The generalized Banach contraction conjecture in \(b\)-metric spaces
type: Item
instance of: scholarly article
title: The generalized Banach contraction conjecture in \(b\)-metric spaces
zbMATH Open document ID: 1398.54081
author: Nguyen Van Dung
author: Vo Thi Le Hang
published in: Journal of Fixed Point Theory and Applications
publication date: 2018-04-11T00:00:00Z
review text: The generalized Banach contraction conjecture on \(b\)-metric spaces states as follows:\ Let \((X,D,K)\) be a complete \(b\)-metric space, \(T: X\to X\) a map, and \(J\) a positive integer. If \(\min\,\{ D(T^kx,T^ky): k=1,\dots,J\} \leq M D(x,y)\) for all \(x,y\in X\) and for some \(M\in (0,1)\), then \(T\) has a fixed point. When \(M\in (0,1/K)\), partial answers to the above conjecture are given, by using combinatorial arguments. First, the conjecture is true in the special case where \(X\) is a complete metric-type space. Next, it is true for the case \(J=2\) and, moreover, also for the case \(J=3\) if \(T\) is assumed to be continuous.
reviewed by: In-Sook Kim
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 47H09
Mathematics Subject Classification ID: 54E99
zbMATH DE Number: 6858693
zbMATH Keywords: fixed point
zbMATH Keywords: contraction
zbMATH Keywords: \(b\)-metric space
Wikidata QID: Q123011135
MaRDI profile type: Publication
full work available at URL: s11784-018-0489-6
OpenAlex ID: W2782829709
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DOI: 10.1007/S11784-018-0489-6
description: scientific article; zbMATH DE number 6858693