Communications on Applied Nonlinear Analysis 

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New Contraction Principle in Revised Fuzzy 𝓴 −Metric Spaces 

  

1A Mohan, 2R Thangathamizh, 3A Muraliraj 

1Urumu Dhanalakshmi College, Bharathidasan University, Trichy, India. 

Email id: appavumohan@gmail.com 

2Jeppiaar Institute of Technology, Sriperumbudur, Kanchipuram, India. 

Email id: thamizh1418@gmail.com 

3Urumu Dhanalakshmi College, Bharathidasan University, Trichy, India. 

Email id: karguzali@gmail.com 

 

Article History: 

Received: 12-01-2025 

Revised: 15-02-2025 

Accepted: 01-03-2025 

Abstract:  

Introduction 

Metric spaces play a crucial role in mathematical analysis and topology. In recent years, 

fuzzy metric spaces have been widely studied due to their applications in various fields. 

Alexander Sostak introduced the concept of revised fuzzy metric spaces, which extends 

traditional fuzzy metric spaces by incorporating revised fuzzy sets. In this paper, we 

introduce a further generalization called revised fuzzy 𝓀 −metric spaces, which allows 

for the involvement of multiple parameters (𝓀), thereby enhancing the flexibility and 

applicability of the framework. 

Objectives 

The primary aim of this study is to define and explore the fundamental properties of 

revised fuzzy 𝓀 −metric spaces. We investigate their topological structure and establish 

significant properties such as first countability and the Hausdorff condition. 

Additionally, we extend existing results in the literature by proving a fixed-point 

theorem in this new setting. 

Method 

We begin by formally defining a revised fuzzy k-metric space and developing its basic 

properties. Using topological arguments, we demonstrate that the topology induced by 

a revised fuzzy 𝓀 −metric is first countable and that the space satisfies the Hausdorff 

condition. Finally, we extend the fixed-point theorem established by Muraliraj and 

Thangathamizh into the context of revised fuzzy 𝓀 −metric spaces, using analytical and 

set-theoretic techniques. 

Result 

Our findings confirm that revised fuzzy k-metric spaces preserve essential topological 

characteristics such as first countability and Hausdorff separation. Furthermore, the 

fixed-point theorem proved in this study generalizes previous results and demonstrates 

the broader applicability of revised fuzzy 𝓀 −metric spaces in fixed-point theory. 

Conclusion 

This study introduces revised fuzzy 𝓀 −metric spaces as a generalization of revised 

fuzzy metric spaces, providing a more comprehensive framework for analyzing metric 

structures with multiple parameters. The established topological properties and fixed-



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point theorem contribute to the further development of fuzzy metric theory, opening 

new avenues for future research in mathematical analysis and its applications. 

Keywords: revised fuzzy 𝓀 −metric spaces, revised fuzzy 2-metric spaces, Hausdorff 

spaces, Contractions, Fixed points 

 

1. Introduction  

 The idea to revise the concept of a fuzzy metric by means of t-conorms instead of t-norms was 

first expressed in [6]. In this paper, we have developed further this approach calling fuzzy metrics 

defined on the base of a t-conorm by t-conorm based fuzzy metrics or by CB-fuzzy metrics for short. 

The three main issues considered in the paper are the following. Construction of revised fuzzy 

𝓀 −metrics from ordinary metrics (Section 4), topological structure induced by CB-fuzzy metrics 

(Section 5), and interrelations between CB-fuzzy metrics and modular metrics (Section 6). 

Additionally, we make some comments concerning the intuitionistic counterpart of a CB-fuzzy metric 

(Section 7).  

Concerning the construction of CB-fuzzy metrics from ordinary metrics we mainly restrict the case of 

fuzzy metrics based on Archimedean t-conorms. Just in this situation we can effectively use the tools 

provided by additive generators of t-conorms. By using additive generators for such CB-fuzzy metrics, 

we presented a scheme for construction of CB-fuzzy metrics from ordinary metrics and illustrated it 

with examples for some concrete t-conorms. We guess that the presented construction will provide a 

scheme allowing to extend some results from the theory of metric spaces to the corresponding results 

for CB-fuzzy metric spaces. Specifically, this can concern the results in the theory of fixed points. 

The motivation in this paper for inventing a new space, which is more general than a revised fuzzy 

metric space due to Alexander Sostack (2018), is given in this paragraph. In a revised fuzzy metric 

space, the fuzzy distance of two points is measured by the degree of the nearness of points with respect 

to a parameter 𝑡 ∈ (0, ∞). For instance, we can think of “t” as the time required to travel between two 

points 𝑥 and 𝑦 in a space. There is an interesting situation of the degree of nearness when we measure 

this degree with respect to different (more than one) parameters. For instance, suppose that we move 

from India, represented by 𝑥, to Serbia, represented by 𝑦, by a plane and measure the degree of the 

nearness of 𝑥 and 𝑦 with respect to time and fuel consumption with planes of different fuel efficiency. 

Then obviously, this degree will be different for distinct planes even for the same time 𝑡, as well as for 

the same plane but for different time intervals. 

 The mentioned situation in the previous paragraph brings the inspiration for introducing the 

notion of revised fuzzy 𝓀 −metric spaces, where 𝓀 ∈ {1, 2, 3, . . . }, which is an extension and 

generalization of the concept of fuzzy metric spaces due to Alexander Sostack (2018). In a revised 

fuzzy 𝓀 −metric spaces, the fuzzy distance of two points is measured by the degree of nearness with 

respect to 𝓀 −parameter(s). Furthermore, fixed point results for contractive mappings in revised fuzzy 

𝓀 −metric spaces are proved. These results generalize the fixed-point results of Muraliraj and 

Thangathamizh (2022) into revised fuzzy 𝓀 −metric spaces. 

 



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2. Preliminaries 

Definition 1[22] (Schweizer and Sklar (1960) A binary operation ⨁: [0, 1]2 → [0, 1] is called a 

triangular conorm (briefly, t-conorm) if the following conditions are satisfied for all 𝔭, 𝔮, 𝔯, 𝔰 ∈ [0, 1]:  

1. ⨁ (𝔭, 𝔮) = ⨁ (𝔮, 𝔭);  

2. if 𝔭 ≤ 𝔯 and 𝔮 ≤ 𝔰, then ⨁ (𝔭, 𝔮) ≤  ⨁ (𝔯, 𝔰);  

3. ⨁ (⨁ (𝔭, 𝔮), 𝔯)  =  ⨁ (𝔭, ⨁ (𝔮, 𝔯));  

4. ⨁ (𝔭, 0) = 𝔭. If ⨁ is continuous, it is called a continuous t-conorm.  

For each t-conorm ⨁: [0, 1]2 → [0, 1] and 𝔭, 𝔮 ∈ [0, 1], instead of ⨁ (𝔭, 𝔮) we will use the infix 

notation 𝔭 ⨁ 𝔮. Three typical examples of continuous t-norms are a product t-conorm ⨁1, a minimum 

t-conorm ⨁2 and a Lukasiewicz t-conorm ⨁3, which are defined for each 𝔭, 𝔮 ∈ [0, 1] by  

𝔭 ⨁1 𝔮 =  𝑚𝑎𝑥{𝔭, 𝔮},  

𝔭 ⨁2 𝔮 =  𝔭 + 𝔮 − 𝔭𝔮, 

𝔭 ⨁3 𝔮 =  𝑚𝑖𝑛{𝑎 + 𝑏 , 1}. 

Remark 2 For each t-norm ⨁: [0, 1]2 → [0, 1], the following assertions hold:  

1. for each 𝔭, 𝔮 ∈ [0, 1] with 𝔭 > 𝑞, there is 𝔯 ∈ (0, 1) such that 𝔭 ⨁ 𝔯 ≥ 𝔮;  

2. for each 𝔰 ∈ (0, 1), there is 𝔱 ∈ (0, 1) such that 𝔱 ⨁ 𝔱 ≥ 𝔰.  

Definition 3[7] An ordered triple (𝔐, 𝔑, ⨁) is called a revised fuzzy metric space if 𝔐  is an arbitrary 

set, ⨁ a continuous t-conorm, 𝔐  is a revised fuzzy set on 𝔐2 × (0, +∞), and thefollowing conditions 

are satisfied for all 𝔭, 𝔮 ∈ 𝔐, 𝔞, 𝔟 > 0 

(RF-1) 𝔑(𝔭, 𝔮, 𝒶) < 1; 

(RF-2) 𝔑(𝔭, 𝔮, 𝒶) = 0 if and only if 𝔭 = 𝔮; 

(RF-3) 𝔑(𝔭, 𝔮, 𝒶) = 𝔑(𝔮, 𝔭, 𝒶); 

(RF-4) 𝔑(𝔭, 𝓇, 𝒶 ) ≤ 𝔑(𝔭, 𝔮, 𝒶)⨁ 𝔑(𝔮, 𝓇, 𝒶); 

(RF-5) 𝔑(𝔭, 𝔮, −): (0, +∞)𝓀 → [0,1]is a right continuous mapping. 

Example 4[7] (Induced revised fuzzy metric)  

Let (𝑋, 𝑑) be a metric space and ⨁ be a product t-conorm. Define a revised fuzzy set 𝔑 on 

𝔐2 × (0, +∞) by  

𝔑(𝔭, 𝔮, 𝔞) =
𝕕(𝔭, 𝔮)

1 + 𝕕(𝔭, 𝔮)
 

for all 𝔭, 𝔮 ∈ 𝔐 and 𝔞 > 0, where 𝑘, 𝑚, 𝑛 > 0. Then, (𝔐, 𝔑, ⨁) is a revised fuzzy metric space called 

the induced revised fuzzy metric.  

In the above example, note that  



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lim
𝑡→+∞

𝔑(𝔭, 𝔮, 𝔞) = 0 for all 𝔭, 𝔮 ∈ 𝔐.           (1)  

As (𝔐, 𝔑, ⨁) represents the degree of the nearness of points 𝔭 and 𝔮 with respect to the parameter 𝔞 

and it is a nondecreasing function of 𝔞 for all 𝔭, 𝔮 ∈ 𝔐; therefore, condition (1) is the most natural 

condition for the degree of the nearness to be perfect (that is, unity). Notice that this is a specific 

condition and may not hold in some fuzzy metric spaces, for instance, in stationary revised fuzzy metric 

spaces. This brings to the following definition:  

Definition 5  

A revised fuzzy metric space (𝔐, 𝔑, ⨁) is called a natural fuzzy metric space if and only if  

lim
𝑡→+∞

𝔑(𝔭, 𝔮, 𝔞) = 0 for all 𝔭, 𝔮 ∈ 𝔐. 

Definition 6  

A 3-tuple (𝔐, 𝔑, ⨁) is said to be a revised fuzzy 2-metric space if 𝔐 is an arbitrary nonempty set, ⨁is 

a continuous t-conorm, and 𝔐  is a revised fuzzy set on𝔐3 × (0, +∞) satisfying  the following 

conditions: For all (𝔭, 𝔮, 𝓇 ∈ 𝔐, 𝒶, 𝒶1, 𝒶2, 𝒶3 ∈ (0, +∞)) 

(RF2M.1) given distinct elements 𝔭, 𝔮 ∈ 𝔐 there is an element𝔯 ∈ 𝔐 such that 𝔑(𝔭, 𝔮, 𝓇, 𝒶) < 1  for 

each 𝔞 > 0;  

(RF2M.2) 𝔑(𝔭, 𝔮, 𝒶) = 0 if at least two of 𝔭, 𝔮, 𝓇 are equal.  

(RF2M.3) 𝔑(𝔭, 𝔮, 𝔯, 𝒶) = 𝔑(𝔭, 𝔯, 𝔮, 𝒶) = 𝔑(𝔯, 𝔭, 𝔮, 𝒶) for all 𝔭, 𝔮, 𝓇 ∈ 𝔐 and all 𝔞 > 0;  

(RF2M.4) 𝔑(𝔭, 𝔮, 𝔯, 𝒶1 +  𝒶2 + 𝒶3) ≤ 𝔑(𝔭, 𝔯, 𝔰, 𝒶1)⨁𝔑(𝔭, 𝔰, 𝔯, 𝒶2)⨁𝔑(𝔰, 𝔮, 𝔯, 𝒶3);  

(RF2M.5)𝔑(𝔭, 𝔮, 𝔯, −) ∶  (0, ∞)  →  (0,1] is a continuous function.  

The pair (𝔑, ⨁) (or only 𝔑) is called a revised fuzzy 2-metric on 𝔐. 

3. Revised fuzzy 𝓴 −metric spaces 

In this section, we introduce the idea of revised fuzzy 𝓀 −metric spaces and investigate the properties 

of such spaces. We begin with the following definition 

Definition 6  

Let 𝔐 be a nonempty set, ⨁a continuous t-conorm, 𝓀a positive integer and 𝔑be a revised fuzzy set 

on 𝔐2 × (0, +∞)𝓀. An ordered triple (𝔐, 𝔑, ⨁)is called a revised fuzzy𝓀 −metric space if the 

following conditions are satisfied for all 𝔭, 𝔮, 𝔯 ∈ 𝔐, 𝔞, 𝔟 > 0 and𝒶1, 𝒶2, . . . , 𝒶𝓀 > 0: 

(RF-k1) 𝔑(𝔭, 𝔮, 𝒶1, 𝒶2, . . . , 𝒶𝓀) < 1; 

(RF-k2) 𝔑(𝔭, 𝔮, 𝒶1, 𝒶2, . . . , 𝒶𝓀) = 0 if and only if 𝔭 = 𝔮; 

(RF-k3) 𝔑(𝔭, 𝔮, 𝒶1, 𝒶2, . . . , 𝒶𝓀)is symmetric. 

(RF-k4) for any𝒿 ∈ {1, 2, 3, . . . , 𝑘}, we have 

𝔑(𝔭, 𝓇, 𝒶1, 𝒶2, . . . , 𝒶𝒿−1, 𝒶 + 𝒷, 𝒶𝒿+1, … , 𝒶𝓀) ≤ {
𝔑(𝔮, 𝓇, 𝒶1, 𝒶2, . . . , 𝒶𝒿−1, 𝒶, 𝒶𝒿+1, … , 𝒶𝒿−1, 𝒶𝓀)

⨁ 𝔑(𝔮, 𝓇, 𝒶1, 𝒶2, . . . , 𝒶𝒿−1, 𝒷, 𝒶𝒿+1, … , 𝒶𝒿−1, 𝒶𝓀)
} 



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(RF-k5) 𝔑(𝔭, 𝔮, −): (0, +∞)𝓀 → [0,1]is a right continuous mapping. 

Remark 7For 𝓀 = 1, the revised fuzzy𝓀 −metric space reduces into the revised fuzzy metric space 

in the sense of Alexander Sostak. 

Example 1 

Let (𝔐, 𝕕)be a metric space, ⨁ the product (maximum) t-conorm, 𝔲 > 0 and 𝓀 be a positive integer. 

Define a revised fuzzy set 𝔑on 𝔐2 × (0, ∞)𝓀by  

𝔑(𝔭, 𝔮, 𝒶1, 𝒶2, . . . , 𝒶𝓀) =
𝕕(𝔭, 𝔮)

𝔲(𝒶1, 𝒶2, . . . , 𝒶𝓀) + 𝕕(𝔭, 𝔮)
 

for all all 𝔭, 𝔮 ∈ 𝔐, 𝔞, 𝔟 > 0 and  𝒶1, 𝒶2, . . . , 𝒶𝓀 > 0. Then, (𝔐, 𝔑, ⨁) is a revised fuzzy 𝓀 −metric 

space. 

From the application point of view, one should define the revised fuzzy 𝓀 −metric with care to the 

physical nature of quantities. For instance, if one considers the degree of the nearness of two points 

𝔭 𝑎𝑛𝑑 𝔮 in a space with respect to time and fuel consumed in moving from 𝔭 𝑡𝑜 𝔮, one cannot use the 

formulae for the degree of the nearness as given in the above examples due to the different dimensions 

of these quantities. In the following example, one such case is presented. 

Example 2 

Let (𝔐, 𝕕) be a metric space, ⨁ the product (maximum) t-conorm, 𝔲 > 0 and 𝓀 be a positive integer. 

Define a revised fuzzy set 𝔑 on 𝔐2 × (0, +∞)𝓀 by  

𝔑(𝔭, 𝔮, 𝒶1, 𝒶2, . . . , 𝒶𝓀) = 1 − 𝔲 [𝔲 + (∑
1

𝒶𝒿

𝓀

𝒿=1

) 𝕕(𝔭, 𝔮)]

−1

 

for all all 𝔭, 𝔮 ∈ 𝔐, 𝔞, 𝔟 > 0 and  𝒶1, 𝒶2, . . . , 𝒶𝓀 > 0. Then, (𝔐, 𝔑, ⨁)is a revised fuzzy 𝓀 −metric 

space. 

Example 3  

Let 𝔐 = ℛ𝓀, where 𝓀 is a positive integer, ⨁ the product t-conorm. Define a revised fuzzy set 𝔑 on 

𝔐2 × (0, +∞)𝓀 by  

𝔑(𝔭, 𝔮, 𝒶1, 𝒶2, . . . , 𝒶𝓀) = 1 − 𝔲 [𝔲 + (∑
|𝔮𝒿 − 𝔭𝒿|

𝒶𝒿

𝓀

𝒿=1

)]

−1

 

for all 𝔭, 𝔮 ∈ 𝔐 and  𝒶1, 𝒶2, . . . , 𝒶𝓀 > 0. Then, (𝔐, 𝔑, ⨁) is a revised fuzzy 𝓀 −metric space. 

In the present paper, we restrict ourselves to only mathematical properties of revised fuzzy 𝓀 −metric 

space. 

Definition 12  

A revised fuzzy 𝓀 −metric space (𝔐, 𝔑, ⨁) is called 𝒯 −natural revised fuzzy 𝓀 −metric space if 

there exists 𝒯 ∈ {1, 2, … , 𝓀} such that 



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lim
𝒶𝓀→+∞

𝔑(𝔭, 𝔮, 𝒶1, 𝒶2, . . . , 𝒶𝓀) = 0, 𝔭, 𝔮 ∈ 𝔐 and  𝒶1, 𝒶2, . . . , 𝒶𝓀 > 0.  

For the rest of this paper, for a given revised fuzzy 𝓀 −metric space(𝔐, 𝔑, ⨁), 𝔭, 𝔮 ∈ 𝔐 and  

𝒶1, 𝒶2, . . . , 𝒶𝓀 > 0, for simplicity, we write 𝔑(𝔭, 𝔮, 𝒶1
𝓀) instead 𝔑(𝔭, 𝔮, 𝒶1, 𝒶2, . . . , 𝒶𝓀).  

Next, we discuss some properties revised fuzzy 𝓀 −metric space and establish the topology of such 

spaces. 

Proposition 13  

Let (𝔐, 𝔑, ⨁) be a revised fuzzy 𝓀 −metric space, 𝒶, 𝒶1, 𝒶2, . . . , 𝒶𝓀. Suppose that 𝒶𝒯 < 𝑎 for some 

𝒯 ∈ {1, 2, … , 𝓀}. Then, 𝔑(𝔭, 𝔮, 𝒶1
𝓀) ≥ 𝔑(𝔭, 𝔮, 𝒶1, 𝒶2, . . , 𝒶𝒯−1, 𝒶𝒯 , 𝒶𝒯+1, . . , 𝒶𝓀) for all 𝔭, 𝔮 ∈ 𝔐. 

Remark 14  

In a revised fuzzy 𝓀 −metric space(𝔐, 𝔑, ⨁), if 𝔑(𝔭, 𝔮, 𝒶1
𝓀) < 휀, where 𝔭, 𝔮 ∈ 𝔐, 𝒶1, 𝒶2, . . . , 𝒶𝓀 > 0 

and 0 < 휀 < 1, then for each 𝒯 ∈ {1, 2, … , 𝓀}, we can find 𝒶 ∈ (0, 𝒶𝓀) such that 

𝔑(𝔭, 𝔮, 𝒶1, 𝒶2, . . , 𝒶𝒯−1, 𝒶𝒯 , 𝒶𝒯+1, . . , 𝒶𝓀) < 휀. 

Definition 15  

Let (𝔐, 𝔑, ⨁) be a revised fuzzy 𝓀 −metric space. An open ball with center 𝔭 ∈ 𝔐 and radius 휀 ∈

(0,1) with respect to parameters𝒶1, 𝒶2, . . . , 𝒶𝓀 > 0, denoted by 𝔅(𝔭, 휀; 𝒶1, 𝒶2, . . . , 𝒶𝓀), is defined by 

𝔅(𝔭, 휀; 𝒶1, 𝒶2, . . . , 𝒶𝓀) = {𝔮 ∈ 𝔐: 𝔑(𝔭, 𝔮, 𝒶1
𝓀) < 휀}. 

Definition 16  

Let (𝔐, 𝔑, ⨁) be a revised fuzzy 𝓀 −metric space. A subset 𝒳 of 𝔐 is called an open set if and only 

if there is an open ball 𝔅 such that 𝔅 ⊆ 𝒳. A subset 𝒴 of 𝔐 is called a closed set if and only if its 

complement is an open set.  

Theorem 17  

Every open ball in a revised fuzzy 𝓀 −metric space is an open set. 

Proof  

Let (𝔐, 𝔑, ⨁) be a revised fuzzy 𝓀 −metric space, 𝔭 ∈ 𝔐, 𝒶1, 𝒶2, . . . , 𝒶𝓀 > 0 and 휀 ∈ (0,1).  

Assume that 𝔮 ∈ 𝔅(𝔭, 휀; 𝒶1, 𝒶2, . . . , 𝒶𝓀). Then, we have 𝔑(𝔭, 𝔮, 𝒶1
𝓀) < 휀.  

Therefore, we can find 𝒯 ∈ {1, 2, … , 𝓀} and 𝒶 ∈ (0, 𝒶𝓀)such that  

휀0 ≔ 𝔑(𝔭, 𝔮, 𝒶1, 𝒶2, . . , 𝒶𝒯−1, 𝒶𝒯 , 𝒶𝒯+1, . . , 𝒶𝓀) < 휀.  

Then, we can find 𝛿 ∈ (0, 1) such that 휀0 < 𝛿 <  휀.  

By Remark 2, there is 휀1 ∈ (0, 1) such that 휀0⨁휀1 ≤ 𝛿.  

Now, we will claim that 𝔅(𝔭, 휀; 𝒶1, 𝒶2, . . . , 𝒶𝓀) ⊆ 𝔅(𝔮, 1 − 휀1; 𝒶1, 𝒶2, . . , 𝒶𝒯−1, 𝒶𝒯 , 𝒶𝒯+1, . . , 𝒶𝓀).  

Assume that 𝔯 ∈ 𝔅(𝔮, 1 − 휀1; 𝒶1, 𝒶2, . . , 𝒶𝒯−1, 𝒶𝒯, 𝒶𝒯+1, . . , 𝒶𝓀).  

Then, 𝔑(𝔮, 𝔯, 𝒶1, 𝒶2, . . , 𝒶𝒯−1, 𝒶𝒯, 𝒶𝒯+1, . . , 𝒶𝓀) < 1 − 휀1. 



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𝔑(𝔭, 𝔯, 𝒶1
𝓀) ≤ 𝔑(𝔭, 𝔮, 𝒶1, 𝒶2, . . , 𝒶𝒯−1, 𝒶𝒯 , 𝒶𝒯+1, . . , 𝒶𝓀)⨁ 𝔑(𝔮, 𝔯, 𝒶1, 𝒶2, . . , 𝒶𝒯−1, 𝒶𝒯 , 𝒶𝒯+1, . . , 𝒶𝓀) 

 ≤ 휀0⨁휀1 ≤  𝛿 < 휀, 

which proves the result.  

From the above theorem, we can directly get the following result: 

Theorem 18  

Let (𝔐, 𝔑, ⨁) be a revised fuzzy 𝓀 −metric space and  

𝜏 = { 𝒳 ⊆ 𝔐 ∶  𝒶 ∈ 𝔐 if and only if there exist     𝒶1, 𝒶2, . . . , 𝒶𝓀 > 0 and 휀 ∈ (0,1) such that 

𝔅(𝔭, 휀; 𝒶1, 𝒶2, . . . , 𝒶𝓀) ⊆  𝒳}. 

Then, 𝜏 is a topology on 𝔐.  

Remark 19  

Let (𝔐, 𝔑, ⨁) be a revised fuzzy 𝓀 −metric space and 𝒶 ∈ 𝔐. Since 

𝔅𝔭 = {𝔅 (𝔭,
1

𝑛
; 𝒶1, 𝒶2, . . . , 𝒶𝓀) : 𝓃 ∈ 𝒩} 

Where 𝒶1 = 𝒶2 = . . . =  𝒶𝓀 =
1

𝓃
 , is a local base at a point 𝒶, the topology 𝜏 given in Theorem 18 is 

first countable. 

Theorem 20 

Every revised fuzzy 𝓀 −metric space is Hausdorff. 

Definition 21  

Let (𝔐, 𝔑, ⨁) be a revised fuzzy 𝓀 −metric space. A sequence {𝔭𝓃} in 𝔐 is said to be convergent and 

converges to 𝔭 ∈ 𝔐 if and only if for every real 𝜖 ∈ (0, 1), there exists 𝑛0 ∈ 𝒩 such that 

𝔑(𝔭𝓃, 𝔭, 𝒶1
𝓀) < 𝜖 for all 𝑛 ∈ 𝑛0 and 𝒶1, 𝒶2, . . . , 𝒶𝓀 > 0.  

The proof of the following lemma is straightforward, so we will omit the proof. 

Lemma 22  

Let (𝔐, 𝔑, ⨁) be a revised fuzzy 𝓀 −metric space. A sequence {𝔭𝓃} in 𝔐 converges to      𝔭 ∈ 𝔐 if 

and only if lim
𝑛→+∞

𝔑(𝔭𝓃, 𝔭, 𝒶1
𝓀) = 0 for all 𝒶1, 𝒶2, . . . , 𝒶𝓀 > 0. 

Definition 23  

Let (𝔐, 𝔑, ⨁) be a revised fuzzy 𝓀 −metric space and {𝔭𝓃} be a sequence in 𝔐.  

1. {𝔭𝓃} is called an 𝔑 −Cauchy sequence if for every 𝜖 ∈ (0, 1), there exists 𝑛0 ∈ 𝒩 such that 

𝔑(𝔭𝓃, 𝔭𝑚, 𝒶1
𝓀) < 𝜖 for all 𝑛, 𝑚 > 𝑛0 and 𝒶1, 𝒶2, . . . , 𝒶𝓀 > 0.  

2. {𝔭𝓃} is called a 𝔾 −Cauchy sequence if lim
𝑛→+∞

𝔑(𝔭𝓃, 𝔭𝓃+𝓍, 𝒶1
𝓀) = 0 for all 𝒶1, 𝒶2, . . . , 𝒶𝓀 > 0 and 

𝓍 > 0. Note that the above definitions of Cauchy sequences are different (for the case 𝓀 = 1. 

Definition 24 Let (𝔐, 𝔑, ⨁) be a revised fuzzy 𝓀 −metric space. 



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1. (𝔐, 𝔑, ⨁) is said to be 𝔑 −complete if every 𝔑 −Cauchy sequence in 𝔐 converges to some 𝔭 ∈

𝔐.  

2. (𝔐, 𝔑, ⨁) is said to be 𝔾 −complete if every G-Cauchy sequence in 𝔐 converges to some 𝔭 ∈ 𝔐. 

4. Fixed point theorems  

In this section, we prove many fixed-point results in revised fuzzy 𝓀 −metric space. For  

simplicity, for a given revised fuzzy 𝓀 −metric space(𝔐, 𝔑, ⨁), 𝒯 ∈ {1, 2, … , 𝓀}, 𝒷 > 0, 𝔭, 𝔮 ∈ 𝔐 

and 𝒶1, 𝒶2, . . . , 𝒶𝓀 > 0, we write 𝔑𝒯
𝒷(𝔭, 𝔮, 𝒶1

𝓀) instead 𝔑 (𝔭, 𝔮, 𝒶1, 𝒶2, . . , 𝒶𝒯−1,
𝒶𝒯

𝒷
, 𝒶𝒯+1, . . , 𝒶𝓀).  

Theorem 26 

Let (𝔐, 𝔑, ⨁) be a 𝔾 −complete revised fuzzy 𝓀 −metric space and 𝔗: 𝔐 → 𝔐 be a mapping  

satisfying the following condition:  

𝔑𝒯

1

𝜆 (𝔗𝔭, 𝔗𝔮, 𝒶1
𝓀) ≤ 𝔑(𝔭, 𝔮, 𝒶1

𝓀)   (2)  

for all 𝔭, 𝔮 ∈ 𝔐 and 𝒶1, 𝒶2, . . . , 𝒶𝓀 > 0, where 𝒯 ∈ {1, 2, … , 𝓀} and 𝜆 ∈ (0,1) is a constant. Suppose 

that (𝔐, 𝔑, ⨁) is an 𝒯 −natural revised fuzzy 𝓀 −metric space. Then, 𝔗 has a unique fixed point.  

Proof  

First, we will show that if a fixed point of 𝔗 exists, then it is unique. Suppose that 𝔵 and 𝔶 are  

fixed points of 𝔗. By (2), we have 

𝔑(𝔵, 𝔶, 𝒶1
𝓀) = 𝔑(𝔗𝔵, 𝔗𝔶, 𝒶1

𝓀) ≤ 𝔑 (𝔵, 𝔶, 𝒶1, 𝒶2, . . , 𝒶𝒯−1,
𝒶𝒯

𝜆
, 𝒶𝒯+1, . . , 𝒶𝓀) = 𝔑𝒯

𝜆 (𝔵, 𝔶, 𝒶1
𝓀) 

By repeating this process, we obtain  

𝔑(𝔵, 𝔶, 𝒶1
𝓀) ≤ 𝔑𝒯

𝜆𝑛
(𝔵, 𝔶, 𝒶1

𝓀)    (3)  

for all 𝓃 ∈ 𝒩. Note that, if {𝔭𝓃} be any sequence such that 𝔭𝓃 > 0 and lim
𝑛→∞

𝔭𝓃 =  0, then since 

(𝔐, 𝔑, ⨁) is 𝒯 −natural, we have lim
 𝑛→+∞

𝔑𝒯
𝒷𝓃(𝔭, 𝔮, 𝒶1

𝓀) = 0 

for all 𝒶1, 𝒶2, . . . , 𝒶𝓀 > 0.  

Using this fact in (3), we obtain 𝔑(𝔵, 𝔶, 𝒶1
𝓀) = 0 for all 𝒶1, 𝒶2, . . . , 𝒶𝓀 > 0, that is, 𝔵 = 𝔶. Therefore, 

the fixed point of 𝔗 is unique.  

For the existence of a fixed point of  𝔗, we choose 𝔭0 ∈ 𝔐 and define an iterative sequence {𝔭𝓃} by 

𝔭𝓃 = 𝔗𝔭𝓃−1for all 𝓃 ∈ 𝒩. If 𝔭𝓃 = 𝔭𝓃−1 for some 𝓃 ∈ 𝒩, then 𝔭𝓃 is the unique fixed point of 𝔗. 

Therefore, we may assume that 𝔭𝓃 ≠ 𝔭𝓃−1 for all 𝓃 ∈ 𝒩. For any 𝓃 ∈ 𝒩 and 𝒶1, 𝒶2, . . . , 𝒶𝓀 > 0, we 

have 

𝔑(𝔭𝓃, 𝔭𝓃+1, 𝒶1
𝓀) = 𝔑(𝔗𝔭𝓃−1, 𝔗𝔭𝓃, 𝒶1

𝓀) ≤ 𝔑 (𝔵, 𝔶, 𝒶1, 𝒶2, . . , 𝒶𝒯−1,
𝒶𝒯

𝜆
, 𝒶𝒯+1, . . , 𝒶𝓀)   

= 𝔑𝒯
𝜆 (𝔭𝓃−1, 𝔭𝓃, 𝒶1

𝓀)  



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By repeating this process, we obtain 

𝔑(𝔭𝓃, 𝔭𝓃+1, 𝒶1
𝓀) ≤ 𝔑𝒯

𝜆𝑛
(𝔭0, 𝔭1, 𝒶1

𝓀)   (4) 

for all 𝓃 ∈ 𝒩. For each 𝓃 ∈ 𝒩 and 𝒶1, 𝒶2, . . . , 𝒶𝓀 > 0 and 𝓍 > 0, we have 

𝔑(𝔭𝓃, 𝔭𝓃+𝓍, 𝒶1
𝓀) ≤ {

𝔑 (𝔭𝓃, 𝔭𝓃+1, 𝒶1, 𝒶2, . . , 𝒶𝒯−1,
𝒶𝒯

2
, 𝒶𝒯+1, . . , 𝒶𝓀)

⨁𝔑 (𝔭𝓃+1, 𝔭𝓃+𝓍, 𝒶1, 𝒶2, . . , 𝒶𝒯−1,
𝒶𝒯

2
, 𝒶𝒯+1, . . , 𝒶𝓀)

}  

≤ {
𝔑𝒯

2 (𝔭𝓃, 𝔭𝓃+1, 𝒶1
𝓀)⨁𝔑 (𝔭𝓃+1, 𝔭𝓃+2, 𝒶1, 𝒶2, . . , 𝒶𝒯−1,

𝒶𝒯

22 , 𝒶𝒯+1, . . , 𝒶𝓀)

⨁𝔑 (𝔭𝓃+2, 𝔭𝓃+𝓍, 𝒶1, 𝒶2, . . , 𝒶𝒯−1,
𝒶𝒯

22 , 𝒶𝒯+1, . . , 𝒶𝓀)
}   

≤ {
𝔑𝒯

2 (𝔭𝓃, 𝔭𝓃+1, 𝒶1
𝓀)⨁𝔑𝒯

22
(𝔭𝓃+1, 𝔭𝓃+2, 𝒶1

𝓀)⨁ … ⨁

𝔑𝒯
2𝓍−1(𝔭𝓃+𝓍−2, 𝔭𝓃+𝓍+1, 𝒶1

𝓀)⨁𝔑2(𝔭𝓃+𝓍−1, 𝔭𝓃+𝓍, 𝒶1
𝓀)

}  

By using (4), we obtain 

𝔑(𝔭𝓃, 𝔭𝓃+𝓍, 𝒶1
𝓀) ≤ 𝔑𝒯

2𝜆𝑛
(𝔭0, 𝔭1, 𝒶1

𝓀) ⨁ 𝔑𝒯
22𝜆𝑛+1

(𝔭0, 𝔭1, 𝒶1
𝓀)⨁ … ⨁𝔑𝒯

2𝓍−1𝜆𝑛+𝓍−1
(𝔭0, 𝔭1, 𝒶1

𝓀) 

Since (𝔐, 𝔑, ⨁) is 𝒯 −natural, it follows from the above inequality that lim
 𝑛→+∞

𝔑(𝔭𝓃, 𝔭𝓃+𝓍, 𝒶1
𝓀) = 0. 

Therefore, {𝔭𝓃} is a 𝔾 −Cauchy sequence. By the 𝔾 −completeness of (𝔐, 𝔑, ⨁), there exists       𝔵 ∈

𝔐 such that 

lim
 𝑛→+∞

𝔑(𝔭𝓃, 𝔵, 𝒶1
𝓀) = 0, for all 𝒶1, 𝒶2, . . . , 𝒶𝓀 > 0.  

We will show that 𝔵 is a fixed point of 𝔗. For each 𝒶1, 𝒶2, . . . , 𝒶𝓀 > 0, we have 

𝔑(𝔵, 𝔗𝔵, 𝒶1
𝓀) ≤ 𝔑𝒯

2 (𝔵, 𝔭𝓃, 𝒶1
𝓀)⨁ 𝔑𝒯

2 (𝔭𝓃, 𝔗𝔵, 𝒶1
𝓀)  

                      = 𝔑𝒯
2 (𝔵, 𝔭𝓃, 𝒶1

𝓀)⨁ 𝔑𝒯
2 (𝔗𝔭𝓃−1, 𝔗𝔵, 𝒶1

𝓀) ≤ 𝔑𝒯
2 (𝔵, 𝔭𝓃, 𝒶1

𝓀)⨁ 𝔑𝒯
2𝜆(𝔭𝓃−1, 𝔵, 𝒶1

𝓀)  

By using (5) in the above inequality, we obtain 𝔑(𝔵, 𝔗𝔵, 𝒶1
𝓀) = 0 for all 𝒶1, 𝒶2, . . . , 𝒶𝓀 > 0, that is, 

𝔵 = 𝔗𝔵. Thus, 𝔵 is the unique fixed point of 𝔗. 

For 𝓀 = 1, the above theorem reduces to the following result of Grabiec (1988). 

Corollary 27  

Let (𝔐, 𝔑, ⨁) be a 𝔾 −complete revised fuzzy metric space such that 

lim
𝑡→+∞

𝔑(𝔭, 𝔮, 𝔞) = 0 for all 𝔭, 𝔮 ∈ 𝔐   (6)  

and 𝔗: 𝔐 → 𝔐be a mapping. Suppose that there exists 𝜆 ∈ (0, 1) such that  

𝔑(𝔗𝔭, 𝔗𝔮, 𝔞) ≤ 𝔑(𝔭, 𝔮, 𝔞)   (7)  

for all 𝔵, 𝔶 ∈ 𝔐. Then, 𝔗 has a unique fixed point.  

Remark 28  

Let (𝔐, 𝔑, ⨁) be a revised fuzzy metric space and 𝔗: 𝔐 → 𝔐 be a mapping. The contractive  



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condition (7) tells that the mapping 𝔗 contract the space with respect to the parameter t in the sense 

that the degree of the nearness of images of any two points under 𝔗 is not less than the degree of the 

nearness of corresponding points (obviously in case of stationery revised fuzzy metric spaces (see 

Gregori and Romaguera 2004) it is not applicable).  

In Theorem 26, the mapping contracts the space with respect to only parameter 𝒶𝒯 for some 

𝒯 ∈ {1, 2, … , 𝓀} and it may not be contractive with respect to other parameters. Similarly, (𝔐, 𝔑, ⨁) 

is assumed l-natural k-fuzzy metric space for at least one 𝒯 ∈ {1, 2, … , 𝓀} only. The following example 

verifies the above remark. 

Example 29  

Let 𝔐 = [0, 1] × [0, 1] and ⨁ be the product t-conorm and the revised fuzzy set 𝔑 on  

𝔐2 × (0, ∞)2 be defined by 

𝔑(𝔭, 𝔮, 𝔞1, 𝔞2) = 1 − [1 +
|𝔮1 − 𝔭1| + |𝔮2 − 𝔭2|

𝔞1
]

−1

 

for all 𝔭 = (𝔭1, 𝔭2), 𝔮 = (𝔮1, 𝔮2) ∈ 𝔐 and 𝔞1, 𝔞2 > 0. Then, (𝔐, 𝔑, ⨁) is a 𝔾 −complete revised fuzzy 

2-metric space (𝓀 = 2). Moreover, lim
𝔞1→+∞

𝔑(𝔭, 𝔮, 𝔞1, 𝔞2) = 0 for all 𝔭, 𝔮 ∈ 𝔐, 𝔞2 > 0,that is, 

(𝔐, 𝔑, ⨁) is a 1 −natural revised fuzzy 2-metric space. Define a mapping 𝔗: 𝔐 → 𝔐 by 

𝔑(𝔗𝔭, 𝔗𝔮, 𝜆𝔞1, 𝔞2) = 1 − [1 +
|𝔮1−𝔭1|+|𝔮2−𝔭2|

2𝜆𝔞1
]

−1

≤ 1 − [1 +
|𝔮1−𝔭1|+|𝔮2−𝔭2|

𝔞1
]

−1

= 𝔑(𝔭, 𝔮, 𝔞1, 𝔞2)  

for 𝜆 ∈ [1/2, 1). By Theorem 26, 𝔗 has a unique fixed point. In this case, a point (0,0) ∈ 𝔐 is a fixed 

point of 𝔗.  

In Theorem 26, corresponding to condition (2), we assume that the space (𝔐, 𝔑, ⨁) is  

𝒯 −natural. Notice that, for the existence of a fixed point, the 𝒯 −naturalness cannot be replaced by 

the 𝔪 −naturalness with 𝔪 ≠ 𝒯. The following example verifies this fact. 

Example 30  

Let 𝔐 = [0, 1] × [0, 1] and ⨁ be the product t-conorm and the revised fuzzy set 𝔑 on  

𝔐2 × (0, ∞)2 be defined by  

𝔑(𝔭, 𝔮, 𝔞1, 𝔞2) = 1 − [1 +
|𝔮1 − 𝔭1| + |𝔮2 − 𝔭2|

𝔞2
]

−1

 

for all 𝔭 = (𝔭1, 𝔭2), 𝔮 = (𝔮1, 𝔮2) ∈ 𝔐 and 𝔞1, 𝔞2 > 0. Then, (𝔐, 𝔑, ⨁) is a 𝔾 −complete revised fuzzy 

2-metric space (𝓀 = 2). Moreover, 

lim
𝔞1→+∞

𝔑(𝔭, 𝔮, 𝔞1, 𝔞2) = 0 for all 𝔭, 𝔮 ∈ 𝔐, 𝔞1 > 0, 

That is, (𝔐, 𝔑, ⨁) is a 2 −natural revised fuzzy 2-metric space. Define a mapping 𝔗: 𝔐 → 𝔐 by 

𝔗(𝔭1, 𝔭2) = (𝔭1, 𝔭2) for all (𝔭1, 𝔭2) ∈ 𝔐. 



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Notice that, for any arbitrary 𝜆 ∈ (0, 1) 

𝔑(𝔗𝔭, 𝔗𝔮, 𝜆𝔞1, 𝔞2) ≤ 𝔑(𝔭, 𝔮, 𝔞1, 𝔞2) 

But the fixed point of 𝔗 is not unique. Indeed, every point (𝔭1, 𝔭2) ∈ 𝔐 is a fixed point of 𝔗. 

Finally, we will prove a fixed-point result for a revised fuzzy 𝓀 −contraction mapping. We begin with 

the definition of a revised fuzzy 𝓀 −contraction mapping as follows: 

Definition 31  

Let (𝔐, 𝔑, ⨁) be a revised fuzzy 𝓀 −metric space. A mapping 𝔗: 𝔐 → 𝔐 is called a revised  

fuzzy 𝓀 −contraction mapping if  

𝔑(𝔗𝔭, 𝔗𝔮, 𝒶1
𝓀) ≤ 𝜆{𝔑(𝔭, 𝔮, 𝒶1

𝓀)}                (8)  

for all (𝔭1, 𝔭2) ∈ 𝔐and𝒶1, 𝒶2, . . . , 𝒶𝓀 > 0, where 𝜆 ∈ [0, 1) is a constant. 

Theorem 32  

Let (𝔐, 𝔑, ⨁) be a 𝔾 −complete revised fuzzy 𝓀 −metric space and 𝔗: 𝔐 → 𝔐 is called a  

revised fuzzy 𝓀 −contraction mapping. Then, 𝔗 has a unique fixed point.  

Proof Let 𝔭0 ∈ 𝔐 and define a sequence {𝔭𝑛} by 𝔭𝑛 = 𝔗𝔭𝑛−1 for all 𝑛 ∈ 𝒩. We will show that this 

sequence is a 𝔾 −Cauchy sequence. For any 𝑛 ∈ 𝒩, we have 

𝔑(𝔭𝑛, 𝔭𝑛+1, 𝒶1
𝓀) = 𝔑(𝔗𝔭𝑛−1, 𝔗𝔭𝑛, 𝒶1

𝓀) ≤ 𝜆{𝔑(𝔭𝑛−1, 𝔭𝑛, 𝒶1
𝓀)} 

By repeating in this manner, we obtain 

𝔑(𝔭𝑛, 𝔭𝑛+1, 𝒶1
𝓀) ≤ 𝜆𝑛{𝔑(𝔭𝑛−1, 𝔭𝑛, 𝒶1

𝓀)}   (9) 

for all 𝑛 ∈ 𝒩. Since 𝜆 ∈ [0, 1), we conclude from (9) that 

lim
𝓃→+∞

{𝔑(𝔭𝑛, 𝔭𝑛+1, 𝒶1
𝓀)} ≥ 1, 

that is, 

lim
𝓃→+∞

𝔑(𝔭𝑛, 𝔭𝑛+1, 𝒶1
𝓀) = 0,              (10) 

for all 𝒶1, 𝒶2, . . . , 𝒶𝓀 > 0. For each 𝑛 ∈ 𝒩, 𝓍 > 0 and 𝒶1, 𝒶2, . . . , 𝒶𝓀 > 0, we have 

𝔑(𝔭𝑛, 𝔭𝑛+1, 𝒶1
𝓀) ≤ 𝔑𝒯

2 (𝔭𝑛, 𝔭𝑛+1, 𝒶1
𝓀)⨁𝔑𝒯

2 (𝔭𝑛, 𝔭𝑛+𝓍, 𝒶1
𝓀)  

 ≤ {
𝔑𝒯

2 (𝔭𝑛, 𝔭𝑛+1, 𝒶1
𝓀)⨁𝔑𝒯

22
(𝔭𝑛+1, 𝔭𝑛+2, 𝒶1

𝓀) ⨁ … ⨁

𝔑𝒯
2𝓍−1

(𝔭𝑛+𝓍−2, 𝔭𝑛+𝓍−1, 𝒶1
𝓀)⨁𝔑𝒯

2𝓍−1
(𝔭𝑛+𝓍−1, 𝔭𝑛+𝓍 , 𝒶1

𝓀)
}             (11) 

From (10), we have, 

lim
𝓃→+∞

𝔑𝓍
𝒯(𝔭𝑛, 𝔭𝑛+1, 𝒶1

𝓀) = 0, for all 𝒶1, 𝒶2, . . . , 𝒶𝓀 > 0 and 𝓍 > 0,which together with inequality (11) 

yields, 

lim
𝓃→+∞

𝔑(𝔭𝑛, 𝔭𝑛+𝓍, 𝒶1
𝓀) ≤ 0 ⨁ 0 ⨁ … ⨁ 0 = 0, for all 𝒶1, 𝒶2, . . . , 𝒶𝓀 > 0 and 𝓍 > 0.  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol X No. Y (2025) 

 

1330 
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Therefore, the sequence  {𝔭𝑛} is a 𝔾 −Cauchy sequence in 𝔐. By the 𝔾 −completeness of  𝔐, there 

exists 𝔵 ∈ 𝔐 such that the sequence  {𝔭𝑛} converges to 𝔵, that is, 

lim
𝓃→+∞

𝔑(𝔭𝑛, 𝔵, 𝒶1
𝓀) = 0,             (12) 

for all 𝒶1, 𝒶2, . . . , 𝒶𝓀 > 0. 

Now, we will show that 𝔵 is a fixed point of 𝔗. For each 𝑛 ∈ 𝒩, we have 

𝔑(𝔭𝑛+1, 𝔗𝔵, 𝒶1
𝓀) = 𝔑(𝔗𝔭𝑛, 𝔗𝔵, 𝒶1

𝓀) ≤ 𝜆{𝔑(𝔭𝑛, 𝔵, 𝒶1
𝓀)} 

By using (12), we have, lim
𝓃→+∞

{𝔑(𝔭𝑛+1, 𝔗𝔵, 𝒶1
𝓀)} = 0,that is, 

lim
𝓃→+∞

𝔑(𝔭𝑛+1, 𝔗𝔵, 𝒶1
𝓀) = 0             (13) 

for all 𝒶1, 𝒶2, . . . , 𝒶𝓀 > 0. For any 𝑛 ∈ 𝒩, we have 

𝔑(𝔵, 𝔗𝔵, 𝒶1
𝓀) ≤ 𝔑2

𝒯(𝔵, 𝔵𝑛+1, 𝒶1
𝓀)⨁𝔑2

𝓀(𝔭𝑛+1, 𝔗𝔵, 𝒶1
𝓀),  

which together with (12) and (13) yields 

𝔑(𝔵, 𝔗𝔵, 𝒶1
𝓀) = 0 for all 𝓌1, 𝓌2, . . . , 𝓌𝓀 > 0. That is, 𝔗𝔵 = 𝔵. Thus, 𝔵 is a fixed point of 𝔗. 

𝔑(𝔵, 𝔶, 𝓌1
𝓀) > 0, that is, 𝔑(𝔵, 𝔶, 𝓌1

𝓀) < 1. 

Now, we have 

𝔑(𝔵, 𝔶, 𝓌1
𝓀) = 𝔑(𝔗𝔵, 𝔗𝔶, 𝓌1

𝓀) ≤ 𝜆{𝔑(𝔵, 𝔶, 𝓌1
𝓀)} 

Since 𝜆 < 1, the above inequality yields a contradiction.  

Therefore, we must have, 𝔵 = 𝔶. Thus, the fixed point of 𝑇 is unique. 

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