# n-Tupled Common Fixed Point Result in Fuzzy b-Metric Spaces

Authors: Badshah-E. Rome; Muhammad Sarwar; Fahd Jarad
Journal: Journal of Function Spaces
 (2022)

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## 1. Introduction

One of the several important settings in which fixed point theory has been explored is the fuzzy context [1, 2]. The introduction of the fuzzy set by Zadeh [3] was a turning point in the landscape of fuzzy mathematics. Fuzzy mathematics has improved enormously in the last two decades. Fuzzy set theory has many important applications in various fields of applied sciences such as neural network theory, stability theory, mathematical programming, modeling theory, engineering sciences, medical sciences (medical genetics and nervous system), image processing, control theory, and communication [4–9]. Kramosil [10] introduced the notion of fuzzy metric space. George and Veeramani [11] later on slightly modified Kramosil’s definition of fuzzy metric space and also proved that every metric induces a fuzzy metric and every fuzzy metric induces Hausdorff topology. Sedghi and Shobe [12] introduced the concept of fuzzy b-metric space. Following this idea, many researchers analysed fixed point theory in fuzzy b-metric space via various contractive conditions [13–17].Sessa [18] explored common fixed points of set-valued as well as single-valued mappings on a complete metric space under a contractive condition along with the commutativity concept. The concept of a common fixed point was later on generalized by many researchers [19–23]. Lakshmikantham and Bhaskar [24] initiated the concept of coupled fixed point. Ćirić and Lakshmikantham [25] established coupled coincidence and coupled common fixed point theorems in partially ordered metric spaces. Subsequently, many researchers explored coupled fixed point theory in various spaces [2, 12, 19, 25–28]. Hu [27] and Zhu and Xiao [2] gave a coupled fixed point theorem for contractions in fuzzy metric spaces. Vasuki [29] obtained common fixed point results in fuzzy metric spaces. Berinde and Borcut [30] presented the idea of a tripled fixed point and obtained some new tripled fixed point results using mixed g-monotone mapping. Their results generalize and extend the Bhaskar and Lakshmikantham’s research for nonlinear mappings. Roldán et al. [31] investigated the multidimensional coincidence points between mappings. Roldán et al. [1] modified the concept of tripled fixed points and generalized the results of Berinde and Borcut [30] and Zhu and Xiao [2].In this paper, we aim to generalize and extend the notion of coupled/tripled common fixed point by introducing the concept ofn-tupled fixed point in fuzzy b-metric space. We established an existence and uniqueness theorem for contractive mapping in fuzzy b-metric space. Our main result generalizes and extends coupled and tripled fixed point theorems appearing in [1, 2, 13] to n-dimensional common fixed points in fuzzy b-metric space. Moreover, it can be particularized to complete metric spaces to obtain an n-tupled Brinde–Borcut type coincidence/fixed point result in a nonfuzzy domain.The paper is organized as follows: Section2 is devoted to recall the basic definitions and lemmas that will be crucial throughout the paper. In Section 3, we introduce the notions of an n-tupled common fixed point and an n-tupled coincidence point. Moreover, an existence and uniqueness theorem for mappings satisfying certain contractive conditions in fuzzy b-metric space has been proved. Section 4 is devoted to generalize the construction presented in Section 3 to nonfuzzy settings. Moreover, the idea is elaborated via a nontrivial example. In section 5, some applications of the main result of the paper are discussed, and a kind of Lipschitzian system and an integral system both for n variables have been solved.


## 2. Preliminaries

In this section, some terms and definitions are provided which will be used in the main work of this manuscript. Henceforth,ℝ and ℕ will denote the set of real numbers and positive integers, respectively, while S will stand for an arbitrary nonempty set. Arguments of a metric d and fuzzy metric T will be represented by subscripts. For example, du,v and Tx,y,δ will be represented by duv and Txyδ, respectively.Definition 1 (see [32]).
A map∗:0,12⟶0,1, such that 0,1,≤,∗ is an ordered abelian topological monoid with unit 1, is called a continuous t-norm.
x∗Ly=maxx+y−1,0, x∗Py=xy, and x∗My=minx,y are examples of some frequently used continuous t-norm that satisfy ∗M≥∗P≥∗L.Definition 2 (see [33]).
A continuoust-norm ∗ is said to be of H-type if the sequence ∗mxm=1∞ is equicontinuous at x=1. That is, for all ξ∈0,1 there exists ς∈0,1 such that 1−ς<x≤1 implies that ∗mx>1−ξ for all m≥1, where the sequence ∗mx is defined as ∗1x=x and ∗nx=∗n−1x∗x.
An important and most commonly used continuoust-norm of H-type is ∗M which satisfies x∗My≥x∗Py,∀x,y∈0,1. The following lemma characterizes continuous t-norm to be of the H-type.Lemma 1 (see [1]).
Let∗ be a t-norm and ϵ∈0,1. If(1)a∗εb=a∗b;ifmaxa,b≤1−ε,mina,b;ifmaxa,b>1−ε.
Then,∗ϵ is a t-norm of theH-type.Definition 3 (see [34]).
The 3-tupleS,T,∗ is called fuzzy metric space if S is a nonempty set, ∗ is a continuous t-norm, and F is a fuzzy set on S×S×0,∞ which satisfies the following conditions, for all u,v,w∈S and δ,ρ>0.(i)
FM1Tuvδ>0(ii)
FM2Tuvδ=1,iffu=v(iii)
FM3Tuvδ=Tvuδ(iv)
FM4Tuwδ+ρ≥Tuvδ∗Tvwρ(v)
FM5Tuv.:0,∞⟶0,1 is continuous(vi)
FM6limδ⟶∞Tuvδ=1Definition 4 (see [12]).
The 3-tupleS,T,∗ is called fuzzy b-metric space (F b MS for short) if S is a nonempty set, ∗ is a continuous t-norm, and F is a fuzzy set on S×S×0,∞ which satisfies the following conditions, for all u,v,w∈S and δ,ρ>0 and a given real number b≥1.(i)
FM1Tuvδ>0(ii)
FM2Tuvδ=1,iffu=v(iii)
FM3Tuvδ=Tvuδ(iv)
FM4Tuwδ+ρ≥Tuvδ∗Tvwρ(v)
FM5Tuv.:0,∞⟶0,1 is continuous(vi)
FM6limδ⟶∞Tuvδ=1Note thatTuvδ represents the degree of closeness between u and v with respect to δ>0. The fuzzy b-metric reduces to a fuzzy metric for b=1. Therefore, the class of fuzzy b-metric spaces is larger than the class of fuzzy metric spaces. The following example shows that a fuzzy b-metric on a nonempty set S need not be a fuzzy metric.Example 1 (see [13]).
LetS=ℝ and Tuvδ=e−u−vp/δ, for allu,v∈S and δ>0 with r∗s=rs. It can be easily verified that S,T,∗ is a F b MS with b=2p−1. But for p=2, S,T,∗ is not a fuzzy metric space.Remark 1 ( see [14]).
Foru≠vandδ>0, it is always true that 0<Tuvδ<1.Lemma 2 (see [35]).
Tuv. is nondecreasing for all u,v∈S.Definition 5(see [36, 37].
In a fuzzyb-metric space S,T,∗:(1)
A sequenceun converges to u∈S if for every ξ∈0,1 and δ>0 there exists nξ∈ℕ such that Tun,u,δ>1−ξ,∀n≥nξ(2)
unn∈ℕ is said to be Cauchy sequence if for every positive real number ξ∈0,1 and δ>0 there exists nξ∈ℕ such that Tun,um,δ>1−ξ,∀m,n≥nξ(3)
A Fb MS is said to be complete if every Cauchy sequence converges in itRemark 2 (see [14]).
In general, a fuzzy b-metric is not continuous.
In a fuzzyb-metric space, we have the following proposition.Proposition 1 (see [37]).
LetS,T,∗ be a F b MS and suppose a sequence un converges to u, then(2)Tu,v,δb≤limsupn⟶∞Tun,v,δ≤Tu,v,bδ,Tu,v,δb≤liminfn⟶∞Tun,v,δ≤Tu,v,bδ.


## 3. Main Results

Definition 6.
LetT:Sn⟶S and θ:S⟶S, a point 1u,2u,3u,…,nu∈Sn is said to be the following:(1)
n-tupled fixed point of T if T1u,2u,3u,…,nu=1u,T2u,3u,…,nu,1u=2u,…,Tnu,1u,2u,…,n−1u=nu(2)
n-tupled coincidence point of T and θ if T1u,2u,3u,…,nu=θ1u,T2u,3u,,…,nu,1u=θ2u,…,Tnu,1u,2u,…,n−1u=θnu(3)
n-tupled common fixed point of T and θ if T1u,2u,3u,…,nu=θ1u,T2u,3u,,…,nu,1u=θ2u=2u,…,Tnu,1u,2u,…,n−1u=θnu=nuDefinition 7.
LetT:Sn⟶S and θ:S⟶S be two mappings. Then, T and θ are said to be commuting if θT1u,2u,3u,…,nu=Tθ1u,θ2u,θ3u,…,θnu.
In our proof of main result, we will use the following lemmas.Lemma 3.
LetT:Sn⟶S and θ:S⟶S be mappings on a F b MS S, such that T≡u is constant and θ is continuous and commuting with T. Then, u,u,u,…,u is a unique n-tupled common fixed point of T and θ.Proof.
AsT is constant on Sn, therefore, there exists u∈S such that T1u,2u,3u,…,nu=u for all 1u,2u,3u,…,nu∈S. Then, from T and θ being commuting, it can be deduced that(3)θu=θT1u,2u,3u,…,nu=Tθ1u,θ2u,θ3u,…,θnu=u.
Therefore,u=θu=Tu,u,u…,u. That is, u,u,u…,u is an n-tupled common fixed point of T and θ. Let 1v,2v,3v,…,nv be another n-tupled common fixed point of T and θ such that u≠kv for k=1,2,…,n. Then,(4)1>Tu,kvδ=TTu,u,u,…,uTkv,k+1v,k+2v,…,n+1−kvδ=Tu,uδ=1.Which is a contradiction. Therefore, u,u,u…,u is a unique n-tupled common fixed point of T and θ.
LetB denote the class of all increasing and continuous functions β:0,1⟶0,1 such that βδ>δ for all δ∈0,1 with β0=0 and β1=1. Then, the following lemma is used.Lemma 4.
LetS,T,∗ be a complete F b MS with b≥1, where a∗b=mina,b for all a,b∈0,1. Let T:Sn⟶S and θ:S⟶S be such that(5)TT1u,2u,3u,…,nuT1v,2v,3v,…,nvδ≥βTθ1u,θ1vbnδ∗Tθ2u,θ2vδ∗Tθ3u,θ3vbnδ∗…∗Tθnu,θnvbnδ. For some β∈B, for all 1u,2u,3u,…,nu,1v,2v,3v,…,nv∈S and δ>0. Suppose 1u,2u,3u,…,nu is an n-tupled coincidence point of T and θ. Then,(6)T1u,2u,3u,…,nuδ=θ1u=θ2u=T2u,3u,…,nu,1uδ=θ3u=T3u,4u,…,nu,1u,2uδ=θ4u=…=θnu=Tnu,1u,2u,…,n−1uδ.Proof.
Suppose on the contrary that there are at least two distinct integersp and q in 1,2,…,n such that θpu≠θqu. Let Tθpu,θqubnδ∗Tθp+1u,θq+1ubnδ∗…∗Tθn+1−pu,θn+1−quδ=Tθiu,θ4ubnδ.. Then,(7)Tθiu,θjuδ=TTiu,i+1u,i+2u,…,n+1−iu,Tju,j+1u,j+2u,…,n+1−juδ≥βTθiu,θjubnδ∗Tθi+1u,θj+1ubnδ∗…∗Tθn+1−iu,θn+1−jubnδ≥βTθju,θjubnδ>Tθju,θjubnδ≥Tθju,θjubδ,
which is a contradiction. Hence,(8)T1u,2u,3u,…,nuδ=θ1u=θ2u=T2u,3u,…,nu,1uδ=θ3u=T3u,4u,…,nu,1u,2uδ=θ4u=…=θnu=Tnu,1u,2u,…,n−1uδ.Theorem 1.
LetS,T,∗ be a complete F b MS, where a∗b=mina,b for all a,b∈0,1. Let T:Sn⟶S and θ:S⟶S be such that TSn⊆θS and θ is continuous and commuting with T. Suppose for all 1u,2u,3u,…,nu,1v,2v,3v,…,nv∈S and δ>0,(9)TT1u,2u,3u,…,nuT1v,2v,3v,…,nvδ≥βTθ1u,θ1vbnδ∗Tθ2u,θ2vδ∗Tθ3u,θ3vbnδ∗…∗Tθnu,θnvbnδ,where β∈B, then T and θ have a unique n-tupled common fixed point.Proof.
IfT is constant, then the proof of the theorem follows from Lemma 2. Suppose T is not constant on Sn. In this case, the proof is divided into three steps.Step 1.
Definition of the sequencesθ1um, θ2um, θ3um,…,θnum.
Let1uo,2uo,3uo,…,nuo be arbitrary points of S. As TSn⊆θS, therefore, there exist 1u1,2u1,3u1,…,nu1∈S such that θ1u1=T1uo,2uo,3uo,…,nuo,θ2u1=T2uo,3uo,…,nuo,1uo\\,θ3u1=T3uo,4uo,…,nuo,1uo,2uo and θnu1,=Tnuo,1uo,2uo,…,n−1uo.
Again, asTSn⊆θS, therefore, there exists 1u2,2u2,3u2,…,nu2∈S such that(10)θ1u2=T1u1,2u1,3u1,…,nu1,θ2u2=T2u1,3u1,…,nu1,1u1,θ3u2=T3u1,4u1,…,nu1,1uo,2u1,…,θnu2=Tnu1,1u1,2u1,…,n−1u1.
Continuing in the same way, sequencesθ1um, θ2um, θ3um,…,θnum can be constructed such that(11)θ1um+1=T1um,2um,3um,…,num,θ2um+1=T2um,3um,…,num,1um,θ3um+1=T3um,4um,…,num,1um,2um,⋮θnum+1=Tnum,1um,2um,…,n−1um,wherem∈N∪0.Step 2.
θ1um, θ2um, θ3um,…,θnum are Cauchy sequences.(12)Letλmδ=Tθ1umθ1um+1δ∗Tθ2umθ2um+1δ∗…∗Tθnumθnum+1δ;∀δ>0.
Using (9), for all δ>0, we have(13)Tθ1umθ1um+1δ=TT1um−1,2um−1,…,num−1T1um,2um,…,numδ,≥βTθ1um−1θ1umbnδ∗Tθ2um−1θ2umbnδ∗…∗Tθnum−1θnumbnδ,=βλm−1bnδ>λm−1bnδ,Tθ2umθ2um+1δ=TT2um−1,3um−1,…,num−1,1um−1T2um,3um,…,num,1umδ,≥βTθ2um−1θ2umbnδ∗Tθ3um−1θ3umbnδ∗…∗Tθnum−1θnumbnδ∗Tθ1um−1θ1umbnδ,=βλm−1bnδ>λm−1bnδ,⋮Tθ2umθ2um+1δ=TTnum−1,1um−1,2um−1,…,n−1um−1Tnum,1um,2um,…,n−1umδ,≥βTθnum−1θnumbnδ∗Tθ1um−1θ1umbnδ∗…∗Tθn−1um−1θn−1umbnδ,=βλm−1bnδ>λm−1bnδ,
(13) implies that for all δ>0 and m≥0,(14)λmδ≥βλm−1bnδ.
Obviously,(15)λmδ≥βλm−1bnδ>λm−1bnδ≥λm−1δ.
It meansλnδ is an increasing sequence in 0,1, and therefore, limsupn⟶∞λnδ=ℓδ≤1 for all δ>0. If ℓδ<1, then by letting n⟶∞ in 6, we get a contradiction ℓδ>ℓδ. Therefore, ℓδ=1, that is, for all m,δ≥0,(16)limsupn⟶∞T1umθ1um+1δ∗T2umθ2um+1δ∗…∗Tnumθnum+1δ=1.
To show that the sequencesθ1um, θ2um, θ3um,…,θnum, where m=1,2,3,⋯, are Cauchy, first we prove that for every ϵ∈0,1 there exist p,q∈N such that \\Tθ1upθ1uqδ∗Tθ2upθ2uqδ∗…∗Tθnupθnuqδ>1−ϵ. Suppose it is not true. Then, there exists some ϵ∈0,1 such that for each r∈N, there exist integers pr and qr with pr>qr≥r such that(17)Tθ1uprθ1uqrδ∗Tθ2uprθ2uqrδ∗…∗Tθnuprθnuqrδ≤1−ϵ.
Letpr be the least such positive integer which exceeds qr and satisfies (17). Then,(18)Tθ1uprθ1uqrδ∗Tθ2uprθ2uqrδ∗…∗Tθnuprθnuqrδ>1−ϵ.
Letmrδ=Tθ1uprθ1uqrδ∗Tθ2uprθ2uqrδ∗…∗Tθnuprθnuqrδ. Using (18) and (19) and properties FbM3 and FbM4, we have(19)1−ϵ≥mrδ≥Tθ1uqrθ1upr−1δ2b∗Tθ1upr−1θ1uprδ2b∗Tθ2uqrθ2upr−1δ2b∗Tθ2upr−1θ2uprδ2b∗…∗Tθnuqrθnupr−1δ2b∗Tθnupr−1θnuprδ2b>1−ϵ∗λrδ2b.
Lettingr⟶∞ we get the contradiction(20)1−ε≥limsupr⟶∞mrδ>1−ε.
Hence,θ1um, θ2um, θ3um,…,θnum are Cauchy sequences.Step 3.
T and θ have an n-tupled common fixed point.
AsS is complete, so there will be some 1u,2u,3u,…,nu∈S such that limm⟶∞θ1um=1u,limm⟶∞θ2um=2u,…,limm⟶∞θnum=nu. Due to continuity of θ, limm⟶∞θθ1um=θ1u,limm⟶∞θθ2um=θ2u,…,limn⟶∞θθnum=θnu. Also, θ commutes with T, therefore, θθ1um+1=θT1um,2um,…,num=Tθ1um,θ2um,…,θnum.
Using (9), we have(21)Tθθ1um+1T1u,2u,3u,…,nuδ==TTθ1um,θ2um,…,θnumT1u,2u,3u,…,nuδ≥βTθθ1um,θ1ubnδ∗Tθθ2um,θ2ubnδ∗Tθθ3um,θ3ubnδ∗…∗Tθθnum,θnubnδ.
Using Proposition1 and (16), we have(22)Tθ1uT1u,2u,3u,…,nuδ≥limsupm⟶∞TTθ1um,θ2um,…,θnumT1u,2u,3u,…,nuδb≥limsupm⟶∞βTθθ1um,θ1ubn−1δ∗Tθθ2um,θ2ubn−1δ∗…∗Tθθnum,θnubn−1δ≥Tθ1u,θ1ubn−1δ∗Tθ2u,θ2ubn−1δ∗Tθ3u,θ3ubn−1δ∗…∗Tθnu,θnubn−1δ=1.
Hence,T1u,2u,3u,…,nuδ=θ1u. Similarly, it can be shown that(23)T2u,3u,…,nu,1uδ=θ2u,T3u,4u,…,nu,1u,2uδ=θ3u,⋮Tnu,1u,2u,…,n−1u=θnu.
Lemma4 implies that(24)T1u,2u,3u,…,nu=θ1u=θ2u=⋯=θnu=Tnu,1u,2u,…,n−1u.
From (9) and (24) along with Proposition 1 and continuity of β, it comes out that(25)T1u,θ1ubδ≥limsupm⟶∞Tθ1um+1,θ1uδ,≥limsupm⟶∞TT1um,2um,…,num,T1u,2u,…,nuδ,≥limsupm⟶∞βTθ1um,θ1ubnδ∗Tθ2um,θ2ubnδ∗…∗Tθnum,θnubnδ,≥limsupm⟶∞Tθ1um,θ1ubnδ∗Tθ2um,θ2ubnδ∗…∗Tθnum,θnubnδ,applying Proposition 1 again, we get(26)T1u,θ1ubδ≥limsupm⟶∞Tθ1u,θ1ubn−1δ∗Tθ2u,θ2ubn−1δ∗Tθ3u,θ3ubn−1δ∗…∗Tθnu,θnubn−1δ=1.
Therefore,θ1u=1u. Similarly, it can be shown that(27)θu2=u2,θu3=u3,…,θun=un.
From (24), it follows that(28)T1u,1u,1u,…,1u=θ1u=1u.
To show uniqueness, letv∈S be another n-tupled common fixed point of T and θ such that v≠1u. Then,(29)T1u,vδ=TT1u,1u,1u,…,1u,Tv,v,…,vδ≥βTθ1u,θvbnδ∗Tθ1u,θvbnδ∗⋯∗Tθ1u,θvbnδ,=βTθ1u,θvbnδ>Tθ1u,θvbnδ=T1u,vbnδ≥T1u,vδ,
which is a contradiction. Thus,T and θ have a unique n-tupled common fixed point. □
Takingn=2 in the abovementioned theorem, we get the following corollary, which is the main result of [13].Corollary 1.
LetS,T,∗ be a complete F b MS, where a∗b=mina,b for all a,b∈0,1. Let T:S2⟶S and θ:S⟶S be such that TS2⊆θS and θ is continuous and commuting with T. Suppose for all 1u,2u,3u,4u∈S and δ>0,(30)TT1u,2uT3u,4uδ≥βTθ1u,θ3ub2δ∗Tθ2u,θ4ub2δ,where β∈B, then there exists a unique u∈S such that u=θu=Tu,u.Corollary 2.
LetS,T,∗ be a complete F b MS, where a∗b=mina,b for all a,b∈0,1. Let T:Sn⟶S and θ:S⟶S be such that TSn⊆θS and θ is continuous and commuting with T. Suppose for all 1u,2u,3u,…,nu,1v,2v,3v,…,nv∈S and δ>0,(31)TT1u,2u,3u,…,nuT1v,2v,3v,…,nvδ≥Tθ1u,θ1vb2δ∗Tθ2u,θ2vb2δ∗…∗Tθnu,θnvb2δα,where β∈B, and α∈0,1, then T and θ have a unique n-tupled common fixed point.Proof.
Proof follows from Theorem1, by setting βt=tα.Corollary 3.
LetS,T,∗ be a complete F b MS, where a∗b=mina,b for all a,b∈0,1. Let T:Sn⟶S and θ:S⟶S be such that TSn⊆θS and θ is continuous and commuting with T. Suppose for all 1u,2u,3u,…,nu,1v,2v,3v,…,nv∈S and δ>0,(32)TT1u,2u,3u,…,nuT1v,2v,3v,…,nvδ≥2Tθ1u,θ1vb2δ∗Tθ2u,θ2vb2δ∗…∗Tθnu,θnvb2δα−Tθ1u,θ1vb2δ∗Tθ2u,θ2vb2δ∗…∗Tθnu,θnvb2δ2,where β∈B, then T and θ have a unique n-tupled common fixed point.Proof.
It is adequate to setβt=2t−t2 in Theorem 1,Remark 3.
Coincidence point ofT and θ is not necessarily unique. For example, if T≡c≡θ, where c∈S is constant, then every point 1u,2u,3u,…,nu∈Sn is coincidence point of T and θ.Example 2.
LetS=ℝ and Tx,y,δ=e−x−y/δfor allx,y∈S and δ>0 with r∗s=minr,s where r,s∈0,1. It can be easily verified that S,T,∗ is a complete F b MS with b=2. Let α,λ,γ>0 be such that maxα,γ≤λ/128 . Define T:S4⟶S,θ:S⟶S as Tx,y,z,w=αx−z+γy−w, θu=λu and βδ=δ for all u,x,y,z,w∈S and δ>0. Obviously θ is continuous, T and θ are commuting and TS4=ℝ=θS. Also,(33)TTxyzwTmnpqδ=eαx−m−αz−p+γy−n−γw−q−1/δ,≥eαx−m+αz−p+γy−n+γw−q−1/δ≥ex−m/δ+z−p/δ+y−n/δ+w−q/δ−4maxα,γ,≥ex−m/δ+z−p/δ+y−n/δ+w−q/δ−λ/128,≥emaxx−m/δ+z−p/δ+y−n/δ+w−q/δ−4λ/128≥mine−λx−λm/16δ,e−λz−λp/16δ,e−λy−λn/16δ,e−λw−λq/16δ=Tθxθm24δ∗Tθyθn24δ∗Tθzθp24δ∗Tθwθq24δ.
That is, all the conditions of Theorem1 are fulfilled. Therefore, 0,0,0,0 is a unique quadrupled common fixed point of T and θ.


## 4. Consequences

It is known that ifS,T,∗ is a F b MS and ⋄ is a continuous t-norm such that u∗v≥u⋄v,∀u,v∈0,1, then S,T,⋄ is F b MS. Since for any t-norm ∗, it is always true that ∗M≥∗, therefore, S,T,∗M is F b MS implies S,T,∗ is F b MS. As Fb MS is a generalization of b-metric space, therefore, from a given b-metric space, a F b MS can be considered in different ways.Example 3 (see [1]).
LetS,d be a b-metric space. For δ>0 and u≠v, the following is defined:Tuvdδ=δ/δ+duvTuveδ=e−duv/δTuvcδ=0,ifδ≤duv;1,ifδ>duv..
It is known that each ofS,Td,∗P, S,Td,∗M, S,Te,∗M, and S,Tc,∗M is a F b MS.
Moreover, the completeness ofS,d implies the completeness of any one of these F b MSs and vice versa. With this approach, many important results for b-metric space can be deduced from the corresponding results in a fuzzy setting. In the following theorem the b-metric space S,d is viewed as the crisp F b MS S,Tc,∗M.Theorem 2.
LetS,d be a complete b-metric space, T:Sn⟶S and θ:S⟶S be given mappings such that TSn⊆θS and θ is continuous and commuting with T. Suppose T and θ satisfy some of the following conditions for all 1u,2u,3u,…,nu,1v,2v,3v,…,nv∈S:(34)dT1u,2u,3u,…,nuT1v,2v,3v,…,nv≤κmaxdθ1uθ1v,dθ2uθ21v,…,dθnuθnvθnvfor someκ∈0,1,dT1u,2u,3u,…,nuT1v,2v,3v,…,nv≤κα1dθ1uθ1v+α2dθ2uθ21v+⋯+αndθnuθnvfor someκ∈0,1and someα1,α2⋯αn∈0,1nwheren∈ℕ.dT1u,2u,3u,…,nuT1v,2v,3v,…,nv≤α1dθ1uθ1v+α2dθ2uθ21v+⋯+αndθnuθnvforα1,α2⋯αn∈0,1such that∑i=1nαi<1.Then, there exists a uniqueu∈S such that u=θu=Tu,u,u,…,u.Proof.
ConsiderTc as defined in example 3. As S,d is complete, S,Tuvc,∗M is complete. We prove (2), for α1=α2=α3=⋯=αn=1/n, δ>0 and ∗=∗M. In case any one of Tθ1uθ1vc, Tθ2uθ21vc,…,Tθnuθnvc is 0, then (2) obvious. Suppose(35)Tθ1uθ1vcδ=Tθ2uθ2vcδ=⋯=Tθnuθnvcδ=1,⇒dθ1uθ1v<δ,dθ2uθ2v<δ,…,dθnuθnv<δ.⇒δ>maxdθ1uθ1v,dθ2uθ2v,…,dθnuθnv.
Now (a)(36)κδ>κmaxdθ1uθ1v,dθ2uθ2v,…,dθnuθnv≥dT1u,2u,3u,…,nuT1v,2v,3v,…,nv.⇒TT1u,2u,3u,…,nuT1v,2v,3v,…,nvcκδ=1.
Hence, (2) is satisfied.
(b) As(37)dT1u,2u,3u,…,nuT1v,2v,3v,…,nv≤κα1dθ1uθ1v+α2dθ2uθ2v+⋯+αndθnuθnv≤κ1ndθ1uθ1v+1ndθ2uθ2v+⋯+1ndθnuθnv,=κndθ1uθ1v+dθ2uθ2v+⋯+dθnuθnv≤κnnmaxdθ1uθ1v,dθ2uθ2v,…,dθnuθnv,=κmaxdθ1uθ1v,dθ2uθ2v,…,dθnuθnv≤κδ,⇒TT1u,2u,3u,…,nuT1v,2v,3v,…,nvcκδ=1.
Hence, (2) is true.
(c) Letκ=∑i=1nαi<1. Then,(38)dT1u,2u,3u,…,nuT1v,2v,3v,…,nv≤α1dθ1uθ1v+α2dθ2uθ2v+⋯+αndθnuθnv≤α1maxdθ1uθ1v,dθ2uθ2v,…,dθnuθnv+α2maxdθ1uθ1v,dθ2uθ2v,…,dθnuθnv+α3maxdθ1uθ1v,dθ2uθ2v,…,dθnuθnv⋮+αnmaxdθ1uθ1v,dθ2uθ2v,…,dθnuθnv=∑i=1nαimaxdθ1uθ1v,dθ2uθ2v,…,dθnuθnv=κmaxdθ1uθ1v,dθ2uθ2v,…,dθnuθnv≤κδ=κmaxdθ1uθ1v,dθ2uθ2v,…,dθnuθnv≤κδ,⇒TT1u,2u,3u,…,nuT1v,2v,3v,…,nvcκδ=1.
Hence, (2) is true.Corollary 4 (see [1], Theorem 15).
Let S,d be a complete metric space and T:S3⟶S and θ:S⟶S be such that TS3⊆θS and θ is continuous and commuting with T. Suppose T and θ satisfy some of the following conditions for all \\1u,2u,3u,1v,2v,3v∈S:(39)dT1u,2u,3u,…,nuT1v,2v,3v≤κmaxdθ1uθ1v+dθ2uθ2v+⋯+dθ3uθ3vfor someκ∈0,1,dT1u,2u,3uT1v,2v,3v≤κα1dθ1uθ1v+α2dθ2uθ2v+α3dθ3uθ3vfor someα1,α2,α3∈0,13andκ∈0,1.dT1u,2u,3uT1v,2v,3v≤α1dθ1uθ1v+α2dθ2uθ2v+α3dθ3uθ3vforα1,α2,α3∈0,1such that∑i=13αi<1.
Then, there exists a uniqueu∈S such that u=θu=Tu,u,u.Proof.
Proof is similar to that of Theorem2.Example 4.
LetS=ℝ, and duv=u−v for all u,v∈ℝ. Let T:Sn⟶S and θ:S⟶S be defined as T1u,2u,3u,…,nu=∑i=1nηiiu+ϵ/K and θu=u for all 1u,2u,3u,…,nu,η1,η2,…,ηn,u,ϵ,K∈ℝ. Then,(40)dT1u,2u,3u,…,nuT1v,2v,3v,…,nv=1K∑i=1nηiiu−iv,≤∑i=1nηiKiu−iv=∑i=1nαiiu−iv,=∑i=1nαiθiu−θiv=∑i=1nαidθiu−θiv.
That is, the condition of Theorem2 (c) is fulfilled.
Moreover, forn-tupple 0u,0u,…,0u, where 0u=ϵ/K−∑i=1nηi, we have(41)T0u,0u,…,0u=∑i=1nηi0u+ϵK=∑i=1nηiϵ/K−∑i=1nηi+ϵK,=ϵ∑i=1nηi+ϵK−∑i=1nηiKK−∑i=1nηi,=ϵK−∑i=1nηi=0u=θ0u.
Hence,0u,0u,…,0u is a unique n-tupled common fixed point of T and θ.


## 5. Applications

### 5.1. Lipschitzian Systems

Letg1,g2,…,gn:ℝ⟶ℝ be Lipschitzian mappings on ℝ (equipped with the Euclidean metric) with Lipschitz constants c1,c2,c3,…,cn, respectively, that is, for each gi where 1≤i≤n, there exists a corresponding real number ci where 1≤i≤n such that giu−giv≤ciu−v for all u,v∈ℝ. Define f:ℝ⟶ℝ as fu=μ1g1u+μ2g2u+⋯+μngnu for all u∈ℝ, where μi∈ℝ for 1≤i≤n. Then,(42)fu−fv=∑i=1nμigiiu−giiv,≤∑i=1nμiciu−v.That is,f is itself a Lipschitzian mapping with cf=∑i=1nμici. If cf=∑i=1nμici<1, then f is contraction and by Banach contraction principle there is a unique uo∈ℝ such that fuo=uo. Define T:Sn⟶S by(43)T1u,2u,3u,…,nu=μ1g11u+μ2g22u+⋯+μngnnu,for all1u,2u,3u,…,nu∈ℝ.Obviously,Tu,u,…,u︷n=fu for all u∈ℝ. Moreover,(44)T1u,2u,3u,…,nu−1v,2v,3v,…,nv=∑i=1nμigiiu−giiv,≤∑i=1nμiciu−v,=∑i=1ncfiu−iv,≤cfmax1≤i≤niu−iv.Ifcf<1, then by Theorem 2 (a), there exists a unique u∈ℝ such that(45)u=fu=Tu,u,…,u︷n.Corollary 5.
Letg1,g2,…,gn:ℝ⟶ℝ be Lipschitzian mappings furnished with the Euclidean metric, and μ1,μ2,…,μn∈ℝ be such that C=∑i=1nμici<1. Then, the system(46)H=μ1g11u+μ2g22u+⋯+μn−1gn−1n−1u+μngnnu=1u,μ1g12u+μ2g23u+⋯+μn−1gn−1nu+μngn1u=2u,μ1g13u+μ2g24u+⋯+μn−1gn−11u+μngn2u=3u,⋮μ1g1nu+μ2g21u+⋯+μn−1gn−1n−2u+μngnn−1u=nu,has a unique solution ou,ou,…,ou︷n, where ou is a unique real solution of ∑i=1nμigiu=u.Example 5.
Consider the following:(47)J=30sinx−281+y2+z+cosw2+150=72x−15tan−1t,30siny−281+z2+w+cost2+150=72y−15tan−1x,30sinz−281+w2+t+cosx2+150=72z−15tan−1y,30sinw−281+t2+x+cosy2+150=72w−15tan−1z,30sint−281+x2+y+cosz2+150=72t−15tan−1w.
Letg1u=5+sinu,g2u=1/1+u2,g3u=u,g4u=cosu2andg5u=tan−1u. g1,g2,g3,g4 and g5 are Lipschitzian mappings with cg1=cg3=cg4=cg5=1 and cg2=33/8. Let μ1=5/12,μ2=−7/18,μ3=μ4=1/72 and μ5=5/24. Then,(48)∑i=15μici=512−338718+172+172+524,=0.40018703501<1.
SystemJ has a unique solution u,u,u,u,u, where u≈2.549220382 is a unique solution of 30sinu−28/1+u2+u+cosu2+150=72u−15tan−1u.


### 5.2. Integral Systems

LetI=a,b, where a,b∈ℝ with a<b and S=ℒ1I with d1g,h=∫Igt−htdt where ∫ is Lebesgue integral. Then, ℒ1I,d1 is a complete metric space. Let κ,λ1,λ2,λ3,…,λn∈ℝ and let H:ℝn⟶ℝ be such that H0,0,0,…,0=0 and satisfying(49)H1u,2u,3u,…,nu−H1v,2v,3v,…,nv≤κ∑i=1nλiiu−ivfor all1u,2u,3u,…,nu,1v,2v,3v,…,nv∈ℝn.IfC∈ℝ, we will look for mappings g1,g2,g3,…,gn∈ℒ1I such that(50)gix=C+∫a,xHg1g2g3⋯gntdt,is satisfied for all x∈I,i=1,2,3,…,n (arguments of H are represented by subscripts).Let for allg1,g2,g3,…,gn∈ℒ1I and all x∈I, J:ℒ1In⟶ℒ1I be defined as follows:(51)Jg1g2g3⋯gnx=C+∫a,xHg1g2g3⋯gntdt.Then,(52)d1Jg1g2g3⋯gnx,Jh1h2h3⋯hnx=∫IJg1g2g3⋯gnx,Jh1h2h3⋯hnxdx,≤∫I∫a,xHg1g2g3⋯gntdt−∫a,xHh1h2h3⋯hntdtdx≤∫I∫a,xκ∑i=1nλigit−hitdtdx≤κ∑i=1nλi∫I∫Igit−hitdtdx=κ∑i=1nλi∫Id1gi,hidx=κb−a∑i=1nλid1gi,hi.If, then by 4.1 (c), (50) has a unique solution of the form g0,g0,g0,…,g0∈ℒ1In.


## 5.1. Lipschitzian Systems

Letg1,g2,…,gn:ℝ⟶ℝ be Lipschitzian mappings on ℝ (equipped with the Euclidean metric) with Lipschitz constants c1,c2,c3,…,cn, respectively, that is, for each gi where 1≤i≤n, there exists a corresponding real number ci where 1≤i≤n such that giu−giv≤ciu−v for all u,v∈ℝ. Define f:ℝ⟶ℝ as fu=μ1g1u+μ2g2u+⋯+μngnu for all u∈ℝ, where μi∈ℝ for 1≤i≤n. Then,(42)fu−fv=∑i=1nμigiiu−giiv,≤∑i=1nμiciu−v.That is,f is itself a Lipschitzian mapping with cf=∑i=1nμici. If cf=∑i=1nμici<1, then f is contraction and by Banach contraction principle there is a unique uo∈ℝ such that fuo=uo. Define T:Sn⟶S by(43)T1u,2u,3u,…,nu=μ1g11u+μ2g22u+⋯+μngnnu,for all1u,2u,3u,…,nu∈ℝ.Obviously,Tu,u,…,u︷n=fu for all u∈ℝ. Moreover,(44)T1u,2u,3u,…,nu−1v,2v,3v,…,nv=∑i=1nμigiiu−giiv,≤∑i=1nμiciu−v,=∑i=1ncfiu−iv,≤cfmax1≤i≤niu−iv.Ifcf<1, then by Theorem 2 (a), there exists a unique u∈ℝ such that(45)u=fu=Tu,u,…,u︷n.Corollary 5.
Letg1,g2,…,gn:ℝ⟶ℝ be Lipschitzian mappings furnished with the Euclidean metric, and μ1,μ2,…,μn∈ℝ be such that C=∑i=1nμici<1. Then, the system(46)H=μ1g11u+μ2g22u+⋯+μn−1gn−1n−1u+μngnnu=1u,μ1g12u+μ2g23u+⋯+μn−1gn−1nu+μngn1u=2u,μ1g13u+μ2g24u+⋯+μn−1gn−11u+μngn2u=3u,⋮μ1g1nu+μ2g21u+⋯+μn−1gn−1n−2u+μngnn−1u=nu,has a unique solution ou,ou,…,ou︷n, where ou is a unique real solution of ∑i=1nμigiu=u.Example 5.
Consider the following:(47)J=30sinx−281+y2+z+cosw2+150=72x−15tan−1t,30siny−281+z2+w+cost2+150=72y−15tan−1x,30sinz−281+w2+t+cosx2+150=72z−15tan−1y,30sinw−281+t2+x+cosy2+150=72w−15tan−1z,30sint−281+x2+y+cosz2+150=72t−15tan−1w.
Letg1u=5+sinu,g2u=1/1+u2,g3u=u,g4u=cosu2andg5u=tan−1u. g1,g2,g3,g4 and g5 are Lipschitzian mappings with cg1=cg3=cg4=cg5=1 and cg2=33/8. Let μ1=5/12,μ2=−7/18,μ3=μ4=1/72 and μ5=5/24. Then,(48)∑i=15μici=512−338718+172+172+524,=0.40018703501<1.
SystemJ has a unique solution u,u,u,u,u, where u≈2.549220382 is a unique solution of 30sinu−28/1+u2+u+cosu2+150=72u−15tan−1u.


## 5.2. Integral Systems

LetI=a,b, where a,b∈ℝ with a<b and S=ℒ1I with d1g,h=∫Igt−htdt where ∫ is Lebesgue integral. Then, ℒ1I,d1 is a complete metric space. Let κ,λ1,λ2,λ3,…,λn∈ℝ and let H:ℝn⟶ℝ be such that H0,0,0,…,0=0 and satisfying(49)H1u,2u,3u,…,nu−H1v,2v,3v,…,nv≤κ∑i=1nλiiu−ivfor all1u,2u,3u,…,nu,1v,2v,3v,…,nv∈ℝn.IfC∈ℝ, we will look for mappings g1,g2,g3,…,gn∈ℒ1I such that(50)gix=C+∫a,xHg1g2g3⋯gntdt,is satisfied for all x∈I,i=1,2,3,…,n (arguments of H are represented by subscripts).Let for allg1,g2,g3,…,gn∈ℒ1I and all x∈I, J:ℒ1In⟶ℒ1I be defined as follows:(51)Jg1g2g3⋯gnx=C+∫a,xHg1g2g3⋯gntdt.Then,(52)d1Jg1g2g3⋯gnx,Jh1h2h3⋯hnx=∫IJg1g2g3⋯gnx,Jh1h2h3⋯hnxdx,≤∫I∫a,xHg1g2g3⋯gntdt−∫a,xHh1h2h3⋯hntdtdx≤∫I∫a,xκ∑i=1nλigit−hitdtdx≤κ∑i=1nλi∫I∫Igit−hitdtdx=κ∑i=1nλi∫Id1gi,hidx=κb−a∑i=1nλid1gi,hi.If, then by 4.1 (c), (50) has a unique solution of the form g0,g0,g0,…,g0∈ℒ1In.


## 6. Conclusion

We generalized the concept of tripled fixed point by introducingn-tupled fixed points and established an n-tupled unique fixed point result in fuzzy b-metric space. This generalization may be helpful for further investigation and applications.We conclude this paper by indicating, in the form of open questions, some directions for further investigation and work.(1)
Can the condition of continuity ofθ in Theorem 1 be relaxed?(2)
If the answer to 1 is yes, then what hypotheses onS and θ are needed to guarantee the existence of the n-tupled common fixed points T and θ?(3)
Can the concept ofn-tupled coincidence point be extended to more than two mappings?



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