Description

• Title:
  A short proof of the converse to the contraction principle and some related results
• Creator:
  Jachymski, Jacek
• Description:
  We simplify a proof of Bessaga's theorem given in the monograph of Deimling. Moreover, our argument let us also obtain the following result. Let $F$ be a selfmap of an arbitrary set $\Omega$ and $\alpha\in (0,1)$. Then $F$ is an $\alpha$-similarity with respect to some complete metric $d$ for $\Omega$ (that is, $d(Fx,Fy)=\alpha d(x,y)$ for all $x,y\in\Omega$) if and only if $F$ is injective and $F$ has a unique fixed point. Finally we present that the converse to the Contraction Principle for bounded spaces is independent of the Axiom of Choice.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2016 Topological Methods in Nonlinear Analysis
• Date:
  2000.03.01
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 15, No 1 (March 2000); 179 - 186;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2000.014/10007
• Subject:
  Fixed point;
  periodic point;
  Banach's contraction;
  similarity;
  Schröder's functional equation and inequality;
  Primary 04A25;
  39B52;
  39B72;
  47H10;
  54H25
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2000.014
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other