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Fixed Point Theorems on Modular Revised Fuzzy Metric Spaces 

 

A. Muraliraj1, P. Shanmugavel2, R. Thangathamizh3* 

1PG & Research Department of Mathematics, 

Urumu Dhanalakshmi College, Bharathidasan University, Trichy, India. 

Email id: karguzali@gmail.com. 

2Department of Mathematics, 

Selvamm arts and science College, Periyar University, Namakkal , India. 

Email id: p.sham1988@gmail.com. 

3Department of Mathematics, K. Ramakrishnan College of Engineering, Trichy, India. 

Email id: thamizh1418@gmail.com. 

*Corresponding Author 

 

Article History: 

Received: 12-04-2024 

Revised: 27-05-2024 

Accepted: 14-06-2024 

Abstract:   

In this paper, we present a new space which is a melange between a revised fuzzy metric 

space and a modular metric space. We state some properties and examples of our new 

space. Then, we formulate and prove the existence and uniqueness results of a fixed point 

for continuous mappings under this new space. To support our results, we introduce some 

examples and an application. 

Keywords: Modular metric space, Revised fuzzy metric space, Modular revised fuzzy 

metric space, fixed point. 

MSC2020: 54H25, 47H10. 

1. Introduction 

In 1965, Zadeh [24] presented the concept of a fuzzy set. Ten years later, Kramosil and Michalek 

[12] stated the definition of fuzzy metric spaces. In 1988, Grabiec [7] implemented the notion of fuzzy 

metric space to extend the Banach contraction theorem over this space. Posteriorly, George and 

Veeramani [6] employed the definition of t-norm to formulate and introduce some results on the notion 

of a fuzzy metric space. Then, several researchers presented different contraction conditions over fuzzy 

metric spaces. 

In 2010, Chistyakov [3-5] introduced the notion of modular metric spaces. Then, numerous 

mathematicians discussed different results in their works over modular metric spaces, for example, 

look at the references [16, 18-19, 21,2-24].Muraliraj and Thangathamizh [17] used the revised fuzzy 

set technique to start a family of revised fuzzy mappings that are extensions of multivalued mappings 

and produced a result in revised fuzzy metric space for these mappings in 2021. Muraliraj and 

Thangathamizh [16] were the first to establish the idea of revised fuzzy contractive mappings and show 

a fixed point theorem in revised fuzzy metric spaces for these mappings. The rational type revised 

fuzzy-contraction condition in RFM spaces was recently established by Muraliraj et al. [14], who also 

proved other FP theorems with an application. The concept of revised fuzzy cone metric (RFCM) 



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space was first presented in 2023 by Thangathamizh et al. [15]. Under the presumption that "the revised 

fuzzy cone contractive sequences are Cauchy," they demonstrated a few fundamental features of FP 

as well as a "revised fuzzy cone Banach contraction theorem." Afterwards, several FP theorems in 

RFCMS were proven by Muraliraj and Thangathamizh [20, 25-27] without requiring that the "Revised 

fuzzy cone contractive sequences are Cauchy. 

In this paper, we introduce a new space named modular revised fuzzy metric space. We launch 

some fixed point results over a modular fuzzy metric spaces. To analyse our work, we state some 

examples, corollaries, and an application. 

2. Preliminaries 

In this section, we will recall some definitions which are crucial in this paper. 

Definition 2.1. [24]  

Let Y be any set. A fuzzy set E in Y is a function with domain Y and values in [0,1]. 

Definition 2.2. [23]  

Given a binary operation ⊛: [0,1]2 →  [0,1].  An operator ⊛ is a continuous t-conorm if 

∀𝛼, 𝛽, 𝛾, 𝛿 ∈ [0,1] satisfy: 

(1) 𝛼 ⊛ 𝛽 = 𝛽 ⊛ 𝛼. 

(2) (𝛼 ⊛  𝛽)  ⊛  𝛾 =  𝛼 ⊛ (𝛽 ⊛  𝛾). 

(3) 𝛼 ⊛  0 =  𝛼. 

(4) If 𝛼 ≤  𝛾 and 𝛽 ≤  𝛿, then 𝛼 ⊛  𝛽 ≤  𝛾 ⊛  𝛿. 

A. Sostak [2] in 2018 introduced the concept of a revised fuzzy metric space using the defintion 

of t-conorm as follows: 

Definition 2.3. [2]  

The triplet (𝑌, 𝛬,⊛) is called a revised fuzzy metric space if  Y is an arbitrary set, ⊛ is a 

continuous   t-conorm and Λ is a revised fuzzy metric on 𝑌2 × (0, ∞) → [0,1]  for all 𝜄, 휂, 𝜗 in 𝑌, and 

for all 𝑠, 𝑡 > 0 satisfying the following conditions: 

(1) 𝛬(𝜄, 휂, 0) = 0, 𝛬(𝜄, 휂, 𝑡) < 1, ∀𝑡 > 0. 

(2) 𝛬(𝜄, 휂, 𝑡) = 0 if and only if 𝜄 = 휂, for all 𝑡 > 0. 

(3) 𝛬(𝜄, 휂, 𝑡) = 𝛬(휂, 𝜄, 𝑡). 

(4) 𝛬(𝜄, 휂, 𝑡) ⊛ 𝛬(휂, 𝜗, 𝑠) ≥ 𝛬(𝜄, 𝜗, 𝑡 + 𝑠).  

(5) 𝛬(𝜄, 휂, . ): (0, ∞) → [0,1] is right continuous. Here, Λ called a revised fuzzy metric on Y. 

Example 2.1. [2] 

Let (𝑌, 𝑑) be a metric space. Define 𝛼 ⊛ 𝛽 =  𝛼 + 𝛽 − 𝛼𝛽 for all 𝛼, 𝛽 ∈ [0,1], and 

𝛬: 𝑌2 × (0, ∞) → [0,1] as 



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𝛬(𝜄, 휂, 𝑡) =
𝑑(𝜄,𝜂)

𝑡+ 𝑑(𝜄,𝜂)
   

∀𝜄, 휂 ∈ 𝛯 and 𝑡 > 0. Then (𝛯, 𝛬,⊛) is a revised fuzzy metric space; called revised fuzzy metric induced 

by the metric 𝑑. 

The notions of convergence, completeness and compactness on revised fuzzy metric spaces were 

presented in [17] as follows: 

Definition 2.4. [17]  

Let (𝛯, 𝛬,⊛) be a revised fuzzy metric space. 

(1) A sequence {𝜄𝜅}𝜅∈𝑁  in 𝑌 is convergent to an element 𝜄 ∈ 𝑌 if lim
𝜅→∞

𝛬(𝜄𝜅 , 𝜄, 𝑡) = 0, for all t > 0. 

(2) A sequence {𝜄𝜅}𝜅∈𝑁  in 𝑌 is Cauchy if for all 0 < 휀 < 1 and for t >0, there exists a number 𝜅0 ∈ 𝑁 

such that   Λ(Lk, lY, t) < ε for each 𝜅, 𝑌 ≥ 𝜅0. 

(3) A revised fuzzy metric space in which every Cauchy sequence is convergent is said to be complete. 

(4) A revised fuzzy metric space in which every sequence has a convergent subsequence is said to be 

compact. 

In 2010, Chistyakov [3-8] defined the notion of modular metric spaces as follows: 

Definition 2.5. [3]  

A modular metric on a nonempty set Y is a function 𝛩: (0, ∞) × 𝑌2 → [0, ∞) that will be written 

as 𝛩%(𝜄, 휂)  =  𝛩(%, 𝜄, 휂); for all 𝜄, 휂, 𝜗 ∈  𝑌 and for all %, 𝜎 > 0, satisfy the following three 

conditions: 

(1) 𝛩%(𝜄, 휂) = 0 if and only if 𝜄 = 휂, ∀ % > 0 and 𝜄, 휂 ∈ 𝑌. 

(2) 𝛩%(𝜄, 휂) = 𝛩%(휂, 𝜄), ∀% >  0 and 𝜄, 휂 ∈ 𝑌. 

(3) 𝛩%+𝜎(𝜄, 휂) ≤ 𝛩%(𝜄, 𝜗) + 𝛩𝜎(𝜗, 휂); for all %, 𝜎 > 0 and 𝜄, 휂, 𝜗 ∈ 𝑌. 

Remark 2.1.  

Let 𝛩 be modular on a set 𝑌. Then for given 𝜄, 휂 ∈ 𝑌, the function 0 < % → 𝛩%(𝜄, 휂) ∈ (0, ∞) is 

non increasing on (0, ∞). In fact if 0 < % < 𝜎, then by above definition  

𝛩𝜎(𝜄, 휂) ≤ 𝛩𝜎−%(𝜄, 𝜄) + 𝛩%(𝜄, 휂) = 𝛩%(𝜄, 휂) for all 𝜄, 휂 ∈ 𝑌. 

Definition 2.6. [5]  

Given a modular 𝛩 on 𝑌, a sequence {𝜄𝜅}𝜅∈𝑁  in 𝑌𝛩  is said to be modular convergent to an element 

𝜄 ∈ 𝑌𝛩  if there exists a number  % > 0, possibly depending on {𝜄𝜅} and 𝜄, such that  

lim
𝜅→∞

𝛩%(𝜄𝜅, 𝜄) = 0. i.e 𝜄𝜅 → 𝜄 as κ → ∞. 

Definition 2.7. [5]  

Given a modular 𝛩 on 𝑌, a sequence {𝜄𝜅}𝜅∈𝑁  in 𝑌𝛩  is said to be modular Cauchy if there exists a 

number  % = %({𝜄𝜅}) > 0 such that lim
𝜅,𝜉→∞

𝛩%(𝜄𝜅 , 𝜄𝜉) = 0 . 



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Definition 2.8. [5]  

A modular space 𝑌𝛩  is said to be modular complete if each Cauchy sequence in 𝑌𝛩  is modular 

convergent. In fact, if {𝜄𝜅} ⊂ 𝑌𝛩  and there exists % = %({𝜄𝜅}) > 0 such that lim
𝜅,𝑌→∞

𝛩%(𝜄𝜅 , 𝜄𝑌) = 0, then 

there exists 𝜄 ∈ 𝑌𝛩, such that lim
𝜅→∞

𝛩%(𝜄𝜅, 𝜄) = 0. 

Definition 2.9. 

A modular 𝛩 on 𝑌 is said to be satisfied the ∆2-condition if  lim
𝑛→∞

𝛩%(𝜄𝜅 , 𝜄) = 0, for some         % >

0 implies that lim
𝑛→∞

𝛩%(𝜄𝜅 , 𝜄) = 0, for all % > 0. 

3. Main results 

In this section, we construct a new space called a modular revised fuzzy metric space. We present 

some examples of this space. Also, we formulate and prove some new fixed point results under this 

space. We start by presenting the following definitions. 

Definition 3.1.  

A modular revised fuzzy metric space is the triplet (𝑌, 휁%,⊛) such that Y is an arbitrary set, (⊛) 

is a continuous t-conorm and 휁% is a revised fuzzy metric on                                                (0, ∞) ×

𝑌2 × (0, ∞) → [0,1]; for all 𝜄, 휂, 𝜗 in 𝑌, and 𝑠, 𝑡 > 0 satisfying the following conditions: 

(1) 휁%(𝜄, 휂, 0) = 0, 휁%(𝜄, 휂, 𝑡) < 1, for all 𝑡, % > 0. 

(2) 휁%(𝜄, 휂, 𝑡) = 0 if and only if 𝜄 = 휂, for all 𝑡, % > 0. 

(3) 휁%(𝜄, 휂, 𝑡) = 휁%(휂, 𝜄, 𝑡), for all 𝑡, % > 0. 

(4) 휁𝜎(𝜄, 휂, 𝑡) ⊛ 휁%(휂, 𝜗, 𝑠) ≤ 휁𝜎+%(𝜄, 𝜗, 𝑡 + 𝑠), for all 𝑡, 𝑠, 𝜎, % > 0. 

(5) 휁%(𝜄, 휂, . ): (0, ∞) → [0,1] is right continuous. Here, 휁% is called a modular revised fuzzy metric. 

Definition 3.2.  

Let (𝑌, 휁%,⊛) be a modular revised fuzzy metric space. 

(1) A sequence {𝜄𝜅}𝜅∈𝑁  in 𝑌 is convergent to an element 𝜄 ∈ 𝑌 𝑖𝑓 lim
𝜅→∞

휁%(𝜄𝜅 , 𝜄, 𝑡) = 0 for all 𝑡 > 0 and 

some % > 0. 

(2) A sequence {𝜄𝜅}𝜅∈𝑁  in 𝑌 is Cauchy if for all 0 < 휀 < 1, there exists a number 𝜅0 ∈ 𝑁 such that 

휁𝜚(𝜄𝜅, 𝜄𝜉 , 𝑡) < 휀, for each 𝜅, 𝑌 ≥ 𝜅0 and some % > 0. 

(3) A modular revised fuzzy metric space in which every Cauchy sequence is convergent is said to be 

complete. 

(4) A modular revised fuzzy metric space in which every sequence has a convergent subsequence is 

said to be compact. 

 

 



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Definition 3.3.  

A revised fuzzy modular metric 휁% on 𝑌 is said to be satisfied the ∆2t-condition if  

lim
𝜅→∞

𝛬(𝜄𝜅 , 𝜄, 𝑡) = 0, for some % > 0 and for some 𝑡 > 0 imply that lim
𝜅→∞

휁%(𝜄𝜅, 𝜄, 𝑡) = 0, for  all % >

0, and for all 𝑡 > 0. 

Example 3.1.  

Let (𝑌, 𝛩%) be a modular metric space. Define 𝛼 ∗ 𝛽 =  𝛼 + 𝛽 −  𝛼𝛽 for all 𝛼, 𝛽 ∈ [0,1], and 

휁%: (0, ∞) × 𝑌2 × (0, ∞) → [0,1] by 

휁𝜚(𝜄, 휂, 𝑡) =
𝛩𝜚(𝜄, 휂)

𝑡 + 𝛩𝜚(𝜄, 휂)
 

Then (𝑌, 휁%,⊛) is a modular revised fuzzy metric space. 

Remark 3.1.  

(1) For 𝛩𝜚(𝜄, 휂) =
|𝜄−𝜂|

𝜚
 , we have휁𝜚(𝜄, 휂, 𝑡) =

|𝜄−𝜂|

𝜚

𝑡+
|𝜄−𝜂|

𝜚

 is a modular revised fuzzy metric. 

(2) For 𝛩𝜚(𝜄, 휂) =
|𝜄−𝜂|

𝜚+1
 , we have 휁𝜚(𝜄, 휂, 𝑡) =

|𝜄−𝜂|

𝜚+1

𝑡+
|𝜄−𝜂|

𝜚+1

 is a modular revised fuzzy metric. 

Example 3.2.  

Let (𝑌, 𝛩%) be a modular metric space. Define 𝛼 ∗ 𝛽 = 𝛼 + 𝛽 − 𝛼𝛽 for all 𝛼, 𝛽 ∈ [0,1], and 

휁%: (0, ∞) × 𝑌2 × (0, ∞) → [0,1] by 

휁𝜚(𝜄, 휂, 𝑡) = 𝑒𝑥𝑝−{𝛩𝜚(𝜄,𝜂)}(1 − 𝑒𝑥𝑝{𝛩𝜚(𝜄,𝜂)}) 

Then (𝑌, 휁%,⊛) is a modular revised fuzzy metric space. 

Remark 3.2.  

(1) For 𝛩𝜚(𝜄, 휂) =
|𝜄−𝜂|

𝜚
, we have 휁𝜚(𝜄, 휂, 𝑡) = 𝑒𝑥𝑝

−{
|𝜄−𝜂|

𝜚
}

(1 − 𝑒𝑥𝑝
{

|𝜄−𝜂|

𝜚
}
)  is a modular revised fuzzy 

metric. 

(2) For 𝛩𝜚(𝜄, 휂) =
|𝜄−𝜂|

𝜚+1
, we have 휁𝜚(𝜄, 휂, 𝑡) = 𝑒𝑥𝑝

−{
|𝜄−𝜂|

𝜚+1
}

(1 − 𝑒𝑥𝑝
{

|𝜄−𝜂|

𝜚+1
}
) is a modular revised fuzzy 

metric. 

Theorem 3.1.  

On a complete modular revised fuzzy metric space (𝑌, 휁%,⊛), consider a continuous mapping 

𝛤: 𝑌 → 𝑌. Suppose there exist a strictly non-decreasing, continuous function        𝛶: (0,1] → [0, ∞) 

with    𝛶(1) = 0 and a real number 𝐻 with 0 < 𝐻 < 1 such that 

𝛶 (
1

𝜁%(𝛤𝜄,𝛤𝜂,𝑡)
− 1)  ≤  𝐻𝛶 (

1

𝜁%(𝜄,𝜂,𝑡)
− 1), for all 𝜄, 휂 ∈ 𝑌, 𝜄 ≠ 휂.                                   (3.1) 

Then 𝛤 has a unique fixed point in 𝑌. 



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Proof.  

Let 𝜄0 be an arbitrary point in 𝑌. Choose 𝜄1 ∈ 𝑌 such that 𝜄1 = 𝛤𝜄0. Continuing this process, we 

construct a sequence (𝜄𝜅) such that 𝜄𝜅+1 = 𝛤𝜄𝜅, for 𝜅 =  0,1,2.. Let 𝜄 = 𝜄𝜅−1 and 휂 = 𝜄𝜅. Replacing this 

in (3.1), we get 

𝛶 (
1

𝜁%(𝛤𝜄,𝛤𝜂,𝑡)
− 1) =  𝛶 (

1

𝜁%(𝛤𝜄𝜅−1 ,𝛤𝜄𝜅 ,𝑡)
− 1) = 𝛶 (

1

𝜁%(𝜄𝜅,𝜄𝜅+1,𝑡)
− 1)   

≤  𝐻𝛶 (
1

휁%(𝜄𝜅−1, 𝜄𝜅 , 𝑡)
− 1) <  𝛶 (

1

휁%(𝜄𝜅−1, 𝜄𝜅 , 𝑡)
− 1). 

Since Υ is a strictly non-decreasing function, we obtain 

          

(3.2) 

1

휁%(𝜄𝜅 , 𝜄𝜅+1, 𝑡)
− 1 <

1

휁%(𝜄𝜅−1, 𝜄𝜅 , 𝑡)
− 1. 

We use the same method for 𝜄 = 𝜄𝜅−2 and 휂 = 𝜄𝜅−1, we get 

(3.3) 

1

휁%(𝜄𝜅 , 𝜄𝜅−1, 𝑡)
− 1 <

1

휁%(𝜄𝜅−1, 𝜄𝜅−2, 𝑡)
− 1. 

(3.4) 

Therefore, (3.3) and (3.4) imply that 휁%(𝜄𝜅 , 𝜄𝜅−1, 𝑡) is a strictly non-increasing sequence of positive 

real numbers in [0,1]. 

Put 𝛴𝜅(%, 𝑡) = 휁%(𝜄𝜅 , 𝜄𝜅+1, 𝑡). Then {𝛴𝜅(%, 𝑡) } is a strictly non-increasing sequence. So ∃ 𝛴(%, 𝑡) 

such that lim
𝜅→∞

𝛴𝜅(%, 𝑡) = 𝛴(%, 𝑡). Assume that 0 < 𝛴(%, 𝑡) < 1. By (3.2), we have 

𝛶(𝛴𝜅(%, 𝑡))  ≤  𝐻𝛶(𝛴𝜅−1(%, 𝑡)). 

So, 

lim
𝜅→∞

 𝛶(𝛴𝜅(%, 𝑡)) ≤ lim
𝜅→∞

 𝐻𝛶(𝛴𝜅−1(%, 𝑡)). 

The continuity of 𝛶 implies that 

𝛶(𝛴(%, 𝑡))  ≤  𝐻𝛶(𝛴(%, 𝑡)), a contradiction. Then 𝛴(%, 𝑡) = 0. 

Now, we will prove that {𝜄𝜅} is a Cauchy sequence. Assume not, then for 0 < 휀 < 1, 

there exist two sub-sequences {𝜄𝑌(𝑖)} and {𝜄𝜅(𝑖)} such that for each 𝑖 ∈ 𝑁. 

let 𝜅(𝑖), 𝑌(𝑖) ∈ 𝑁 satisfying 𝜅(𝑖), 𝑌(𝑖) ≥ 𝜅 and 𝜅(𝑖) < 𝑌(𝑖) < 𝑖, such that 

1

𝜁𝜚(𝜄𝜅(𝑖),𝜄𝜉(𝑖),𝑡)
− 1 ≥ 휀,

1

𝜁𝜚(𝜄𝜅(𝑖)−1,𝜄𝜉(𝑖)−1,𝑡)
− 1 < 휀,

1

𝜁𝜚(𝜄𝜅(𝑖)−1,𝜄𝜉(𝑖),𝑡)
− 1 < 휀.        (3.5) 

Consider 

휀 ≤
1

𝜁𝜚(𝜄𝜅(𝑖),𝜄𝜉(𝑖),𝑡)
− 1 ≤

1

𝜁𝜚
2

(𝜄𝜅(𝑖),𝜄𝜉(𝑖)−1,
𝑡

2
)

− 1 ⊛
1

𝜁𝜚
2

(𝜄𝜅(𝑖)−1,𝜄𝜉(𝑖),
𝑡

2
)

− 1.  

By definition of ∆2t-condition on Y and (3.5), we have 



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1

𝜁𝜚
2

(𝜄𝜅(𝑖)−1,𝜄𝜉(𝑖),
𝑡

2
)

− 1 < 휀. 

Hence, 

휀 ≤
1

𝜁𝜚(𝜄𝜅(𝑖),𝜄𝜉(𝑖),𝑡)
− 1 ≤

1

𝜁𝜚
2

(𝜄𝜅(𝑖),𝜄𝜅(𝑖)−1,
𝑡

2
)

− 1 ⊛ 휀. 

If 𝑖 → ∞, we have 𝛴𝜅(𝑖) (
𝜚

2
,

𝑡

2
) =

1

𝜁𝜚
2

(𝜄𝜅(𝑖),𝜄𝜅(𝑖)−1,
𝑡

2
)

− 1 → 0. 

So, 
1

𝜁𝜚(𝜄𝜅(𝑖),𝜄𝜉(𝑖),𝑡)
− 1 → 휀. 

Then by (3.1), we have 

𝛶 (
1

𝜁%(𝜄𝜅(𝑖),𝜄𝑌(𝑖),𝑡)
− 1) ≤ 𝐻𝛶 (

1

𝜁%(𝜄𝜅(𝑖)−1,𝜄𝑌(𝑖)−1,𝑡)
− 1) <  𝛶 (

1

𝜁%(𝜄𝜅(𝑖)−1,𝜄𝑌(𝑖)−1,𝑡)
− 1). Thus 

휀 ≤
1

𝜁𝜚(𝜄𝜅(𝑖),𝜄𝜉(𝑖),𝑡)
− 1 <

1

𝜁𝜚(𝜄𝜅(𝑖),𝜄𝜉(𝑖)−1,
𝑡

2
)

− 1 < 휀, 

which is impossible.  

Hence {𝜄𝜅} is a Cauchy sequence in a complete modular revised fuzzy metric space.  

So ∃$ ∈ 𝑌 such that 𝑙𝑖𝑚 𝜄𝜅 = $, that means lim
 𝜅→∞

휁%(𝜄𝜅 , $, 𝑡) = 0. 

To show $ is a fixed point of 𝛤, we have : 

𝛤 is continuous: 𝜄𝜅 → $ ⇒ 𝛤𝜄𝜅 → 𝛤$. By (3.1), we have 

𝛶 (
1

𝜁%(𝜄𝜅,𝛤𝜄𝜅,𝑡)
− 1) ≤ 𝐻𝛶 (

1

𝜁%(𝜄𝜅−1,𝜄𝜅,𝑡)
− 1).  

Since 𝛶(1) = 0 and for 𝜅 → ∞, we get 

𝛶 (
1

𝜁%($,𝛤$,𝑡)
− 1) ≤ 𝐻𝛶 (

1

𝜁%($,$,𝑡)
− 1) = 𝐻𝛶(1) = 0. So 휁%($, 𝛤$, 𝑡) = 0. 

Hence, 휁%($, 𝛤$, 𝑡) = 0 ⇒ 𝛤$ = $. Thus $ is a fixed point of 𝛤. 

Now, we will prove that $ is unique.  

Assume not, ∃𝜔 ∈ 𝑌, such that 𝛤𝜔 = 𝜔 where 𝜔 ≠ $ and lim
𝜅→∞

𝜄𝜅 = 𝜔. Then  

𝛶 (
1

𝜁𝜚(𝜔,𝜛,𝑡)
− 1) = 𝛶 (

1

𝜁𝜚(𝛤𝜔,𝛤𝜛,𝑡)
− 1) ≤ 𝐻𝛶 (

1

𝜁𝜚(𝜔,𝜛,𝑡)
− 1)  

≤ 𝐻𝛶 (휁𝜚

2
(𝜔, 𝜄𝜅 ,

𝑡

2
) ⊛ 휁𝜚

2
(𝜄𝜅 , 𝜛,

𝑡

2
)). 

Since 𝛶(1) = 0 and for 𝜅 → ∞ on both sides, we have 

𝛶 (
1

𝜁𝜚(𝜔,𝜛,𝑡)
− 1) ≤ 𝐻𝛶 (휁𝜚

2
(𝜔, 𝜄𝜅 ,

𝑡

2
) ⊛ 휁𝜚

2
(𝜄𝜅 , 𝜛,

𝑡

2
)) = 𝐻𝛶(1) = 0.  



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So, 𝛶 (
1

𝜁%(𝜔,$,𝑡)
− 1) = 0. 

Hence, 휁%(𝜔, $, 𝑡) = 0 ⇒ 𝑤 = $. Thus Γ has a unique fixed point $.  

Theorem 3.2.  

On a complete modular revised fuzzy metric space (𝑌, 휁%,⊛), consider a continuous mapping 

𝛤: 𝑌 → 𝑌. Suppose there exist a strictly non-decreasing, continuous function 𝛶: (0,1] → [0, ∞) with 

𝛶(1) = 0 and a real number 𝐻 with 0 < 𝐻 < 1 such that 

𝛶 (
1

𝜁%(𝛤𝜄,𝛤𝜂,𝑡)
− 1) ≤ 𝐻 (

1
𝜁𝜚(𝜄,𝜂,𝑡)+𝜁𝜚(𝛤𝜄,𝜄,𝑡)

4

− 1 +
1

𝜁𝜚(𝜂,𝛤𝜂,𝑡)

2

− 1)        (3.6) 

for all 𝜄, 휂 ∈ 𝑌, 𝜄 ≠ 휂. Then 𝛤 has a unique fixed point in 𝑌. 

Proof.  

Let 𝜄0 be an arbitrary point in Y. Choose 𝜄1 ∈ 𝑌 such that 𝜄1 =  𝛤𝜄0. Continuing this process, we 

construct a sequence (𝜄𝜅) such that 𝜄𝜅+1 = 𝛤𝜄𝜅, for κ = 0,1,2.. 

Let 𝜄 = 𝜄𝜅−1 and 휂 = 𝜄𝜅. Replacing this in (3.6), we get 

𝛶 (
1

𝜁%(𝛤𝜄,𝛤𝜂,𝑡)
− 1) = 𝛶 (

1

𝜁%(𝛤𝜄𝜅−1,𝛤𝜄𝜅,𝑡)
− 1) = 𝛶 (

1

𝜁%(𝜄𝜅,𝜄𝜅+1,𝑡)
− 1)  

≤ 𝛶𝐻 (
1

𝜁𝜚(𝜄𝜅−1,𝜄𝜅,𝑡)+𝜁𝜚(𝛤𝜄𝜅−1,𝜄𝜅−1,𝑡)

4

− 1 +
1

𝜁𝜚(𝜄𝜅,𝛤𝜄𝜅,𝑡)

2

− 1)  

< 𝛶 (
1

𝜁𝜚(𝜄𝜅−1,𝜄𝜅,𝑡)+𝜁𝜚(𝛤𝜄𝜅−1,𝜄𝜅−1,𝑡)

4

− 1 +
1

𝜁𝜚(𝜄𝜅,𝛤𝜄𝜅,𝑡)

2

− 1)  

= 𝛶 (
1

𝜁𝜚(𝜄𝜅−1,𝜄𝜅,𝑡)+𝜁𝜚(𝜄𝜅,𝜄𝜅−1,𝑡)

4

− 1 +
1

𝜁𝜚(𝜄𝜅,𝜄𝜅+1,𝑡)

2

− 1)            (3.7) 

Since 𝛶 is a strictly non-decreasing function, we obtain 

1

𝜁𝜚(𝜄𝜅,𝜄𝜅+1,𝑡)
− 1 < 𝛶 (

1
𝜁𝜚(𝜄𝜅−1,𝜄𝜅,𝑡)+𝜁𝜚(𝜄𝜅,𝜄𝜅−1,𝑡)

4

− 1 +
1

𝜁𝜚(𝜄𝜅,𝜄𝜅+1,𝑡)

2

− 1).  

Hence  

1

𝜁%(𝜄𝜅,𝜄𝜅+1,𝑡)
− 1 <

1

𝜁%(𝜄𝜅,𝜄𝜅−1,𝑡)
− 1.  

We use the same method for 𝜄 = 𝜄𝜅−2 and 휂 = 𝜄𝜅−1, we get 

                   (3.8) 

1

𝜁%(𝜄𝜅,𝜄𝜅−1,𝑡)
− 1 <

1

𝜁%(𝜄𝜅−1,𝜄𝜅−2,𝑡)
− 1.                    (3.9) 

Therefore, (3.8) and (3.9) imply that {휁%(𝜄𝜅 , 𝜄𝜅+1, 𝑡)} is a strictly non-increasing sequence of positive 

real numbers in [0,1]. 

Put 𝛴𝜅(%, 𝑡) = 휁%(𝜄𝜅 , 𝜄𝜅+1, 𝑡). Then {𝛴𝜅(%, 𝑡)} is a strictly non-increasing sequence.  



Communications on Applied Nonlinear Analysis 

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So ∃ 𝛴(𝑡, %) such that lim 𝛴𝜅(%, 𝑡) = 𝛴(𝑡, %).  

Assume that  

0 < 𝛴(𝑡, %)  <  1. 

By (3.7), we have 

 𝛶(𝛴𝜅(𝜚, 𝑡)) ≤ 𝛶𝐻 (
𝛴𝜅−1(𝜚,𝑡)

2
+

𝛴𝜅(𝜚,𝑡)

2
). 

So, 

lim
𝜅→∞

𝛶(𝛴𝜅(𝜚, 𝑡)) ≤ lim
𝜅→∞

𝛶𝐻 (
𝛴𝜅−1(𝜚,𝑡)

2
+

𝛴𝜅(𝜚,𝑡)

2
)  

By the continuity of 𝛶, we have 

𝛶(𝛴(𝜚, 𝑡)) ≤ 𝛶𝐻 (
𝛴(𝜚,𝑡)

2
+

𝛴(𝜚,𝑡)

2
), a contradiction. Then 𝛴(%, 𝑡) = 0. 

Now, we will prove that {𝜄𝜅} is a Cauchy sequence.  

Assume not, then for 0 < 휀 < 1, there exists two sub-sequences {𝜄𝑌(𝑖)} and {𝜄𝜅(𝑖)} such that for each 

𝑖 ∈ 𝑁, let 𝜅(𝑖), 𝑌(𝑖) ∈ 𝑁 satisfying 𝜅(𝑖), 𝑌(𝑖) ≥ 𝜅 and 𝜅(𝑖) > 𝑌(𝑖) > 𝑖, such that 

1

𝜁𝜚(𝜄𝜅(𝑖),𝜄𝜉(𝑖),𝑡)
− 1 ≥ 휀,

1

𝜁𝜚(𝜄𝜅(𝑖)−1,𝜄𝜉(𝑖)−1,𝑡)
− 1 < 휀,

1

𝜁𝜚(𝜄𝜅(𝑖)−1,𝜄𝜉(𝑖),𝑡)
− 1 < 휀.                  

(3.10) 

Consider 

휀 ≤ (
1

𝜁𝜚(𝜄𝜅(𝑖),𝜄𝜉(𝑖),𝑡)
− 1) ≤ (

1

𝜁𝜚
2

(𝜄𝜅(𝑖),𝜄𝜅(𝑖)−1,
𝑡

2
)

− 1) ⊛ (
1

𝜁𝜚
2

(𝜄𝜅(𝑖)−1,𝜄𝜉(𝑖),
𝑡

2
)

− 1). 

By definition of ∆2t-condition on Y and (3.10), we have 

휁𝜚

2
(𝜄𝜅(𝑖)−1, 𝜄𝜉(𝑖),

𝑡

2
) < 휀. 

Thus 

휀 ≤ (
1

𝜁𝜚(𝜄𝜅(𝑖),𝜄𝜉(𝑖),𝑡)
− 1) ≤ (

1

𝜁𝜚
2

(𝜄𝜅(𝑖),𝜄𝜅(𝑖)−1,
𝑡

2
)

− 1) ⊛ 휀. 

If i → ∞, we have 𝛴𝜅(𝑖) (
𝜚

2
,

𝑡

2
) ≤

1

𝜁𝜚
2

(𝜄𝜅(𝑖),𝜄𝜅(𝑖)−1,
𝑡

2
)

− 1 → 0. . 

So 

휁𝜚(𝜄𝜅(𝑖), 𝜄𝜉(𝑖), 𝑡) → 휀. 

Then by (3.6), we have 

𝛶 (
1

𝜁𝜚(𝜄𝜅(𝑖),𝜄𝜉(𝑖),𝑡)
− 1) ≤ 𝛶𝐻 (

1

𝜁𝜚(𝜄𝜅(𝑖)−1,𝜄𝜉(𝑖)−1,𝑡)+𝜁𝜚(𝛤𝜄𝜅(𝑖)−1,𝜄𝜅(𝑖)−1,𝑡)

4

− 1 +
1

𝜁𝜚(𝜄𝜉(𝑖)−1,𝛤𝜄𝜉(𝑖)−1,𝑡)

2

− 1)  



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< 𝛶 (
1

𝜁𝜚(𝜄𝜅(𝑖)−1,𝜄𝜉(𝑖)−1,𝑡)+𝜁𝜚(𝛤𝜄𝜅(𝑖)−1,𝜄𝜅(𝑖)−1,𝑡)

4

− 1 +
1

𝜁𝜚(𝜄𝜉(𝑖)−1,𝛤𝜄𝜉(𝑖)−1,𝑡)

2

− 1)  

= 𝛶 (
1

𝜁𝜚(𝜄𝜅(𝑖)−1,𝜄𝜉(𝑖)−1,𝑡)+𝜁𝜚(𝜄𝜅(𝑖),𝜄𝜅(𝑖)−1,𝑡)

4

− 1 +
1

𝜁𝜚(𝜄𝜉(𝑖)−1,𝜄𝜉(𝑖),𝑡)

2

− 1)  

Since 𝛶 is a strictly decreasing function, (3.10) implies that 

휀 <
𝜀+0

4
+

1

2
=

𝜀

4
+

1

2
< 휀,  

which is impossible.  

Hence {𝜄𝜅} is a Cauchy sequence in a complete modular revised fuzzy metric space.  

So ∃$ ∈ 𝑌 such that lim
𝜅→∞

𝜄𝜅 = $, that means lim
𝜅→∞

 휁%(𝜄𝜅 , $, 𝑡) = 0. 

To show $ is a fixed point of 𝛤,  

we have : 

𝛤 is continuous: 𝜄𝜅 → $ ⇒ 𝛤𝜄𝜅 → 𝛤$. 

By (3.6), we have 

𝛶 (
1

𝜁𝜚(𝜄𝜅,𝛤𝜄𝜅,𝑡)
− 1) ≤ 𝛶𝐻 (

1
𝜁𝜚(𝜄𝜅−1,𝜄𝜅,𝑡)+𝜁𝜚(𝛤𝜄𝜅−1,𝜄𝜅−1,𝑡)

4

− 1 +
1

𝜁𝜚(𝜄𝜅,𝛤𝜄𝜅,𝑡)

2

− 1)  

= 𝐻𝛶 (
1

𝜁𝜚(𝜄𝜅−1,𝜄𝜅,𝑡)+𝜁𝜚(𝜄𝜅,𝜄𝜅−1,𝑡)

4

− 1 +
1

𝜁𝜚(𝜄𝜅,𝛤𝜄𝜅,𝑡)

2

− 1)  

Since Υ(1) = 0 and for κ → ∞, we get 

𝛶 (
1

𝜁𝜚(𝜔,𝛤𝜛,𝑡)
− 1) = 𝐻𝛶 ((

1
𝜁𝜚(𝜔,𝜛,𝑡)+𝜁𝜚(𝜔,𝜛,𝑡)

4

− 1) + (
1

𝜁𝜚(𝜔,𝜛,𝑡)

2

− 1))  

= 𝐻𝛶 (
1

𝜁𝜚(𝜔,𝛤𝜛,𝑡)
− 1) = 𝐻𝛶(1) = 0. 

Hence, 휁%($, 𝛤$, 𝑡) = 0 ⇒ 𝛤$ = $. Thus $ is a fixed point of 𝛤. 

Now, we will prove that $ is unique.  

Assume not, ∃𝑤 ∈ 𝑌, such that 𝛤𝑤 = 𝑤 where 𝑤 ≠ $ and lim
𝜅→∞

𝜄𝜅 =  𝑤. By (3.6), we have  

𝛶 (
1

𝜁𝜚(𝜔,𝛤𝜛,𝑡)
− 1) = 𝛶 (

1

𝜁𝜚(𝛤𝜔,𝛤𝜛,𝑡)
− 1)  

≤ 𝐻𝛶 ((
1

𝜁𝜚(𝜔,𝜛,𝑡)+𝜁𝜚(𝛤𝜔,𝜛,𝑡)

4

− 1) + (
1

𝜁𝜚(𝜔,𝛤𝜛,𝑡)

2

− 1)) = 𝐻𝛶 (
1

𝜁𝜚(𝜔,𝛤𝜛,𝑡)

4

−
1

2
)  



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Hence, 휁%(𝑤, $, 𝑡) ≤ 0. 

Thus, 휁%(𝑤, $, 𝑡) = 0 ⇒ 𝑤 = $. So 𝛤 has a unique fixed point $.  

The two following examples satisfy Theorem (3.1). 

Example 3.3.  

Let 𝑌 = [0,1] and 휁𝜚(𝜄, 휂, 𝑡) =

|𝜄−𝜂|

𝜚

𝑡+
|𝜄−𝜂|

𝜚

 . Define 𝛤: [0,1] → [0,1] via 𝛤(𝜄) =
𝜄

3
. 

Also, define 𝛶: (0,1] → [0, ∞) via 𝛶(𝜄) =
1

𝜄
− 1. Note that 𝛶 is a strictly non-decreasing, continuous 

function and 𝛶(1) = 0. Now, we have: 

휁𝜚(𝛤𝜄, 𝛤휂, 𝑡) =
|𝜄−𝜂|

3𝜚𝑡+|𝜄−𝜂|
, 휁𝜚(𝜄, 휂, 𝑡) =

|𝜄−𝜂|

3𝜚𝑡+|𝜄−𝜂|
   

𝛶 (
1

𝜁𝜚(𝛤𝜄,𝛤𝜂,𝑡)
− 1) = 휁𝜚(𝛤𝜄, 𝛤휂, 𝑡) =

|𝜄−𝜂|

3𝜚𝑡
; 𝛶 (

1

𝜁𝜚(𝜄,𝜂,𝑡)
− 1) = 휁𝜚(𝜄, 휂, 𝑡) =

|𝜄−𝜂|

𝜚𝑡
 

So,  

Hence, for 𝐻 =
1

3
, we get 𝛶 (

1

𝜁%(𝛤𝜄,𝛤𝜂,𝑡)
− 1) = 𝐻𝛶 (

1

𝜁%(𝜄,𝜂,𝑡)
− 1).  

Thus, Theorem (3.1) implies that 𝛤 has a unique fixed point 0 ∈ 𝑌. 

Example 3.4.  

Let 𝑌 = [0,1] and 휁𝜚(𝜄, 휂, 𝑡) = 𝑒𝑥𝑝
{−

|𝜄−𝜂|

𝑡𝜚
}

(𝑒𝑥𝑝
{

|𝜄−𝜂|

𝑡𝜚
}

− 1) .  

Define 𝛤: [0,1] → [0,1] via 𝛤(𝜄) =
𝜄

5
. Also, define 𝛶: (0,1] → [0, ∞) via 𝛶(𝜄) = −𝑙𝑛𝜄.  

Note that Υ is a strictly non-decreasing, continuous function and 𝛶(1) = 0.  

Now, we have: 

휁𝜚(𝛤𝜄, 𝛤휂, 𝑡) = 𝑒𝑥𝑝
{−

|𝜄−𝜂|

5𝑡𝜚
}

(𝑒𝑥𝑝
{

|𝜄−𝜂|

5𝑡𝜚
}

− 1) , 휁𝜚(𝜄, 휂, 𝑡) = 𝑒𝑥𝑝
{−

|𝜄−𝜂|

𝑡𝜚
}

(𝑒𝑥𝑝
{

|𝜄−𝜂|

𝑡𝜚
}

− 1). 

So, Υ(
1

𝜁𝜚(𝛤𝜄,𝛤𝜂,𝑡)
− 1) =

|𝜄−𝜂|

5𝑡𝜚
; Υ(

1

𝜁𝜚(𝜄,𝜂,𝑡)
− 1) =

|𝜄−𝜂|

𝑡𝜚
. 

Hence, for 𝐻 =
1

5
, we get 𝛶 (

1

𝜁%(𝛤𝜄,𝛤𝜂,𝑡)
− 1) = 𝐻𝛶 (

1

𝜁%(𝜄,𝜂,𝑡)
− 1). 

Thus Theorem (3.1) implies that 𝛤 has a unique fixed point 0 ∈ 𝑌. 

The following example satisfies Theorem (3.2). 

 

 

 



Communications on Applied Nonlinear Analysis 

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223 
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Example 3.5.  

Let 𝑌 = [0,1] and 휁𝜚(𝜄, 휂, 𝑡) =

|𝜄−𝜂|

𝜚

𝑡+
|𝜄−𝜂|

𝜚

. Define 𝛤 ∶ [0,1] → [0,1] via Γ(𝜄) = 𝑐, 𝑐 ∈ [0,1]. Also, 

define 𝛶: (0,1] → [0, ∞) via 𝛶(𝜄) =
1

𝜄
− 1. note that 𝛶 is a strictly non- decreasing, continuous 

function and 𝛶(1) = 0. Now, we have: 

휁𝜚(Γ𝜄, Γ휂, 𝑡) =
|𝑐 − 𝑐|

𝑡 + |𝑐 − 𝑐|
= 0;   

휁𝜚(𝜄, 휂, 𝑡)

4
=

1

4
×

|𝜄 − 휂|

𝜚𝑡 + |𝜄 − 휂|
 

휁𝜚(Γ𝜄, 𝜄, 𝑡) =
1

4
×

|𝑐−𝜄|

𝜚𝑡+|𝑐−𝜄|
  and   휁𝜚(휂, Γ휂, 𝑡) =

1

2
×

|𝑐−𝜂|

𝜚𝑡+|𝑐−𝜂|
. 

So, 

(
1

𝜁𝜚(𝜄,𝜂,𝑡)+𝜁𝜚(Γ𝜄,𝜄,𝑡)

4

− 1) + (
1

𝜁𝜚(𝜂,Γ𝜂,𝑡)

2

− 1) = 0 = 휁𝜚(Γ𝜄, Γ휂, 𝑡). 

So, for 𝐻 such that 0 < 𝐻 < 1, we have 

(
1

𝜁𝜚(𝜄,𝜂,𝑡)+𝜁𝜚(Γ𝜄,𝜄,𝑡)

4

− 1) + (
1

𝜁𝜚(𝜂,Γ𝜂,𝑡)

2

− 1) ≤ 0 = 휁𝜚(Γ𝜄, Γ휂, 𝑡). 

Thus Theorem (3.2) implies that 𝛤 has a unique fixed point 𝑐 ∈ 𝑌. 

4. Application 

In this section, we use our obtained results to show that the following integral equation has a 

solution: 

𝜄(𝜅) = ℎ(𝜅) + ∫ Ω(𝑘, 𝑠)𝜍
1

0
(𝑠, 𝜄(𝑠))𝑑𝑠, 𝜅 ∈ [0,1].  (4.1) 

Let 𝑌 = 𝐶([0,1]) be the space of all continuous functions defined on [0,1].  

Define a modular revised fuzzy metric: 

휁%(𝜄, 휂, 𝑡) ∶  (0, ∞)2 × 𝐶([0,1])2 → [0,1],  

By 

휁𝜚(𝜄, 휂, 𝑡) =

𝑠𝑢𝑝

𝜅 ∈ [0,1]
|𝜄(𝜅) − 휂(𝜅)|

𝜚

𝑡 +
𝑠𝑢𝑝

𝜅 ∈ [0,1]
|𝜄(𝜅) − 휂(𝜅)|

𝜚

 

Then (𝑌, 휁%,⊛) is a complete modular revised fuzzy metric space. 

Theorem 4.1.  

Suppose we have the following hypotheses: 

(1) ∃𝑎 continuous function 𝑔: [0,1] → [0,1] such that 

|(𝑠, 𝜄) − (𝑠, 휂)| ≤ 𝑔(𝑠)|𝜄 − 휂|.        (4.2)  



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And 

∫ 𝑔(𝑘)𝑑𝑘 ≤
1

3

1

0
            (4.3) 

(2) Ω(𝑘, 𝑠) ≥ 0, ∀ 𝑘, 𝑠 ∈ [0,1]  

(4.4) 

Then the integral equation (4.1) has a solution 𝜄⊛ ∈ 𝐶2([0,1]).  

Proof.  

Take the operator: 

Γ𝜄(𝜅) = ℎ(𝜅) + ∫ Ω(𝑘, 𝑠)𝜍
1

0
(𝑠, 𝜄(𝑠))𝑑𝑠, 𝜅 ∈ [0,1]  

For all 𝜄, 휂 ∈ 𝐶([0,1]), we have 

휁𝜚(𝜄, 휂, 𝑡) =

𝑠𝑢𝑝

𝜅∈[0,1]
|𝜄(𝜅)−𝜂(𝜅)|

𝜚

𝑡+
𝑠𝑢𝑝

𝜅∈[0,1]
|𝜄(𝜅)−𝜂(𝜅)|

𝜚

 =

𝑠𝑢𝑝

𝜅∈[0,1]
1

𝜚
|ℎ(𝜅)+∫ Ω(𝑘,𝑠)𝜍

1
0

(𝑠,𝜄(𝑠))𝑑𝑠−ℎ(𝜅)−∫ Ω(𝑘,𝑠)𝜍
1

0
(𝑠,𝜂(𝑠))𝑑𝑠|

𝑡+
𝑠𝑢𝑝

𝜅∈[0,1]
1

𝜚
|ℎ(𝜅)+∫ Ω(𝑘,𝑠)𝜍

1
0

(𝑠,𝜄(𝑠))𝑑𝑠−ℎ(𝜅)−∫ Ω(𝑘,𝑠)𝜍
1

0
(𝑠,𝜂(𝑠))𝑑𝑠|

  

=

𝑠𝑢𝑝

𝜅∈[0,1]
1

𝜚
|∫ Ω(𝑘,𝑠)𝜍

1
0

(𝑠,𝜄(𝑠))𝑑𝑠−𝜍(𝑠,𝜂(𝑠))𝑑𝑠|

𝑡+
𝑠𝑢𝑝

𝜅∈[0,1]
1

𝜚
|∫ Ω(𝑘,𝑠)𝜍

1
0

(𝑠,𝜄(𝑠))−𝜍(𝑠,𝜂(𝑠))𝑑𝑠|
  

By (4.2) and (4.4), we have 

|∫ Ω(𝑘, 𝑠)𝜍
1

0
(𝑠, 𝜄(𝑠)) − 𝜍(𝑠, 휂(𝑠))𝑑𝑠| ≤ ∫ Ω(𝑘, 𝑠)

1

0
|𝜍(𝑠, 𝜄(𝑠)) − 𝜍(𝑠, 휂(𝑠))𝑑𝑠|  

≤ ∫ 𝑔(𝑠)
1

0
|𝜄(𝑠) − 휂(𝑠)|𝑑𝑠. 

Hence by (4.3), we get 

𝑠𝑢𝑝

𝜅∈[0,1]
1

𝜚
|∫ Ω(𝑘,𝑠)𝜍

1
0

(𝑠,𝜄(𝑠))−𝜍(𝑠,𝜂(𝑠))𝑑𝑠|

𝑡+
𝑠𝑢𝑝

𝜅∈[0,1]
1

𝜚
|∫ Ω(𝑘,𝑠)𝜍

1
0

(𝑠,𝜄(𝑠))−𝜍(𝑠,𝜂(𝑠))𝑑𝑠|
≤

𝑠𝑢𝑝

𝜅∈[0,1]
1

𝜚
∫ 𝑔(𝑠)

1
0

|𝜄(𝑠)−𝜂(𝑠)|𝑑𝑠

𝑡+
𝑠𝑢𝑝

𝜅∈[0,1]
1

𝜚
∫ 𝑔(𝑠)

1
0

|𝜄(𝑠)−𝜂(𝑠)|𝑑𝑠
 =

𝑠𝑢𝑝

𝜅∈[0,1]
1

3𝜚
|𝜄(𝜅)−𝜂(𝜅)|

𝑡+
𝑠𝑢𝑝

𝜅∈[0,1]
1

3𝜚
|𝜄(𝜅)−𝜂(𝜅)|

  

Thus 

𝑡
𝑠𝑢𝑝

𝜅∈[0,1]
1

3𝜚
|𝜄(𝜅)−𝜂(𝜅)|

≤
𝑡

𝑠𝑢𝑝

𝜅∈[0,1]
1

𝜚
|∫ Ω(𝑘,𝑠)𝜍

1
0

(𝑠,𝜄(𝑠))−𝜍(𝑠,𝜂(𝑠))𝑑𝑠|
  

So 

3𝑡
𝑠𝑢𝑝

𝜅∈[0,1]
1

𝜚
|𝜄(𝜅)−𝜂(𝜅)|

≤
𝑡

𝑠𝑢𝑝

𝜅∈[0,1]
1

𝜚
|∫ Ω(𝑘,𝑠)𝜍

1
0

(𝑠,𝜄(𝑠))−𝜍(𝑠,𝜂(𝑠))𝑑𝑠|
  

Define 𝛶: (0,1] → [0, +∞) by 𝛶(𝑡) =
1

𝜄
− 1.  

Then 𝛶 is a strictly non-decreasing, continuous function and 𝛶(1) = 0.  

For 𝐻 =
1

3
, we obtain 𝛶 (

1

𝜁%(𝛤𝜄,𝛤𝜂,𝑡)
− 1)  ≤  𝐻𝛶 (

1

𝜁%(𝜄,𝜂,𝑡)
− 1).  

Therefore theorem (3.1) implies 𝛤 has a unique fixed point and hence the integral equation (4.1) has a 

solution 𝜄⊛ ∈ 𝐶2([0,1]).  



Communications on Applied Nonlinear Analysis 

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Conclusion:  

In this paper, we defined a new space called modular revised fuzzy metric space and stated 

some examples of this space. Also, we formulate and prove some new fixed point results under this 

space. In addition, we provided some examples and an application for showing the validity of our 

results. 

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