# Common Fixed Point Theorems of New Contractive Conditions in Fuzzy Metric Spaces

Authors: Jiang Zhu; Yuan Wang; Shin Min Kang
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

Some new limit contractive conditions in fuzzy metric spaces are introduced,  by using property (E.A), some common fixed point theorems for four maps are proved in GV-fuzzy metric spaces. As an application of our results, some new contractive conditions are presented, and some common fixed point theorems are proved under these contractive conditions. The contractive conditions presented in this paper contain or generalize many contractive conditions that appeared in the literatures. Some examples are given to illustrate that our results are real generalizations for the results in the references and to show that our limit contractive conditions are important for the existence of fixed point.


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## Body

## 1. Introduction

The theory of fuzzy sets was first introduced by Zadeh [1], after many authors introduced the notion of fuzzy metric spaces in different ways (see [2–5]). In particular, Kramosil and Michálek [4] generalized the concept of probabilistic metric space given by Menger [6] to the fuzzy framework. Later on, George and Veeramani [2] modified the concept of fuzzy metric space introduced by Kramosil and Michálek and defined the Hausdorff and first countable topology on the modified fuzzy metric space. Actually, this topology can also be constructed on each fuzzy metric space in the sense of Kramosil and Michálek and it is metrizbale [2, 7]. Other recent contributions to the study of fuzzy metric spaces in the sense of [2] may be found in [8, 9]. Since then, many authors have proved fixed point and common fixed point theorems in fuzzy metric spaces in the sense of [2]. Especially, we want to emphasize that some common fixed point theorems for φ-type contraction maps in fuzzy metric spaces have been recently obtained in [10–16].Quite recently, Miheţ [13] proved some existence theorems of common fixed point for two self-mappings f, g of a fuzzy metric space (X,M,*) under the following contractive condition:
(1)M(fx,fy,t)≥φ(min{M(gx,gy,t),M(fx,gx,t),M(fy,gy,t),M(fx,gy,t),M(fy,gx,t)})
for all x,y∈X and t>0, where φ:[0,1]→[0,1] is continuous and nondecreasing on [0,1], and φ(x)>x for all x∈[0,1].C. Vetro and P. Vetro [15] proved some existence theorems of common fixed point for two self-mappings f, g of a fuzzy metric space (X,M,*) under the following contractive condition:
(2)1M(fx,fy,t)-1≤r(1min{M(gx,gy,t),M(fx,gx,t),M(fy,gy,t)}-1)
for all x,y∈X and t>0, where r:[0,+∞)→[0,+∞) with r(τ)<τ for every τ>0, an upper semicontinuous function.Gopal et al. [10] proved some existence theorems of common fixed point for four self-mappings A, B, S and T of a fuzzy metric space (X,M,*) under the following contractive condition:
(3)1M(Ax,By,t)-1≤r(1min{M(Sx,Ty,t),M(Ax,Sx,t),M(By,Ty,t)}-1)
for all x,y∈X and t>0, where r:[0,+∞)→[0,+∞) with r(τ)<τ for every τ>0, an upper semicontinuous function.Imdad and Ali [11], Vijayaraju and Sajath [16] proved some existence theorems of common fixed point for four self-mappings A, B, S, and T of a fuzzy metric space (X,M,*) under the following contractive condition:
(4)M(Ax,By,t)≥ϕ(min{M(Sx,Ty,t),M(Sx,Ax,t),M(By,Ty,t)})
for all x,y∈X and t>0, where ϕ:[0,1]→[0,1] with ϕ(s)>s whenever 0<s<1 is a continuous or increasing and left-continuous function.Imdad et al. [12] proved some existence theorems of common fixed point for four self-mappings f, g, S, and T of a fuzzy metric space (X,M,*) under the following contractive condition:
(5)M(fx,gy,t)≥φ(min{M(Sx,Ty,t),M(fx,Sx,t),M(fy,Ty,t),M(fx,Ty,t),M(fy,Sx,t)})
for all x,y∈X and t>0, where φ:[0,1]→[0,1] is continuous and nondecreasing on [0,1], and φ(x)>x for all x∈[0,1].Shen et al. [14] proved an existence theorem of fixed point for self-mapping T of a fuzzy metric space (X,M,*) under the following contractive condition:
(6)φ(M(Tx,Ty,t))≤k(t)·φ(M(x,y,t))
for all x,y∈X and t>0, where φ:[0,1]→[0,1] satisfies the following properties:(P1)
φ is strictly decreasing and left continuous;(P2)
φ(λ)=0 if and only if λ=1.Furthermore, letk be a function from (0,∞) into (0,1).The purpose of this paper is to present limit contractive conditions to unify all of theseφ-type nonlinear contractive conditions. Then, by using property (E.A), some common fixed point theorems for four maps are proved in GV-fuzzy metric spaces. As an application of our limit contraction condition, we present some new φ-type integral contractive conditions and some common fixed point theorems for four maps in GV-fuzzy metric spaces under these contractive conditions. Our results generalize the corresponding results in [10–16]. Some examples are given to illustrate that our results are real generalizations for the results in the references and show that our limit contractive conditions are important for the existence of fixed point.For the reader’s convenience, we recall some terminologies from the theory of fuzzy metric spaces, which will be used in what follows.Definition 1 (see [4]).
Acontinuoust-norm in the sense of Kramosil and Michálek is a binary operation * on [0,1] satisfying the following conditions:(1)
* is associative and commutative,(2)
a*1=a for all a∈[0,1],(3)
a*b≤c*d whenever a≤c and b≤d for all a,b,c,d∈[0,1],(4)
the mapping*:[0,1]×[0,1]→[0,1] is continuous.Three typical examples of continuoust-norm are a*b=max{a+b-1,0}, a*b=ab,  and a*b=min{a,b}.Definition 2 (see [2]).
Afuzzy metric space in the sense of George and Veeramani is a triple (X,M,*), where X is a nonempty set, M is a fuzzy set on X2×(0,∞), and * is a continuous t-norm such that the following conditions are satisfied for all x,y,z∈X and t,s>0(GV-1)
M(x,y,t)>0;(GV-2)
M(x,y,t)=1 if and only if x=y;(GV-3)
M(x,y,t)=M(y,x,t);(GV-4)
M(x,y,t)*M(y,z,s)≤M(x,z,t+s);(GV-5)
M(x,y,·):(0,∞)→[0,1] is continuous.In what follows, fuzzy metric spaces in the sense of George and Veeramani will be calledGV-fuzzy metric spaces.Lemma 3 (see [17]).
Let(X,M,*) be a GV-fuzzy metric space. Then M(x,y,t) is non-decreasing with respect to t for all x,y∈X.Definition 4 (see [13]).
Let(X,M,*) be a GV-fuzzy metric space. Then one has the following:(1)
a sequence{xn} in X is said to be convergent to x∈X if limn→∞M(xn,x,t)=1 for all t>0.(2)
a sequence{xn} in X is said to be Cauchy sequence if limn→∞M(xn,xn+p,t)=1 for all t>0 and p∈ℕ.(3)
a fuzzy metric space is calledcomplete if every Cauchy sequence converges in X.Lemma 5 (see [5]).
Let(X,M,*) be a GV-fuzzy metric space. Then M is a continuous function on X2×(0,∞).Definition 6 (see [18]).
Let(X,M,*) be a GV-fuzzy metric space. Then two self-mappings A and S of (X,M,*) satisfy property(E.A) if there exists a sequence {xn} in X and z in X such that {Axn} and {Sxn} converge to z that is, for any t>0,
(7)limn→∞M(Axn,z,t)=limn→∞M(Sxn,z,t)=1.Definition 7 (see [18]).
Let(X,M,*) be a GV-fuzzy metric space. Then two pairs of self-mappings A,S and B,T of (X,M,*) are said to share common property(E.A) if there exist sequences {xn} and {yn} in X such that, for any t>0,
(8)limn→∞M(Axn,z,t)=limn→∞M(Sxn,z,t)=limn→∞M(Byn,z,t)=limn→∞M(Tyn,z,t)=1
for some z∈X.Definition 8 (see [19]).
Let(X,M,*) be a GV-fuzzy metric space. Then two self-mappings A and S of (X,M,*) are said to be weak compatible if they commute at their coincidence point; that is,
(9)Ax=SximpliesSAx=ASx.


## 2. Main Results

Let(X,M,*) be a GV-fuzzy metric space, and let A, B, S, and T be self-mappings of (X,M,*). For any x,y∈X and t>0, we define (10)Min(x,y,t)=min{M(Sx,Ty,t),M(Sx,Ax,t),M(Ty,By,t),M(Sx,By,t),M(Ty,Ax,t)}.Consider that(C1)0<lim_n→∞Min(xn,yn,t)=L(t)<1 implies that lim¯n→∞M(Axn,Byn,t)>L(t) for all t>0 and any sequence {xn} and {yn} in X.Theorem 9.
Let(X,M,*) be a GV-fuzzy metric space, and A, B, S, and T be self-mappings of (X,M,*) such that one has the following:(1)
(C1) holds;(2)
A, S are weakly compatible and B, T are weakly compatible;(3)
A, S satisfy property (E.A) or B, T satisfy property (E.A);(4)
AX⊂TX and BX⊂SX;(5)
one of the range of the mappingsA, B, S,  or T is a closed subspace of X.Then A, B, S, and T have a unique common fixed point.Proof.
Suppose thatB,T satisfy the property (E.A). Then there exists a sequence {xn} in X such that
(11)limn→∞M(Bxn,z,t)=limn→∞M(Txn,z,t)=1
for all t>0 and some z∈X.
SinceBX⊂SX, there exists a sequence {yn} in X such that Bxn=Syn. Hence limn→∞M(Syn,z,t)=1 for all t>0.
Suppose thatSX is a closed subspace of X. Then z=Su for some u∈X. Subsequently, we have that
(12)limn→∞M(Bxn,Su,t)=limn→∞M(Txn,Su,t)=limn→∞M(Syn,Su,t)=1
for all t>0. Then by using Lemma 5 we have that
(13)limn→∞M(Txn,Bxn,t)=1,limn→∞M(Txn,Au,t)=M(Su,Au,t).
Thus, we have that
(14)lim_n→∞Min(u,xn,t)=lim_n→∞min{M(Su,Txn,t),M(Su,Au,t),M(Txn,Bxn,t),M(Su,Bxn,t),M(Txn,Au,t)}=M(Su,Au,t)=limn→∞M(Bxn,Au,t)
for any t>0; that is,
(15)lim_n→∞Min(u,xn,t)=lim¯n→∞M(Au,Bxn,t)=M(Su,Au,t).
If Su≠Au, then there exists t>0 such that 0<M(Su,Au,t)<1. This and (C1) imply that
(16)lim¯n→∞M(Au,Bxn,t)>lim_n→∞Min(u,xn,t).
This is a contradiction. Thus, we have that Su=Au. The weak compatibility of A and S implies that ASu=SAu, and then AAu=ASu=SAu=SSu.
On the other hand, sinceAX⊂TX, there exists v∈X such that Au=Tv. Since for any t>0(17)Min(u,v,t)=min{M(Su,Tv,t),M(Su,Au,t),M(Tv,Bv,t),M(Su,Bv,t),M(Tv,Au,t)}=M(Au,Bv,t),
by (C1), we get  M(Au,Bv,t)=1, which implies that Au=Bv; that is, Au=Su=Tv=Bv. The weak compatibility of B and T implies that BTv=TBv and TTv=TBv=BTv=BBv.
Let us show thatAu is a common fixed point of A, B, T, and S. Since
(18)Min(Au,v,t)=min{M(SAu,Tv,t),M(SAu,AAu,t),M(Tv,Bv,t),M(SAu,Bv,t),M(Tv,AAu,t)}=min{M(AAu,Bv,t),1,1,M(AAu,Bv,t),M(Tv,AAu,t)}=M(AAu,Bv,t),
by (C1), we get that M(AAu,Bv,t)=1, which implies that AAu=Bv. Therefore, AAu=SAu=Bv=Au, and Au is a common fixed point of A and S. Similarly, we can prove that Bv is a common fixed point of B and T. Noting that Au=Bv, we conclude that Au is a common fixed point A,B,T, and S. The proof is similar when TX is assumed to be a closed subspace of X. The cases in which AX or BX is a closed subspace of X are similar to the cases in which TX or SX, respectively, is closed since AX⊂TX and BX⊂SX. If Au=Bu=Tu=Su=u and Av=Bv=Tv=Su=v, then
(19)Min(u,v,t)=min{M(Su,Tv,t),M(Su,Au,t),M(Tv,Bv,t),M(Su,Bv,t),M(Tv,Au,t)}=min{M(Au,Tv,t),1,1,M(Su,Bv,t),M(Tv,Au,t)}=M(Au,Bv,t).
Therefore, by (C1), we have u=v; that is, the common fixed point is unique. This completes the proof.To introduce some integral contractive conditions, leth:[0,1]→ℝ+ be nonnegative, Lebesgue integrable, and satisfy
(20)∫0ε‍h(t)dt>0
for each 0<ε≤1. We denote d=∫01‍h(t)dt.Theorem 10.
Let(X,M,*) be a GV-fuzzy metric space, and A, B, S, and T be self-mappings of (X,M,*). If one of the following conditions is satisfied
(
C
2
) There exists a function ω:[0,d]→[0,d] such that for any 0<s<d, ω(s)<s,  lim¯u→sω(u)<s and for any x,y∈X, t∈(0,∞), 0<Min(x,y,t)<1 implies that
(21)ω(∫0M(Ax,By,t)‍h(s)ds)≥∫0Min(x,y,t)‍h(s)ds.
(
C
3
) There exists a function ϕ:[0,d]→[0,d] such that, for any 0<s<d, ϕ(s)>s, lim_t→sϕ(t)>s, and for any x,y∈X, t∈(0,∞), 0<Min(x,y,t)<1 implies that
(22)∫0M(Ax,By,t)‍h(s)ds≥ϕ(n∫0Min(x,y,t)‍h(s)ds).
Then (C1) holds.Proof.
(
C
2
)
⇒
(
C
1
). Assume that (C2) holds. If {xn} and {yn} in X and 0<lim_n→∞Min(xn,yn,t)=L(t)<1, then
(23)lim¯n→∞ω(∫0M(Axn,Byn,t)‍h(τ)dτ)≥lim¯n→∞∫0Min(xn,yn,t)‍h(τ)dτ≥∫0lim_n→∞Min(xn,yn,t)‍h(τ)dτ=∫0L(t)‍h(τ)dτ.
If lim¯n→∞M(Axn,Byn,t)=1, then lim¯n→∞M(Axn,Byn,t)>L(t). If lim¯n→∞M(Axn,Byn,t)=r(t)<1, then there exists a subsequence {M(Axni,Byni,t)} such that
(24)limi→∞M(Axni,Byni,t)=lim¯n→∞M(Axn,Byn,t)=r(t).
Thus,
(25)lim¯n→∞∫0M(Axn,Byn,t)‍h(τ)dτ=limi→∞∫0M(Axni,Byni,t)‍h(τ)dτ=∫0r(t)‍h(τ)dτ>lim¯u→∫0r(t)‍h(τ)dτω(u)≥lim¯i→∞ω(∫0Min(xni,yni,t)‍h(τ)dτ)≥∫0L(t)‍h(τ)dτ.
This implies that
(26)lim¯n→∞M(Axn,Byn,t)=r(t)>L(t).
Thus, (C1) holds.
(
C
3
)
⇒
(
C
1
). Assume that (C3) holds. If {xn} and {yn} in X and 0<lim_n→∞Min(xn,yn,t)=L(t)<1, then there exists a subsequence {Min(xni,yni,t)} such that limi→∞Min(xni,yni,t)=L(t). This implies that
(27)lim_i→∞ϕ(∫0Min(xni,yni,t)‍h(τ)dτ)≥lim_t→∫0L(t)‍h(τ)dτϕ(t)>∫0L(t)‍h(τ)dτ=limi→∞∫0Min(xni,yni,t)‍h(τ)dτ.
It follows from
(28)∫0M(Ax,By,t)‍h(τ)dτ≥ϕ(∫0Min(x,y,t)‍h(τ)dτ)
that we can get
(29)lim¯n→∞∫0M(Axn,Byn,t)‍h(τ)dτ=∫0lim¯n→∞M(Axn,Byn,t)‍h(τ)dτ≥lim¯n→∞ϕ(∫0Min(xn,yn,t)‍h(τ)dτ)≥lim¯i→∞ϕ(∫0Min(xni,yni,t)‍h(τ)dτ)≥lim_i→∞ϕ(∫0Min(xni,yni,t)‍h(τ)dτ)>∫0L(t)‍h(τ)dτ.
This implies that lim¯n→∞M(Axn,Byn,t)>L(t);  that is, (C1) holds.It follows from Theorems9 and 10 that we have the following fixed point theorems for integral type contractive mappings.Theorem 11.
Let(X,M,*) be a GV-fuzzy metric space, and let A, B, S, and T be self-mappings of (X,M,*) such that one has the following(1)
one of(C2)-(C3) holds;(2)
A, S are weakly compatible and B, T are weakly compatible;(3)
A, S satisfy the property (E.A) or B, T satisfy the property (E.A);(4)
AX⊂TX and BX⊂SX;(5)
one of the range of the mappingsA, B, S, or T is a closed subspace of X. Then A, B, S, and T have a unique common fixed point.In(C2) and (C3), by taking h(t)≡1, we have the following contractive conditions.(
C
4
) There exists a function ω:[0,1]→[0,1] such that for any 0<s<1, ω(s)<s, lim¯u→sω(u)<s and for any x,y∈X, t∈(0,∞), 0<Min(x,y,t)<1 implies that
(30)ω(M(x,y,t))≥Min(x,y,t).(
C
5
) There exists a function ϕ:[0,1]→[0,1] such that, for any 0<s<1,ϕ(s)>s,lim_t→sϕ(t)>s,and for any x,y∈X, t∈(0,∞), 0<Min(x,y,t)<1 implies that
(31)M(Ax,By,t)≥ϕ(Min(x,y,t)).Corollary 12.
Let(X,M,*) be a GV-fuzzy metric space, and let A, B, S, and T be self-mappings of (X,M,*) such that one has the following:(1)
one of(C4)-(C5) holds;(2)
A, S are weakly compatible and B, T are weakly compatible;(3)
A, S satisfy property (E.A) or B, T satisfy the property (E.A);(4)
AX⊂TX and BX⊂SX;(5)
one of the range of the mappingsA, B, S, or T is a closed subspace of X. Then A, B, S and T have a unique common fixed point.Remark 13.
As a special case of(C4), we can take function ω:[0,1]→[0,1] as one of the following:(1)
ω is nonincreasing, for any 0<s<1, ω(s)<s, lim¯u→s-ω(u)<s;(2)
ω is an upper semicontinuous function such that, for any 0<s<1, ω(s)<s;(3)
ω is non-increasing and left-upper semicontinuous such that ω(s)<s for any 0<s<1.
As a special case of(C5), we can take function ϕ:[0,1]→[0,1] as one of the follows:(1)
ϕ is a nondecreasing and left-continuous function such that, for any 0<s<1, ϕ(s)>s;(2)
ϕ is a a lower semi-continuous function such that for any 0<s<1, ϕ(s)>s;(3)
ϕ(s)=s+ψ(s), where ψ:[0,1]→[0,1] is a continuous function with for any 0<s<1, ψ(s)>0.From Corollary12, we have the following corollaries.Corollary 14.
Let(X,M,*) be a GV-fuzzy metric space, and let A, B, S, and T be self-mappings of (X,M,*) such that one has the following:(1)
there exists an upper semicontinuous functionr:[0,+∞)→[0,+∞) with r(s)<s for any s>0 such that for all x,y∈X and t>0(32)1M(Ax,By,t)-1≤r(1Min(x,y,t)-1);(2)
A, S are weakly compatible and B, T are weakly compatible;(3)
A, S satisfy property (E.A) or B, T satisfy property (E.A);(4)
AX⊂TX and BX⊂SX;(5)
one of the range of the mappingsA, B, S or T is a closed subspace of X. Then A, B, S, and T have a unique common fixed point.Proof.
Define functionϕ:[0,1]→[0,1] by
(33)ϕ(t)={0ift=0;11+r((1/t)-1)ift∈(0,1].
Then, for any s>0,
(34)lim_t→sϕ(t)=11+lim¯t→sr((1/t)-1)=11+r((1/s)-1)>11+(1/s)-1=s,
and (32) can be rewritten as: for any x,y∈X, t∈(0,∞), 0<Min(x,y,t)<1 implies that
(35)M(Ax,By,t)≥ϕ(Min(x,y,t)).
That is, (C5) holds. Then the conclusion can be deduced from Corollary 12. This completes the proof.Corollary 15.
Let(X,M,*) be a GV-fuzzy metric space, and A,B,S, and T be self-mappings of (X,M,*) such that one has the following:(1)
there exists a strictly decreasing and left continuous functionφ:[0,1]→[0,1] with φ(λ)=0 if and only if λ=1 and function k:(0,∞)→(0,1) such that for all x,y∈X(x≠y) and t>0(36)φ(M(Ax,By,t))≤k(t)·φ(Min(x,y,t));(2)
A, S are weakly compatible and B, T are weakly compatible;(3)
A, S satisfy property (E.A) or B, T satisfy property (E.A);(4)
AX⊂TX and BX⊂SX;(5)
one of the range of the mappingsA, B, S, or T is a closed subspace of X.Then, A, B, S and T have a unique common fixed point.Proof.
Define functionϕ:[0,1]×(0,+∞)→[0,1] by
(37)ϕ(s,t)=φ-1(k(t)φ(s)).
Since φ is strictly decreasing and left continuous, we have that φ-1 is strictly decreasing and right continuous and ϕ(s,t) is increasing in s. Then we have that
(38)lim_u→s-ϕ(u,t)=φ-1(k(t)φ(s)).
Also we have that ϕ(u,t)≥ϕ(s,t) for u>s; this shows that
(39)lim_u→s+ϕ(u,t)≥φ-1(k(t)φ(s)).
That is, we can get that
(40)lim_u→sϕ(u,t)≥ϕ(s,t)=φ-1(k(t)φ(s))>φ-1(φ(s))=s,
and (36) can be rewritten as follows: for any x,y∈X, t∈(0,∞), 0<Min(x,y,t)<1 implies that
(41)M(Ax,By,t)≥ϕ(Min(x,y,t),t).
If {xn} and {yn} in X and 0<lim_n→∞Min(xn,yn,t)=L(t)<1, then there exists a subsequence {Min(xni,yni,t)} such that limi→∞Min(xni,yni,t)=L(t). This shows that
(42)lim_i→∞ϕ(Min(xni,yni,t),t)≥lim_s→L(t)ϕ(s,t)>L(t)=limi→∞Min(xni,yni,t).
It follows from M(Ax,By,t)≥ϕ(Min(x,y,t),t) that we can get
(43)lim¯n→∞M(Axn,Byn,t)≥lim¯n→∞ϕ(Min(xn,yn,t),t)≥lim¯i→∞ϕ(Min(xni,yni,t),t)≥lim_i→∞ϕ(Min(xni,yni,t),t)>L(t).
Thus, we have that (C1) holds. The conclusion can be deduced from Theorem 9. This completes the proof.Remark 16.
The main result Theorems 3.1 and 2.1 in [13] are the special cases of our Theorem 9 and Corollary 14 for A=B=f and S=T=g. Especially, it follows from Corollary 12 that the condition “φ is nondecreasing” is not needed. Therefore, our results improve and generalize the results in [13].Let(X,M,*) be a GV-fuzzy metric space, and let A, B, S, and T be self-mappings of (X,M,*). For any x,y∈X and t>0, we define
(44)Min~(x,y,t)=min{M(Sx,Ty,t),M(Sx,Ax,t),M(Ty,By,t)}.Consider that (C~1) 0<lim_n→∞Min~(xn,yn,t)=L~(t)<1 implies lim¯n→∞M(Axn,Byn,t)>L~(t) for all t>0 and any sequence {xn} and {yn} in X.By (44), we can write the conditions (C~2)–(C~5) which correspond to the conditions (C2)–(C5) in Theorem 9, Theorem 10 and Corollary 12 by replacing Min(x,y,t) with Min~(x,y,t). It is similar to the proof of Theorem 10, we can know that one of (C~4)-(C~5) can imply (C~1) holds.Corollary 17.
Let(X,M,*) be a GV-fuzzy metric space, and A, B, S and T be self-mappings of (X,M,*) such that one has the following:(1)
(C~1) or one of (C~4)-(C~5) holds;(2)
A, S are weakly compatible and B, T are weakly compatible;(3)
A, S satisfy property (E.A) or B, T satisfy property (E.A);(4)
AX⊂TX and BX⊂SX;(5)
one of the range of the mappingsA, B, S, or T is a closed subspace of X. Then A,B,S, and T have a unique common fixed point.Proof.
It is clear that we only need to prove Corollary17 for (C~1). Assume that sequence {xn} and {yn} in X, 0<lim_n→∞Min(xn,yn,t)=L(t)<1. Then
(45)lim_n→∞Min~(xn,yn,t)=L~(t)≥lim_n→∞Min(xn,yn,t)=L(t).
If lim_n→∞Min~(xn,yn,t)=L~(t)<1, then by (C~1) we have that
(46)lim¯n→∞M(Axn,Byn,t)>L~(t)≥L(t).
If lim_n→∞Min~(xn,yn,t)=L~(t)=1, then, for any n, either Min~(xn,yn,t)<1 or Min~(xn,yn,t)=1. If Min~(xn,yn,t)<1, then by (C~1) we have that M(Axn,Byn,t)>Min~(xn,yn,t). If Min~(xn,yn,t)=1, then by (44) we have that Axn=Byn. By (GV-2) we get that M(Axn,Byn,t)=1=Min~(xn,yn,t). Thus, (C~1) implies that M(Axn,Byn,t)≥Min~(xn,yn,t) for any n. This implies that
(47)lim¯n→∞M(Axn,Byn,t)≥lim_n→∞Min~(xn,yn,t)=L~(t)=1>L(t).
Therefore, (C~1) implies that (C1) holds. Then by Theorem 9 we know that A, B, S, and T have a unique common fixed point. This completes the proof.Remark 18.
In Theorem9 and Corollaries 12–17, conditions (3), (4), and (5) can be replaced by the following conditions:(3′)
A, S and B, T share the common property (E.A);(4′)
the range, of the mappingsS and T are closed subspaces of X.
The proof can be got by properly modifying the proof of Theorem9, Corollaries 12–17. Thus, from Theorem 9, Corollaries 12–17 and Remark 13, we can see that our results generalize and improve the results in [10–16].Example 19.
LetX=[0,1], M(x,y,t)=t/(t+|x-y|) for every x,y∈X and t>0 and a*b=min{a,b} for all a,b∈[0,1]. Then (X,M,*) is a fuzzy metric space. Define A=B and T=S:X→X by Ax=sin(x/(1+x)) and Tx=sinx for all x∈X. Then we have the following:(1)
A and S satisfy the property (E.A) for the sequence xn=1/n,n=1,2,…,(2)
A and S are weakly compatible,(3)
AX=[0,sin(1/2)]⊂TX=[0,sin1], and AX,TX are closed;(4)
(C1) holds. In fact, if {xn} and {yn} in X and 0<lim_n→∞Min(xn,yn,t)=L(t)<1, then there exists a subsequence {Min(xni,yni,t)} such that limi→∞Min(xni,yni,t)=L(t). Since {xn} and {yn} in X, {xn} and {yn} have convergent subsequence. With out of generality, assume that limi→∞xni=x0, and limi→∞yni=y0. Then we have that
(48)limi→∞M(Axni,Byni,t)=tt+|sin(x0/(1+x0))-sin(y0/(1+y0))|,limi→∞Min(xni,yni,t)=min{tt+|sinx0-siny0|,tt+|sinx0-sin(x0/(1+x0))|,tt+|siny0-sin(y0/(1+y0))|,tt+|sinx0-sin(y0/(1+y0))|,tt+|siny0-sin(x0/(1+x0))|}=L(t).
If0≤x0<y0, then
(49)L(t)=min{tt+siny0-sinx0,tt+sinx0-sin(x0/(1+x0)),tt+siny0-sin(y0/(1+y0)),tt+|sinx0-sin(y0/(1+y0))|,tt+siny0-sin(x0/(1+x0))}.
It is clear that
(50)siny0-sinx01+x0>siny0-sinx0,siny0-sinx01+x0>sinx0-sinx01+x0,siny0-sinx01+x0>siny0-siny01+y0.
It follows from
(51)siny0+sinx0>sinx01+x0+siny01+y0
that
(52)siny0-sinx01+x0>siny01+y0-sinx0.
On the other hand,
(53)siny0-sinx01+x0>sinx0-sinx01+x0>sinx0-siny01+y0.
Thus, we have that
(54)siny0-sinx01+x0>|sinx0-siny01+y0|.
It follows from those inequalities that
(55)L(t)=tt+siny0-sin(x0/(1+x0)).
Similarly, if 0≤y0<x0, then we can get that
(56)L(t)=tt+sinx0-sin(y0/(1+y0)),
and if x0=y0>0, then we have that
(57)L(t)=tt+sinx0-sin(x0/(1+x0)).
Thus, we have that
(58)L(t)={tt+siny0-sin(x0/(1+x0))if0≤x0<y0;tt+sinx0-sin(y0/(1+y0))if0≤y0<x0;tt+sinx0-sin(x0/(1+x0))ifx0=y0>0.
It is clear that
(59)tt+sin(y0/(1+y0))-sin(x0/(1+x0))>tt+siny0-sin(x0/(1+x0))if0≤x0<y0;(60)tt+sin(x0/(1+x0))-sin(y0/(1+y0))>tt+sinx0-sin(y0/(1+y0))if0≤y0<x0;(61)1>tt+sinx0-sin(x0/(1+x0))ifx0=y0>0.
This shows that
(62)lim¯n→∞M(Axn,Byn,t)≥limi→∞M(Axni,Byni,t)>L(t)=limi→∞Min(xni,yni,t)=lim_n→∞Min(xn,yn,t).
Thus, (C1) holds.(5) By Theorem9, A and T have common fixed point.Example 20.
LetX=[0,1], M(x,y,t)=t/(t+|x-y|) for every x,y∈X and t>0 and a*b=min{a,b} for all a,b∈[0,1]. Then (X,M,*) is a fuzzy metric space. Define A=B and T=S:X→X by
(63)Ax={12ifx=0;0ifx=1;x2ifx∈(0,1);
and Tx=x for all x∈X. Then we have the following:(1)
A and S satisfy the property (E.A) for the sequence xn=1/n, n=1,2,…;(2)
A and S are weakly compatible;(3)
AX=[0,1/2]⊂TX=[0,1], and AX, TX are closed;(4)
(C1) does not hold. In fact, for xn=1/n and yn=0, n=1,2,…, we have that
(64)Min(xn,yn,t)=min{M(1n,0,t),M(1n,12n,t),M(0,12,t),M(1n,12,t),M(0,1n,t)}=min{tt+(1/n),tt+(1/2n),tt+(1/2),tt+(1/2)-(1/n),tt+(1/n)}→tt+(1/2)(n→∞),lim¯n→∞M(Axn,Byn,t)=lim¯n→∞M(12n,12,t)=lim¯n→∞tt+(1/2)-(1/2n)=tt+(1/2).
Thus, (C1) does not hold.
(5)A and T have no common fixed point.Example19 does not satisfy the φ-type contractive conditions used in [10–16]. Example 20 shows that if (C1) does not hold, then A and T may have no common fixed point. Thus, (C1) is important for the existence of common fixed point.



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*Source: 145190-2013-05-30.xml*