# Comment on “Common Fixed Point Theorems for Commutating Mappings in Fuzzy Metric Spaces”

Authors: Yonghong Shen; Wei Chen; Sanfu Wang
Journal: Abstract and Applied Analysis
 (2012)

---

## Abstract

In the recent paper “common fixed point theorems for commutating mappings in fuzzy metric spaces,” the authors proved that a common fixed point theorem for commutating mappings inG-complete fuzzy metric spaces and gave an example to illustrate the main result. In this note, we point out that the above example is incorrect because it does not satisfy the condition of G-completeness, and then two appropriate examples are given. In addition, we prove that the theorem proposed by Zheng and Lian actually holds in an M-complete fuzzy metric space. Our results improve and extend some existing results in the relevant literature.


---

## Body

## 1. Introduction

In [1], Zheng and Lian extended Jungck's theorem in [2] to fuzzy metric spaces and obtained the following fixed point theorem for commutative mappings in fuzzy metric spaces in the sense of Kramosil and Michálek [3].Theorem 1.1 (Zheng and Lian [1]).
Let(X,M,*) be a complete fuzzy metric space and let f:X→X be a continuous map and g:X→X a map. If(i)
g(X)⊆f(X),(ii)
g commutes with f,(iii)
andM(g(x),g(y),t)≥M(f(x),f(y),ψ(t)) for all x,y∈X and t>0, where ψ:[0,+∞)→(0,+∞) is an increasing and left-continuous function with ψ(t)>t for all t>0.Then f and g have a unique common fixed point.Remark 1.2.
It can be seen from the proof of Theorem1.1 that the fuzzy metric space (X,M,*) is complete in the sense of G-completeness.Furthermore, the authors constructed the following example to illustrate the above theorem.Example 1.3 (Zheng and Lian [1]).
LetX=[0,+∞) be endowed with the usual metric d(x,y)=|x-y|. For all x,y∈X and t≥0, define
(1.1)M(x,y,t)={0,t=0,e-|x-y|/t,t>0.
Setf(x)=A-1(ex-1) and g(x)=ln(1+Ax), ψ(t)=A-2t, where A∈(0,1) is a constant.In the Example1.3, the authors claimed that (X,M,*) is a complete fuzzy metric space in the sense of G-completeness (now known as a G-complete fuzzy metric space) with respect to t-norm a*b=ab, and then checked all the conditions of Theorem 1.1. Therefore, they concluded that f and g have the unique common fixed point 0.However, we note that Example1.3 is incorrect for Theorem 1.1, because X=[0,+∞) is not G-complete regarding the fuzzy metric M. For details, the reader can refer to [4–6].In fact,G-completeness is a very strong kind of completeness. For instance, George and Veeramani [4] found that even ℝ is not G-complete with respect to the standard fuzzy metric induced by Euclidean metric, and then proposed another kind of completeness (now known as an M-complete fuzzy metric space) by modifying the definition of Cauchy sequence. For these two types of completeness, it is easy to see that every G-complete fuzzy metric space is M-complete. Therefore, the construction of fixed point theorems in M-complete fuzzy metric spaces is more valuable and reasonable.The main purpose of this note is to provide two appropriate examples for Theorem1.1 and prove that this theorem does hold even if G-completeness of the fuzzy metric space is replaced by M-completeness. Our results not only improve and generalize Theorem 1.1, but also extend some main results of [2, 7].


## 2. Preliminaries

For completeness and clarity, in this section, some related concepts and conclusions are summarized below. Letℕ denote the set of all positive integers.Definition 2.1 (Schweizer and Sklar [8]).
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)
anda*b≤c*d whenever a≤c, b≤d and a,b,c,d∈[0,1].Definition 2.2 (Kramosil and Michálek [3]).
The triple(X,M,*) is called a fuzzy metric space if X is an arbitrary 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 and s,t>0,(FM-1)
M(x,y,0)=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)
andM(x,y,·):[0,+∞)→[0,1] is left-continuous.Remark 2.3.
According to (FM-2) and (FM-4), it can easily be seen thatM(x,y,·) is non-decreasing for all x,y∈X (see Lemma  4 in [9]).Similar to the case in [1], in this note, we suppose that (X,M,*) is a fuzzy metric space with the following additional condition:(FM-6)
limt→+∞M(x,y,t)=1, for all x,y∈X.Definition 2.4 (Grabiec [9], George and Veeramani [4]).
Let(X,M,*) be a fuzzy metric space. Then(i)
a sequence{xn} in X is said to be convergent to a point x∈X, denoted by limn→∞xn=x, if limn→∞M(xn,x,t)=1, for any t>0;(ii)
a sequence{xn} in X is called a G-Cauchy sequence if and only if limn→∞M(xn+p,xn,t)=1 for any t>0 and p>0;(iii)
a sequence{xn} in X is called an M-Cauchy sequence if and only if for each ϵ∈(0,1) and t>0, there exists n0∈ℕ such that M(xm,xn,t)>1-ϵ, for any m,n≥n0;(iv)
a fuzzy metric space(X,M,*) is said to be G-complete (M-complete) if every G-Cauchy sequence (M-Cauchy sequence) is convergent;(v)
a mapf:X→X is said to be continuous at x0∈X if {f(xn)} converges to f(x0) for each {xn} converging to x0.The authors have proved the following conclusion (see the proof of Theorem  2.2 in [1]).Lemma 2.5 (Zheng and Lian [1]).
Letψ:[0,+∞)→(0,+∞) be an increasing and left-continuous function with ψ(t)>t for all t>0. Then
(2.1)limn→+∞ψn(t)=+∞,
for any t>0, where ψn(t) denotes the composition of ψ(t) with itself n times.


## 3. Two Appropriate Examples

In this section, we will construct two appropriate examples for Theorem1.1.Example 3.1.
LetX={1/n:n∈ℕ}∪{0} be equipped with the usual metric d(x,y)=|x-y|, ψ(t)=k·t, for all t∈[0,+∞), where k∈(1,4] is a constant. For all x,y∈X and t≥0, define
(3.1)M(x,y,t)={0,t=0,tt+|x-y|,t>0.
Clearly,(X,M,*) is a G-complete fuzzy metric space with regard to t-norm a*b=ab.
Setf(x)=x/2 and g(x)=x/8 for all x∈X. It is obvious that g(X)⊆f(X). For any x,y∈X and t>0, we have
(3.2)M(f(x),f(y),ψ(t))=k·tk·t+|x-y|/2=2k·t2k·t+|x-y|≤8·t8·t+|x-y|=M(g(x),g(y),t).
Thus, all the conditions of Theorem1.1 are satisfied and f and g have a unique fixed point, that is, x=0.Example 3.2.
LetX be the subset of ℝ2 defined by
(3.3)X={A,B,C,D,E,F},
where A=(0,0), B=(1,0), C=(2,0), D=(0,2), E=(1,1), and F=(2,2). ψ(t)=2t, for all t∈[0,+∞). For all x,y∈X and t≥0, define
(3.4)M(x,y,t)={0,t=0,e-d(x,y)/t,t>0,
where d(x,y) denotes the Euclidean distance of ℝ2.
Clearly,(X,M,*) is also a G-complete fuzzy metric space with regard to t-norm a*b=ab.
Letf:X→X and g:X→X be given by
(3.5)f(A)=f(B)=D,f(C)=F,f(D)=A,f(E)=E,f(F)=B,g(A)=g(B)=B,g(C)=g(D)=g(E)=g(F)=E.
Obviously,g(X)⊆f(X). Furthermore, it is easy to see that M(f(x),f(y),ψ(t))≤M(g(x),g(y),t) for any x,y∈X and t>0. Hence, all the conditions of Theorem 1.1 are satisfied and x=E is the unique common fixed point of f and g.


## 4. Main Results

Now, we will prove that Theorem1.1 does hold even if G-completeness is replaced by M-completeness.Theorem 4.1.
Let(X,M,*) be an M-complete fuzzy metric space and let f:X→X be a continuous map and g:X→X a map. If(i)
g(X)⊆f(X),(ii)
g commutes with f,(iii)
andM(g(x),g(y),t)≥M(f(x),f(y),ψ(t)) for all x,y∈X and t>0, where ψ:[0,+∞)→(0,+∞) is an increasing and left-continuous function with ψ(t)>t for all t>0.Then f and g have a unique common fixed point.Proof.
Letx0∈X. From (i), we can find x1 such that f(x1)=g(x0). By induction, we can find a sequence {xn}⊆X such that f(xn)⊆g(xn-1). For any t>0, we have
(4.1)M(f(xn),f(xn+1),t)=M(g(xn-1),g(xn),t)≥M(f(xn-1),f(xn),ψ(t))≥⋯≥M(f(x0),f(x1),ψn(t)).
Aslimn→+∞ψn(t)=+∞, for any t>0, it follows by (FM-6) that limn→+∞M(f(x0),f(x1),ψn(t))=1. Hence, limn→+∞M(f(xn),f(xn+1),t)=1, for any t>0.
Next, we claim that{f(xn)} is an M-Cauchy sequence. Suppose that it is not. Then there exist ϵ∈(0,1) and two sequences {p(n)}, {q(n)} such that for every n∈ℕ and t>0, and then we can obtain that
(4.2)p(n)>q(n)≥n,M(f(xp(n)),f(xq(n)),t)≤1-ϵ.
Moreover, for everyn∈ℕ, we can choose the two smallest numbers p(n) and q(n) such that
(4.3)M(f(xp(n)-1),f(xq(n)-1),t)>1-ϵ.
For everyn∈ℕ, we can obtain
(4.4)1-ϵ≥M(f(xp(n)),f(xq(n)),t)=M(g(xp(n)-1),g(xq(n)-1),t)≥M(f(xp(n)-1),f(xq(n)-1),ψ(t))≥M(f(xp(n)-1),f(xq(n)-1),t)>1-ϵ.
Clearly, this leads to a contradiction.
Here, we also consider another particular case. That is, for eacht>0, there exist ϵ∈(0,1) and n0∈ℕ such that M(f(xm),f(xn),t)≤1-ϵ, for all m,n≥n0. Then, for any p∈ℕ, we know that M(f(xn0+p+2),f(xn0+p+1),t)≤1-ϵ. Since
(4.5)M(f(xn0+p+2),f(xn0+p+1),t)=M(g(xn0+p+1),g(xn0+p),t)≥M(f(xn0+p+1),f(xn0+p),ψ(t))≥M(f(xn0+p+1),f(xn0+p),t),
we can conclude that {M(f(xn0+p+2),f(xn0+p+1),t)} is a monotone and bounded sequence with respect to p. Therefore, there exists γ∈[0,1-ϵ] such that limp→+∞M(f(xn0+p+2),f(xn0+p+1),t)=γ. In addition, according to the foregoing inequality, we can obtain
(4.6)M(f(xn0+p+2),f(xn0+p+1),t)≥M(f(xn0+1),f(xn0),ψp+1(t)).
By supposing thatp→+∞, it follows that γ≥1, which is also a contradiction.
Hence,{f(xn)} is an M-Cauchy sequence in the M-complete fuzzy metric space X. Furthermore, we conclude that there exists a point y∈X such that limn→+∞f(xn)=y. So limn→+∞g(xn)=limn→+∞f(xn+1)=y.
By (iii), it can be seen that the continuity off implies that of g. Consequently, we obtain that limn→+∞g(f(xn))=g(y). According to the commutativity of f and g, we know that limn→+∞f(g(xn))=limn→+∞g(f(xn))=g(y). Because of the uniqueness of limits, it follows immediately that f(y)=g(y). So f(f(y))=f(g(y))=g(f(y))=g(g(y)). Thus, we have
(4.7)M(g(y),g(g(y)),t)≥M(f(y),f(g(y)),ψ(t))=M(g(y),g(g(y)),ψ(t))≥⋯≥M(g(y),g(g(y)),ψn(t)).
Lettingn→+∞, we obtain that limn→+∞M(g(y),g(g(y)),ψn(t))=1, for any t>0. So M(g(y),g(g(y)),t)=1. By (FM-2), we conclude that g(g(y))=g(y). Hence, g(y)=g(g(y))=f(g(y)), that is, g(y) is a common fixed point of f and g.
Furthermore, we show thatg(y) is the unique common fixed point of f and g. Assume that x and z are two common fixed point of f and g, for any t>0, we then obtain
(4.8)M(x,z,t)=M(g(x),g(z),t)≥M(f(x),f(z),ψ(t))=M(x,z,ψ(t))≥⋯≥M(x,z,ψn(t)).
Asn→+∞, we have limn→+∞M(x,z,ψn(t))=1. Thus M(x,z,t)=1 for any t>0. Furthermore, we can obtain x=z. This completes the proof.Remark 4.2.
It should be pointed out that the foregoing three examples are suitable for Theorem4.1.Corollary 4.3.
Let(X,M,*) be an M-complete fuzzy metric space and let f:X→X be a continuous map and g:X→X a map. If(i)
g(X)⊆f(X),(ii)
g commutes with f,(iii)
andM(g(x),g(y),kt)≥M(f(x),f(y),t) for all x,y∈X and t>0, where 0<k<1.
Thenf and g have a unique common fixed point.Remark 4.4.
Corollary4.3 is the immediate consequence of Theorem 4.1, which can be regarded as an improvement of Theorem  2 in [7].



---

*Source: 142858-2012-11-19.xml*