A short proof of the converse to the contraction principle and some related results
type: Item
instance of: scholarly article
title: A short proof of the converse to the contraction principle and some related results
zbMATH Open document ID: 0967.47035
DOI: 10.12775/TMNA.2000.014
published in: Topological Methods in Nonlinear Analysis
publication date: 2000-12-19T00:00:00Z
review text: This article deals with some reasoning in connection with Bessaga's well-known theorem about the converse to the constraction principle. The author offers a simpler proof of this theorem and gives the following modification: the map \(F\) of an arbitrary set \(\Omega\) into itself is an \(\alpha\)-similarity with respect to some complete metric \(d\) for \(\Omega\) (that means \(d(Fx, Fy)= \alpha d(x,y)\), \((x,y\in\Omega)\), \(\alpha\) is a fixed number from \((0,1)\)) if and only if \(F\) is injective and \(F\) has a unique fixed point. The situations when this result can be proved without the axiom of choice is also described.
Mathematics Subject Classification ID: 47H09
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 39B52
Mathematics Subject Classification ID: 39B72
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 03E25
zbMATH DE Number: 1545396
zbMATH Keywords: converse to the contraction principle
zbMATH Keywords: Bessaga's theorem
author: J. Jachymski
reviewed by: Peter P. Zabreĭko
MaRDI profile type: Publication
full work available at URL: tmna.2000.014
OpenAlex ID: W2547413383
description: scientific article; zbMATH DE number 1545396