# Fixed Point Theorems in Fuzzy Metric Spaces

Authors: Ismat Beg; Shaban Sedghi; Nabi Shobe
Journal: International Journal of Analysis
 (2013)

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## Abstract

We prove a fixed point theorem for mappings satisfying an implicit relation in a complete fuzzy metric space.


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## Body

## 1. Introduction and Preliminaries

The concept of a fuzzy set was introduced by Zadeh [1] in 1965. This concept was used in topology and analysis by many authors. George and Veeramani [2] modified the concept of fuzzy metric space introduced by Kramosil and Michalek [3] and defined the Hausdorff topology of fuzzy metric spaces, which have important applications [4] in quantum particle physics. Afterwards, Beg and Abbas [5], Vasuki [6], Popa [7], and Imad et al. [8, 9] obtained results about fixed point and invariant approximation in fuzzy metric spaces. The aim of this paper is to obtain fixed point of mapping satisfying an implicit relation on fuzzy metric spaces. Our results generalize several known results in the literature.First, we give some basic definitions.Definition 1 (see [10]).
A binary operation*:[0,1]2→[0,1] is called a continuous t-norm if ([0,1],*) is an abelian topological monoid, that is,(1)
* is associative and commutative,(2)
* is continuous,(3)
a*1=a for all a∈[0,1],(4)
a*b≤c*d whenever a≤c and b≤d, for each a,b,c,d∈[0,1].Four typical examples of a continuoust-norms are a*1b=min{a,b}, a*2b=ab/max{a,b,λ} for 0<λ<1 and a*3b=ab, a*4b=max{a+b-1,0}.Definition 2 (see [2]).
The 3-tuple(X,M,*) is called a fuzzy metric space if X is an arbitrary nonempty set, * is a continuous t-norm, and M is a fuzzy set on X2×[0,∞) satisfying the following conditions, for each x,y,z∈X and t,s>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.Let(X,M,*) be a fuzzy metric space. For t>0, the open ball B(x,r,t) with a center x∈X and a radius 0<r<1 is defined by
(1)B(x,r,t)={y∈X:M(x,y,t)>1-r}.A subsetA⊂X is called open if, for each x∈A, there exist t>0 and 0<r<1 such that B(x,r,t)⊂A. Let τ denote the family of all open subsets of X. Then, τ is called the topology on X induced by the fuzzy metric M. This topology is Hausdorff and first countable. A subset A of X is said to be F-bounded if there exist t>0 and 0<r<1 such that M(x,y,t)>1-r for all x,y∈A.Example 3.
LetX=ℝ. Denote a*b=ab for all a,b∈[0,1]. For each t∈(0,∞), define
(2)M(x,y,t)=tt+|x-y|,
for all x,y∈X. Then, (X,M,*) is a fuzzy metric space.Example 4.
Let(X,d) be an ordinary metric space and ψ be an increasing and continuous function from ℝ+ into (0,1) such that limt→∞ψ(t)=1. Four typical examples of these functions are ψ(x)=x/(x+1), ψ(x)=sin(πx/(2x+1)), ψ(x)=1-e-x, and ψ(x)=e-1/x. Let a*b≤ab for all a,b∈[0,1]. For each t∈(0,∞), define
(3)M(x,y,t)=[ψ(t)]d(x,y),
for all x,y∈X. It is easy to see that (X,M,*) is a fuzzy metric space.Definition 5 (see [2]).
Let(X,M,*) be a fuzzy metric space.(i)
A sequence{xn} in X is said to be convergent to a point x∈X if limn→∞M(xn,x,t)=1 for all t>0.(ii)
A sequence{xn} in X is called Cauchy sequence if, for each 0<ϵ<1 and t>0, there exits n0∈ℕ such that M(xn,xm,t)>1-ϵ for each n,m≥n0.(iii)
A fuzzy metric space in which every Cauchy sequence is convergent is said to be complete.(iv)
A fuzzy metric space in which every sequence has a convergent subsequence is said to be compact.Lemma 6 (see [11]).
For allx,y∈X, M(x,y,·) is a nondecreasing function.Definition 7.
Let(X,M,*) be a fuzzy metric space. M is said to be continuous on X2×(0,∞) if
(4)limn→∞M(xn,yn,tn)=M(x,y,t),
whenever {(xn,yn,tn)} is a sequence in X2×(0,∞) which converges to a point (x,y,t)∈X2×(0,∞), that is,
(5)limn→∞M(xn,x,t)=limn→∞M(yn,y,t)=1,limn→∞M(x,y,tn)=M(x,y,t).Lemma 8 (see [11]).
M is a continuous function on X2×(0,∞).


## 2. Main Results

Let(X,M,*) be a fuzzy metric space and ∅≠S⊆X. Define δM(S,t)=inf{M(x,y,t):x,y∈S} for all t>0. For an An={xn,xn+1,…} in fuzzy metric space (X,M,*), let rn(t)=δM(An,t). Then, rn(t) is finite for all n∈N, {rn(t)} is nonincreasing, rn(t)→r(t) for some 0≤r(t)≤1, and also rn(t)≤M(xl,xk,t) for all l,k≥n.Letℱ be the set of all continuous functions F:[0,1]3×[0,1]→[-1,1] such that F is nondecreasing on [0,1]3 satisfying the following conditions.(
F
1
)
:
F
(
(
u
,
u
,
u
)
,
v
)
≤
0 implies that v≥γ(u) where γ:[0,1]→[0,1] is a nondecreasing continuous function with γ(s)>s for s∈[0,1).Example 9.
F
:
[
0,1
]
3
×
[
0,1
]
→
[
-
1,1
], defined the following:(i)
F((t1,t2,t3),t4)=γ(min{t1,t2,t3})-t4,(ii)
F((t1,t2,t3),t4)=γ(∑i=13aiti)-t4 such that for every ai≥0, ∑i=13ai=1, where γ(s)=sh for 0<h<1.
In this paper, our main result is the following theorem.Theorem 10.
Let(X,M,*) be a complete bounded fuzzy metric space and T a self map of X satisfying for all x,y∈X the implicit relation(i)
(6)F(M(x,y,t),M(Tx,x,t),M(Tx,y,t),M(Tx,Ty,t))≤0,where F∈ℱ.
Then,T has a unique fixed point p in X, and T is continuous at p.Proof.
Letx0∈X and Txn=xn+1. Let rn(t)=δM(An,t), where An={xn,xn+1,…}. Then, we know limn→∞rn(t)=r(t) for some 0≤r(t)≤1. If xn+1=xn for some n∈N, then T has a fixed point, p∈X. Assume that xn+1≠xn for each n∈N. Let k∈N be fixed. Taking x=xn-1, y=xn+m-1 in (i) where n≥k and m∈N, we have
(7)F(M(xn-1,xn+m-1,t),M(Txn-1,xn-1,t),M(Txn-1,xn+m-1,t),M(Txn-1,Txn+m-1,t))=F(M(xn-1,xn+m-1,t),M(xn,xn-1,t),M(xn,xn+m-1,t),M(xn,xn+m,t))≤0.
Thus, we have(8)F(rn-1(t),rn-1(t),rn(t),M(xn,xn+m,t))≤0,
since F is nondecreasing on [0,1]3. Also, since rn(t) is nonincreasing, we have
(9)F(rk-1(t),rk-1(t),rk-1(t),M(xn,xn+m,t))≤0,
which implies that
(10)M(xn,xn+m,t)≥γ(rn-1(t)).
Thus, for alln≥k, we have infn≥k{M(xn,xn+m,t)}=rk(t)≥γ(rn-1(t)). Letting k→∞, we get r(t)≥γ(r(t)). If r(t)≠1, then r(t)≥γ(r(t))>r(t), which is a contradiction. Thus r(t)=1 and hence limn→∞γn(t)=1. Thus, given ϵ>0, there exists an n0∈N such that rn(t)>1-ϵ. Then, we have for n≥n0 and m∈N, M(xn,xn+m,t)>1-ϵ. Therefore, {xn} is a Cauchy sequence in X. By the completeness of X, there exists a p∈X such that limn→∞Txn=limn→∞xn+1=p. Taking x=xn,y=p in (i), we have
(11)F(M(xn,p,t),M(Txn,p,t),M(Txn,xn,t),M(Txn,Tp,t))=F(M(xn,p,t),M(xn+1,p,t),M(xn+1,xn,t),M(xn+1,Tp,t))≤0.
Takinglimn→∞, we have
(12)F(M(p,p,t),M(p,p,t),M(p,p,t),M(p,Tp,t))≤0,
which implies that M(p,Tp,t)≥γ(M(p,p,t))=γ(1)=1. Hence, Tp=p. For the uniqueness, let p and w be fixed points of  T. Taking x=p, y=w in (i), we have
(13)F(M(p,w,t),M(Tp,p,t),M(Tp,w,t),M(Tp,Tw,t))=F(M(p,w,t),M(p,p,t),M(p,w,t),M(p,w,t))≤0.
SinceF is nondecreasing on [0,1]3, we have
(14)F(M(p,w,t),M(p,w,t),M(p,w,t),M(p,w,t))≤0,
which implies that M(p,w,t)≥γ(M(p,w,t))>M(p,w,t), which is a contradiction. Thus, we have p=w. Now, we show that T is continuous at p. Let {yn} be a sequence in X and limn→∞yn=p. Taking x=p,y=yn in (i), we have
(15)F(M(p,yn,t),M(Tp,p,t),M(Tp,yn,t),M(Tp,Tyn,t))=F(M(p,yn,t),M(p,p,t),M(p,yn,t),M(p,Tyn,t))≤0,
which implies that γ(M(p,Tyn,t))≥γ(M(p,yn,t)). Taking limit inf, we have
(16)limninf{γ(M(p,Tyn,t))}≥limninfγ(M(p,yn,t))=γ(1)=1.
Thus,limn→∞Tyn=p=Tp. Hence, T is continuous at p.Corollary 11.
Let(X,M,*) be a complete bounded fuzzy metric space, m∈N, and T a self map of X satisfying for all x,y∈X,(i)
(17)F(M(x,y,t),M(Tmx,x,t),M(Tmx,y,t),M(Tmx,Tmy,t))≤0,where F∈ℱ.
Then,T has a unique fixed point p in X, and Tm is continuous at p.Proof.
From Theorem10, Tm has a unique fixed point p in X and Tm is continuous at p. Since Tp=TTmp=TmTp, Tp is also a fixed point of Tm. By the uniqueness, it follows Tp=p.In Theorem10 (Corollary 11, resp.), If we take F((t1,t2,t3),t4)=γ(min{t1,t2,t3})-t4, then we have the next result.Corollary 12.
Let(X,M,*) be a complete bounded fuzzy metric space and T a self map of X satisfying for all x,y∈X,(i)
(18)M(Tx,Ty,t)≥γ(min{M(x,y,t),M(Tx,x,t),M(Tx,y,t)}).
Then,T has a unique fixed point p in X, and T is continuous at p.Example 13.
LetX=R. Define M:X2×[0,∞)→R+ by M(x,y,t)=e-|x-y|/t for all x,y∈X and t>0. Then, (X,M,*) is a complete fuzzy metric space where a*b=ab. Define map T:X→X by T(x)=x/2 for x∈X and let γ:[0,1]→[0,1] defined by γ(s)=s1/2. It is easy to see that
(19)M(Tx,Ty,t)=e-|x/2-y/2|/t=e-|x-y|/t=M(x,y,t)≥min{M(x,y,t),M(Tx,x,t),M(Tx,y,t).
Thus,T satisfy all the hypotheses of Corollary 12, and hence T has a unique fixed point. Indeed, 0 is the unique fixed point of T.Corollary 14.
Let(X,M,*) be a complete bounded fuzzy metric space, m∈N, and T a self map of X satisfying for all x,y∈X,(i)
(20)M(Tmx,Tmy,t)≥γ(min{M(x,y,t),M(Tmx,x,t),M(Tmx,y,t)}).
Then,T has a unique fixed point p in X, and Tm is continuous at p.Corollary 15.
Let(X,M,*) be a complete bounded fuzzy metric space and T a self map of X satisfying for all x,y∈X,(i)
(21)M(Tx,Ty,t)≥a1M(x,y,t)+a2M(Tx,x,t)+a3M(Tx,y,t),such that for every ai≥0, ∑i=13ai=1. Then, T has a unique fixed point p in X, and T is continuous at p.
Letℱ be the class of all continuous functions F:[0,1]3×[0,1]→[-1,1] such that F is nondecreasing on [0,1]3 satisfying F((u,u,u),v)<0 implies that v>u.Example 16.
LetF:[0,1]3×[0,1]→[-1,1] defined by F((t1,t2,t3),t4)=(min{t1,t2,t3})h-t4 for some 0<h<1. Then, F is continuous on [0,1]3 and nondecreasing on [0,1]3. Also, we have that F((u,u,u),v)<0 implies that v>u. Thus F∈ℱ.Theorem 17.
Let(X,M,*) be a compact fuzzy metric space and T a continuous self map of X satisfying for all x,y∈X with(i)
(22)F(M(x,y,t),M(Tx,x,t),M(Tx,y,t),M(Tx,Ty,t))<0,where F∈ℱ.
Then,T has a unique fixed point p in X.Proof.
We know that forn=1,2,…,TnX is compact and Tn+1X⊂TnX. Let X0=⋂n=1∞TnX. Then, X0 is a nonempty compact subset of X, and TX0=X0. We claim that X0 is a singleton set. Suppose X0 is not singleton. Then, we know that the function M:X2×(0,∞)→[-1,1] has a minimum value. That is, there exists a (x0,y0,t)∈X02×(0,∞) such that M(x0,y0,t)≤M(x,y,t) for all x,y∈X0. Since TX0=X0, there exist x1,y1∈X0 such that Tx1=x0,Ty1=y0. Thus, we have
(23)F(M(x1,y1,t),M(x0,x1,t),M(x0,y1,t),M(x0,y0,t))<0,
and so F(M(x0,y0,t),M(x0,y0,t),M(x0,y0,t),M(x0,y0,t))<0 which implies that M(x0,y0,t)>M(x0,y0,t). This is a contradiction. Thus, X0 is singleton, and hence T has a fixed point in X. Uniqueness of fixed point of T follows from (i).Corollary 18.
Let(X,M,*) be a compact fuzzy metric space, m∈N, and Tm a continuous self map of X satisfying for all x,y∈X,(i)
(24)F(M(x,y,t),M(Tmx,x,t),M(Tmx,y,t),M(Tmx,Tmy,t))<0,where F∈ℱ.
Then,T has a unique fixed point p in X.Proof.
From Theorem17, Tm has a unique fixed point p in X. Since Tp=TTmp=TmTp, Tp is also a fixed point of Tm. By the uniqueness, it follows Tp=p.



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*Source: 934145-2013-01-08.xml*