# Fixed Point Results for Generalized Fuzzy Contractive Mappings in Fuzzy Metric Spaces with Application in Integral Equations

Authors: Babak Mohammadi; Azhar Hussain; Vahid Parvaneh; Naeem Saleem; Rogheieh Jalal Shahkoohi
Journal: Journal of Mathematics
 (2021)

---

## Abstract

In this paper, we introduce generalizedα-η-fuzzy contractive mappings and generalized β-ζ-fuzzy contractive mappings and prove existence of fixed point for such mappings. Our results generalize and improve the recent work of Gopal and Vetro (Iranian journal of fuzzy systems, 11 (2014), 95–107). Some equivalent conditions of our results are presented. Also, an example is given to support our new results.


---

## Body

## 1. Introduction

The notion of fuzzy sets was introduced initially by Zadeh [1] in 1965. Afterwards, Kramosil and Michalek [2] introduced the notion of fuzzy metric space (FMS). In 1994, George and Veeramani [3] modified the notion of FMS, introduced by Kramosil and Michalek [2], and presented a Hausdorff topology of a FMS. Recently, Samet et al. [4] introduced a new concept of α-admissible and α-ζ-contractive-type mappings which is described below.LetΠ,d be a metric space. A mapping Q:Π⟶Π is said to be an α−ζ-contractive mapping if there exist two functions α:Π2⟶0,+∞ and ζ:0,+∞⟶0,+∞ so that ζ is nondecreasing and Σn=1∞ζnt<∞, for all t>0, such that(1)αι,κdQι,Qκ≤ζdι,κ,for all ι,κ∈Π. They established and proved some fixed point theorems for such mappings in complete metric spaces. Very recently, motivated by Samet et al. [4], the concept of α-admissible and α-η-fuzzy contractive mappings and β-ζ-fuzzy contractive mappings in a FMS was discussed by Gopal and Vetro [5]. Gopal and Vetro introduced the concept of α-η-fuzzy contractive mappings which is described below.LetΠ,ℱ,∗ be a FMS. A mapping Q:Π⟶Π is called an α-η-fuzzy contractive mapping if there exist two functions α:Π2×0,+∞⟶0,+∞ and η:0,+∞⟶0,+∞ such that η is right continuous and ηt<t, for all t>0, and(2)αι,κ,t1ℱQι,Qκ,t−1≤η1ℱι,κ,t−1,for all ι,κ∈Π and for all t>0.In [6], the authors studied the existence of common α-fuzzy fixed points for fuzzy mappings via F-contractions on a metric space. They obtained some common fixed points of fuzzy (multivalued) mappings satisfying an F-contraction associated with the Hausdorff metric.In this paper, we introduce generalizedα-η-fuzzy contractive and generalized β-ζ-fuzzy contractive mappings, which are motivated by Gopal and Vetro [5] and show that the fixed point exists for the presented mappings. Our results generalize and improve the results of Gopal and Vetro [5]. To show the usefulness of our results, some equivalent conditions are presented.


## 2. Preliminaries

In this section, we recall some definitions and notions which will be used in this paper.Definition 1.
(see [7]). A binary operation ∗:0,12⟶0,1 is a continuous t-norm if it satisfies the following conditions:(Q1) ∗ is associative and commutative(Q2) ∗ is continuous(Q3) a∗1=a for all a∈0,1(Q4) a∗b≤c∗d whenever a≤c and b≤d for all a,b,c,d∈0,1
Three examples of a continuous t-norm are Lukasievicz t-norm, that is,a∗Lb=maxa+b−1,0, product t-norm, that is, a∗Pb=ab, and minimum t-norm, that is, a∗Mb=mina,b. The concept of FMS which was defined by George and Veeramani [3] which is as follows.Definition 2 (see [3]).
LetΠ be an arbitrary nonempty set, ∗ be a continuous t-norm, and ℱ be a fuzzy set on Π2×0,∞. The triple Π,ℱ,∗ is called a FMS if, for each ι,κ,z∈Π and for all t,s>0,(ℱ1) ℱι,κ,t>0(ℱ2) ℱι,κ,t=1 if and only if ι=κ(ℱ3) ℱι,κ,t=ℱκ,ι,t(ℱ4) ℱι,z,t+s≥ℱι,κ,t∗ℱκ,z,s(ℱ5) ℱι,κ,.:0,∞⟶0,1 is continuous
If property (ℱ4) be replaced by ℱι,z,maxt,s≥ℱι,κ,t∗ℱκ,z,s, for each ι,κ,z∈Π and t,s>0, then the triple Π,ℱ,∗ is called a non-Archimedean FMS.Example 1.
LetΠ,d be a metric space. Define a∗b=ab (or a∗b=mina,b) for all a,b∈0,1, and define ℱ:Π2×0,∞⟶0,1 as ℱι,κ,t=t/t+dι,κ for all ι,κ∈Π and t>0. Then, Π,ℱ,∗ is a FMS. We call this induced fuzzy metric by the metric d the standard fuzzy metric.Definition 3 (see [3, 8, 9]).
LetΠ,ℱ,∗ be a FMS.(i)
A sequenceιn in Π is said to be convergent to a point ι∈Π whenever limn⟶∞ℱιn,ι,t=1 for all t>0; in this case, we write limn⟶∞ιn=ι(ii)
A sequenceιn in Π is called ℱ-Cauchy if, for each 0<ε<1 and t>0, there exists n0∈ℕ such that ℱιn,ιm,t>1−ε, for each n,m≥n0(iii)
A sequenceιn in Π is called G-Cauchy if limn⟶∞ℱιn,ιn+p,t=1, for each p∈ℕ and t>0(iv)
A FMS in which everyℱ-Cauchy (G-Cauchy) sequence is convergent is said to be ℱ-complete (G-complete)Definition 4 (see [10]).
LetΠ,ℱ,∗ be a FMS. The fuzzy metric ℱ is said to be triangular if the following condition holds,(3)1ℱι,z,t−1≤1ℱι,κ,t−1+1ℱκ,z,t−1,for all ι,κ,z∈Π and for all t>0.


## 3. Main Results

In this section, we give our main results of this study. Firstly, we give some notions that will be needed in the sequel.Denote byΦ the collection of all nondecreasing right continuous functions η:0,∞⟶0,∞ such that ηt<t for all t>0 and η−10=0. It is well known that limn⟶∞ηnt=0 for all t>0, where ηn denotes the nth iterate of η.Proposition 1 (see [11]).
LetΠ,ℱ,∗ be a FMS. Then, ℱ is a continuous function on Π2×0,∞.The notion ofα-admissibility for self-mappings in a FMS is given in [5] as follows.Definition 5.
LetΠ,ℱ,∗ be a FMS, α:Π2×0,∞⟶0,∞, and Q:Π⟶Π. Q is called α-admissible whenever, for all t>0,(4)ι,κ∈Π,αι,κ,t≥1⟹αQι,Qκ,t≥1.Definition 6.
LetΠ,ℱ,∗ be a FMS. Q:Π⟶Π is called a generalized α-η-fuzzy contractive mapping whenever there exist two functions η∈Φ and α:Π2×0,∞⟶0,∞ satisfying(5)αι,κ,t1ℱQι,Qκ,t−1≤η1mι,κ,t−1,for all ι,κ∈Π and t>0, where(6)mι,κ,t=minℱι,κ,t,ℱι,Qι,t,ℱκ,Qκ,t,2ℱι,Qκ,tℱκ,Qι,tℱι,Qκ,t+ℱκ,Qι,t.
Now, we are ready to state and prove our first main result.Theorem 1.
LetΠ,ℱ,∗ be a G-complete FMS. Let ℱ be triangular and Q:Π⟶Π be a generalized α-η-fuzzy contractive mapping so that(i)
Q is α-admissible,=(ii)
There existsι0∈Π such that αι0,Qι0,t≥1, for all t>0(iii)
Q is continuous
Then,Q possesses a fixed point.Proof.
Define the sequenceιn in Π by ιn=Qιn−1, for all n∈ℕ. If ιn=ιn−1 for some n∈ℕ, then ι∗=ιn−1 is a fixed point of Q. Assume that ιn≠ιn−1 for all n∈ℕ. Since αι0,ι1,t=αι0,Qι0,t≥1 and Q is α-admissible, so we get αι1,ι2,t=αQι0,Qι1,t≥1 for all t>0. Continuing this process we get αιn−1,ιn,t≥1 for all n∈ℕ and for all t>0. By (5), for any n∈ℕ and t>0, we have(7)1ℱιn,ιn+1,t−1=1ℱQιn−1,Qιn,t−1≤αιn−1,ιn,t1ℱQιn−1,Qιn,t−1≤η1mιn−1,ιn,t−1,where(8)mιn−1,ιn,t=minℱιn−1,ιn,t,ℱιn−1,Qιn−1,t,ℱιn,Qιn,t,2ℱιn−1,Qιn,tℱQιn−1,ιn,tℱQιn−1,ιn,t+ℱιn−1,Qιn,t=minℱιn−1,ιn,t,ℱιn,ιn+1,t,2ℱιn−1,ιn+1,t1+ℱιn−1,ιn+1,t.
On the contrary,(9)2ℱιn−1,ιn+1,t1+ℱιn−1,ιn+1,t=21/ℱιn−1,ιn+1,t+1≥21/ℱιn−1,ιn,t+1/ℱιn,ιn+1,t≥minℱιn−1,ιn,t,ℱιn,ιn+1,t.
From (8) and (9), we get mιn−1,ιn,t=minℱιn−1,ιn,t,ℱιn,ιn+1,t. Substituting in (7), we obtain(10)1ℱιn,ιn+1,t−1≤η1minℱιn−1,ιn,t,ℱιn,ιn+1,t−1.
Now, ifminℱιn−1,ιn,t,ℱιn,ιn+1,t=ℱιn,ιn+1,t, by the property ηt<t, then(11)1ℱιn,ιn+1,t−1≤η1ℱιn,ιn+1,t−1<1ℱιn,ιn+1,t−1,which is a contradiction. So, we have minℱιn−1,ιn,t,ℱιn,ιn+1,t=ℱιn−1,ιn,t and so(12)1ℱιn,ιn+1,t−1≤η1ℱιn−1,ιn,t−1<1ℱιn−1,ιn,t−1.
Hence,ℱιn,ιn+1,t>ℱιn−1,ιn,t. So, the sequence ℱιn,ιn+1,t is strictly increasing in the interval 0,1, for all t>0 . Let St=limn⟶∞ℱιn,ιn+1,t, for all t>0. We claim that St=1, for all t>0. Suppose the contradiction, St0<1, for some t0>0. Taking limit in both sides of (12), we obtain(13)1St0−1≤η1St0−1<1St0−1,which is a contradiction. Thus, we have limn⟶∞ℱιn,ιn+1,t=1, for all t>0. Now, for a fixed p∈ℕ, we have(14)ℱιn,ιn+p,t≥ℱιn,ιn+1,tp∗ℱιn+1,ιn+2,tp∗…∗ℱιn+p−1,ιn+p,tp⟶1∗1∗…∗1=1,as n⟶∞. So, ιn is a G-Cauchy sequence. Since Π,ℱ,∗ is G-complete, there exists ι∗∈Π such that ιn⟶ι∗, for all t>0. Since Q is continuous, so we get Qιn⟶Qι∗, for all t>0. Now, we have limn⟶∞ℱιn+1,Qι∗,t=limn⟶∞ℱQιn,Qι∗,t=1, for all t>0, that is, ιn⟶Qι∗. Uniqueness of the limit implies that Qι∗=ι∗.
In the above theorem, the fixed point is unique ifαι,κ,t≥1 for all t>0, whenever ι and κ are fixed points of Q.
In the following theorem, we omit the continuity condition of the mappingQ.Theorem 2.
We haveΠ,ℱ,∗ as a G-complete FMS, ℱ a triangular fuzzy set, and Q:Π⟶Π as a generalized α-η-fuzzy contractive mapping such that(i)
Q is α-admissible(ii)
There existsι0∈Π such that αι0,Qι0,t≥1 for all t>0(iii)
For each sequenceιn in Π with αιn,ιn+1,t≥1, for all n∈ℕ and t>0 such that ιn⟶ι, we have αιn,ι,t≥1, for all n∈ℕ and t>0
Then,Q admits a fixed point.Proof.
Following the proof of Theorem1, we obtain a sequence ιn in Π which is G-Cauchy, αιn,ιn+1,t≥1, for all n∈ℕ, and there exists ι∗∈Π such that ιn⟶ι∗. From condition (iii), we get αιn,ι∗,t≥1, for all n∈ℕ and t>0. Now, we have(15)1ℱQι∗,ι∗,t−1≤1ℱQι∗,Qιn,t−1+1ℱιn+1,ι∗,t−1≤αιn,ι∗,t1ℱQιn,Qι∗,t−1+1ℱιn+1,ι∗,t−1≤η1mιn,ι∗,t−1+1ℱιn+1,ι∗,t−1.
On the contrary,(16)ℱι∗,Qι∗,t≥mιn,ι∗,t=minℱιn,ι∗,t,ℱιn,Qιn,t,ℱι∗,Qι∗,t,2ℱιn,Qι∗,tℱQιn,ι∗,tℱιn,Qι∗,t+ℱQιn,ι∗,t=minℱιn,ι∗,t,ℱιn,ιn+1,t,ℱι∗,Qι∗,t,2ℱιn,Qι∗,tℱιn+1,ι∗,tℱιn,Qι∗,t+ℱιn+1,ι∗,t,where(17)2ℱιn,Qι∗,tℱιn+1,ι∗,tℱιn,Qι∗,t+ℱιn+1,ι∗,t=21/ℱtn+1,ι∗,t+1/ℱtn,Qι∗,t≥21/ℱtn+1,ι∗,t+1/ℱtn,ι∗,t+1/ℱι∗,Qι∗,t−1⟶21+1/ℱι∗,Qι∗,t≥21/ℱι∗,Qι∗,t+1/ℱι∗,Qι∗,t=ℱι∗,Qι∗,t.
Thus, taking limit on both sides of (16) as n⟶∞, we get limn⟶∞mιn,ι∗,t=ℱι∗,Qι∗,t. Now, taking limit on both sides of (15) as n⟶∞ and from right continuity of η, we obtain 1/ℱQι∗,ι∗,t−1≤η1/ℱQι∗,ι∗,t−1. This implies 1/ℱQι∗,ι∗,t−1=0. Therefore, ℱQι∗,ι∗,t=1 and so Qι∗=ι∗.Remark 1.
Sincemι,κ,t≤ℱι,κ,t implies 1/ℱι,κ,t−1≤1/mι,κ,t−1, for all ι,κ∈Π and t>0, thus any α-η-fuzzy contractive mapping in the sense of [5] which is a generalized α-η-fuzzy contractive mapping. Therefore, Theorems 1 and 2 are generalizations of Theorems 3.5 and 3.6 in [5], respectively.
The following example shows that Theorem2 is a real generalization of Theorem 3.6 in [5].Example 2.
Let(18)A1=mn:m=0,1,3,9,…,n=1,4,…,3k+1,…,A2=mn:m=1,3,9,…,n=2,5,…,3k+2,…,A3=2k:k∈ℕ,and Π=A1∪A2∪A3. Let a∗b=ab, for all a,b∈0,1, and ℱι,κ,t=t/t+ι−κ, for all ι,κ∈Π and t>0. Define Q:Π⟶Π by(19)Qι=3ι11,ι∈A1,ι8,ι∈A2,2ι,ι∈A3,and α:Π2×0,∞⟶0,∞ by(20)αι,κ,t=1,ι,κ∈A1∪A2,0,otherwise.
Ifι,κ∈A1, then(21)1ℱQι,Qκ,t−1=3ι/11−3κ/11t=311ι−κt=3111ℱι,κ,t−1≤6111mι,κ,t−1.
Ifι,κ∈A2, then(22)1ℱQι,Qκ,t−1=ι/8−κ/8t=18ι−κt=181ℱι,κ,t−1≤6111mι,κ,t−1.
Ifι,κ∈A3, then αι,κ,t=0, and clearly (8) holds true.
Ifι∈A1 and κ∈A2, then(23)1ℱQι,Qκ,t−1=3ι/11−κ/8t=311ι−11/24κt.
So, ifι>11/24κ, then(24)1ℱQι,Qκ,t−1=311ι−11/24κt≤311ι−1/8κt=611121ℱι,Qκ,t−1=611121ℱι,Qκ,t+1−1≤611121ℱι,Qκ,t+1ℱκ,Qι,t−1=61112ℱι,Qκ,tℱκ,Qι,t/ℱι,Qκ,t+ℱκ,Qι,t−1≤6111mι,κ,t−1,and if ι<11/24κ, then(25)1ℱQι,Qκ,t−1=31111/24κ−ιt≤311κ−ιt≤3111ℱι,κ,t−1≤6111mι,κ,t−1.
We see that1/ℱQι,Qκ,t−1≤6/111/mι,κ,t−1, for all ι,κ∈A1∪A2. So, by definition of the mapping α, we get αι,κ,t1/ℱQι,Qκ,t−1≤6/111/mι,κ,t−1 for all ι,κ∈Π. Thus, Q is a generalized α-η-fuzzy contractive mapping with ηt=6/11t. Also, for ι0=1, we have αι0,Qι0,t=α1,3/11,t=1. It is easy to check that Q is α-admissible and the condition (iii) in Theorem 2 holds. So, by Theorem 2, Q possesses a fixed point. Here, Q0=0. Now, let ι=1,κ=27/29. Then, we have(26)αι,κ,t1ℱQι,Qκ,t−1=3/11−1/827/29t=3992552t>229t=1−27/29t=1ℱι,κ,t−1>η1ℱι,κ,t−1.
Thus,Q is not an α-η-fuzzy contractive mapping. Therefore, the main result of [5] is not applicable in this example.
LetΠ,≤¯ be a poset. A mapping Q:Π⟶Π is called nondecreasing if ι≤¯κ implies Qι≤¯Qκ, for all ι,κ∈Π.Definition 7.
LetΠ,≤¯ be a poset and Π,ℱ,∗ be a FMS. We say that Q:Π⟶Π is a generalized fuzzy ordered η -contractive mapping whenever there exists η∈Φ such that(27)1ℱQι,Qκ,t−1≤η1mι,κ,t−1,for all ι,κ∈Π with ι≤¯κ and t>0, where(28)mι,κ,t=minℱι,κ,t,ℱι,Qι,t,ℱκ,Qκ,t,2ℱι,Qκ,tℱκ,Qι,tℱι,Qκ,t+ℱκ,Qι,t.Corollary 1.
LetΠ,≤¯ be a poset and Π,ℱ,∗ be a G-complete FMS. Let ℱ be triangular and Q:Π⟶Π be a generalized fuzzy ordered η-contractive such that(i)
Q is nondecreasing(ii)
There existsι0∈Π such that ι0≤¯Qι0(iii)
Q is continuous
Then,Q admits a fixed point.Proof.
Define the functionα:Π2×0,∞⟶0,∞ by αι,κ,t=1 if ι≤¯κ and αι,κ,t=0 otherwise. Then, apply Theorem 1.Corollary 2.
LetΠ,≤¯ be a poset and Π,ℱ,∗ be a G-complete FMS. Let ℱ be triangular and Q:Π⟶Π be a generalized fuzzy ordered η-contractive mapping such that(i)
Q is nondecreasing(ii)
There existsι0∈Π such that ι0≤¯Qι0(iii)
For each nondecreasing sequenceιn in Π which ιn⟶ι, we have ιn≤¯ι, for all n∈ℕ
Then,Q possesses a fixed point.Proof.
Define the functionα:Π2×0,∞⟶0,∞ by αι,κ,t=1 if ι≤¯κ and αι,κ,t=0 otherwise. Then, apply Theorem 2.
Using equivalent conditions from [12], we can find also some equivalent results for Theorems 1 and 2.Proposition 2.
LetΠ,ℱ,∗ be a G-complete FMS. Let ℱ be triangular and Q:Π⟶Π be a mapping such that there exist a mapping α:Π2×0,∞⟶0,∞, a lower semicontinuous function ζ:0,∞⟶0,∞ and a function η:0,∞⟶0,∞ with η−10=0 and liminft⟶∞ηt>0 such that(29)ζαι,κ,t1ℱQι,Qκ,t−1≤ζ1mι,κ,t−1−η1mι,κ,t−1,for all ι,κ∈Π and t>0. Moreover, assume that(i)
Q is α-admissible(ii)
There existsι0∈Π such that αι0,Qι0,t≥1, for all t>0(iii)
Q is continuous or for each sequence ιn in Π with αιn,ιn+1,t≥1, for all n∈ℕ and t>0, for which ιn⟶ι, and we have αιn,ι,t≥1 for all n∈ℕ and t>0
Then,Q admits a fixed point.Proposition 3.
LetΠ,ℱ,∗ be a G-complete FMS. Let ℱ be triangular and Q:Π⟶Π be a mapping such that there exist a mapping α:Π2×0,∞⟶0,∞, a continuous from right function ζ:0,∞⟶0,∞ with Σn=1∞ζnt<∞ for all t>0 and a function η:0,∞⟶0,∞ with ηt<t for all t>0 such that(30)ζαι,κ,t1ℱQι,Qκ,t−1≤ηζ1mι,κ,t−1,for all ι,κ∈Π and t>0. Moreover, let(i)
Q is α-admissible(ii)
There existsι0∈Π such that αι0,Qι0,t≥1 for all t>0(iii)
Q is continuous or for each sequence ιn in Π with αιn,ιn+1,t≥1 for all n∈ℕ and t>0 for which ιn⟶ι, and we have αιn,ι,t≥1 for all n∈ℕ and t>0
Then,t>0 possesses a fixed point.


## 4. Generalizedβ-ζ-Fuzzy Contractive Mappings

In this section, we introduce generalizedβ-ζ-fuzzy contractive mappings and prove existence of fixed point for such mappings. Denote by Ψ the collection of all nondecreasing left continuous functions ζ:0,1⟶0,1 such that ζr>r, for all r∈0,1. It is well known that limn⟶∞ζnr=1 and ζ1=1, for all r∈0,1, where ζn denotes the nth iterate of ζ.Definition 8 (see [5]).
LetΠ,ℱ,∗ be a FMS, β:Π2×0,∞⟶0,∞, and Q:Π⟶Π be a mapping. It is called that Q is β-admissible whenever, for all t>0,(31)ι,κ∈Π,βι,κ,t≤1⟹βQι,Qκ,t≤1.Definition 9.
LetΠ,ℱ,∗ be a FMS. We say that Q:Π⟶Π is a generalized β-ζ-fuzzy contractive mapping whenever there exist two functions ζ∈Ψ and β:Π2×0,∞⟶0,∞ satisfying(32)βι,κ,tℱQι,Qκ,t≥ζmι,κ,t,for all ι,κ∈Π and t>0, where(33)mι,κ,t=minℱι,κ,t,ℱι,Qι,t,ℱκ,Qκ,t,2ℱι,Qκ,tℱκ,Qι,tℱι,Qκ,t+ℱκ,Qι,t.Theorem 3.
LetΠ,ℱ,∗ be a ℱ-complete non-Archimedean FMS and Q:Π⟶Π be a generalized β-ζ-fuzzy contractive mapping so that(i)
Q is β-admissible(ii)
There existsι0∈Π such that βι0,Qι0,t≤1 for all t>0(iii)
For each sequenceιn in Π with βιn,ιn+1,t≤1, for all n∈ℕ and t>0, we have βιm,ιn,t≤1, for all m,n∈ℕ, with n>m and t>0(iv)
For each sequenceιn in Π with βιn,ιn+1,t≤1, for all n∈ℕ and t>0, in which ιn⟶ι, and we have βιn,ι,t≤1, for all n∈ℕ and t>0
Then,Q admits a fixed point.Proof.
Define the sequenceιn in Π by ιn=Qιn−1, for all n∈ℕ. If ιn=ιn−1 for some n∈ℕ, then ι∗=ιn−1 is a fixed point of Q. Assume that ιn≠ιn−1, for all n∈ℕ. Since βι0,ι1,t=βι0,Qι0,t≤1 and Q is β-admissible, so we get βι1,ι2,t=βQι0,Qι1,t≤1 for all t>0. Continuing this process, we get βιn−1,ιn,t≤1 for all n∈ℕ and for all t>0. By (32), for any n∈ℕ and t>0, we have(34)ℱιn,ιn+1,t=ℱQιn−1,Qιn,t≥βιn−1,ιn,tℱQιn−1,Qιn,t≥ζmιn−1,ιn,t,where, as in the proof of Theorem 1, we have(35)mιn−1,ιn,t=minℱιn−1,ιn,t,ℱιn,ιn+1,t.
Substituting in (34), we obtain(36)ℱιn,ιn+1,t≥ζminℱιn−1,ιn,t,ℱιn,ιn+1,t.
Now, ifminℱιn−1,ιn,t,ℱιn,ιn+1,t=ℱιn,ιn+1,t, then(37)ℱιn,ιn+1,t≥ζℱιn,ιn+1,t>ℱιn,ιn+1,t,which is a contradiction. So, we have(38)minℱιn−1,ιn,t,ℱιn,ιn+1,t=ℱιn−1,ιn,t,so(39)ℱιn,ιn+1,t≥ζℱιn−1,ιn,t.
From (39), we obtain(40)ℱιn,ιn+1,t≥ζnℱι0,ι1,t⟶1,as n⟶∞. So, we have limn⟶∞ℱιn,ιn+1,t=1, for all t>0. Now, if ιn is not an ℱ-Cauchy sequence, then there are ε∈0,1, t>0 and k0∈ℕ such that, for each k∈ℕ with k≥k0, there exist mk,nk∈ℕ with mk>nk≥k satisfying(41)ℱιmk,ιnk,t≤1−ε,βιmk,ιnk,t≤1.
Let, for eachk, mk be the least positive integer exceeding nk satisfying the above property, that is,(42)ℱιmk−1,ιnk,t>1−ε,ℱιmk,ιnk,t≤1−ε.
Now, we have(43)1−ε≥ℱιmk−1,ιnk,t≥ℱιmk−1,ιnk,t∗ℱιmk−1,ιmk,t≥1−εℱιmk−1,ιmk,t.
Thus,limk⟶∞ℱιmk−1,ιnk,t=1−ε, for all t>0. Also, we have(44)ℱιmk−1,ιnk,t≥ℱιmk−1,ιmk+1,t∗ℱιmk−1,ιnk+1,t∗ℱιnk+1,ιnk,t≥ℱιmk−1,ιmk+1,t∗βιmk−1,ιnk+1,tℱQιmk,Qιnk,t∗ℱιnk+1,ιnk,t≥ℱιmk−1,ιmk+1,t∗ζmιmk,ιnk,t∗ℱιnk+1,ιnk,t.
On the contrary,(45)ℱιmk,ιnk,t≥mιmk,ιnk,t=minℱιmk,ιnk,t,ℱιmk,ιmk+1,t,ℱιnk,ιnk+1,t,2ℱιmk,ιnk+1,tℱιmk+1,ιnk,tℱιmk,ιnk+1,t+ℱιmk+1,ιnk,t=minℱιmk,ιnk,t,ℱιmk,ιmk+1,t,ℱιnk,ιnk+1,t,21/ℱιmk,ιnk+1,t+1/ℱιmk,ιnk,t≥minℱιmk,ιnk,t,ℱιmk,ιmk+1,t,ℱιnk,ιnk+1,t,21/ℱιmk,ιnk,t∗ℱιnk,ιnk+1,t+1/ℱιmk+1,ιmk,t∗ℱιmk,ιnk,t.
Taking the limit in the above inequality, we obtain(46)1−ε≥limk⟶∞mιmk,ιnk,t≥min1−ε,1,1,1−ε=1−ε,which implies(47)limk⟶∞mιmk,ιnk,t=1−ε.
Taking limit in (44) and from left continuity of ζ, we obtain(48)1−ε≥ζ1−ε>1−ε.
This contradiction shows thatιn is a ℱ-Cauchy sequence. Let ιn⟶ι∗. Then, from assumption (iv), βιn,ι∗,t≤1, for all n∈ℕ and t>0. Now, we have(49)ℱQι∗,ι∗,t≥ℱQιn,Qι∗,t∗ℱιn+1,ι∗,t≥βιn,ι∗,tℱQιn,Qι∗,t∗ℱιn+1,ι∗,t≥ζmιn,ι∗,t∗ℱιn+1,ι∗,t.
On the contrary, as in the proof of Theorem2, we have(50)limn⟶∞mιn,ι∗,t=ℱQι∗,ι∗,t.
Taking limit in (49) and from left continuity of ζ, we obtain(51)ℱQι∗,ι∗,t≥ζℱQι∗,ι∗,t,which implies ℱQι∗,ι∗,t=1, so Qι∗=ι∗.Remark 2.
Sinceℱι,κ,t≥mι,κ,t, for all ι,κ∈Π and t>0, thus any β-ζ-fuzzy contractive mapping in the sense of [5] which is a generalized β-ζ-fuzzy contractive mapping. Therefore, Theorem 3 is a generalization of Theorem 4.4 in [5].


## 5. Application

This section is devoted to the application for the uniqueness of the solution of an integral equation. Consider the integral equation,(52)ιr=gr+∫0rFr,s,ιsds,for allr∈0,I,I>0,and Banach space C0,I,ℝ of all continuous functions defined on 0,I equipped with supremum norm,(53)ι=supr∈0,1ιr,ι∈C0,I,ℝ,with induced metric(54)dι,κ=supr∈0,Iιr−κr.Now, consider the FMS with productt−norm as(55)ℱι,κ,t=tt+dι,κ,for allι,κ∈C0,I,ℝ,t>0.According to [3], standard FMS and the corresponding metric space have same topologies. So, FMS defined in (55) is complete.Theorem 6.
Consider the integral operatorQ in C0,I,ℝ as(56)Qιr=gr+∫0rFr,s,ιsds,where F satisfies the following conditions: there exists f:0,I×0,I⟶0,∞ such that f∈L10,I,ℝ and for all ι,κ∈C0,I,ℝ, r,s∈0,I, we have(57)Fs,r,ιr−Fs,r,κr≤fr,smaxιs−κs,ιs−Qκs,κs−Qκs,2ιs−Qκsκs−Qιsιs−Qκs|+ιs−κs,where(58)supr∈0,1∫0rfr,sds≤λ<1.Then, the integral equation (52) possesses a unique solution.Proof.
Letι,κ∈C0,I,ℝ and consider(59)Qιr−Qκr≤∫0rFr,s,ιs−Fr,s,κsds≤∫0rfr,smaxιs−κs,ιs−Qκs,κs−Qκs,2ιs−Qκsκs−Qιsιs−Qκs|+ιs−κsds≤∫0rfr,smaxsups∈0,1ιs−κs,sups∈0,1ιs−Qκs,sups∈0,1κs−Qκs,2sups∈0,1ιs−Qκssups∈0,1κs−Qιssups∈0,1ιs−Qκs+sups∈0,1ιs−κsds≤maxdι,κ,dι,Qκ,dκ,Qκ,2dι,Qκdκ,Qι/dι,Qκ+dι,κ∫0rfr,sds≤λmaxdι,κ,dι,Qκ,dκ,Qκ,2dι,Qκdκ,Qι/dι,Qκ+dι,κ.
Consequently,(60)dQι,Qκ≤λmaxdι,κ,dι,Qκ,dκ,Qκ,2dι,Qκdκ,Qιdι,Qκ+dι,κ.
Using (55), we can write(61)mι,κ,t=mintt+dι,κ,tt+dι,Qι,tt+dκ,Qκ,tt+2dι,QκdQι,κ/dι,Qκ+dQι,κ=tt+maxdι,κ,dι,Qι,dκ,Qκ,2dι,QκdQι,κ/dι,Qκ+dQι,κ1mι,κ,t−1=t+maxdι,κ,dι,Qι,dκ,Qκ,2dι,QκdQι,κ/dι,Qκ+dQι,κt−1=maxdι,κ,dι,Qι,dκ,Qκ,2dι,QκdQι,κ/dι,Qκ+dQι,κt.
Therefore,(62)1ℱQι,Qκ,t−1=dQι,Qκt≤λmaxdι,κ,dι,Qκ,dκ,Qκ,2dι,Qκdκ,Qι/dι,Qκ+dι,κt=λ1mι,κ,t−1=λ1mι,κ,t−1.
So, (5) holds for ηk=λk for all k>0 with αι,κ,t=1, for all ι,κ∈C0,I,ℝ. As C0,I,ℝ is complete, so all the remaining conditions of Theorem 1 hold. Hence, (52) admits a unique solution.



---

*Source: 9931066-2021-05-17.xml*