# κG-Contractive Mappings in Fuzzy Metric Spaces

Authors: Maria Samreen; Sharafat Hussain; Hassen Aydi; Yaé Ulrich Gaba
Journal: Journal of Mathematics
 (2022)

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## Abstract

This article is concerned with a class of contractions in the framework of a fuzzy metric space endowed with a graph. The main results obtained not only broaden but also generalize a number of relevant results in the literature. Particularly, we get some results for cyclic contractions in fuzzy metric spaces. We also consider an integral equation.


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## Body

## 1. Introduction

In recent years, fixed-point theory has become a tantalizing field of research in nonlinear analysis due to its wide range of applications in computer sciences, game theory, physics, economics, and various other fields. It has become a tool to find solutions for nonlinear equations such as differential equations, integrodifferential equations, and matrix equations [1, 2]. After the fundamental Banach’s theorem [3] in 1922, a huge amount of literature appeared for the investigation of fixed points of self-operators. Among those results, two aspects must be considered: underlying geometric structure on which self-operator is defined and an appropriate contraction condition on operator.The investigation of existence of a fixed point of self-mappings on a metric spaceN,d equipped with a partial ordering was originated by the works of Ran and Reurings [4]. By replacing a partial order with a graph, Jachymski [5] studied some prolific results about the existence of a fixed point involving the graph concept. Later on, several researchers conducted further studies in this area by relaxing the contractive condition and analyzing the connectivity condition of a graph. For more details, see [6–8].On the other hand, Kramosil and Michalek [9] gave the concept of a fuzzy metric space. Afterward Grabiec [10] extended the Banach contraction principle (BCP) to fuzzy metric spaces in the sense of Kramosil and Michalek. Many authors undertook further advancements in this direction by generalizing the contraction inequalities and by providing the underlying space with rich geometric structure to make the results more viable. In this context, recently a number of authors contributed new and exciting developments in fixed-point theory using fuzzy metric spaces; see [9–18] and the references included therein. For some very recent studies on the topic, we refer the reader to [19–21].In particular, Altun [11] proved some useful results about the existence of fixed points of fuzzy order κ-contractive mappings with the setting of non-Archimedean fuzzy metric spaces. The work of Jachymski motivated us to introduce the notion of fuzzy order κG-contractive mappings and study related fixed-point results in the context of a fuzzy metric space equipped with a graph. Our introduced notion unifies the concept of Altun and Mihet, and Jachymski. By a successful application of our main result, we also ensure the existence of a solution for an integral equation.


## 2. Preliminaries

In this section, we recall some basic notions and results which are used in the sequel. Throughout this article, we useJ=0,1.Definition 1 (see [9]).
A binary operationθ:J×J⟶J is a continuous t-norm if, for all u,v,w,z∈J, it satisfies the following conditions:(1)
θ is continuous,(2)
uθvθw=uθvθw and uθv=vθu,(3)
uθ1=u,(4)
w≥u and z≥v imply that wθz≥uθv.
If there exists a strictly increasing sequencevn⊂0,1 such that vnθvn=vn for all n∈ℕ, then the continuous t-norm θ is known as Hadzic type.Definition 2 (see [9]).
LetN be an arbitrary set, let Γ:N×N×0,∞⟶J be a fuzzy set, and let θ be a continuous t-norm. N,Γ,θ is said to be a fuzzy metric space if, for all α,β,γ∈N and ι,λ>0, the following conditions hold:(1)
FM1Γα,β,0=0,(2)
FM2Γα,β,ι=1 if and only if α=β,(3)
FM3Γα,β,ι=Γβ,α,ι,(4)
FM4Γα,β,ιθΓα,γ,λ≤Γα,γ,ι+λ,(5)
FM5Γα,β,⋅:0,∞⟶J is left continuous.Remark 1 (see [11]).
If we take the inequalityNA rather than FM4 of Definition 2,(1)Γα,β,ιθΓα,γ,λ≤Γα,γ,maxι,λ,for allα,β,γ∈Nandι,λ>0,NA,then we call N,Γ,θ a non-Archimedean fuzzy metric space. It is worthwhile to note that every non-Archimedean fuzzy metric space is a fuzzy metric space. From now on, we denote a fuzzy metric space simply by N.Definition 3 (see [9, 11]).
A sequenceαn in N is called(i)
Cauchy iflimn⟶∞Γαn,αn+p,ι=1 for each ι>0 and p>0;(ii)
convergent toα∈N if Γn⟶∞αn,α,ι=1 for all ι>0;(iii)
M-Cauchy if for each ε∈0,1 and ι>0 there is a natural number n0 such that Γαn,αm,ι>1−ε for all m,n>n0;(iv)
G-Cauchy if limn⟶∞Γαn,αn+1,ι=1,for allι>0.N is called complete if every Cauchy sequence is convergent. Additionally, it is called M-complete if every M-Cauchy sequence is convergent and G-complete if every G-Cauchy sequence is convergent.
In the following, we take into account the main results of [11]. A self-map T on an ordered set N,≺¯ is named nondecreasing (nonincreasing) w.r.t ≺¯ if α≺¯β implies Tα≺¯TβTα≽Tβ.Definition 4 (see [11]).
LetN,≺¯ be a partially order set and let N,Γ,θ be a fuzzy metric space. Let κ:J⟶J be a function. T:N⟶N is called a fuzzy order κ-contractive mapping if the following holds:(2)α,β∈N,α≺¯β⇒ΓTα,Tβ,ι≥κΓα,β,ι,for allι>0.Theorem 1 (see [11], Theorem 2.3).
LetN,Γ,θ be an M-complete non-Archimedean fuzzy metric space with the t-norm θ of Hadzic type, in which N is a set with the partial order ≺¯. Let T:N⟶N be a fuzzy order κ-contractive mapping, where κ:J⟶J is a continuous, nondecreasing function. Assume that(i)
T is nondecreasing;(ii)
either for every nondecreasing sequenceαn in N with αn⟶α∈N implies αn≺¯α for all n or T is continuous;(iii)
there existsα0∈N such that α0≺¯Tα0 and limn⟶∞κnΓα0,Tα0,ι=1 for each ι>0.Then there isω∈N such that Tω=ω.Theorem 2 (see [11], Theorem 2.4).
LetN,Γ,θ be an M-complete non-Archimedean fuzzy metric space with the t-norm θ of Hadzic type in which N is a set with the partial order ≺¯. Let T:N⟶N be a fuzzy order κ-contractive mapping, where κ:J⟶J is a continuous mapping such that κι>ι for all ι∈0,1. Assume that(i)
T is nondecreasing;(ii)
either for every nondecreasing sequenceαn in N with αn⟶α∈N implies αn≺¯α for all n or T is continuous;(iii)
there isα0∈N such that α0≺¯Tα0, Γα0,Tα0,ι>0 for all ι>0.Then there existsω∈N such that Tω=ω.Following [5], we include basic notions about graph which are used throughout this paper. Let N,Γ,θ be a fuzzy metric space. The diagonal of N×N is represented by Δ. Consider a directed graph G such that its diagonal is a subset of EG and VG is equal to the whole set N. We assume that G does not have any edges which are parallel and describe G with the pair VG,EG. By G−1, we mean the converse of graph G and it is defined as(3)EG−1=α,β∈N×N:β,α∈EG.The undirected graph is denoted byG˜ and it is obtained by neglecting the direction of edges. EG˜ is given as(4)EG˜=EG−1∪EG.IfV′ is a subset of VG, E′ is a subset of EG and, for any edge α,β∈E′,α,β∈V; then V′,E′ is called a subgraph of G. The sequence αjj=0p of p+1 vertices, such that(5)α=α0,αp=βandαj−1,αj∈EG,wherej∈1,2,…,p,is called the path of length p between two vertices α and β in G, where p is a nonnegative integer. If between any two vertices there is always a path, then we call G a connected graph. G is said to be weakly connected if G˜ is connected. Let α be a vertex in G and let EG be symmetric; then the set of vertices and edges that are comprised in certain path starting at α is known as component of G which contains α; it is a subgraph of G and is denoted by Gα. The set of all edges in Gα is denoted by EGα and VGα=αG. It is worthwhile to indicate that Gα is connected. Here, αG is the equivalence class of R and the relation R on vertices of V is defined as(6)βRγ,if there is a path amidstβandγinG.Letαn be any sequence in N. If for αn with αn⟶α and αn,αn+1∈EG for n∈ℕ implies that there exists a subsequence αnk of αn such that αnk,α∈EG for k∈ℕ, then G is called C-graph.


## 3. Main Results

From now on, we denote a fuzzy metric space endowed with a graphG simply by N.Definition 5.
A mappingT:N⟶N is called fuzzy order κG-contractive if, for all α,β∈N, we have(7)α,β∈EGimpliesTα,Tβ∈EG,(8)ΓTα,Tβ,ι≥κΓα,β,ι,for allι>0wheneverα,β∈EG,where κ:J⟶J is a continuous and nondecreasing function.Example 1.
Suppose thatG is any graph such that VG=N. Define T:N⟶N by Tα=u for all α∈N, where u is any constant; then T is a fuzzy order κG-contractive mapping.Example 2.
LetG0 be a graph such that EG0:=N×N. Then any fuzzy order κG-contractive mapping is a fuzzy order κG0-contractive mapping.Definition 6.
Two sequencesαn and βn in N are said to be equivalent if limn⟶∞Γαn,βn,ι=1 for all ι>0. In addition, αn and βn are called Cauchy equivalent if each of them is Cauchy.Lemma 1.
Letαn and βn be equivalent sequences in N. Then the following hold:(i)
ifαn⟶α, then βn⟶α and its converse holds;(ii)
ifαn is a Cauchy sequence, then βn is also a Cauchy sequence and vice versa.Proof.
By using conditionFM4 of Definitions 2 and 3, one can easily invoke the conclusions.Proposition 1.
LetT:N⟶N be a fuzzy order κG-contractive mapping, where κ:J⟶J is a continuous, nondecreasing function. Then the following hold:(i)
T is both a fuzzy order κG˜ contractive mapping and a fuzzy order κG−1-contractive mapping,(ii)
α0G˜ is T-invariant, and T|α0G˜ is a fuzzy order κG˜α-contractive mapping provided that α0∈N is such that Tα0∈α0G˜.Proof.
(i) It could be proved easily by taking into accountFM3 of Definition 3. (ii) Let α∈α0G˜. Then there exists a path between α and α0; that is, there exists a sequence zjj=0p of p+1 vertices such that α=z0,zp=α0andzj−1,zj∈EG,forj=1,2,…,p. Since T is fuzzy order κG-contractive, one writes Tzj−1,Tzj∈EG for j∈1,2,…,p. Thus Tα∈Tα0G˜=α0G˜. Now, suppose that α,β∈EG˜α0. Since T is fuzzy order κG-contractive, we have Tα,Tβ∈EG. But α0G˜ is T-invariant, which implies that Tα,Tβ∈EG˜α0. As G˜α0 is a subgraph of G, we have(9)ΓTα,Tβ,ι≥κΓα,β,ι,for allι>0wheneverα,β∈EG˜α0.Lemma 2.
LetT:N⟶N be a fuzzy order κG-contractive mapping, where κ:J⟶J is a continuous, nondecreasing function. Assume that, for α∈N and β∈αG˜, we have(10)limn⟶∞κnΓzj−1,zj,ι=1,for allι>0,where zjj=0p is any path between α and β. Then Tnα and Tnβ are equivalent.Proof.
Letα∈N and β∈αG˜; then a path α=α0,α1,…,αp=β exists between α and β with αj−1,αj∈EG˜ for j∈1,2,…,p. From Proposition 1, T is a fuzzy order κG˜-contractive mapping. So, for all n∈ℕ and j∈1,2,…,p, we obtain(11)ΓTnαj−1,Tnαj,ι≥κΓTn−1αj−1,Tn−1αj,ι,for allι>0,whenever Tnαj−1,Tnαj∈EG˜. Now, for all n∈ℕ and j∈1,2,…,p, we obtain(12)ΓTnαj−1,Tnαj,ι≥κΓTn−1αj−1,Tn−1αj,ι≥⋯≥κnΓαj−1,αj,ι.
It yields thatTnαjj=0p is a path in G˜ from Tnα to Tnβ. Thus,(13)ΓTnα,Tnβ,ι≥ΓTnα0,Tnα1,ιpθ⋯ptimesθΓTnαp−1,Tnαp,ιp≥κnΓα0,α1,ιpθ⋯ptimesθκnΓαp−1,αp,ιp.
By lettingn⟶∞, we conclude limn⟶∞ΓTnα,Tnβ,ι=1.Proposition 2.
LetT:N⟶N be a fuzzy order κG-contractive mapping, where κ:J⟶J is a continuous, nondecreasing function. Assume that, for α0∈N, we have Tα0∈α0G˜ and(14)limn⟶∞κnΓzj−1,zj,ι=1,for allι>0,where zjj=0p is any path between α0 and Tα0. Suppose that, for any sequence Tnα0 in N with Tnα0∈α0G˜,n≥1 and ΓTnα0,Tn+1α0,ι⟶1, there is n0∈ℕ satisfying Tmα0,Tnα0∈EG for each m,n≥n0.
ThenTnα0 is an M-Cauchy sequence in N.Proof.
SinceTα0∈α0G˜, from Proposition 1, it follows that Tnα0∈α0G˜ for all n≥1. Now, we suppose that βjj=0p is a path connecting α0 and Tα0. Using the similar reasoning like in the proof of Lemma 2, we have(15)limn⟶∞ΓTnα0,Tn+1α0,ι=1,for alln∈ℕ.
Now, we show thatTnα0 is an M-Cauchy sequence. Suppose on the contrary that Tnα0 is not an M-Cauchy sequence; then there exist ε∈0,1 and two subsequences Tmkα0 and Tnkα0 of Tnα0 with nk>mk>k such that(16)ΓTmkα0,Tnkα0,ι≤1−ε.
Let, for eachk, mk be the least integer exceeding nk satisfying (16), but(17)ΓTmk−1α0,Tnkα0,ι>1−ε.
Now, for eachk∈ℕ, we obtain(18)1−ε≥ΓTmkα0,Tnkα0,ι≥ΓTmk−1α0,Tnkα0,ι2θΓTmk−1α0,Tmkα0,ι2>1−εθΓTmk−1α0,Tmkα0,ι2.
By takingk⟶∞ and using (15), we obtain(19)limk⟶∞ΓTmkα0,Tnkα0,ι=1−ε.
Sincelimn⟶∞ΓTnα0,Tn+1α0,ι=1,for alln∈ℕ and ι≥0, by condition (i), there exists a positive integer n0 such that Tmα0,Tnα0∈EG for each m,n∈ℕ with m,n>n0. Now, by using triangular inequality for all k≥n0, we obtain(20)ΓTmkα0,Tnkα0,ι≥ΓTmkα0,Tmk+1α0,ι3θΓTmk+1α0,Tnk+1α0,ι3θΓTnk+1α0,Tnkα0,ι3≥ΓTmkα0,Tmk+1α0,ι3θκΓTmkα0,Tnkα0,ι3θΓTnk+1α0,Tnkα0,ι3.
On lettingk⟶∞ and by using (15) and (19), we acquire(21)1−ε≥1θκ1−εθ1=κ1−ε>1−ε.
Hence,Tnα0 is an M-Cauchy sequence in N.Remark 2.
It is important to note that assertion of Proposition2 still holds if we replace the condition Tα0∈α0G˜ with α0,Tα0∈EG. In that case, condition (i) is also replaced by the following: “for any sequence Tnα0 in N with Tnα0,Tn+1α0∈EG and ΓTnα0,Tn+1α0,ι⟶1, there exists n0∈ℕ such that Tmα0,Tnα0∈EG for all m,n≥n0.”
Below we extend the definitions ofHf- and Cf-graphs to the setting of fuzzy metric spaces.Definition 7.
Suppose thatT:N⟶N, β∈N, and the sequence Tnβ in N is such that Tnβ⟶α∈N.(i)
If there exists a subsequenceTnkβ of Tnβ such that Tnkβ,α∈EG for all k≥p∈ℕ, then G is called a Cf-graph.(ii)
A graphG is called an Hf-graph if Tnβ∈αG˜ for n≥1; then rTnβ,α,ι⟶1asn⟶∞, where rTnβ,α,ι=Γz0,z1,ι/pθ⋯p−timesθΓzp−1,zp,ι/p and zjj=0p is a path from Tnβ to α.
It is evident that everyC-graph is a Cf-graph for the identity mapping I, but the other side is not always true. Similarly, every H-graph is an Hf-graph for the identity mapping I, but the other side is not always true (see [8], Examples 2.4 and 2.5).Lemma 3.
LetT:N⟶N be fuzzy order κG-contractive, where κ:J⟶J is a continuous, nondecreasing function. Then, for any α∈N and β∈αG˜ and for each n∈ℕ, there is rnα,β,ι∈J such that(22)ΓTnα,Tnβ,ι≥rnα,β,ι.Furthermore, iflimn⟶∞κnΓzj−1,zj,ι=1 for all ι>0, where zji=1p is any path between α and β, then(23)limn⟶∞ΓTnα,Tnβ,ι=limn⟶∞rnα,β,ι=1.Proof.
Letα∈N and β∈αG˜; then there exists a path α=α0,α1,…,αp=β between α and β with αj−1,αj∈EG˜ for j∈1,2,…,p. From Proposition 1 for all ι>0, we have(24)ΓTnαj−1,Tnαj,ι≥κΓTn−1αj−1,Tn−1αj,ι.
SinceT is a fuzzy order κG˜ contraction such that Tnαj−1,Tnαj∈EG˜ for each natural number n and j varies from 1 to p, for a positive integer n and j∈1,2,…,p, we have(25)ΓTnαj−1,Tnαj,ι≥κΓTn−1αj−1,Tn−1αj,ι≥⋯≥κnΓαj−1,αj,ι.
Note thatTnαjj=0p is a path in G˜ from Tnα to Tnβ. Thus,(26)ΓTnα,Tnβ,ι≥ΓTnα0,Tnα1,ιpθ⋯ptimesθΓTnαp−1,Tnαp,ιp≥κnΓα0,α1,ιpθ⋯ptimesθκnΓαp−1,αp,ιp.
It suffices to set(27)rnα,β,ι=κnΓα0,α1,ιpθκnΓα1,α2,ιpθ⋯ptimesθκnΓαp−1,αp,tp.
On lettingn⟶∞, we obtain(28)limn⟶∞ΓTnα,Tnβ,ι=limn⟶∞rnα,β,ι=1.
Now, we proceed to establish our first main result. From now on,N represents an M-complete fuzzy metric space endowed with a graph G.Theorem 3.
LetT:N⟶N be a fuzzy order κG-contractive mapping, where κ:J⟶J is a continuous, nondecreasing function. Assume that, for a given α0∈N, α0,Tα0∈EG and(29)limn⟶∞κnΓα0,Tα0,ι=1,for allι>0.Moreover, if(i)
G is a Cf-graph,(ii)
for any sequenceTnα0 in N with Tnα0,Tn+1α0∈EG,n≥1 and ΓTnα0,Tn+1α0,ι⟶1, then there is a natural number n0 so that Tmα0,Tnα0∈EG for every m,n≥n0.Then there isω∈α0G˜ in such a way that Tω=ω. In addition, if, for every β∈α0G˜, we have(30)limn⟶∞κnΓzj−1,zj,ι=1,for allι>0,where zjj=0p is any path between α and β, then T|α0G˜ is a Picard operator.Proof.
Letα0,Tα0∈EG; then Tα0∈α0G˜. From Proposition 2, we infer that Tnα0 is an M-Cauchy sequence. Since N is an M-complete space, there exists ω∈N such that Tnα0⟶ω. Since G is a Cf-graph, there are Tnkα0, a subsequence of Tnα0, and a positive integer q in such a way that Tnkα0,ω∈EG for every positive integer k and q≤k. Thus, for all p≤k, we obtain(31)ΓTω,ω,ι≥ΓTω,Tnk+1α0,ι2θΓTnkα0,ω,ι2≥κΓω,Tnkα0,ι2θΓTnkα0,ω,ι2.
Using the continuity ofκ and taking n⟶∞, one gets(32)ΓTω,ω,ι=1.
This infers thatTω=ω. We observe that α0,Tα0,…,Tn1α0,…,Tpα0,ω is a path between α0 and ω in G˜. Thus, ω∈α0G˜. Now, take a random element β∈α0G˜ and, using Lemma 2, we obtain(33)limn⟶∞ΓTnβ,Tnα0,ι=1.
Thus,limn⟶∞Tnβ=ω. To ensure the uniqueness, we argue by contradiction. Suppose that ω and π are two distinct fixed points of T in α0G˜, so(34)Γω,π,ι=ΓTnω,Tnπ,ι≥ΓTnω,Tnα0,ι2θΓTnα0,Tnπ,ι2.
On lettingn⟶∞, we obtain ω=π.Remark 3.
Observe that, in Theorem3, T is a Picard operator on N if the underlying graph G is weakly connected such that α0G˜=N.
By analyzing condition on graphG and imposing a strict condition on κ, we have the following variant.Theorem 4.
LetT:N⟶N be a fuzzy order κG-contractive mapping, where κ:J⟶J is a continuous, nondecreasing function such that κι>ι for all ι∈0,1. Assume that for α0∈N and β∈α0G˜ we get(35)limn⟶∞κnΓzj−1,zj,ι=1,for allι>0,where zjj=0p is any path between α0 and β. Moreover, if(i)
G is an Hf-graph,(ii)
for any sequenceTnα0 in N with Tnα0,Tn+1α0∈EG,n≥1 and ΓTnα0,Tn+1α0,ι⟶1, then there is n0∈N in such a way that Tmα0,Tnα0∈EG,∀m,n≥n0;(iii)
G is weakly connected.ThenT is a Picard operator.Proof.
Assume thatα0∈N and condition (iii) holds. Then Tα0∈α0G˜; that is, there is a path between α0 and Tα0 in G˜. Take into account Proposition 2; we infer that the sequence Tnα0 is an M-Cauchy sequence. Since N is an M-complete fuzzy metric space, there exists ω∈N such that Tnα0⟶ω. Since condition (iii) holds, we have, for each natural number n, Tnα0∈ωG˜. Then there exists a path zjnj=0pn of finite length pn between Tnα0 and ω with z0n=Tnα0,zpnn=ω. Using the conditions that T is a κG-contractive mapping and κι>ι, we obtain(36)Γω,Tω,ι≥Γω,Tn+1α0,ι2θΓTn+1α0,Tω,ι2≥Γω,Tn+1α0,ι2θΓTz0n,Tz1n,ι2lnθ⋯pntimesθΓTzj−1n,Tzjn,ι2ln≥Γω,Tn+1α0,ι2θκΓz0n,z1n,ι2lnθ⋯pntimesθκΓzj−1n,zjn,ι2ln=Γω,Tn+1α0,ι2θrTnα0,ω,ι.
By lettingn⟶∞ and using condition (i), we get Γω,Tn+1α0,ι/2θrTnα0,ω,ι⟶1. Hence, we have Γω,Tω,ι=1. Thus, ω=Tω. Now, let β be a random element of α0G˜; then Lemma 2 implies(37)limn⟶∞ΓTnβ,Tnα0,ι=1.
Thus,limn⟶∞Tnβ=ω. Now, for uniqueness, suppose that ω≠π are two fixed points of T in α0G˜; we have(38)Γω,π,ι=ΓTnω,Tnπ,ι≥ΓTnω,Tnα0,ι2θΓTnα0,Tnπ,ι2.
On lettingn⟶∞, we obtain ω=π.Definition 8.
LetT:N⟶N. If, for all α,β∈N and some sequence pjj∈ℕ of a positive integer, Tpjα⟶β implies TTpjα⟶Tβ as j⟶∞, then T is called orbitally continuous. If, for each α,β∈N and some sequence pjj∈ℕ of a positive integer Tpjα⟶β and Tpjα,Tpj+1α∈EG for each positive integer n implies TTpjα⟶Tβ, then T is known as orbitally G-continuous.Theorem 5.
LetT:N⟶N be a fuzzy order κG-contractive mapping, where κ:J⟶J is a continuous and nondecreasing function. Assume that, for given α0∈N,(39)α0,Tα0∈EGandlimn⟶∞κnΓα0,Tα0,ι=1,for allι>0.Moreover, if(i)
T is orbitally G-continuous or T is orbitally continuous,(ii)
for any sequenceTnα0 in N with Tnα0,Tn+1α0∈EG,n≥1 and ΓTnα0,Tn+1α0,ι⟶1, there is n0∈ℕ in such a way that Tmα0,Tnα0∈EG,∀m,n≥n0.Then there is an elementω in α0G˜ such that Tω=ω. In addition, if, for every β∈α0G˜, we have(40)limn⟶∞κnΓzj−1,zj,ι=1,for allι>0,where zjj=0p is any path between α and β, then T|α0G˜ is a Picard operator.Proof.
Sinceα0,Tα0∈EG, we have Tα0∈α0G˜. From Proposition 2, Tnα0 is an M-Cauchy sequence. Since N is an M-complete fuzzy metric space, there exists ω∈N such that Tnα0⟶ω. As Tnα0,Tn+1α0∈EG, for all n≥1, and T is orbitally G-continuous, we obtain TTnα0⟶Tω. Hence, Tω=ω. Let β∈α0G˜ be arbitrary, so Lemma 3 yields Tnβ⟶ω. Uniqueness can be easily followed.
Now, we include another result when the underlying fuzzy metric space is endowed with a partial order.Corollary 1.
LetN,≺¯ be a partially ordered set and let N,Γ,θ be an M-complete fuzzy metric space. Let T:N⟶N be a fuzzy order κ-contractive mapping, where κ:J⟶J is a continuous and nondecreasing function. Assume that(i)
T is nondecreasing (or nonincreasing),(ii)
either for every nondecreasing (nonincreasing) sequenceαn in N with αn⟶α∈N implies αn≺¯α (or αn≽α) for all n or T is continuous, and(iii)
there existsα0∈N such that α0≺¯Tα0α0≽Tα0 and limn⟶∞κnΓα0,Tα0,ι=1 for each ι>0.ThenT has a fixed point.Proof.
DefineG by VG=N and EG=α,β∈N×N:β≺¯α.
SinceT is nondecreasing, we have that (7) is satisfied and (8) holds in the light of condition (2). Thus, T is a κG-contractive mapping. Also, note that condition (ii) implies that G is a Cf-graph. Thus, taking into account Theorem 3, we invoke the result.Remark 4.
Note that Corollary1 extends/generalizes ([22], Theorem 2.3) to non-Archimedean M-complete fuzzy metric spaces.Example 3.
LetN=0,∞ be equipped with a fuzzy metric Γα,β,ι=minα,β/maxα,β so that αθβ=αβ. Then N,Γ,θ is an M-complete fuzzy metric space (see [23]). Define T:N⟶N by(41)Tα=2α,0<α<1,α+53,1≤α≤4,2α−5,α>4.
LetG be a graph in such a way that VG=N and EG=Δ∪α,β:α,β∈1,4,α≤β. Condition (7) is trivial. We show that T is fuzzy order κG-contractive with κu=u. Indeed, let α,β∈N with α,β∈EG. Now, if α≤β, then(42)ΓTα,Tβ,ι=1≥κ1=κΓα,β,ι.
Ifα,β∈1,4 with α≤β , then(43)ΓTα,Tβ,ι=minα+5/3,β+5/3maxα+5/3,β+5/3=α+5β+5≥αβ=κΓα,β,ι.
Therefore,T is a fuzzy order κG-contractive mapping with κu=u. Moreover, for α0=1, α0,Tα0∈EG. Hence, all the conditions of Theorem 3 are satisfied and T has a fixed point.


## 4. Application

In [24], the following definition was initiated: let N be a nonempty set and let Gjj=1k be nonempty closed subsets of N, where n is any positive integer. Then, D=∪j=1kGj is called a cyclic representation of N with respect to the mapping T:D⟶D if(44)TG1⊂G2,…,TGm−1⊂Gm,TGm⊂G1.ThenT is called a cyclic operator.Theorem 6.
LetN,Γ,θ be an M-complete fuzzy metric space. Let Gjj=1k be nonempty closed subsets of N, D=∪j=1kGj, and T:D⟶D. Further, suppose the following:(i)
∪j=1kGj is a cyclic representation of D with respect to T.(ii)
There existsκ:J⟶J, where κ is a continuous and nondecreasing function such that, for α∈Gj,β∈Gj+1, we have(45)ΓTα,Tβ,ι≥κΓα,β,ι,∀ι>0.(iii)
Letα0∈D; for a sequence Tnα0 in D with ΓTnα0,Tn+1α0,ι⟶1, there are a natural number n0 and some j∈1,2,…,k such that Tmα0∈Gj,Tnα0∈Gj+1, for all m,n≥n0 and(46)limn⟶∞κnΓα0,Tα0,ι=1,∀ι>0.ThenT has a fixed point w∈∩j=1kGj and Tnα0⟶w.Proof.
SinceGjj=1k are closed subsets, one can assert that D,Γ,θ is an M-complete fuzzy metric space. Let G be a graph such that(47)VG=DandEG=Δ∪α,β∈D×D:α∈Gj,β∈Gj+1;j=1,2,…,k.
AsD=∪j=1kGj is a cyclic representation of D with respect to T, it follows that T preserves edges. Also, observe that, in light of inequality (45), T is a κG-contractive mapping. Now, for a sequence Tnα0 in D, we have Tnα0,Tn+1α0∈EG,∀n≥1 and if ΓTnα0,Tn+1α0,ι⟶1, then, from condition (iii), there is n0∈ℕ such that Tmα0,Tnα0∈EG for all m,n≥n0. Let Tnα⟶w in D be such that Tnα,Tn+1α∈EG for all n≥1; then, from (44) in every Gj, the sequence Tnα has infinitely many terms; thus we have a subsequence of Tnα which converges to w in each Gj. As Gj’s are all closed subsets, we get w∈∩j=1kGj. Thus, there exists a subsequence Tnrα in some Gj, j∈1,2,…,k, such that Tnrα,w∈EG for r≥1, which shows that G is a weakly connected Cf-graph. Taking into account Theorem 3, we invoke the conclusion.
Now, we present an application of our result to find a solution of an integral equation. Consider the below integral equation:(48)αr=∫0rKr,s,αsds+gr,∀r∈0,IwhereI>0.
LetN=C0,I be the set of all real valued continuous functions defined on 0,I. Then N,d is a complete metric space endowed with the metric(49)dα,β=supr∈0.Iαr−βrα,β∈N.
Foraθb=ab for all a,b∈J, define a fuzzy metric by(50)Γα,β,ι=ιι+dα,β,for allα,β∈Nandι>0.
ThenN,Γ,θ is an M-complete fuzzy metric space.
In the following, we discuss the existence of the solution for the integral equation (48).Theorem 7.
LetN be the set of all real valued continuous functions defined on 0,I. Let T:N⟶N be the integral operator given by(51)Tαr=∫0rKr,s,αsds+gr,∀r∈0,Iandg∈N,where K:0,I×0,I×ℝ⟶ℝ is continuous. Assume that(i)
there exist a nondecreasing, continuous functionκ:J⟶J and a continuous function T:0,I×0,I⟶ℝ+ such that(52)Kr,s,αs−Kr,s,βs≤fr,sκαs−βs,for eachr,s∈0,I and αr≤βr for all r∈0,1;(ii)
Kr,s,.:ℝ⟶ℝ is nondecreasing for each r,s∈0,I;(iii)
supr∈0,I∫0rfr,sds≤1;(iv)
there existsα0∈N such that α0r≤∫0rKr,s,α0sds+gr for all r∈0,I;(v)
for any sequenceαn in N with Γαn,αn+1,ι⟶1 there exists n0∈ℕ such that αmι≤αnι for all m,n∈ℕ, m,n≥n0.Then the integral equation (48) has a unique solution in the set α∈N:αr≤α0r,orαr≥α0rfor allr∈0,I.Proof.
LetG be a graph such that VG=N and EG=α,β∈N×N:αr≤βrfor allr∈0,I. It follows from (ii) that T is nondecreasing and thus preserves edges. Also, for every nondecreasing sequence αn⊂N which converges to ω∈N, it implies αnr≤ωr for every r∈0,I, which shows that G is a Cf-graph. Now, for all α,β∈N, we obtain(53)Tαr−Tβr≤∫0rKr,s,αs−Kr,s,βsds≤∫0rfr,sαs−βsds≤καs−βs∫0rfr,sds.
Hence,dTα,Tβ≤κdα,β. Thus,(54)ιι+dTα,Tβ≥κdα,β≥κιt+dα,β.
It follows that(55)ΓTα,Tβ,ι≥κΓα,β,ι.
From (iv), we haveα0,Tα0∈EG and the conclusion follows from Theorem 3. Moreover, the solution lies in α0G˜=α∈N:αr≤α0,orαr≥α0for allr∈0,I.



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*Source: 5920124-2022-02-26.xml*