# On Fixed Point Theorems in Intuitionistic Fuzzy Metric Spaces

Authors: Zhenhua Jiao
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

The author extends two fixed point theorems (due to Gregori, Sapena, and Žikić, resp.) in fuzzy metric spaces to intuitionistic fuzzy metric spaces.


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## Body

## 1. Introduction

In this paper, we pay our attention to the fixed point theory on intuitionistic fuzzy metric spaces. Since Zadeh [1] introduced the theory of fuzzy sets, many authors have studied the character of fuzzy metric spaces in different ways [2–5]. Among others, fixed point theorem was an important subject. Gregori and Sapena [6] investigated fixed point theorems for fuzzy contractive mappings defined on fuzzy metric spaces. Recently, Žikić [7] proved a fixed point theorem for mappings on fuzzy metric space which improved the result of Gregori and Sapena. As further development, Atanassov [8] introduced and studied the concept of intuitionistic fuzzy sets as a generalization of fuzzy sets, and later there has been much progress in the study of intuitionistic fuzzy sets [9, 10]. Using the idea of intuitionistic sets, Park [11] defined the notion of intuitionistic fuzzy metric spaces with the help of continuous t-norms and continuous t-conorms as a generalization of fuzzy metric space. Recently, several authors studied the structure of intuitionistic fuzzy metric spaces and fixed point theorems for the mappings defined on intuitionistic fuzzy metric spaces. We refer the reader to [11–13] for further details. In this paper, we will prove the following two fixed point theorems.The first theorem extends Gregori-Sapena's fixed point theorem [6] in fuzzy metric spaces to complete intuitionistic fuzzy metric spaces. As preparation, we recall the definition of s-increasing sequence [6]. A sequence {tn} of positive real numbers is said to be an s-increasing sequence if there exists m0∈ℕ such that tm+1≤tm+1, for all m≥m0.Theorem 1.1.
Let(X,M,N,*,◊) be a complete intuitionistic fuzzy metric space such that for every  s-increasing sequence  {tn} and arbitrary  x,y∈X,
(1.1)lim⁡n→∞∏i=n∞M(x,y,ti)=1,lim⁡n→∞∏i=n∞N(x,y,ti)=0
hold.
Letk∈(0,1) and T:X→X be a mapping satisfying M(Tx,Ty,kt)≥M(x,y,t) and N(Tx,Ty,kt)≤N(x,y,t) for all x,y∈X. Then T has a unique fixed point.The second theorem extends Žikić’s fixed point theorem [7] in fuzzy metric space to intuitionistic fuzzy metric space.Theorem 1.2.
Let(X,M,N,*,◊) be a complete intuitionistic fuzzy metric space such that for some  σ0∈(0,1) and  x0∈X,
(1.2)lim⁡n→∞∏i=n∞M(x0,Tx0,1σ0i)=1,lim⁡n→∞∏i=n∞N(x0,Tx0,1σ0i)=0
hold.
Letk∈(0,1) and T:X→X be a mapping satisfying M(Tx,Ty,kt)≥M(x,y,t) and N(Tx,Ty,kt)≤N(x,y,t) for all x,y∈X. Then T has a unique fixed point.


## 2. Basic Notions and Preliminary Results

For the sake of completeness, in this section we will recall some definitions and preliminaries on intuitionistic fuzzy metric spaces.Definition 2.1 (see [14]).
LetX be a nonempty fixed set. An intuitionistic fuzzy setA is an object having the form
(2.1)A={〈x,μA(x),νA(x)〉:x∈X},
where the functions μA:X→[0,1] and νA:X→[0,1] denote the degree of membership and the degree of nonmembership of each element x∈X to the set A, respectively, and 0≤μA(x)+νA(x)≤1 for each x∈X.For developing intuitionistic fuzzy topological spaces, in [10], Çoker introduced the intuitionistic fuzzy sets 0~ and 1~ in X as follows.Definition 2.2 (see [10]).
0~={〈x,0,1〉:x∈X} and 1~={〈x,1,0〉:x∈X}.By Definition2.2, Çoker defined the notion of intuitionistic fuzzy topological spaces.Definition 2.3 (see [10]).
Anintuitionistic fuzzy topology on a nonempty set X is a family τ of intuitionistic fuzzy sets in X satisfying the following axioms:
(T1)
0~,1~∈τ;(T2)
G1∩G2∈τ for any G1,G2∈τ;(T3)
⋃Gi∈τ for any arbitrary family {Gi:i∈J}⊆τ.
In this case, the pair(X,τ) is called an intuitionistic fuzzy topological space.Definition 2.4 (see [15]).
A binary operation*:[0,1]×[0,1]→[0,1] is a continuoust-norm (triangular norm) if * satisfies the following conditions:  (a)
* is associative and commutative;(b)
* is continuous;(c)
a*1=a for all a∈[0,1];(d)
a*b≤c*d whenever a≤c and b≤d, and a,b,c,d∈[0,1].
By this definition, it is easy to see that1*1=1. According to condition (a), the following product is well defined: M(x1,y1,t1)*M(x2,y2,t2)*⋯*M(xn,yn,tn), and we will denote it by ∏i=1i=nM(xi,yi,ti).Definition 2.5 (see [15]).
A binary operation◊:[0,1]×[0,1]→[0,1] is a continuoust-conorm (triangular conorm) if ◊ satisfies the following conditions: (e)
◊ is associative and commutative;(f)
◊ is continuous;(g)
a◊0=a for all a∈[0,1];(h)
a◊b≤c◊d whenever a≤c and b≤d, and a,b,c,d∈[0,1].
By this definition, it is easy to see that0◊0=0. According to condition (e), the following product is well defined: N(x1,y1,t1)◊N(x2,y2,t2)◊⋯◊N(xn,yn,tn), and we also denote this product by ∏i=1i=nN(xi,yi,ti).Remark 2.6.
The origin of concepts oft-norms and related t-conorms was in the theory of statistical metric spaces in the work of Menger [5]. These concepts are known as the axiomatic skeletons that we use for characterizing fuzzy intersections and unions, respectively. Basic examples of t-norms are a*b=ab and a*b=min{a,b}, and basic examples of t-conorms are a◊b=max{a,b} and a◊b=min{1,a+b}.Definition 2.7 (see [13]).
A 5-tuple(X,M,N,*,◊) is said to be an intuitionistic fuzzy metric space if X is an arbitrary set, * is a continuous t-norm, ◊ is a continuous t-conorm, and M,N are fuzzy sets on X×X×[0,∞) satisfying the following conditions:  (IFm 1)
M(x,y,t)+N(x,y,t)≤1;(IFm 2)
M(x,y,0)=0;(IFm 3)
M(x,y,t)=1 for all t>0 if and only if x=y;(IFm 4)
M(x,y,t)=M(y,x,t);(IFm 5)
M(x,y,t)*M(y,z,s)≤M(x,z,t+s) for all x,y,z∈X,s,t>0;(IFm 6)
M(x,y,·):[0,∞)→[0,1] is left continuous;(IFm 7)
limt→∞M(x,y,t)=1 for all x,y∈X;(IFm 8)
N(x,y,0)=1;(IFm 9)
N(x,y,t)=0 for all t>0 if and only if x=y;(IFm 10)
N(x,y,t)=N(y,x,t);(IFm 11)
N(x,y,t)◊N(y,z,s)≥N(x,z,t+s) for all x,y,z∈X,s,t>0;(IFm 12)
N(x,y,·):[0,∞)→[0,1] is right continuous;(IFm 13)
limt→∞N(x,y,t)=0 for all x,y∈X.
We denote by(M,N) the intuitionistic fuzzy metric on X. In intuitionistic fuzzy metric space X, it is easy to see M(x,y,·) is nondecreasing and N(x,y,·) is nonincreasing for all x,y∈X. We also note that the successive product ∏ with respect to M(x,y,t) is in the sense of * and the successive product ∏ with respect to N(x,y,t) is in the sense of ◊ throughout this paper.Definition 2.8.
Let(X,M,N,*,◊) be an intuitionistic fuzzy metric space. Then  (I)
a sequence{xn} in X is Cauchy sequence if and only if for each t>0 and p>0,
(2.2)lim⁡n→∞M(xn,xn+p,t)=1,lim⁡n→∞N(xn,xn+p,t)=0,(II)
a sequence{xn} in X is convergent tox∈X if and only if  for each t>0,
(2.3)lim⁡n→∞M(xn,x,t)=1,lim⁡n→∞N(xn,x,t)=0.Definition 2.9.
An intuitionistic fuzzy metric space is said to becomplete if and only if every Cauchy sequence is convergent.


## 3. Proof of Theorem1.1

In this section, we will give the proof of Theorem1.1 of the present paper.Proof.
Select an arbitrary pointx∈X. Let xn=Tn(x),n∈ℕ. We have
(3.1)M(x1,x2,t)=M(T(x),T2(x),t)≥M(x,T(x),tk)=M(x,x1,tk),N(x1,x2,t)=N(T(x),T2(x),t)≤N(x,T(x),tk)=N(x,x1,tk).
By induction it follows that M(xn,xn+1,t)≥M(x,x1,t/kn) and N(xn,xn+1,t)≤N(x,x1,t/kn).
Lett>0. For m,n∈ℕ, without loss of generality, we suppose n<m; if we choose si>0,i=n,…,m-1, satisfying sn+sn+1+⋯+sm-1≤1, then we have
(3.2)M(xn,xm,t)≥M(xn,xn+1,snt)*⋯*M(xm-1,xm,sm-1t)≥M(x,x1,sntkn)*⋯*M(x,x1,sm-1tkm-1),N(xn,xm,t)≤N(xn,xn+1,snt)◊⋯◊N(xm-1,xm,sm-1t)≤N(x,x1,sntkn)◊⋯◊N(x,x1,sm-1tkm-1).
In particular, since ∑n=1∞1/n(n+1)=1, taking si=1/i(i+1),i=n,…,m-1, one achieves
(3.3)M(xn,xm,t)≥M(x,x1,tn(n+1)kn)*⋯*M(x,x1,t(m-1)mkm-1),(3.4)N(xn,xm,t)≤N(x,x1,tn(n+1)kn)◊⋯◊N(x,x1,t(m-1)mkm-1).
We define tn=t/n(n+1)kn. It is preliminary to show that (tn+1-tn)→∞, as n→∞, so {tn} is an s-increasing sequence, and hence we get
(3.5)lim⁡m→∞∏n=m∞M(x,x1,tn(n+1)kn)=1,lim⁡m→∞∏n=m∞N(x,x1,tn(n+1)kn)=0.
The combination of (3.3), (3.4), and (3.5) implies limn→∞M(xn,xm,t)=1 and limn→∞N(xn,xm,t)=0 for m>n. Hence {xn} is a Cauchy sequence. Since X is complete, there is y∈X such that limn→∞xn=y. We claim y is a fixed point of T. In fact, it is easy to see
(3.6)M(T(y),y,t)≥{lim⁡n→∞M(T(y),T(xn),t2)}*{lim⁡n→∞M(xn+1,y,t2)}≥{lim⁡n→∞M(y,xn,t2k)}*{lim⁡n→∞M(xn+1,y,t2)}=1*1,N(T(y),y,t)≤{lim⁡n→∞N(T(y),T(xn),t2)}◊{lim⁡n→∞N(xn+1,y,t2)}≤{lim⁡n→∞N(y,xn,t2k)}◊{lim⁡n→∞N(xn+1,y,t2)}=0◊0.
Thus M(T(y),y,t)=1 and N(T(y),y,t)=0, and we obtain T(y)=y. In the sequel, we show the uniqueness of the fixed point. We assume T(z)=z for some z∈X. We have
(3.7)1≥M(y,z,t)=M(Ty,Tz,t)≥M(y,z,tk)=M(T(y),T(z),tk)≥M(y,z,tk2)=M(T(y),T(z),tk2)⋯≥lim⁡n→∞M(y,z,tkn)=1,0≤N(y,z,t)=N(Ty,Tz,t)≤N(y,z,tk)=N(T(y),T(z),tk)≤N(y,z,tk2)=N(T(y),T(z),tk2)⋯≤lim⁡n→∞N(y,z,tkn)=0.
Thus we get M(y,z,t)=1 and N(y,z,t)=0, and hence y=z. The proof is complete.


## 4. Proof of Theorem1.2

In this section, we will give the proof of Theorem1.2 by three lemmas.Lemma 4.1.
For any monotonely nondecreasing functionF:(0,∞)→[0,1], the following implication holds:
(4.1)lim⁡n→∞∏i=n∞F(σ0i)=0⟹lim⁡n→∞∏i=n∞F(σi)=0
for all σ∈(0,1), where the infinite product ∏ is in the sense of ◊.Proof
Case 1 (σ<σ0).
Fori∈ℕ, σi<σ0i, and since F is nondecreasing, F(σi)≤F(σ0i) hold. And hence ∏i=n∞F(σi)≤∏i=n∞F(σ0i),n∈ℕ. So implication (4.1) holds.Case 2 (σ≥σ0).
Ifσ=σ0, it follows
(4.2)∏i=2m∞F(σi)=[∏i=m∞F(σ2i)]◊[∏i=m∞F(σ2i+1)]≤[∏i=m∞F(σ0i)]◊[∏i=m∞F(σ0i)].
Then we have limm→∞∏i=2m∞F(σi)≤0◊0=0. And limm→∞∏i=2m+1∞F(σi)≤limm→∞∏i=2m+2∞F(σi)=0. Thus it follows that  limm→∞∏i=m∞F(σi)=0 for σ=σ0. Since F is nondecreasing, it is easy to show limm→∞∏i=m∞F(σi)=0 for σ<σ0.
For an arbitraryσ>σ0, there exists m∈ℕ such that σ<σ0[(1/2)m], and we can repeat the above process m-times to get limm→∞∏i=m∞F(σi)=0.Lemma 4.2.
For any monotonely nonincreasing functionG:(0,∞)→[0,1], the following implication holds:
(4.3)lim⁡n→∞∏i=n∞G(σ0i)=1⟹lim⁡n→∞∏i=n∞G(σi)=1
for all σ∈(0,1), where the infinite product ∏ is in the sense of *.Proof.
One can take a similar procedure as in the proof of Lemma4.1 to complete the proof of this lemma. For simplicity, we omit the detailed argument. We refer the reader to [7] for further details.Lemma 4.3.
We definexn=Tn(x0)(n∈ℕ). Then  {xn} is a Cauchy sequence.Proof.
We assumeF(x)=N(x0,T(x0),1/x) and G(x)=M(x0,T(x0),1/x) for x>0. Then F(x) (G(x)) is nondecreasing (nonincreasing) mapping from (0,∞) into [0,1]. Taking 1>σ>k, by Lemmas 4.1 and 4.2, we have
(4.4)lim⁡n→∞∏i=n∞M(x0,T(x0),1(k/σ)i)=1,lim⁡n→∞∏i=n∞N(x0,T(x0),1(k/σ)i)=0.
Since σ<1, ∑n=1∞σn<∞, for any ɛ0>0 there exists n0 such that ∑n=n0∞σn<ɛ0. For the above ɛ0>0, if m>n>n0 and t>ɛ0,
(4.5)M(xn,xm,t)≥M(xn,xm,ɛ0)≥∏i=nm-1M(xi,xi-1,σi)≥∏i=nm-1M(x0,Tx0,σiki)=∏i=nm-1M(x0,Tx0,1(k/σ)i),N(xn,xm,t)≤N(xn,xm,ɛ0)≤∏i=nm-1N(xi,xi-1,σi)≤∏i=nm-1N(x0,Tx0,σiki)=∏i=nm-1N(x0,Tx0,1(k/σ)i)
hold.
And according to (4.4), we have limn→∞M(xn,xm,t)=1 and limn→∞N(xn,xm,t)=0 for m>n. So {xn} is Cauchy sequence.
SinceX is complete, there exists some y∈X such that limn→∞xn=y. One can prove y is the unique fixed point of T by repeating the same process as in the proof of Theorem 1.1. Thus, we complete the proof of Theorem 1.2.



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