A continuation method for weakly contractive mappings under the interior condition
type: Item
instance of: scholarly article
title: A continuation method for weakly contractive mappings under the interior condition
zbMATH Open document ID: 1181.54046
DOI: 10.1155/2009/809315
author: David Ariza-Ruiz
author: Antonio Jiménez-Melado
published in: Fixed Point Theory and Applications
publication date: 2009-11-10T00:00:00Z
full work available at URL: https://eudml.org/doc/45704
review text: Let \(X\) be a complete metric space and \(U\) open in \(X\). In this paper, it is shown that the property of having a fixed point is invariant by a certain class of homotopies for weakly contractive maps \(H : \bar{U}\times [0,1] \to X\) satisfying the interior condition: \[ \begin{aligned}\text{(I-C)}\quad &\text{there exists \(r > 0\) such that \(x \neq H(x,t)\) for every \(t \in [0,1]\), \(f(x) \not\in \bar{U}\),}\\ &\text{and \(x \in U_r = \{y \in U : d(y,\partial{U}) < r\}\).} \end{aligned} \] The notion of weakly contractive maps was introduced by \textit{J.\,Dugundji} and \textit{A.\,Granas} [Bull.\ Greek Math.\ Soc.\ 19, 141--151 (1978; Zbl 0417.54010)]. An analogous result is also obtained for a class of homotopies satisfying a generalized contractive condition which was introduced by \textit{G.\,E. Hardy} and \textit{T.\,D. Rogers} [Can.\ Math.\ Bull.\ 16, 201--206 (1973; Zbl 0266.54015)].
reviewed by: Marlène Frigon
Mathematics Subject Classification ID: 54H25
zbMATH DE Number: 5627868
zbMATH Keywords: fixed point
zbMATH Keywords: weakly contractive maps
zbMATH Keywords: continuation method
zbMATH Keywords: interior condition
Wikidata QID: Q59219949
MaRDI profile type: Publication
OpenAlex ID: W2021484915
cites work: Fixed Point Theorems for Various Classes of 1-Set-Contractive and 1-Ball- Contractive Mappings in Banach Space
cites work: Fixed points of condensing functions
cites work: Continuation method for contractive maps
cites work: Fixed point theory and applications
cites work: Q4356004
cites work: Q4786432
cites work: Fixed point theorems under the interior condition
cites work: A Mönch type fixed point theorem under the interior condition
cites work: Q5324474
cites work: Q4844908
cites work: Fixed point theory for generalized contractions on spaces with two metrics
cites work: Q3207726
cites work: A Comparison of Various Definitions of Contractive Mappings
cites work: A Generalization of a Fixed Point Theorem of Reich
description: scientific article; zbMATH DE number 5627868