Remarks on b-metrics, ultrametrics, and metric-preserving functions

In the paper, we continue the research of Borsík and Doboš on functions which allow us to introduce a metric to the product of metric spaces. We extend their scope to a broader class of spaces which usually fail to satisfy the triangle inequality, albeit they tend to satisfy some weaker form of this axiom. In particular, we examine the behavior of functions preserving b-metric inequality. We provide analogues of the results of Borsík and Doboš adjusted to the new broader setting. The results we obtained are illustrated with multitude of examples. Furthermore, the connections of newly introduced families of functions with the ones already known from the literature are investigated.

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“…We can now move on to extending the results obtained in [4,7,11,42,46] to a multidimensional scope. Theorem 3.1.…”

Section: Resultsmentioning

confidence: 97%

“…These triplets could cause additional issues in formulation of some theorems as well as in the proofs, therefore we have decided to exclude them from the above definition. We will use the symbol G to denote the set of all G-triangle triplets (similar to the notions used in [42] as well as in their previous papers). Definition 14.…”

Section: Resultsmentioning

confidence: 99%

On Functions Preserving Products of Certain Classes of Semimetric Spaces

Lichman

Nowakowski

Turoboś

2021

Tatra Mountains Mathematical Publications

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“…We will use the symbol △ G to denote the set of all G-triangle triplets (similar to the notions used in [42] as well as in their previous papers). Definition 14.…”

Section: Resultsmentioning

confidence: 99%

“…conducted by Borsík, Doboš and Piotrowski (see [3,4,5,6,10,11,12,13,14]), Corazza (see [7]), Das (see [9]), Dovgoshey (see [15,16,17,49]), Jůza (see [25]), Khemaratchatakumthorn, Pongsriiam, Termwuttipong and Samphavat (see [27,28,29,40,41,42,44]), Pokorny (see [37,38,39]), Vallin (see [47,48,49]) and recently also by Jachymski and Turoboś (see [22,46]).…”

Section: Introductionmentioning

confidence: 99%

On functions preserving products of certain classes of semimetric spaces

Lichman¹,

Nowakowski²,

Turoboś³

2020

Preprint

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In the paper we continue the research of Borsík and Doboš on functions which allow us to introduce a metric to the product of metric spaces. In this paper we extend their scope on broader class of spaces which usually fail to satisfy the triangle inequality, albeit they tend to satisfy some weaker form of this axiom. In particular, we examine the behavior of functions preserving b-metric inequality. We provide analogues of the results of Borsík and Doboš, adjusted to the new, broader setting. The results we obtained are illustrated with multitude of examples. Furthermore, the connections of newly introduced families of functions with the ones already known from the literature are investigated.

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“…These distances and generalized metrics also have many applications. Examples and applications of ultrametrics can be found in [2,5,6,14,15,18,19,21,26,28]. We also refer the reader to [8,9,12,20,[22][23][24] for examples and applications of w-distances.…”

Section: Introductionmentioning

confidence: 99%

On ultrametrics, b-metrics, w-distances, metric-preserving functions, and fixed point theorems

Samphavat,

Prinyasart

2024

Fixed Point Theory Algorithms Sci Eng

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In this article, new classes of functions based on new variations of metric-preserving functions are defined. Necessary and sufficient conditions for functions to be in these classes are also provided. As a result, we can explain relations between all classes and learn that all functions in the classes are weakly separated from 0. We can extend fixed point theorems, which were originally provided by Kirk and Shahzad and were later extended by Pongsriiam and Termwuttipong, in this journal by considering all functions that are weakly separated from 0.

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Remarks on b-metrics, ultrametrics, and metric-preserving functions

Abstract: We introduce new classes of functions related to metric-preserving functions, b-metrics, and ultrametrics. We investigate their properties and compare them to those of metric-preserving functions.

Cited by 11 publications

References 26 publications

On Functions Preserving Products of Certain Classes of Semimetric Spaces

On Functions Preserving Products of Certain Classes of Semimetric Spaces

On functions preserving products of certain classes of semimetric spaces

On ultrametrics, b-metrics, w-distances, metric-preserving functions, and fixed point theorems

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