Fixed points and completeness in metric and in generalized metric spaces
Abstract
The famous Banach Contraction Principle holds in complete metric spaces, but completeness is not a necessary condition -- there are incomplete metric spaces on which every contraction has a fixed point. The aim of this paper is to present various circumstances in which fixed point results imply completeness. For metric spaces this is the case of Ekeland variational principle and of its equivalent - Caristi fixed point theorem. Other fixed point results having this property will be also presented in metric spaces, in quasi-metric spaces and in partial metric spaces. A discussion on topology and order and on fixed points in ordered structures and their completeness properties is included as well.
Keywords
Cite
@article{arxiv.1508.05173,
title = {Fixed points and completeness in metric and in generalized metric spaces},
author = {S. Cobzaş},
journal= {arXiv preprint arXiv:1508.05173},
year = {2019}
}
Comments
89 pages, Additions in v5: paper reorganized, new subsections: Takahashi min. princ, strong EkVP, Aryutunov princ, weak sharp minima, Bao-Cobzas-Soubeyran. Published in Fundamental'naya i Prikladnaya Matematica vol. 22 (2018), no. 1, 127--215 (in Russian)