# On Some Fixed Point Results inE−Fuzzy Metric Spaces

Authors: Rachid Mecheraoui; Zoran D. Mitrović; Vahid Parvaneh; Hassen Aydi; Naeem Saleem
Journal: Journal of Mathematics
 (2021)

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## Abstract

In the existing literature, Banach contraction theorem as well as Meir-Keeler fixed point theorem were extended to fuzzy metric spaces. However, the existing extensions require strong additional assumptions. The purpose of this paper is to determine a class of fuzzy metric spaces in which both theorems remain true without the need of any additional condition. We demonstrate the wide validity of the new class.


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## Body

## 1. Introduction

The well-known Banach’s contraction principle (BCP) is a classic method in nonlinear analysis and is one of the most important and heavily researched fixed point theorems. It states that ifℒ:O⟶O is a contraction on the complete metric space O, thenℒ has a unique fixed point u in O, andlimnℒnκ=u for all κ∈O.There exist many different concepts of a fuzzy metric space (cf. [1–5]). However, the definition of George and Veeramani [4] is reverently cited in most references in this area. The authors of [4] updated Kramosil and Michalek’s definition of fuzzy metric space and obtained a Hausdorff topology for this kind of fuzzy metric space. The topology induced by a fuzzy metric space in the sense of George and Veeramani has recently been demonstrated to be metrizable in [6]. It would be very useful if the BCP remains true in fuzzy metric spaces. This has been discussed in [7, 8] and then in numerous other directions over the years (see, for example, [9–11]). In [12], the generalized altering distance function has been defined and the Banach contraction principal in complete fuzzy metric spaces using altering distance has been extended.In [7], in order to give the fuzzy version of the BCP, Grabiec introduced a new definition of a Cauchy sequence in fuzzy metric spaces as follows.Definition 1.
A sequenceκnn in a fuzzy metric space O,Q,∗ is Cauchy if limnQκn+p,κn,t=1 for each t>0 and p>0.
It can be seen easily that this definition of a Cauchy sequence is incorrect, for more details see [13–15], whereas, in [8], Gregori and Sapena extended the Banach fixed point theorem to fuzzy version stating the following theorem.Theorem 1.
LetO,Q,∗ be a complete fuzzy metric space in which fuzzy contractive sequences are Cauchy. Let ℒ:O⟶O be a fuzzy contractive mapping with k serving as the contractive constant. Then, ℒ has a unique fixed point.Here, authors added a fairly strong assumption, that is, “every contractive sequence is Cauchy.”In this paper, we prove that the Banach contraction theorem as well as the Meir-Keeler fixed point theorem remain true in fuzzy metric spaces with only a slight modification in the definition of fuzzy spaces given by George and Veeramani. Last, in this paper, we give some results to illustrate the broad validity of our results.Before stating the main results, we need the following definitions.Definition 2 (Schweizer and Sklar [16]).
A binary operation∗:0,1×0,1⟶0,1 is called a continuous t-norm if it satisfies the following assertions:(T1) ∗ is commutative and associative;(T2) ∗ is continuous;(T3) a∗1=a for all a∈0,1;(T4) a∗b≤c∗d when a≤c and b≤d, with a,b,c,d∈0,1.
Here is the definition of a fuzzy metric space given by George and Veeramani:Definition 3 (George and Veeramani [4]).
A fuzzy metric space is an ordered tripleO,Q,∗ in which O is a nonempty set, ∗ is a continuous t-norm, and Q is a fuzzy set on O×O×0,∞⟶0,1 such that(T5) Qκ,ω,t>0;(T6) Qκ,ω,t=1 if and only if κ=ω;(T7) Qκ,ω,t=Qω,κ,t;(T8) Qκ,ω,t∗Qω,z,s≤Qκ,z,t+s;(T9) Qκ,ω,⋅:0,+∞⟶0,1 is left continuous,for all κ,ω,z∈O and t,s>0.
In this paper, we will consider the following class of fuzzy metric spaces.Definition 4 (E−fuzzy metric space).
LetO denote a nonempty set, ∗ refers to a continuous t-norm, and Q serves as a fuzzy set on O×O×0,∞⟶0,1 such that(F1) Qκ,ω,t>0;(F2) Qκ,ω,t=1 if and only if κ=ω;(F3) Qκ,ω,t=Qω,κ,t;(F4) Qκ,ω,t∗Qω,z,s≤Qκ,z,t+s;(F5) Qκ,ω,⋅:0,+∞⟶0,1 is left continuous.(F6) For some r>0, the family Qκ,ω,⋅:0,r⟶0,1;κ,ω∈O2 is uniformly equicontinuous,for all κ,ω,z∈O, and t,s>0. Then, the triple O,Q,∗ is called an E−fuzzy metric space.Remark 1.
Obviously, allE−fuzzy metric space is a fuzzy metric space. So, all properties in fuzzy metric spaces remain true in E−fuzzy metric spaces.Definition 5 (George and Veeramani [4]).
LetO,Q,∗ be a fuzzy metric space. Then,(i)
A sequenceκnn converges to κ∈O if and only if Qκn,κ,t⟶1 as n⟶+∞ for all t>0;(ii)
A sequenceκnn in O is a Cauchy sequence if and only if for all ε∈0,1 and t>0, there exists n0 such that Qκn,xm,t>1−ε for all m,n≥n0;(iii)
The fuzzy metric space is complete if every Cauchy sequence converges to somex∈O.
In the sequel, we use the following essential technical lemma.Lemma 1.
LetO,Q,∗ be an E−fuzzy metric space, Q˜ be the continuous extension of Q up to 0,∞, and κnn be a sequence in O such that limn⟶∞Qκn,κn+1,t=1, for all t>0. Then,(1)limn⟶∞Q˜κn,κn+1,0=1.Proof.
For allx,ω∈O, function t↦Qx,ω,t is positive, continuous, and nondecreasing on 0,+∞, so Q˜ is well defined. Let tnn be a monotonically decreasing sequence of positive numbers, converging to 0, and κnn be a sequence in O such that limn⟶∞Qκn,κn+1,t=1, for all t>0, i.e., for allt>0and for allε>0,there  existsn0∈ℕ:for alln≥n0;1−Qκn,κn+1,t<ε.
From which it follows thatfor allt>0and for allε>0,there existsn0∈ℕ:for alln≥n0,and for allk∈ℕ,(2)1−Q˜κn,κn+1,0+Q˜κn,κn+1,0−Qκn,κn+1,tk+Qκn,κn+1,tk−Qκn,κn+1,t<ε2.
Therefore,for allt>0and for allε>0,there existsn0∈ℕ:for alln≥n0and for allk∈ℕ,(3)1−Q˜κn,κn+1,0<ε2+Q˜κn,κn+1,0−Qκn,κn+1,tk+Qκn,κn+1,tk−Qκn,κn+1,t.
On the other hand, by the fact thatlimkQκn,κn+1,tk=Q˜κn,κn+1,0 and assumption (F6), we deduce that there existst0>0:for allε>0,there existsn0∈ℕ,for alln≥n0,there existsk0∈ℕ, such that(4)Q˜κn,κn+1,0−Qκn,κn+1,tk≤ε4,(5)Qκn,κn+1,tk−Qκn,κn+1,t≤ε4,for all k>k0 and t<t0. Hence, by relations (3)–(5), it yields for allε>0,there existsn0∈ℕ:for alln≥n0;1−Q˜κn,κn+1,0<ε, and this means(6)limnQ˜κn,κn+1,0=1,which achieves the proof of the lemma.


## 2. Main Results

Now, we will present our key finding.Theorem 2.
LetO,Q,∗ be a complete E−fuzzy metric space. Let ℒ:O⟶O be a fuzzy contractive mapping with the contractive constant k, i.e., there exists k∈0,1 such that(7)1Qℒκ,ℒω,t−1≤k1Qκ,ω,t−1,for all κ,ω in O and for all t>0. Then, ℒ has a unique fixed point κ∗. Furthermore, for all κ∈O, the sequence ℒnκ converges to κ∗.Proof.
Letκ in O and κn=ℒnκn∈ℕ. Let t>0 and n∈ℕ. By inequality (7), we obtain(8)1Qκn+1,κn+2,t−1≤k1Qκn,κn+1,t−1,for all t>0 and for all n in ℕ, which deduce that(9)limn⟶∞Qκn,κn+1,t=1,for all t>0. Now, to prove that κnn is a Cauchy sequence, we assume to the contrary. Since t↦Qκ,ω,t is a nondecreasing function, there existsε∈0,1and there existsξ>0such that for allp∈ℕ,there existsnp≥p<mp∈ℕ so that(10)Qκmp,κnp,t≤1−ε,for all t<ξ. Let t0<minξ,r. By virtue of limit (9) and the last relation, we can write that there existsε∈0,1;for allp∈ℕ,there existsnp≥p<mp∈ℕ:(11)Qκmp,κnp,t0≤1−ε,Qκmp−1,κnp,t0>1−ε.
Taking into account the continuity of the functiont↦Qκ,ω,t and the fact that Qκmp−1,κnp,t0>1−ε, we can choose q0∈ℕ such that(12)Qκmp−1,κnp,t0−1q0>1−ε.
By virtue of assumptions (T4) and (F4) and relations (11) and (12), it follows that(13)1−ε≥Qκmp,κnp,t0≥Qκmp,κmp−1,1q0∗Qκmp−1,κnp,t0−1q0≥Q˜κmp,κmp−1,0∗1−ε.
So, according to assumptions (T2)-(T3), limit (9), and Lemma 1, one has(14)limp⟶∞Qκmp,κnp,t0=1−ε.
Suppose that for allp1≥0, there exists p≥p1 such that Qκmp+1,κnp+1,t0≤1−ε means, having in mind relations (7) and (14), that the sequence κnn has two subsequences κnpp and κmpp verifying(15)limp⟶∞Qκmp,κnp,t0=limp⟶∞Qκmp+1,κnp+1,t0=1−ε,(for the sake of simplicity, we have saved the same notation for the subsequence).
Now, we suppose that there existsp1≥0 such that Qκmp+1,κnp+1,t0>1−ε for all p≥p1. We claim that limpQκmp+1,κnp+1,t0=1−ε. Suppose not, i.e., there exists α>0 and two subsequences κnpp and κmpp verifying(16)Qκmp+1,κnp+1,t0>α+1−ε,for all p∈ℕ.
Havingq∈ℕ satisfying Qκmp+1,κnp+1,t0−1/q>α+1−ε, we obtain(17)1−ε≥Qκmp,κnp,t0≥Qκmp,κmp+1,12q∗Qκmp+1,κnp+1,t0−1q∗Qκnp+1,κnp,12q≥Q˜κmp,κmp+1,0∗α+1−ε∗Q˜κnp+1,κnp,0⟶α+1−ε,as p⟶∞.
This is a contradiction. Then,(18)limpQκmp+1,κnp+1,t0=1−ε.
Relations (14), (15), and (18) drive to a clear contradiction with condition (7). So, κnn is a Cauchy sequence in the complete fuzzy metric space O and we deduce that there exists κ∗∈O such that(19)limnQκn,κ∗,t=1,for all t>0, and by relation (7), we obtain(20)1Qℒκn,ℒκ∗,t−1≤k1Qκn,κ∗,t−1,for all n∈ℕ and for all t>0. Passing to the limit, having in mind the limit in (19), it follows that Qκ∗,ℒκ∗,t=1, which, with assumption (F2) and relation (7), means that κ∗ is the unique fixed point of mapping ℒ. This achieves the proof.Theorem 3.
(fuzzy Meir-Keeler fixed point theorem). LetO,Q,∗ be a complete E−fuzzy metric space. Let ℒ:O⟶O be a fuzzy Meir-Keeler type mapping, i.e., for all ε∈0,1, there exists δ>0 such that(21)ε−δ<Qκ,ω,t≤ε⟹Qℒκ,ℒω,t>ε,for all κ,ω in O and for all t>0. Then, ℒ has a unique fixed point κ∗. Furthermore, for all κ∈O, the sequence ℒnκ converges to κ∗.Proof.
Letκ∈O and κn=ℒnκn∈ℕ and t>0. Obviously, we have(22)Qκ,ℒκ,t−δ<Qκ,ℒκ,t≤Qκ,ℒκ,t,for all δ>0, and due to relation (21), we obtain Qℒ2κ,ℒκ,t>Qκ,ℒκ,t. Recursively, we obtain a sequence Qκn,κn+1,tn in 0,1 verifying(23)Qκn,κn+1,t<Qκn+1,κn+2,t,for all n in ℕ. It is a bounded increasing sequence. Then, there exists a function u:0,∞⟶0,1 such that(24)limn⟶+∞Qκn,κn+1,t=supn∈ℕQκn,κn+1,t=ut,for all t>0. We claim that ut=1, for all t>0. Suppose not, i.e., there exists t0>0 such that ut0∈0,1. By the limit in (24), for all δ∈0,ut0, there exists n0∈ℕ such that(25)ut0−δ<Qκn,κn+1,t0≤ut0,for all n≥n0, which, with condition (21), implies that Qκn+1,κn+2,t0>ut0. This is a clear contradiction with (24). Therefore,(26)limnQκn,κn+1,t=1,for all t>0. Now, we follow, exactly, the same lines as in the proof of Theorem 2 to deduce that κnn is a Cauchy sequence in the complete fuzzy metric space O, which deduce that there exists κ∗∈O such that(27)limnQκ∗,κn,t=1.
On the other hand, for alln∈ℕ and all δ∈0,Qκ∗,κn,t, we have(28)Qκ∗,κn,t−δ<Qκ∗,κn,t≤Qκ∗,κn,t.
Condition (21) assures that(29)1≥Qℒκ∗,ℒκn,t>Qκ∗,κn,t,which, with the limit in (27), gives limnQℒκ∗,κn,t=1, and finally(30)κ∗=ℒκ∗.
For the uniqueness, we assume that there existsω∗≠κ∗∈O such that ω∗=ℒω∗. It is clear that for allδ∈0,Qκ∗,ω∗,t,Qκ∗,ω∗,t−δ<Qκ∗,ω∗,t≤Qκ∗,ω∗,t.
Hence, by (21), Qℒκ∗,ℒω∗,t>Qκ∗,ω∗,t or Qκ∗,ω∗,t>Qκ∗,ω∗,t, a contradiction, and this achieves the proof.
Now, we give the following corollary.Corollary 1.
LetO,d be a complete metric space, and ℒ a Meir-Keeler mapping on O, i.e., for each ε>0, there exists δ>0 such that for all κ,ω∈O,(31)ε≤dκ,ω<ε+δ⟹dℒκ,ℒω<ε.LetQ be a function on O×O×0,+∞ defined by(32)Qκ,ω,t=t+1t+1+dκ,ω.Then,(1)
O,Q,⋅ is an E−fuzzy metric space, where · is the product t-norm.(2)
For allε>0, there exists δ>0 such that(33)ε−δ<Qκ,ω,t≤ε⟹Qℒκ,ℒω,t>ε,for all κ,ω in O and for all t>0.Proof.
O,Q,⋅ is a fuzzy metric space (see [4]) and t↦Qκ,ω,t;κ,ω∈O is a set of functions with common Lipschitz constant “1.” So, it is uniformly equicontinuous. This means that O,Q,⋅ is an E−fuzzy metric space. For the second assumption, it suffices to see that
For allε>0,δ∈0,ε,t>0 and all κ,ω∈O, we have(34)ε−δ<Qκ,ω,t≤ε⟺ε−δ<t+1t+1+dκ,ω≤ε⟺ε−δ<11+1/t+1dκ,ω≤ε⟺t+11ε−1≤dκ,ω<t+11ε−δ−1.
Letε0=t+11/ε−1 and δε0>0 such that(35)ε0≤dκ,ω<ε0+δε0⟹dℒκ,ℒω<ε0.
Now, we chooseδ in (34) such that t+11/ε−δ−1<t+11/ε−1+δε0. Therefore, using relations (34) and (35), it follows that(36)ε−δ<Qκ,ω,t≤ε⟹t+11ε−1<dκ,ω≤t+11ε−1+δε0⟹dℒκ,ℒω<t+11ε−1⟹t+1t+1+dℒκ,ℒω>ε⟹Qℒκ,ℒω,t>ε,and this achieves the proof.


## 3. Application

The purpose of this section is to give an example of the existence of a solution for an integral equation, where we can apply Theorem2 to get its solution. For such integral equations, we refer the reader to [17] where the authors provide a common solution for a system of two integral equations.Consider the integral equation,(37)κr=gr+∫0rFr,s,κsds,for allr∈0,I,I>0,and Banach space C0,I,ℝ of all continuous functions defined on 0,I equipped with supremum norm(38)κ=supr∈0,1κr,κ∈C0,I,ℝ,with induced metric(39)dκ,ω=supr∈0,Iκr−ωr.Now, consider the fuzzy metric space with productt−norm as(40)Qκ,ω,t=tt+dκ,ω,for allκ,ω∈C0,I,ℝ,t>0.According to George and Veeramani, standard fuzzy metric space and the corresponding metric space have same topologies. So, fuzzy metric space defined in (40) is complete.Theorem 4.
Consider the integral operatorℒ on C0,I,ℝ as(41)ℒκr=gr+∫0rFr,s,κsds.Suppose that there existsf:0,I×0,I⟶0,∞ such that f∈L10,I,ℝ and suppose that F satisfies the following condition:(42)Fs,r,κr−Fs,r,ωr≤fr,sκs−ωs,for all κ,ω∈C0,I,ℝ and for all r,s∈0,I where(43)supr∈0,1∫0rfr,sds≤k<1.Then, the integral equation (37) has a unique solution.Proof.
Letκ,ω∈C0,I,ℝ and consider(44)ℒκr−ℒωr≤∫0rFr,s,κs−Fr,s,ωsds≤∫0rfr,sκs−ωsds≤dκ,ω∫0rfr,sds≤kdκ,ω.
So,(45)dℒκ,ℒω≤kdκ,ω.
Using (40), we can write(46)1Qκ,ω,t−1=dκ,ωt1Qℒκ,ℒω,t−1=t+dℒκ,ℒω−tt=dℒκ,ℒωt≤kdκ,ωt≤k1Qκ,ω,t−1.
Since all the conditions of Theorem2 hold, (37) has a unique solution.



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