Remarks on w-distances and metric-preserving functions

Abstract

In this paper, we define new classes of functions related to metrics and w-distances. We also provide characterizations of functions in these classes. As a consequence, we obtain relations between all classes.

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

“…Throughout this article, for any function f : [0, ∞) → [0, ∞), we denote by f 0 a function from [0, ∞) to [0, ∞) such that f 0 (0) = 0 and f 0 (t) = f (t) for all t > 0. The following lemma gives relations between f • d and f 0 • d, where d is a semimetric and f satisfies f -1 ({0}) ⊆ {0}, called weakly amenable [20,Definition 6].…”

“…From the definitions and the facts that every ultrametric is a metric, and every metric is a b-metric, we obtain the following inclusions: BW * ⊆ MW * ⊆ UW * and MW ⊆ UW ⊆ UW * . So, we can extend the diagram showing the relations between W, W * MW, and MW * in [20,Sect. 3.6] by adding UW * , UW, and BW * to the diagram.…”

“…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. Some results concerning b-metrics can be found in [1,10,11,21].…”

“…The idea of metric-preserving functions was modified by replacing metrics with ultrametrics in 2014 by Pongsriiam and Termwuttipong [18]. Later, Khemaratchatakumthorn and Pongsriiam [10] introduced a variation involving b-metrics, and Prinyasart and Samphavat [20] investigated a variation related to w-distances. In this article, we introduce new variations of the concept of metric-preserving functions concerning ultrametrics, bmetrics, and w-distances as follows.…”