Quasisymmetric mappings in b-metric spaces
type: Item
instance of: scholarly article
title: Quasisymmetric mappings in b-metric spaces
zbMATH Open document ID: 1471.54018
published in: Journal of Mathematical Sciences (New York)
publication date: 2021-08-04T00:00:00Z
review text: Let \((X,d)\) and \((Y,\rho)\) be \(b\)-metric spaces with the coefficients \(K_1\) and \(K_2\), respectively and let \(f:X\to Y\) be an \(\eta\)-quasisymmetric embedding (i.e., \(\eta:\mathbb R^+\to \mathbb R^+\) is a homeomorphism so that \(d(x,a)\le td(x,b)\) implies \(\rho(f(x),f(a))\le\eta(t)\rho(f(x),f(b))\) for all triples \(a,b,x\in X\) and for every \(t\in \mathbb R^+\)). The authors prove that \(f\) maps bounded subspaces to bounded subspaces and, if \(A\subseteq B\subseteq X\), \(0<\operatorname{diam} A,\operatorname{diam} B<\infty\), then \(\operatorname{diam} f(B)\) is finite and the following holds: \[ \frac1{2K_2\eta(\frac{\operatorname{diam} B}{\operatorname{diam} A})}\le\frac{\operatorname{diam} f(A)}{\operatorname{diam} f(B)}\le\eta\Bigl(\frac{2K_1\operatorname{diam} A}{\operatorname{diam} B}\Bigr), \] thus adapting a result known for metric spaces. They also describe \(\eta\)-quasisymmetric embeddings \(f\) for which the image \(f(X)\) of a \(b\)-metric space \(X\) with coefficient \(K_1\) is again a \(b\)-metric space with coefficient \(K_2\). Finally, a similar question is investigated for additive metric spaces.
reviewed by: Zoran Kadelburg
Mathematics Subject Classification ID: 54E99
Mathematics Subject Classification ID: 47H99
zbMATH DE Number: 7378023
zbMATH Keywords: b-metric space
zbMATH Keywords: qusisymmetric mapping
zbMATH Keywords: additive metric space
author: E. A. Petrov
author: R. R. Salimov
MaRDI profile type: Publication
full work available at URL: s10958-021-05459-3
OpenAlex ID: W3183688441
cites work: The boundary correspondence under quasiconformal mappings
cites work: Quasisymmetric embeddings of metric spaces
cites work: Q5228545
cites work: Q4354145
cites work: Quasihyperbolic metric and Quasisymmetric mappings in metric spaces
cites work: Quasisymmetric rigidity of square Sierpiński carpets
cites work: Weakly quasisymmetric maps and uniform spaces
cites work: Q5818996
cites work: Q4884981
cites work: Q4225413
cites work: KKM mappings in metric type spaces
cites work: A fixed point theorem for set-valued quasi-contractions in \(b\)-metric spaces
cites work: A common fixed point for weak \(\varphi \)-contractions on \(b\)-metric spaces
cites work: On the topology and \(wt\)-distance on metric type spaces
cites work: Fixed points for multivalued contractions in \(b\)-metric spaces with applications to fractals
cites work: Fixed Point Theory in Distance Spaces
cites work: KKM mappings in cone \(b\)-metric spaces
cites work: A generalized Ostrowski-Grüss type inequality for bounded differentiable mappings and its applications
cites work: Partial b-metric spaces and fixed point theorems
cites work: Various generalizations of metric spaces and fixed point theorems
cites work: Lectures on analysis on metric spaces
cites work: A note on metric preserving functions
cites work: Functions transferring metrics to metrics
cites work: Remarks on b-Metric and metric-preserving functions
cites work: Remarks on b-metrics, ultrametrics, and metric-preserving functions
cites work: Q5812325
cites work: Encyclopedia of Distances
cites work: A note on the metric properties of trees
cites work: Q4792088
cites work: Fitting Tree Metrics: Hierarchical Clustering and Phylogeny
DOI: 10.1007/S10958-021-05459-3
description: scientific article; zbMATH DE number 7378023