Two refinements of Frink's metrization theorem and fixed point results for Lipschitzian mappings on quasimetric spaces
type: Item
instance of: scholarly article
title: Two refinements of Frink's metrization theorem and fixed point results for Lipschitzian mappings on quasimetric spaces
zbMATH Open document ID: 1416.54013
author: Katarzyna Chrząszcz
author: Filip Turoboś
author: J. Jachymski
published in: Aequationes Mathematicae
publication date: 2019-03-13T00:00:00Z
review text: A pair $(X,d)$ consisting of a nonempty set $X$ and a function $d : X \times X \to \mathbb{R}_+$ is called a quasimetric space if $d$ satisfies two first axioms of a metric and the triangle inequality in the form \[ d(x,z) \leq K \cdot \max \{d(x,y), d(y,z)\}, \quad \forall x,y,z \in X \] with $K \geq 1.$ \par Some results of [\textit{A. H. Frink}, Bull. Am. Math. Soc. 43, 133--142 (1937; Zbl 0016.08205)] concerning the construction of a metric in $X$ equivalent, in a certain sense, to a given quasimetric are improved. The authors present analogs of the Caristi fixed point theorem and the Cantor intersection theorem from which they deduce a version of the Banach fixed point theorem in quasimetric spaces which is due to \textit{I. A. Bakhtin} [Funkts. Anal. 30, 26--37 (1989; Zbl 0748.47048)].
reviewed by: Valerii V. Obukhovskij
Mathematics Subject Classification ID: 54E25
Mathematics Subject Classification ID: 47H09
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 54E15
Mathematics Subject Classification ID: 54E35
Mathematics Subject Classification ID: 54H25
zbMATH DE Number: 7035785
zbMATH Keywords: semimetric space
zbMATH Keywords: quasimetric space
zbMATH Keywords: metrization
zbMATH Keywords: metric
zbMATH Keywords: fixed point
zbMATH Keywords: contraction
zbMATH Keywords: Cantor intersection theorem
zbMATH Keywords: Caristi fixed point theorem
zbMATH Keywords: Banach fixed point theorem
MaRDI profile type: Publication
full work available at URL: s00010-018-0597-9
OpenAlex ID: W2889235981
cites work: Stone-type theorem on \(b\)-metric spaces and applications
cites work: Topological and geometrical properties of spaces with symmetric and nonsymmetric \(f\)-quasimetrics
cites work: Q3983330
cites work: A contraction principle in semimetric spaces
cites work: On Nonlinear Contractions
cites work: Q5522002
cites work: The solution by iteration of linear functional equations in Banach spaces
cites work: Fixed Point Theorems for Mappings Satisfying Inwardness Conditions
cites work: On characterizations and topology of regular semimetric spaces
cites work: Q4098245
cites work: On relaxations of contraction constants and Caristi's theorem in \(b\)-metric spaces
cites work: Comparing Top k Lists
cites work: Q3998720
cites work: Metrization of Quasi-Metric Spaces.
cites work: Caristi's fixed point theorem and selections of set-valued contractions
cites work: On Kirk's asymptotic contractions
cites work: Fixed Point Theory in Distance Spaces
cites work: Q3337413
cites work: On quasi-metric and metric spaces
cites work: Quasi-metric and metric spaces
cites work: On Semi-Metric Spaces
DOI: 10.1007/S00010-018-0597-9
description: scientific article; zbMATH DE number 7035785