George-Veeramani Fuzzy Metrics Revised

Abstract

In this note, we present an alternative approach to the concept of a fuzzy metric, calling it a revised fuzzy metric. In contrast to the traditional approach to the theory of fuzzy metric spaces which is based on the use of a t-norm, we proceed from a t-conorm in the definition of a revised fuzzy metric. Here, we restrict our study to the case of fuzzy metrics as they are defined by George-Veeramani, however, similar revision can be done also for some other approaches to the concept of a fuzzy metric.

The First Rational Type Revised Fuzzy-Contractions in Revised Fuzzy Metric Spaces with an Applications

“…Definition 1 [1]. A binary operation of the form : [0, 1] 2 → [0, 1] is said to be a t-conorm if it satisfies the following conditions:…”

“…is associative and commutative, continuous. (b) p 0 = p, f or all p ∈ [0, 1], (c) p q ≤ u v. Whenever, p ≤ u and q ≤ v. For all p, q, u, v ∈ [0, 1].…”

“…In the year 2018, Alexander Sostak [1][2][3] introduced the idea of revised fuzzy metrics, which allow for the progressive evaluation of an element's inclusion in a collection. Revised fuzzy contraction mappings were described by Muraliraj and Thangathamizh [4][5][6][7], and the existence of fixed points was established for it.…”

Introduction on Revised Fuzzy Modular Spaces

“…After that Yonghong shen [15] et.al introduced the concept of fuzzy modular space in 2013. In 2018, Alexander sostak [2] defined revised fuzzy metric spaces and further in 2020 established metric spaces based on extended t-conorms [3]. Olga Grigorenka [13] et al introduced the t-conorm based space) if 𝑋 is a real or complex vector space, ⨁ is a continuous 𝑡-conorm, and 𝜇 is a fuzzy set on 𝑋 × (0, ∞) satisfying the following conditions, for all 𝑥, 𝑦 ∈ 𝑋, 𝑠, 𝑡 > 0 and 𝛼, 𝛽 ≥ 0 with 𝛼 + 𝛽 = 1:…”

The Continuity of Fuzzy Metric Spaces

“…Kaleva and Seikkala in 1984 [2] generalized the idea of a metric space to introduce the definition of the fuzzy metrics space by specifying the distance between two points (rather than the probabilistic metric space) to be a non-negative fuzzy number. Sostak in [3] presented an alternative method to introduce a new version of the concept of a fuzzy metric called revised fuzzy metric. A t-conorm binary operation is used in the definition of the revised fuzzy metric to assess the degree of proximity of two points.…”