# On Some New Common Fixed Point Results for Finite Number of Mappings in Fuzzy Metric Spaces

Authors: Ayush Bartwal; Junaid Ahmad; R. C. Dimri; Gopi Prasad; Ebenezer Bonyah
Journal: Advances in Mathematical Physics
 (2022)

---

## Abstract

We essentially suggest the concept of mutual sequences and Cauchy mutual sequence and utilize the same to prove the existence and uniqueness of common fixed point results for finite number of self- and non-self-mappings using fuzzyℤ∗-contractive mappings in fuzzy metric spaces. Our main result was obtained under generalized contractive condition in the fuzzy metric spaces. We provide examples to vindicate the claims and usefulness of such investigations. In this way, the present results generalize and enrich the several existing literature of the fuzzy metric spaces.


---

## Body

## 1. Introduction

In 1975, Kramosil and Michalek [1] introduced the notion of fuzzy metric spaces using the theory of fuzzy sets, which generalizes the metric spaces. Later on, many authors have introduced the notion of fuzzy metric spaces in different ways (see [2–5]). The widely accepted definition is given by George and Veeramani [6]. They presented slight modification on the definition of fuzzy metric spaces initiated by the respective authors by obtaining Hausdorff topology on the same setting. Utilizing the notion of the fuzzy metric, many authors proved various interesting common fixed point result for self- and non-self-mappings using different contraction in this setting. In 1984, Hadžić [7] proved some common fixed point theorems for family of mapping. After that, Bari and Vetro [8] also proved theorems for family of mappings in fuzzy metric spaces. In 1994, Subrahmanyam [9] generalized Jungck’s theorem [10] in the setting of fuzzy metric spaces introduced by Kramosil and Michalek [1]. Vasuki [11] proved common fixed point theorems in the same setting. In 2002, Rhoades [12] proved common fixed point theorems for non-self-mappings using quasicontraction.Jungck and Rhoades [13] introduced the concept of weak compatibility in metric spaces, which was further studied by Singh and Jain [14] in the fuzzy metric settings. Sedghi et al. [15] proved common fixed point theorems for four weakly compatible mappings. For the common fixed point using the notion of common limit range property, we refer common fixed point theorems by Chauhan et al. [16]. In this continuation, Imdad et al. [17] proved common fixed point theorems in fuzzy metric spaces using common property (E.A) and Prasad et al. [18] presented some coincidence point theorems via contractive mappings.Recently, Roldan and Sintunavarat [19] introduced an important concept of fuzzy metric spaces on the product space XN which is induced by a simple fuzzy metric structure and compare the convergence, Cauchy, and completeness between these two structures. They also proved common fixed point results using CLRg property in the same metric setting. On the other hand, Shukla et al. [20] unify classes of different fuzzy contractive mappings presented in [21–24] and introduced a new class of fuzzy Z-contractive mapping and notions of properties S and S′ to prove fixed point results in the fuzzy metric spaces.In this paper, firstly we define the mutual sequences and Cauchy mutual sequences. The idea behind defining Cauchy mutual sequence is to collect those Cauchy sequences which are converging to the same limit. After that, we utilize this idea to find common fixed points. Indeed, Cauchy mutual sequences in a fuzzy metric spaceX,M,∗ are the special type of Cauchy sequences in fuzzy metric space XN,MN,∗ which converge to the same limit ϱ∈X, if they are convergent. We also generalize the ℤ-contraction for finite number of mappings; using these contractive mappings, we will prove some unique common fixed point theorems in fuzzy metric spaces. The main aim of this paper is to prove unique common fixed point theorems using ℤ∗-contraction (which is the extension of ℤ-contraction for finite number of mappings) in fuzzy metric spaces for self- and non-self-mappings with the help of mutual sequences. We also give examples for validity of our claims. In this way, our results generalize and improve several existing results.


## 2. Preliminaries

Throughout this paper,N,n,m are natural numbers, i∈1,2,..,N; XN will denote Cartesian product of N-copies of X and X is any nonempty set. In the sequel, sometimes Tϱ will be denoted by Tϱ.Definition 1 (see [6]).
An ordered tripleX,M,∗ is called a fuzzy metric space if X is a (nonempty) set, M is a fuzzy set on X2×0,∞, and ∗ is a continuous t-norm satisfying the following conditions, for all ϱ,ρ,z∈X and t,s>0:
(1)
Mϱ,ρ,t>0(2)
Mϱ,ρ,t=1, if and only if ϱ=ρ(3)
Mϱ,ρ,t=Mρ,ϱ,t(4)
Mϱ,z,t+s≥Mϱ,ρ,t∗Mρ,z,s(5)
Mϱ,ρ,.:0,∞⟶0,1 is continuousDefinition 2 (see [6]).
(i)
LetX,M,∗ be a fuzzy metric space. A sequence ϱn is said to be converge to ϱ in X if and only if limn⟶∞Mϱn,ϱ,t=1 for all t>0, i.e., for each r∈0,1 and t>0, there exists n0∈ℕ such that Mϱn,ϱ,t>1−r for all n≥n0(ii)
A sequenceϱn in a fuzzy metric space X,M,∗ is a Cauchy sequence if and only if for each ϵ>0, t>0 there exists n0∈N such that Mϱn,ϱm,t>1−ϵ for all n,m>n0. On the other hand, ϱn is called a Cauchy sequence if limn⟶∞Mϱn,ϱn+m,t=1 for all m∈ℕ and t>0(iii)
A fuzzy metric spaceX,M,∗ is said to be complete if every Cauchy sequence in X is convergent to some ϱ∈XLemma 3 (see [19]).
LetX,M,∗ be a fuzzy metric space and XN=X×X×⋯×X⏟Ntimes, where N∈ℕ and define a fuzzy set MN on XN×XN×0,∞⟶0,1 such that
(1)MNP,Q,t=∗i=1NMpi,qi,t,forallP=p1,p2,⋯,pN,Q=q1,q2,⋯,qN∈XN,t>0.
Then, the following hold:(i)
XN,MN,∗ is also a fuzzy metric space(ii)
LetPn=pn1,pn2,⋯,pnN be a sequence on XN and P=p1,p2,⋯,pN∈XN; then, sequence Pn converges to P on XN,MN,∗ if and only if all sequences pni converge to pi on X,M,∗, for all i∈1,2,⋯,N(iii)
LetPn=pn1,pn2,⋯,pnN be a sequence on XN; then, Pn is Cauchy sequence on XN,MN,∗ if and only if pni is Cauchy sequence on X,M,∗, for all i∈1,2,⋯,N(iv)
X,M,∗ is complete if and only if XN,MN,∗ is completeDefinition 4 (see [20]).
LetZ denote the family of all functions ζ:0,1×0,1⟶ℝ satisfying the following condition: ζt,s>s,forallt,s∈0,1.Definition 5 (see [20]).
LetT be a self-mapping and X,M,∗ a fuzzy metric space. If there exists ζ∈Z such that
(2)MTϱ,Tρ,t≥ζMTϱ,Tρ,t,Mϱ,ρ,t,for all ϱ,ρ∈X, Tϱ≠Tρ, t>0, then T is called a fuzzy Z-contractive mapping with respect to the function ζ∈Z.Definition 6 (see [20]).
LetT be any self-mapping in X, ζ∈Z and X,M,∗ a fuzzy metric space then quadruplet X,M,T,ζ has the property (S′), if there exists ϱn∈X such that ϱn=Tnϱ, for all n∈ℕ and infm>nMϱn,ϱm,t≤infm>nMϱn+1,ϱm+1,t, for all n∈ℕ,t>0 and 0<limn⟶∞infm>nMϱn,ϱm,t<1, for all t>0 implies that
(3)limn⟶∞infm>nζMϱn+1,ϱm+1,t,Mϱn,ϱm,t=1,forallt>0.


## 3. Proposed Results

For brevity, we observe that Definitions4 and 6 can be unified as follows.Definition 7.
Letℤ∗ denote the set of all functions ζ:0,1×0,1 to ℝ satisfying the following conditions:
(i)
ζp,q>q, for all~p,q∈0,1(ii)
Letpn and qn be two sequences in 0,1 such that pn≤qn, for all n∈ℕ and limn⟶∞pn∈0,1 and then limn⟶∞infm>nζqn,pn=1Example 1.
Defineζ:0,1×0,1⟶ℝ such that ζp,q=q/p and qn=1/2−1/3n, pn=1/2+1/3n, we observe that ζ∈ℤ∗ as limn⟶∞1/2−1/3n/1/2+1/3n=1.Definition 8.
LetX,M,∗ be a fuzzy metric space and T1,T2,⋯,TN are N-mapping on X satisfying following condition:
(4)MNT1ϱ1,T2ϱ2,⋯,TNϱN,T2ρ2,T3ρ3,⋯,TNρN,T1ρ1,t≥ζMNT1ϱ1,T2ϱ2,⋯,TNϱN,T2ρ2,T3ρ3,⋯,TNρN,T1ρ1,t,MNϱ1,ϱ2,⋯,ϱN,ρ2,ρ3,⋯,ρN,ρ1,t,for all t>0, ϱ1,ϱ2,⋯,ϱN≠ρ1,ρ2,⋯,ρN∈X, T1ϱ1,T2ϱ2,⋯,TNϱN≠T1ρ1,T2ρ2,⋯,TNρN, where N∈ℕ,ζ∈ℤ∗ and XN,MN,∗ is a fuzzy metric space induced by X,M,∗. Then, T1,T2,⋯,TN are said to be fuzzy ℤ∗-contractive mappings.
Observe that for “N=1,” Definition 5 is a particular case of Definition 8.Definition 9.
A sequenceϱn1,ϱn2,⋯,ϱnN∈XN=X×X×⋯×X⏟Ntimes is said to be a mutual sequence in a fuzzy metric space X,M,∗.Definition 10.
Letϱn1,ϱn2,⋯,ϱnN∈XN=X×X×⋯×X⏟Ntimes be a mutual sequence in a fuzzy metric space X,M,∗ and all sequences ϱni, i∈1,2,⋯N converge, then the sequence ϱn1,ϱn2,⋯,ϱnN is said to be convergent mutual sequence. If the sequence ϱni, for all i∈1,2,⋯,N converge to the unique limit ϱ∈X, then the mutual sequence is said to be coconvergent mutual sequence and limit ϱ is said to be the mutual limit.Definition 11.
A mutual sequenceϱn1,ϱn2,⋯,ϱnN∈XN=X×X×⋯×X⏟Ntimes is said to be a Cauchy mutual sequence in a fuzzy metric space X,M,∗, if for each ϵ>0, there exists n0∈ℕ such that for all n,m≥n0n,m∈ℕ, we have
(5)MNϱn1,ϱn2,⋯,ϱnN,ϱm2,ϱm3,⋯,ϱmN,ϱm1,t>1−ϵ,for all t>0, i.e., a mutual sequence is said to be a Cauchy mutual sequence if as n,m⟶∞,
(6)MNϱn1,ϱn2,⋯,ϱnN,ϱm2,ϱm3,⋯,ϱmN,ϱm1,t⟶1,for all t>0, where XN,MN,∗ is fuzzy metric spaces induced by X,M,∗.
Now, we are considering the following lemmas to prove the existence and uniqueness of common fixed points.Lemma 12.
Every Cauchy mutual sequence inX,M,∗ is a Cauchy sequence in XN,MN,∗.Proof.
Letϱn1,ϱn2,⋯,ϱnN be a mutual sequence. By Definition 1 and Lemma 3, we have
(7)MNϱn1,ϱn2,⋯,ϱnN,ϱm1,ϱm2,⋯,ϱmN,t≥MNϱn1,ϱn2,⋯,ϱnN,ϱk2,ϱk3,⋯,ϱkN,ϱk1,t2∗MNϱk2,ϱk3,⋯,ϱkN,ϱk1,ϱm1,ϱm2,⋯,ϱmN,t2.
Now, we have sequenceϱn1,ϱn2,⋯,ϱnN a Cauchy mutual sequence. So, as n,m,k⟶∞, we get
(8)MNϱn1,ϱn2,⋯,ϱnN,ϱm1,ϱm2,⋯,ϱmN,t⟶1.
Hence,ϱn1,ϱn2,⋯,ϱnN is a Cauchy sequence in XN,MN,∗.
The following example shows that the converse of Lemma12 may not be true.Example 2.
LetX=1,∞ define a fuzzy metric space X,M,∗, where
(9)Mϱ,ρ,t=1,ifϱ=ρ22+maxϱ,ρ,otherwiseforallϱ,ρ∈ℝ+,t>0,and ∗ be a continuous t-norm defined as a∗b=mina,b. Consider a mutual sequence 1−1/n,2−1/n,3−1/n on fuzzy metric space XN,MN,∗, then mutual sequence is a Cauchy sequence in XN,MN,∗ but not Cauchy mutual sequence because as n,m⟶∞, we have
(10)M31−1n,2−1n,3−1n,2−1m,3−1m,1−1m,t⟶25,for all t>0.Lemma 13.
In a fuzzy metric space, every convergent Cauchy mutual sequence is coconvergent.Proof.
Letϱn1,ϱn2,⋯,ϱnN be a convergent mutual sequence which converges to ϱ1,ϱ2,⋯,ϱN, where ϱi∈X, for i∈1,2,⋯,N. Since ϱn1,ϱn2,⋯,ϱnN is a convergent Cauchy mutual sequence, as n,m⟶∞, we have
(11)MNϱn1,ϱn2,⋯,ϱnN,ϱm2,ϱm3,⋯,ϱmN,ϱm1,t⟶1,for all t>0, which implies that
(12)MNϱ1,ϱ2,⋯,ϱN,ϱ2,ϱ3,⋯,ϱN,ϱ1,t=1,for all t>0. Hence, ϱ1=ϱ2=⋯=ϱN.Theorem 14.
LetX,M,∗ be a complete fuzzy metric space; T1,T2,⋯,TN are N-self mappings on X satisfying
(a)
fuzzyℤ∗-contraction(b)
limn⟶∞infm>nMNT1mϱ,T2mϱ,⋯,TNnϱ,T2mϱ,T3mϱ,⋯,TNnϱ,T1mϱ,t>0, for all t>0, ϱ∈X
Then,T1,T2,⋯,TN have unique common fixed point.Proof.
Letϱ0i∈X and Tiϱni=ϱn+1i, for all n∈ℕ∪0 and i∈1,2,⋯N. We get ϱn1,ϱn2,⋯,ϱnN as a mutual sequence on X,M,∗ and according to Lemma 3, XN,MN,∗ is also a fuzzy metric space.
Ifϱni=ϱn−1i, for all i∈1,2,⋯,N and for any n∈ℕ, then Tiϱni=ϱn−1i=ϱni, i.e., ϱni is the fixed point of Ti′s, for every i and a fixed n. Suppose that ϱn1≠ϱn2≠⋯≠ϱnN, for fixed n. From, Definitions 8 and 7(13)MNϱn1,ϱn2,⋯,ϱnN,ϱn−12,ϱn−13,⋯,ϱn−1N,ϱn−11,t=MNT1ϱn1,T2ϱn2,⋯,TNϱnN,T2ϱn−12,T3ϱn−13,⋯,TNϱn−1N,T1ϱn−11,t≥ζMNT1ϱn1,T2ϱn2,⋯,TNϱnN,T2ϱn−12,T3ϱn−13,⋯,TNϱn−1N,T1ϱn−11,t,MNϱn1,ϱn2,⋯,ϱnN,ϱn−12,ϱn−13,⋯,ϱn−1N,ϱn−11,t>MNϱn1,ϱn2,⋯,ϱnN,ϱn−12,ϱn−13,⋯,ϱn−1N,ϱn−11,t,for all t>0. So, ϱn1=ϱn2=⋯=ϱnN=ϱsay is a common fixed point of Ti′s.
Now, assume that no consecutive terms of the sequenceϱn1,ϱn2,⋯,ϱnN are the same and ϱn1,ϱn2,⋯,ϱnN=ϱm1,ϱm2,⋯,ϱmN for some n<m, i.e.,
(14)ϱn+1i=Tiϱni=Tiϱmi=ϱm+1i,for some n<m and for all i. From Definition 8 and 7, we have
(15)MNϱn+21,ϱn+22,⋯,ϱn+2N,ϱn+12,ϱn+13,⋯,ϱn+1N,ϱn+11,t=MNT1ϱn+11,T2ϱn+12,⋯,TNϱn+1N,T2ϱn2,T3ϱn3,⋯,TNϱnN,T1ϱn1,t≥ζMNT1ϱn+11,T2ϱn+12,⋯,TNϱn+1N,T2ϱn2,T3ϱn3,⋯,TNϱnN,T1ϱn1,t,MNϱn+11,ϱn+12,⋯,ϱn+1N,ϱn2,ϱn3,⋯,ϱnN,ϱn1,t>MNϱn+11,ϱn+12,⋯,ϱn+1N,ϱn2,ϱn3,⋯,ϱnN,ϱn1,t,for all t>0. Similarly, we get
(16)MNϱn+11,ϱn+12,⋯,ϱn+1N,ϱn2,ϱn3,⋯,ϱnN,ϱn1,t<MNϱn+21,ϱn+22,⋯,ϱn+2N,ϱn+12,ϱn+13,⋯,ϱn+1N,ϱn+11,t<⋯<MNϱm+11,ϱm+12,⋯,ϱm+1N,ϱm2,ϱm3,⋯,ϱmN,ϱm1,t,for all t>0, which is a contradiction in light of the inequality (14). Therefore, ϱn1,ϱn2,⋯,ϱnN≠ϱm1,ϱm2,⋯,ϱmN, for some n<m.
Now consider,ϱn1,ϱn2,⋯,ϱnN≠ϱm1,ϱm2,⋯,ϱmN, for all n≠m∈ℕ. Then, from Definitions 8 and 7, we have
(17)MNϱm+11,ϱm+12,⋯,ϱm+1N,ϱn+12,ϱn+13,⋯,ϱn+1N,ϱn+11,t=MNT1ϱm1,T2ϱm2,⋯,TNϱmN,T2ϱn2,T3ϱn3,⋯,TNϱnN,T1ϱn1,t≥ζMNT1ϱm1,T2ϱm2,⋯,TNϱmN,T2ϱn2,T3ϱn3,⋯,TNϱnN,T1ϱn1,tMNϱm1,ϱm2,⋯,ϱmN,ϱn2,ϱn3,⋯,ϱnN,ϱn1,t>MNϱm1,ϱm2,⋯,ϱmN,ϱn2,ϱn3,⋯,ϱnN,ϱn1,t,for all t>0 and n<m. Taking infimum (over m>n) in the above inequality, we have
(18)infm>nMNϱm1,ϱm2,⋯,ϱmN,ϱn2,ϱn3,⋯,ϱnN,ϱn1,t≤infm>nMNϱm+11,ϱm+12,⋯,ϱm+1N,ϱn+12,ϱn+13,⋯,ϱn+1N,ϱn+11,t,for all t>0. Therefore, infm>nMNϱm1,ϱm2,⋯,ϱmN,ϱn2,ϱn3,⋯,ϱnN,ϱn1,t is a monotonic and bounded sequence, for all t>0. So, there exist some st≤1 such that
(19)limn⟶∞infm>nMNϱm1,ϱm2,⋯,ϱmN,ϱn2,ϱn3,⋯,ϱnN,ϱn1,t=st,for all t>0.
Denote(20)pn=MNϱm1,ϱm2,⋯,ϱmN,ϱn2,ϱn3,⋯,ϱnN,ϱn1,t,(21)qn=MNϱm+11,ϱm+12,⋯,ϱm+1N,ϱn+12,ϱn+13,⋯,ϱn+1N,ϱn+11,t,
for allt>0.
Now, our claim isst=1, for every t>0. Letting on contrary that st1<1, for some t1>0. In light of (3), we have pn≤qn and by condition (b), limn⟶∞pn∈0,1. Applying Definition 7, we get
(22)limn⟶∞infm>nζMNϱm1,ϱm2,⋯,ϱmN,ϱn2,ϱn3,⋯,ϱnN,ϱn1,t1,MNϱm+11,ϱm+12,⋯,ϱm+1N,ϱn+12,ϱn+13,⋯,ϱn+1N,ϱn+11,t1=1.
From (2), we have
(23)infm>nMNϱm+11,ϱm+12,⋯,ϱm+1N,ϱn+12,ϱn+13,⋯,ϱn+1N,ϱn+11,t1≥infm>nζMNT1ϱm1,T2ϱm2,⋯,TNϱmN,T2ϱn2,T3ϱn3,⋯,TNϱnN,T1ϱn1,t1,MNϱm1,ϱm2,⋯,ϱmN,ϱn2,ϱn3,⋯,ϱnN,ϱn1,t1>infm>nMNϱm1,ϱm2,⋯,ϱmN,ϱn2,ϱn3,⋯,ϱnN,ϱn1,t1.
By (22) and as n⟶∞, we get
(24)infm>nMNϱm1,ϱm2,⋯,ϱmN,ϱn2,ϱn3,⋯,ϱnN,ϱn1,t1=st1=1,which is a contradiction. We conclude that
(25)limn⟶∞limn,m⟶∞MNϱm1,ϱm2,⋯,ϱmN,ϱn2,ϱn3,⋯,ϱnN,ϱn1,t=1,for all t>0. Hence, the mutual sequence ϱn1,ϱn2,⋯,ϱnN is a Cauchy mutual sequence. Completeness of X and Lemma 12 ensure that there exists ϱ1,ϱ2,⋯,ϱN∈X such that
(26)limn⟶∞MNϱn1,ϱn2,⋯,ϱnN,ϱ1,ϱ2,⋯,ϱN,t=1,for all t>0. From Lemma 13, sequence ϱn1,ϱn2,⋯,ϱnN is coconvergent sequence, i.e., ϱ1=ϱ2=⋯=ϱN=ϱsay. Now, we have to prove that ϱ is a common fixed point of T1,T2,⋯,TN. Suppose that Tiϱ≠ϱ, i∈1,2,⋯,N. Without loss of generality, let us assume that ϱn1,ϱn2,⋯,ϱnN≠ϱ,ϱ,⋯,ϱ and ϱn1,ϱn2,⋯,ϱnN≠T2ϱ,T3ϱ,⋯,TNϱ,T1ϱ, for all n∈N. So, there exists t1>0 such that MNϱ,ϱ,⋯,ϱ,T2ϱ,T3ϱ,⋯,TNϱ,T1ϱ,t1<1, MNϱn1,ϱn2,⋯,ϱnN,ϱ,ϱ,⋯,ϱ,t1<1 and
(27)MNT1ϱn1,T2ϱn2,⋯,TNϱnN,T2ϱ,T3ϱ,⋯,TNϱ,T1ϱ,t1=MNϱn+11,ϱn+12,⋯,ϱn+1N,T2ϱ,T3ϱ,⋯,TNϱ,T1ϱ,t1<1,for all n∈ℕ. Then, we have
(28)MNϱn1,ϱn2,⋯,ϱnN,ϱ,ϱ,⋯,ϱ,t1<ζMNT1ϱn1,T2ϱn2,⋯,TNϱnN,T2ϱ,T3ϱ,⋯,TNϱ,T1ϱ,t1,MNϱn1,ϱn2,⋯,ϱnN,ϱ,ϱ,⋯,ϱ,t1=MNT1ϱn1,T2ϱn2,⋯,TNϱnN,T2ϱ,T3ϱ,⋯,TNϱ,T1ϱ,t1=MNϱn+11,ϱn+12,⋯,ϱn+1N,T2ϱ,T3ϱ,⋯,TNϱ,T1ϱ,t1,as n⟶∞; from (5) and Lemma 13, we get
(29)MNT1ϱ,T2ϱ,⋯TNϱ,ϱ,ϱ,⋯,ϱ,t1≥1,which is a contradiction. Hence, MNT1ϱ,T2ϱ,⋯TNϱ,ϱ,ϱ,⋯,ϱ,t=1, for all t>0. Hence, ϱ is the common fixed point of Ti′s, for all i∈1,2,⋯N.
Now, we have to prove the uniqueness of the common fixed point ofTi′s. Assume on contrary that ϱ,ρ∈X be two distinct common fixed points of Ti′s, for all i∈1,2,⋯,N and there exists t1>0 such that MNϱ,ϱ,⋯,ϱ,ρ,ρ,⋯,ρ,t1<1. Then, from Definitions 8 and 7, we get
(30)MNϱ,ϱ,⋯,ϱ,ρ,ρ,⋯,ρ,t1=MNT1ϱ,T2ϱ,⋯,TNϱ,T2ρ,T3ρ,⋯,TNρ,T1ρ,t1=ζMNT1ϱ,T2ϱ,⋯,TNϱ,T2ρ,T3ρ,⋯,TNρ,T1ρ,t1,MNϱ,ϱ,⋯,ϱ,ρ,ρ,⋯,ρ,t1>MNϱ,ϱ,⋯,ϱ,ρ,ρ,⋯,ρ,t1,implying thereby MNϱ,ϱ,⋯,ϱ,ρ,ρ,⋯,ρ,t=1, for all t>0. Hence, ϱ=ρ.Remark 15.
On puttingN=1, in Theorem 14, it reduces to Theorem 3.19 presented in [20].Example 3.
LetX=0,1/5,1/3,1/2,1,2,12,17,31,45,60,71,91,100,111 and X,M,∗ be a fuzzy metric space in which M is a fuzzy set defined on X2×0,∞ such that M=t/t+ϱ−ρ for all ϱ,ρ∈X and t>0, ∗ is a continuous t-norm defined as ϱ∗ρ=ϱ.ρ, then X,M,∗ is a complete metric space. Now, let us define 5-maps T1,T2,⋯T5:X⟶M as
(31)T1ϱ=1,ifϱ∈12,17,31,0,otherwise,T2ϱ=12,ifϱ∈45,60,71,0,otherwise,T3ϱ=2,ifϱ∈91,100,111,0,otherwise,T4ϱ=0,ϱ∈1,12,13,15,0,13,otherwise,T5ϱ=0,ifϱ∈12,13,15,0,15,otherwise,and a function ζ:0,1×0,1⟶ℝ such that ζp,q=q/p for all p,q∈0,1.Letϱ01,ϱ02,ϱ03,ϱ04,ϱ05=31,60,91,1,2 and Tiϱni=ϱn+1i for all n∈ℕ∪0, we get 31,60,91,1,2,1,1/2,2,0,1/5,0,0,0,0,0,⋯ as a mutual sequence. We can easily observe that conditions a and b of Theorem 14 are satisfied. Hence, 0 is the unique common fixed point of T1,T2,⋯T5.Now, we present the fixed point theorem for non-self-mappings.Theorem 16.
LetX,M,∗ be a fuzzy metric space, Y1,Y2,⋯,YN are N subset of X. Let T1:Y1⟶Y2,T2:Y2⟶Y3,⋯,TN−1:YN−1⟶YN and TN:YN⟶Y1 are N mappings satisfying the following conditions:
(i)
TiYi are complete subspace of X(ii)
MNT1ϱ1,T2ϱ2,⋯,TNϱN,T2ρ2,T3ρ3,⋯,TNρN,T1ρ1,t≥ζMNT1ϱ1,T2ϱ2,⋯,TNϱN,T2ρ2,T3ρ3,⋯,TNρN,T1ρ1,t,MNϱ1,ϱ2,⋯,ϱN,ρ2,ρ3,⋯,ρN,ρ1,t for all t>0, ϱi≠ρi∈Yi, i∈1,2,⋯N, T1ϱ1,T2ϱ2,⋯,TNϱN≠T1ρ1,T2ρ2,⋯,TNρN, where N∈ℕ,ζ∈ℤ∗ and XN,MN,∗ is fuzzy metric spaces induced by X,M,∗(iii)
limn⟶∞infm>nMNT1mϱ1,T2mϱ2,⋯,TNnϱN,T2mϱ2,T3mϱ3,⋯,TNnϱN,T1mϱ1,t>0 for all t>0, ϱi∈Yi, i∈1,2,⋯,N
Then,T1,T2,⋯,TN have unique common fixed point.Proof.
Letϱ11∈Y1 and T1ϱn1=ϱn2, T2ϱn2=ϱn3,..., TN−1ϱnN−1=ϱnN and TNϱnN=ϱn+11, for all n∈ℕ. We get ϱm1,ϱm2,⋯,ϱmN∈XN as a mutual on X,M,∗.
Ifϱni=ϱn+1i, for all 1≤i≤N∈ℕ and for any n∈ℕ, then T1ϱn1=ϱn2=ϱn+12, T2ϱn2=ϱn3=ϱn+13, ..., TN−1ϱnN−1=ϱnN=ϱn+1N and TNϱnN=ϱn+11=ϱn+11. Now, from Lemma 3, Definition 7, and condition (ii), we have
(32)MNϱn1,ϱn2,⋯,ϱnN,ϱn+12,ϱn+13,⋯,ϱn+1N,ϱn+11,t=MNϱn+11,ϱn2,⋯,ϱnN,ϱn+12,ϱn+13,⋯,ϱn+1N,ϱn+11,t=MNTNϱnN,T1ϱn1,⋯,TN−1ϱnN−1,T1ϱn+11,T2ϱn+12,⋯,TN−1ϱn+1N−1,TNϱnN,t≥ζMNTNϱnN,T1ϱn1,⋯,TN−1ϱnN−1,T1ϱn+11,T2ϱn+12,⋯,TN−1ϱn+1N−1,TNϱnN,t,MNϱnN,ϱn1,⋯,ϱnN−1,ϱn+11,ϱn+12,⋯,ϱn+1N−1,ϱnN,t>MNϱnN,ϱn1,⋯,ϱnN−1,ϱn+11,ϱn+12,⋯,ϱn+1N−1,ϱnN,t=MNϱn1,ϱn2,⋯,ϱnN,ϱn+12,ϱn+13,⋯,ϱn+1N,ϱn+11,t,for all t>0, a contradiction, which implies that MNϱn1,ϱn2,⋯,ϱnN,ϱn+12,ϱn+13,⋯,ϱn+1N,ϱn+11,t=1, i.e., ϱn1=ϱn2=⋯=ϱnN=ϱsay is a common fixed point of Ti’s.
From Lemma3, Definition 7, and condition (ii), for all t>0, we have
(33)MNϱn+21,ϱn+22,⋯,ϱn+2N,ϱn+12,ϱn+13,⋯,ϱn+1N,ϱn+11,t=MNTNϱn+1N,T1ϱn+21,⋯,TN−1ϱn+2N−1,T1ϱn+11,T2ϱn+12,⋯,TN−1ϱn+1N−1,TNϱnN,t≥ζMNTNϱn+1N,T1ϱn+21,⋯,TN−1ϱn+2N−1,T1ϱn+11,T2ϱn+12,⋯,TN−1ϱn+1N−1,TNϱnN,t,MNϱn+1N,ϱn+21,⋯,ϱn+2N−1,ϱn+11,ϱn+12,⋯,ϱn+1N−1,ϱnN,t>MNϱn+1N,ϱn+21,⋯,ϱn+2N−1,ϱn+11,ϱn+12,⋯,ϱn+1N−1,ϱnNt=MNTN−1ϱn+1N−1,TNϱn+1N,⋯,TN−2ϱn+2N−2,TNϱnN,T1ϱn+11,⋯,TN−1ϱnN−1,t≥ζMNTN−1ϱn+1N−1,TNϱn+1N,⋯,TN−2ϱn+2N−2,TNϱnN,T1ϱn+11,⋯,TN−1ϱnN−1,t,MNϱn+1N−1,ϱn+1N,⋯,ϱn+2N−2,ϱnN,ϱn+11,⋯,ϱn+1N−2,ϱnN−1,t>MNϱn+1N−1,ϱn+1N,⋯,ϱn+2N−2,ϱnN,ϱn+11,⋯,ϱn+1N−2,ϱnN−1,t>⋯>MNϱn+11,ϱn+12,⋯,ϱn+1N,ϱn2,ϱn3,⋯,ϱnN,ϱn1,t.
In the light of inequality (6), we observe that the behavior of mutual sequence in the proof of Theorem 14 and mutual sequence as above is alike. The proof of sequence ϱm1,ϱm2,⋯,ϱmN to be a Cauchy mutual sequence is immediate from Theorem 14.
From (i) and Lemma12, there exist ϱi∈TiYi such that ϱm1,ϱm2,⋯,ϱmN converges to ϱ1,ϱ2,⋯,ϱN, and from Lemma 13, sequence ϱm1,ϱm2,⋯,ϱmN is coconvergent to some point ϱ∈X.
Now, we have to prove thatϱ is a common fixed point of Ti’s. Without loss of generality, let us assume that ϱn1,ϱn2,⋯,ϱnN≠ϱ,ϱ,⋯,ϱ and
(34)ϱn1,ϱn2,⋯,ϱnN≠T2ϱ,T3ϱ,⋯,TNϱ,T1ϱ,foralln∈N.
So, there existst1>0 such that MNϱ,ϱ,⋯,ϱ,T2ϱ,T3ϱ,⋯,TNϱ,T1ϱ,t1<1,
(35)MNϱn1,ϱn2,⋯,ϱnN,ϱ,ϱ,⋯,ϱ,t1<1,(36)MNT1ϱ,T2ϱ,⋯,TNϱ,T2ϱn+12,T3ϱn+13,⋯,TNϱn+1N,T1ϱn1,t1=MNT1ϱ,T2ϱ,⋯,TNϱ,ϱn+11,ϱn+12,⋯,ϱn+1N,t1<1,for all n∈ℕ. Then, we have
(37)MNϱ,ϱ,⋯,ϱ,ϱn+12,⋯,ϱn+1N,ϱn1,t1<ζMNT1ϱ,T2ϱ,⋯,TNϱ,T2ϱn+12,T3ϱn+13,⋯,TNϱn+1N,T1ϱn1,t1,MNϱ,ϱ,⋯,ϱ,ϱn+12,ϱn+13,⋯,ϱn+1N,ϱn1,t1=MNT1ϱ,T2ϱ,⋯,TNϱ,T2ϱn+12,T3ϱn+13,⋯,TNϱn+1N,T1ϱn1,t1=MNT1ϱ,T2ϱ,⋯,TNϱ,ϱn+11,ϱn+12,⋯,ϱn+1N,t1,as n⟶∞, and from (5), we get MNT1ϱ,T2ϱ,⋯TNϱ,ϱ,ϱ,⋯,ϱ,t1≥1, a contradiction. Hence, MNT1ϱ,T2ϱ,⋯TNϱ,ϱ,ϱ,⋯,ϱ,t=1, for all t>0. Therefore, ϱ is the common fixed point of Ti′s, for all i∈1,2,⋯N.
The proof of uniqueness of common fixed point runs similar to the proof of Theorem14. Hence, we are through.Example 4.
LetX=0,1 and X,M,∗ be a fuzzy metric space where
(38)Mϱ,ρ,t=1,ifϱ=ρminϱ,ρ,otherwiseforallϱ,ρ∈X,t>0,
∗ is continuous t-norm defined as a∗b=mina,b. Let
(39)Y1=11000,1600,1200,170,1,Y2=1900,1500,1100,150,1,Y3=1800,1400,190,130,1,Y4=1700,1300,180,120,1,be subset of X; we define T1:Y1⟶Y2,T2:Y2⟶Y3,T3:Y3⟶Y4 and T4:Y4⟶Y1 such that
(40)T11=T21=T31=T41=1,T111000=1900,T11600=1500,T11200=1100,T1170=150,T21900=1800,T21500=1400,T21100=190,T2150=130,T31800=1700,T31400=1300,T3190=180,T3130=120,T41700=1600,T41300=1200,T4180=170,T4120=1.
Now, define functionζ similar to Example 3. We can easily verify that all conditions of Theorem 16 are satisfied. If ϱ11=1/1000 and T1ϱn1=ϱn2, T2ϱn2=ϱn3,..., TN−1ϱnN−1=ϱnN and TNϱnN=ϱn+11 for all n∈N. We get 1/1000,1/900,1/800,1/700,1/600,1/500,1/400,1/300,⋯ as mutual sequence, 1 as unique common fixed point of T1,T2,T3, and T4.


## 4. Conclusion

In this paper, the concept of mutual sequences inX,M,∗ is given, and with the help of induced fuzzy metric structure XN,MN,∗, we define Cauchy mutual sequences in simple fuzzy metric structure X,M,∗. For brevity, Definitions 2.6 and 2.4 (presented in [20]) are unified as Definition 7. We also present ℤ∗ contraction, which is an extension of ℤ-contraction for finite number of mappings. With the help of mutual sequences, we proved unique common fixed point theorems for finite numbers of mappings using ℤ∗ contraction. We also provide many examples to show that our results are meaningful and to support our theorems. The given results generalize and extend several results in the existing literature. As perspectives, it would be interesting that the results presented in this paper proved for other contractive conditions and extend to other nonclassical metric structure, like bipolar fuzzy metric spaces [5] and relational fuzzy metric spaces [25].



---

*Source: 1550332-2022-08-17.xml*