# On Convergence of Fixed Points in Fuzzy Metric Spaces

Authors: Yonghong Shen; Dong Qiu; Wei Chen
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We mainly focus on the convergence of the sequence of fixed points for some different sequences of contraction mappings or fuzzy metrics in fuzzy metric spaces. Our results provide a novel research direction for fixed point theory in fuzzy metric spaces as well as a substantial extension of several important results from classical metric spaces.


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## Body

## 1. Introduction

Fixed point theory of classical metric spaces plays an important role in general topology. In 1988, Grabiec [1] first extended fixed point theorems of Banach and Edelstein to fuzzy metric spaces in the sense of Kramosil and Michalek. Since then, many authors had dedicated themselves to the study of fixed point theory in fuzzy metric spaces [2–18]. Besides, some authors extended fixed point theory to other types of fuzzy metric spaces in recent years. For instance, Alaca et al. [19] extended the well-known fixed point theorems of Banach and Edelstein to intuitionistic fuzzy metric spaces with the help of Grabiec's work. Simultaneously, Mohamad [20] and Razani [21] proved the existence of fixed point for a nonexpansive mapping of intuitionistic fuzzy metric spaces and the intuitionistic Banach fixed point theorem in complete intuitionistic fuzzy metric spaces, respectively. Later, Ćirić et al. [22] investigated the existence of fixed points for a class of asymptotically nonexpansive mappings in an arbitrary intuitionistic fuzzy metric space. On the other hand, Adibi et al. [23] extended a common fixed point theorem to L-fuzzy metric spaces and proved a coincidence point theorem and a fixed point theorem for compatible mappings of type (P) in these spaces. In 2008, Ješić and Babačev [24] further studied some common fixed point theorems for a pair of R-weakly commuting mappings with nonlinear contractive condition in intuitionistic fuzzy metric spaces and L-fuzzy metric spaces. In the same year, Park et al. [25] extended some common fixed point theorems for five mappings to M-fuzzy metric spaces. Up to now, one can see that the majority of papers mainly focus on the existence of fixed points for different mappings in different fuzzy metric spaces. However, the aim of this paper is to show that the convergence of the sequence of fixed points to some sequences of contraction mappings or fuzzy metrics satisfies certain conditions in fuzzy metric spaces.


## 2. Preliminaries

Now, we begin with some basic concepts and lemmas. Letℕ denote the set of all positive integers.Definition 1 (Schweizer and Sklar [26]).
A binary operation*:[0,1]×[0,1]→[0,1] is called a continuous triangular norm (shortly, continuous t-norm) if it satisfies the following conditions:(TN-1)
* is commutative and associative;(TN-2)
* is continuous;(TN-3)
a*1=a for every a∈[0,1];(TN-4)
a*b≤c*d whenever a≤c, b≤d, and a,b,c,d∈[0,1].In particular, at-norm * is said to be positive [27] if a*b>0 whenever a,b∈(0,1].We redefine the notion of a fuzzy metric space by appending the following condition (FM-6) based on the one in the sense of George and Veeramani [2].Definition 2.
Afuzzy metric space is an ordered triple (X,M,*) such that X is a (nonempty) set, * is a continuous t-norm, and M is a fuzzy set on X×X×(0,∞) satisfying the following conditions, for all x,y,z∈X,  s,t>0:(FM-1)M(x,y,t)>0;(FM-2)M(x,y,t)=1 if and only if x=y;(FM-3)M(x,y,t)=M(y,x,t);(FM-4)M(x,y,t)*M(y,z,s)≤M(x,z,t+s);(FM-5)M(x,y,·):(0,∞)→(0,1] is continuous;(FM-6)limt→∞M(x,y,t)=1.Definition 3 (Grabiec [1] and Vasuki and Veeramani [17]).
Let(X,M,*) be a fuzzy metric space. Then(a)
a sequence{xn} is said to converge tox in X, denoted by xn→x, if and only if limn→∞M(xn,x,t)=1 for all t>0; that is, for each r∈(0,1) and t>0, there exists an n0∈ℕ such that M(xn,x,t)>1-r for all n≥n0;(b)
a sequence{xn} in X is a G-Cauchy sequence if and only if limn→∞M(xn+p,xn,t)=1 for any p>0 and t>0;(c)
the fuzzy metric space(X,M,*) is called G-complete if every G-Cauchy sequence is convergent.Definition 4 (Grabiec [1]).
Let(X,M,*) be a fuzzy metric space. A mapping T:X→X is called a contraction mapping if there exists k∈(0,1) such that
(1)M(Tx,Ty,kt)≥M(x,y,t)
for every x,y∈X and t>0.According to fuzzy Banach contraction theorem of complete fuzzy metric space in the sense of Grabiec [1], we can obtain the following lemma.Lemma 5.
Let(X,M,*) be a G-complete fuzzy metric space. If T:X→X is a contraction mapping, then T has a unique fixed point.Definition 6.
Let(X,M,*) be a fuzzy metric space and let {Tn} be a sequence of self-mappings on X. T0:X→X is a given mapping. The sequence {Tn} is said to converge pointwise toT0 if for each r∈(0,1) and x0∈X, there exists an n0∈ℕ such that
(2)M(Tnx0,T0x0,t)>1-r
for all n≥n0 and t>0.Definition 7.
Let(X,M,*) be a fuzzy metric space and let {Tn} be a sequence of self-mappings on X. T0:X→X is a given mapping. The sequence {Tn} is said to converge uniformly toT0 if for each r∈(0,1) and t>0, there exists an n0∈ℕ such that
(3)M(Tnx,T0x,t)>1-r
for all n≥n0 and x∈X.Definition 8.
Let(X,M,*) be a fuzzy metric space. A sequence of self-mappings {Tn} is uniformly equicontinuous if for each r∈(0,1), there exists an ϵ∈(0,1) such that M(x,y,s)>1-ϵ implies M(Tnx,Tny,t)>1-r for every x,y∈X,  n∈ℕ, and s,t>0.Definition 9 (George and Veeramani [2]).
Let(X,M,*) be a fuzzy metric space. The open ballB(x,r,t) and closed ballB[x,r,t] with center x∈X and radius r, 0<r<1, t>0, respectively, are defined as follows:
(4)B(x,r,t)={y∈X:M(x,y,t)>1-r},B[x,r,t]={y∈X:M(x,y,t)≥1-r}.Lemma 10 (George and Veeramani [2]).
Every open (closed) ball is an open (a closed) set.Definition 11 (Gregori and Romaguera [3]).
A fuzzy metric space(X,M,*) is a compact space if (X,τM) is a compact topological space, where τM is a topology induced by the fuzzy metric M.Based on the corresponding conclusions stated in [2], we can easily obtain the following lemma.Lemma 12.
Every closed subsetA of a compact fuzzy metric space (X,M,*) is compact.Lemma 13.
Let(X,M,*) be a fuzzy metric space and let {Tn} be a sequence of self-mappings on X. T0:X→X is a contraction mapping of X; that is, there exists k∈(0,1) such that M(T0x,T0y,kt)≥M(x,y,t) for every x,y∈X. A is a compact subset of X. If {Tn} converges pointwise to T0 in A and it is a uniformly equicontinuous sequence, then the sequence {Tn} converges uniformly to T0 in A.Proof.
For eachr-∈(0,1), we may choose an appropriate r such that (1-r)*(1-r)*(1-r)>1-r-. Since {Tn} is uniformly equicontinuous, there exists ϵ∈(0,1)(ϵ≤r) such that M(x,y,s)>1-ϵ⇒M(Tnx,Tny,t)>1-r for every x,y∈X, s,t>0, and n∈ℕ. For the foregoing ϵ, we fix s>0. Define 𝒞={B(x,ϵ,s):x∈A}. By Lemma 10, 𝒞 is a family of open sets of A. Obviously, 𝒞 constitutes an open covering of A; that is, A⊂⋃‍B(x,ϵ,s). Since A is compact, there exist x1,x2,…,xm∈A such that A⊂⋃i=1mB(xi,ϵ,s). For every xi∈A (i=1,2,…,m), since {Tn} converges pointwise to T0 in A, for r∈(0,1), there exist ni∈ℕ (i=1,2,…,m) such that M(Tnxi,T0xi,t)>1-r for all n≥ni. Set n*=max{ni:i=1,2,…,m}. Clearly, n* depends only on r. For every x∈X, there is an i0∈{1,2,…,m} such that x∈B(xi0,ϵ,s). Then we have M(x,xi0,s)>1-ϵ⇒M(Tnx,Tnxi0,t)>1-r for all n∈ℕ. Thus, for all n≥n*,
(5)M(Tnx,T0x,(2t+ks))≥M(Tnx,Tnxi0,t)*M(Tnxi0,T0x,t+ks)≥M(Tnx,Tnxi0,t)*M(Tnxi0,T0xi0,t)*M(T0xi0,T0x,ks)≥M(Tnx,Tnxi0,t)*M(Tnxi0,T0xi0,t)*M(xi0,x,s)≥(1-r)*(1-r)*(1-ϵ)≥(1-r)*(1-r)*(1-r)>1-r-.
Hence, the sequence {Tn} converges uniformly to T0 in A.Definition 14.
A fuzzy metric space(X,M,*) in which every point has a compact neighborhood is called locally compact.Definition 15.
Let(X,M0,*) be a fuzzy metric space and let {Mn} be a sequence of fuzzy metrics on X. The sequence {Mn} is said to upper semiconverge uniformly toM0 if for each r∈(0,1) and t>0, there exists an n0∈ℕ such that Mn(x,y,t)≥M0(x,y,t) and M0(x,y,t)/Mn(x,y,t)>1-r for all n≥n0, x,y∈X.


## 3. Main Results

Theorem 16.
Let(X,M,*) be a G-complete fuzzy metric space and let {Tn} be a sequence of self-mappings on X where t-norm a*b=min{a,b}. T0 is a contraction mapping of X; that is, there exists k0∈(0,1) such that M(T0x,T0y,k0t)≥M(x,y,t) for all x,y∈X, t>0, and satisfying T0x0=x0. If there exists at least a fixed point xn for each Tn (n∈ℕ) and the sequence {Tn} converges uniformly to T0, then xn→x0.Proof.
Suppose thatxn↛x0; namely, there exist t0>0 and r0∈(0,1) such that for any n∈ℕ there is a k(n)>n satisfying M(xk(n),x0,t0)<1-r0. Fix a number h∈(k0,1). According to the condition (FM-6) of Definition 2, for t0>0, we can find an appropriate p∈ℕ such that M(xn,x0,t0(h/k0)p)>1-r0 for any n∈ℕ. Since the sequence {Tn} converges uniformly to T0, we can make n0 sufficiently large such that M(Tnxn,T0x,t)>1-r0 for all n≥n0, t>0. Now for n≥n0, we have
(6)1-r0>M(xk(n),x0,t0)=M(Tk(n)xk(n),T0x0,t0)≥M(Tk(n)xk(n),T0xk(n),(1-h)t0)*M(T0xk(n),T0x0,ht0)≥M(Tk(n)xk(n),T0xk(n),(1-h)t0)*M(xk(n),x0,t0hk0)≥M(Tk(n)xk(n),T0xk(n),(1-h)t0)*M(Tk(n)xk(n),T0xk(n),t0(1-h)hk0)*M(xk(n),x0,t0(hk0)2)≥M(Tk(n)xk(n),T0xk(n),(1-h)t0)*M(Tk(n)xk(n),T0xk(n),t0(1-h)hk0)*⋯*M(Tk(n)xk(n),T0xk(n),t0(1-h)(hk0)p-1)*M(xk(n),x0,t0(hk0)p)≥(1-r0)*(1-r0)*⋯*(1-r0)︸p*(1-r0)=1-r0.
This leads to a contradiction. Hence, xn→x0.Theorem 17.
Let(X,M,*) be a G-complete fuzzy metric space where t-norm is positive. If T0:X→X is a self-mapping of X and T0m is a contraction mapping for a certain positive integer m, then T0 has a unique fixed point.Proof.
First of all, ifm=1, the theorem is evident. In addition, if m≥2, according to Lemma 5, we need only to prove that T0 is a contraction mapping. Since T0m is a contraction mapping, there is k0∈(0,1) such that M(T0mx,T0my,k0mt)≥M(x,y,t) for every x,y∈X, and t>0. Define another fuzzy metric M~(x,y,t) on X using M(x,y,t) as follows:
(7)M~(x,y,t)=M(x,y,t)*M(T0x,T0y,kt)*M(T02x,T02y,k2t)*⋯*M(T0m-1x,T0m-1y,km-1t).
Actually, it is easy to verify that the foregoing two fuzzy metrics are equivalent. Meantime, we claim that T0 is a contraction mapping with respect to the fuzzy metric M~(x,y,t), since
(8)M~(T0x,T0y,k0t)=M(T0x,T0y,kt)*M(T02x,T02y,k2t)*M(T03x,T03y,k3t)*⋯*M(T0mx,T0my,kmt)≥M(T0x,T0y,kt)*M(T02x,T02y,k2t)*M(T03x,T03y,k3t)*⋯*M(x,y,t)=M~(x,y,t).Corollary 18.
Let(X,M,*) be a G-complete fuzzy metric space and let {Tn} be a sequence of self-mappings on X where t-norm a*b=min{a,b}. T0:X→X is a self-mapping of X, and T0m is a contraction mapping for a certain positive integer m. If there exists at least a fixed point xn for each Tn(n∈ℕ) and the sequence {Tn} converges uniformly to T0, then xn→x0=T0x0.Proof.
It follows from Theorems16 and 17.Theorem 19.
Let(X,M,*) be a locally compact fuzzy metric space and let {Tn} be a sequence of self-mappings on X. T0:X→X is a contraction mapping; that is, there exists a k0∈(0,1) such that M(T0x,T0y,k0t)≥M(x,y,t) for all x,y∈X, t>0. If the following conditions are satisfied:(i)
Tnm is a contraction mapping for a certain m=m(n),(ii)
{Tn} converges pointwise to T0 and {Tn} is a uniformly equicontinuous sequence,(iii)
Tnxn=xn, n=0,1,2,3,…,then the sequence {xn} converges to x0; that is, xn→x0.Proof.
For eachϵ∈(0,1), we choose r∈(0,1) such that (1-r)*(1-r)≥1-ϵ. If given x0∈X, we may assume that r is sufficiently small such that K(x0,r)={x:M(x,x0,t)≥1-r} is a compact subset of X. By Lemma 13, since {Tn} is uniformly equicontinuous and pointwise convergent on K(x0,r), we know that {Tn} converges uniformly to T0 on the compact subset K(x0,r). Then, for the foregoing r, there exists nϵ∈ℕ such that M(Tnx,T0x,(1-k0)t)>1-r for all n≥nϵ, t>0, and x∈K(x0,r). In addition, since T0 is a contraction mapping, we have M(T0x,T0y,k0t)≥M(x,y,t) for all x,y∈K(x0,r). Thus, for all n≥nϵ and x∈K(x0,r), we can obtain
(9)M(Tnx,x0,t)=M(Tnx,T0x0,t)≥M(Tnx,T0x,(1-k0)t)*M(T0x,T0x0,k0t)≥M(Tnx,T0x,(1-k0)t)*M(x,x0,t)≥(1-r)*(1-r)≥1-ϵ.
Therefore, for all n≥nϵ, K(x0,r) is an invariant set for Tn. Since Tnm is a contraction mapping for a certain positive integer m=m(n), it follows that the fixed point xn of Tn is contained in the set K(x0,r), when n≥nϵ. By the definition of K(x0,r), we have M(xn,x0,t)≥1-r for all n≥nϵ. In fact, although r should satisfy the foregoing condition, it may be sufficiently small. Hence, we can obtain xn→x0.In addition, ift-norm a*b=a·b, then we can obtain the following some important conclusions.Lemma 20.
Let(X,M0,*) be a G-complete fuzzy metric space and let A be a compact subset of X where t-norm a*b=a·b. {Mn} and {Tn} are a sequence of fuzzy metrics and a sequence of self-mappings on X, respectively. If they satisfy the following conditions:(i)
{Mn} upper semiconverges uniformly to M0,(ii)
Tn is a contraction mapping for the fuzzy metric Mn, n=0,1,2,…,(iii)
{Tn} converges pointwise to T0,then {Tn} converges uniformly to T0 in A with regard to the fuzzy metric M0.Proof.
For eachϵ∈(0,1), choose r∈(0,1) such that (1-r)*(1-r)>1-ϵ. Since {Mn} upper semiconverges uniformly to M0, there exists nr∈ℕ such that Mn(x,y,t)≥M0(x,y,t) and M0(x,y,t)/Mn(x,y,t)>1-r for all n≥nr, t>0. Choose x,y in X such that M0(x,y,t)>1-r for each t>0. Then, for all n≥nr, we have
(10)M0(Tnx,Tny,t)=M0(Tnx,Tny,t)Mn(Tnx,Tny,t)*Mn(Tnx,Tny,t)≥(1-r)*Mn(Tnx,Tny,t)≥(1-r)*Mn(x,y,tkn)(kn∈(0,1))≥(1-r)*M0(x,y,tkn)≥(1-r)*(1-r)>1-ϵ.
Therefore, the sequence {Tn}(n≥nr) is uniformly equicontinuous in A with regard to the fuzzy metric M0. Since {Tn} is pointwise convergent and A is a compact subset of X, according to Lemma 13, it follows that the subsequence {Tn} (n≥nr) converges uniformly to T0 in A. Hence, {Tn} converges uniformly to T0 in A.Theorem 21.
Let(X,M0,*) be a locally compact fuzzy metric space where t-norm a*b=a·b. If {Mn} and {Tn} satisfy the following conditions:(i)
{Mn} upper semiconverges uniformly to M0,(ii)
Tn is a contraction mapping for the fuzzy metric Mn, n=0,1,2,…,(iii)
{Tn} converges pointwise to T0,(iv)
Tnxn=xn, n=0,1,2,…,then the sequence of fixed points {xn} of {Tn} converges to the fixed point x0 of T0; that is, xn→x0.Proof.
For eachϵ∈(0,1), choose r∈(0,1) such that (1-r)*(1-r)≥1-ϵ. Meantime, for x0∈X, we may make r sufficiently small such that K(x0,r)={x:M(x,x0,t)≥1-r} is compact in X for each t>0. By Lemma 20, we know that {Tn} converges uniformly to T0 in K(x0,r) with respect to the fuzzy metric M0. Then, for every x∈X, there exists an nr∈ℕ such that M0(Tnx,T0x,t)>1-r for all n≥nr, t>0. Thus, when n≥nr, for all x∈K(x0,r), we have
(11)M0(Tnx,x0,(1+k0)t)≥M0(Tnx,T0x,t)*M0(T0x,x0,k0t)≥M0(Tnx,T0x,t)*M0(T0x,T0x0,k0t)≥M0(Tnx,T0x,t)*M0(x,x0,t)≥(1-r)*(1-r)≥1-ϵ.
Therefore, K(x0,r) is an invariant set in X with regard to M0. Since Tn is still a contraction mapping restricted to K(x0,r) concerning on Mn, one can see that the fixed point is also included in K(x0,r). Apparently, for all n≥nr, we can obtain M(xn,x0,t)≥1-r. Since r is sufficiently small, it can easily be shown that {xn} converges to x0; that is, xn→x0. This completes the proof.Theorem 22.
Let(X,M0,*) be a compact fuzzy metric space where t-norm a*b=a·b. The sequences {Mn} and {Tn} satisfy the following conditions:(i)
{Mn} upper semiconverges uniformly to M0;(ii)
Tn is a contraction mapping for the fuzzy metric Mn, n=0,1,2,…;(iii)
{Tn} converges pointwise to T0.
If every mappingTn (n∈ℕ) has a fixed point xn and there is a subsequence {xnk} of {xn} which converges to x0, then T0x0=x0.Proof.
LetK denote the closure of the set {xnk}. By Lemma 12, we can easily know that K is a compact set. According to Lemma 20, it follows that the subsequence {Tnk} converges uniformly to T0 in K with regard to M0. Obviously, {Tnkxnk} converges to T0x0. Hence, T0x0=x0.Theorem 23.
Let(X,M,*) be a fuzzy metric space. {Tn} is a sequence of contraction mappings and satisfying Tnxn=xn (n=1,2,3,…). T0:X→X is a contraction mapping. If {Tn} is a pointwise convergent sequence with respect to T0 and the subsequence {xnk} of {xn} converges to x0, then T0 has a fixed point x0=T0x0.Proof.
For eachϵ∈(0,1), choose r∈(0,1) such that (1-r)*(1-r)≥1-ϵ. Since {xnk} is a convergent subsequence and {Tn} is a pointwise convergent sequence, for a given x0, we may choose Kr∈ℕ such that M(xnk,x0,t)≥1-r and M(Tnkx0,T0x0,t)≥1-r for all k≥Kr, t>0. For every n∈ℕ∪{0}, we denote by ln (ln∈(0,1)) the contraction constant of Tn. Thus, for all k≥Kr, we have
(12)M(xnk,T0x0,(lnk+1)t)=M(Tnkxnk,T0x0,(lnk+1)t)lnk∈(0,1)≥M(Tnkxnk,Tnkx0,lnkt)*M(Tnkx0,T0x0,t)≥M(xnk,x0,t)*M(Tnkx0,T0x0,t)≥(1-r)*(1-r)≥1-ϵ.
Therefore, the subsequence {xnk} converges to T0x0. According to the uniqueness of limit, it follows that x0=T0x0; that is, x0 is a fixed point of T0.



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