-Keeler contraction iff (1) there exists an \(x_0\in X\) such that the sequence \((T^nx_0)_n\) is asymptotically regular, (2) for any sequences \((x_n)\) and \((y_n)\) such that \((d(x_n,y_n))\) and \((d(Tx_n,Ty_n))\) converge to the same limit \(L\) with \(L\leq d(Tx_n,Ty_n)\) for all \(n\) we have that \(L=0\), (3) \(T\) is continuous.
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 6941002
zbMATH Keywords: Meir-Keeler contraction
zbMATH Keywords: fixed point
zbMATH Keywords: asymptotically regular sequence
reviewed by: Christian Fenske
MaRDI profile type: Publication
full work available at URL: s12215-017-0312-z
OpenAlex ID: W2625223266
cites work: Topics in Fixed Point Theory
cites work: Remarks on some recent fixed point theorems
cites work: Q5628889
cites work: New results and generalizations for approximate fixed point property and their applications
cites work: Fixed point theorems for multivalued mappings on complete metric spaces
cites work: On Fixed and Periodic Points Under Contractive Mappings
cites work: On characterizations of Meir-Keeler contractive maps
cites work: On Contractive Mappings
cites work: Equivalent conditions and the Meir-Keeler type theorems
cites work: A new approach to the study of fixed point theory for simulation functions
cites work: New fixed point theorem under \(R\)-contractions
cites work: A theorem on contraction mappings
cites work: Some theorems on weakly contractive maps.
DOI: 10.1007/S12215-017-0312-Z
published in: Rendiconti del Circolo Matematico di Palermo
Recommended article: On characterizations of Meir-Keeler contractive maps
Recommended article: Q4397528
Recommended article: A generalization of the Meir-Keeler type contraction
Recommended article: Analytical Meir-Keeler type contraction mappings and equivalent characterizations
Recommended article: On generalizations of the Meir-Keeler type contraction maps
Recommended article: Some classes of Meir-Keeler contractions
Recommended article: Q5455203
Recommended article: Q3211973
Recommended article: A generalized Meir-Keeler-type contraction on partial metric spaces
Recommended article: Q5039955
description: scientific article; zbMATH DE number 6941002