Weakly Compatible Mappings along with $CLR_{S}$ property in Fuzzy Metric Spaces

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Available online at www.ispacs.com/jnaa Volume 2013, Year 2013 Article ID jnaa-00206, 12 Pages

doi:10.5899/2013/jnaa-00206 Research Article

Weakly Compatible Mappings along with

CLR

S

property in

Fuzzy Metric Spaces

Saurabh Manro1∗_{, S. S. Bhatia}1_{, Sanjay Kumar}2_{, Poom Kumam}3_{, Sumitra Dalal}4

(1)Thapar University School of Mathematics and Computer Applications Patiala, Punjab, India (2)Deenbandhu Chhotu Ram University of Science and Technology Murthal (Sonepat), India

(3)Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bang Mod, Thung khru, Bangkok 10140, Thailand

(4)Department of Mathematics, Faculty of Science, Jazan University, Saudi Arabia

Copyright 2013 c⃝Saurabh Manro, S. S. Bhatia, Sanjay Kumar, Poom Kumam and Sumitra Dalal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The aim of this work is to use newly introduced property, which is so called ”common limit in the range (CLRS)” for

four self-mappings, and prove some theorems which satisfy this property. Moreover, we establish some new existence of a common fixed point theorem for generalized contractive mappings in fuzzy metric spaces by using this new prop-erty and give some examples to support our results. Ours results does not require condition of closeness of range and so our theorems generalize, unify, and extend many results in literature. Our results improve and extend the results of Cho et al. [4], Pathak et al. [20] and Imdad et. al.[10] besides several known results.

Keywords: Weakly Compatible Maps, Fuzzy metric space, property (E.A), Common property (E.A),CLRgproperty,CLRS

prop-erty,CLRSTproperty, Common limit in range property.

AMS (2010) Subject Classification:47H10, 54H25

1 Introduction

The concept of fuzzy sets was introduced by Zadeh [28] in 1965. In 1975, Kramosil and Michalek [14] gave the notion of fuzzy metric spaces, which could be considered as a reformulation, in the fuzzy context, of the notion of probabilistic metric spaces due to Menger [17]. On the other hand, fixed point theory is one of the most famous mathematical theories with application in several branches of science. Fixed point theory in fuzzy metric spaces has been developing since the work of Heilpern [9]. He introduced the concept of fuzzy contraction mappings and proved some fixed point theorems for fuzzy contraction mappings in metric linear spaces, which is a fuzzy extension of the Banach contraction principle. In [[6], [7]], George and Veeramani introduced and studied the notion of fuzzy metric spaces that constitutes a modification of the one due to Kramosil and Michalek. Many authors have contributed to the development of this theory and apply to fixed point theory, for instance [1,3-5,8,10,11,15,16,18-20,22,24-27]. In 1976, Jungck [12] introduced the notion of commuting mappings. Afterward, Sessa [23] gave the notion of weakly

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commuting mappings. Jungck [13] defined the notion of compatible mappings to generalize the concept of weak commutativity and showed that weakly commuting mappings are compatible but the converse is not true. The concept of property (E.A) in metric space has been recently introduced by Aamri and El Moutawakil [2]. The concept of property (E.A) allows replacing the completeness requirement of the space with a more natural condition of closeness of the range. In 2009, M. Abbas et. al.[1] introduced the notion of common property (E.A).

Recently in 2011, Sintunavarat and Kumam [25] introduced the concept of the common limit in the range property and also established existence of a common fixed point theorems for generalize contractive mappings satisfy this property in fuzzy metric spaces.

The aim of this work is to use newly introduced property [15], which is so called ”common limit in the range (CLRS)”

for four self-mappings, and establish some common fixed point theorem using this property for generalized contrac-tive mappings in fuzzy metric spaces that does not require condition of closeness of range and give some examples Moreover, Ours results generalize, unify, and extend many results in literature.

2 Preliminaries

The concept of triangular norms (t-norms) is originally introduced by Menger [17] in study of statistical metric spaces.

Definition 2.1(Schweizer and Sklar [21]). A binary operation∗:[0,1]×[0,1]→[0,1]is called a continuous

trian-gular norm (t-norm) if it satisfies the following conditions:

(i) ∗is associative and commutative;

(ii) ∗is continuous;

(iii) a∗1=a for all a∈[0,1];

(iv) a∗b≤c∗d, whenever a≤c and b≤d, for all a,b,c,d∈[0,1].

Three basic examples of continuoust-norms area∗1b=min{a,b},a∗2b=abanda∗3b=max{a+b−1,0}.

Definition 2.2(Kramosil and Michalek [14]). A fuzzy metric space is a triple(X,M,∗), where X is a non-empty set,

∗is a continuous t-norm and M is a fuzzy set on X×X×[0,+∞), satisfying the following properties:

(KM-1) M(x,y,0) =0for all x,y∈X ;

(KM-2) M(x,y,t) =1for all t>0 iff x=y;

(KM-3) M(x,y,t) =M(y,x,t)for all x,y∈X and for all t>0;

(KM-4) M(x,y,·):[0,+∞)→[0,1]is left continuous for all x,y∈X ;

(KM-5) M(x,z,t+s)≥M(x,y,t)∗M(y,z,s)for all x,y,z∈X and for all t,s>0.

We denote such space asKM-fuzzy metric space.

Lemma 2.1([14]). In a KM-fuzzy metric space(X,M,∗), M(x,y,·)is non-decreasing for all x,y∈X .

Definition 2.3([6]). Let(X,M,∗)be a fuzzy metric space. Then a sequence{xn}n∈Nis said to be

(i) convergent to x∈X , that is,limn→+∞xn=x, iflimn→+∞M(xn,x,t) =1for all t>0,

(ii) Cauchy sequence iflimn→+∞M(xn+p,xn,t) =1for all t>0and p>0.

Definition 2.4([6]). A fuzzy metric space is said to be complete if and only if every Cauchy sequence in X is

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Definition 2.5([13]). Two self-mappings f and g of a fuzzy metric space (X,M,∗) are said to be compatible if limn→+∞M(f gxn,g f xn) =1for all t>0, whenever{xn}is a sequence in X such thatlimn→+∞f xn=limn→+∞gxn=z

for some z∈X .

Definition 2.6([13]). Two self-mappings f and g of a fuzzy metric space(X,M,∗)are said to be non-compatible if

there exists at least one sequence{xn}in X such thatlimn→+∞f xn=limn→+∞gxn=z for some z∈X , but for some

t>0, eitherlimn→+∞M(f gxn,g f xn)̸=1or the limit does not exist.

Definition 2.7([8]). A pair(f,g)of self-mappings of a non-empty set X is said to be weakly compatible (or

coinci-dentally commuting) if they commute at their coincidence points, that is, if f z=gz for some z∈X , then f gz=g f z.

If two self-mappings f andgof a fuzzy metric space(X,M,∗)are compatible then they are weakly compatible but the converse need not be true.

Definition 2.8([2]). A pair(f,g)of self-mappings of a fuzzy metric space(X,M,∗)is said to satisfy the property

(E.A) if there exists a sequence{xn}in X such that

lim

n→+∞f xn=n→lim+∞gxn=z,

From Definition 2.8, it is easy to see that any two non-compatible self-mappings of a fuzzy metric space(X,M,∗) satisfy the property (E.A) but the reverse need not be true.

Definition 2.9([1]). Two pairs(A,S)and(B,T)of self-mappings of a fuzzy metric space(X,M,∗)are said to satisfy

the common property (E.A) if there exist two sequences{xn},{yn}in X such that

lim

n→+∞Axn=n→lim+∞Sxn=n→lim+∞Byn=n→lim+∞Tyn=z,

Definition 2.10([25]). A pair of self-mappings(f,g)of a fuzzy metric space(X,M,∗)is said to satisfy the common

limit in the range of g property (CLRg) if there exists a sequence{xn}in X such that

lim

n→+∞f xn=n→lim+∞gxn=gz,

With a view to extend the (CLRg) property to two pair of self mappings, very recently Imdad et. al. [10] define the

(CLRST) property (with respect to mappings S and T) as follows:

Definition 2.11. [10] Two pairs (A,S) and (B,T ) of self mappings of a fuzzy metric space(X,M,∗)are said to satisfy

the (CLRST) property (with respect to mappings S and T) if there exist two sequences{xn},{yn}in X such that

lim

n→+∞Axn=n→lim+∞Sxn=n→lim+∞Byn=n→lim+∞Tyn=Sz,

for some z∈S(X)and z∈T(X).

Inspired by Sintunavarat et al. [25] and Imdad et. al. [10], Manro et al. [15] introduced the following notion:

Definition 2.12([15]). Two pairs(A,S)and(B,T)of self-mappings of a fuzzy metric space(X,M,∗)are said to share

the common limit in the range of S property if there exist two sequences{xn},{yn}in X such that

lim

n→+∞Axn=n→lim+∞Sxn=n→lim+∞Byn=n→lim+∞Tyn=Sz,

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Example 2.1([8]). Let(X,M,∗)be a fuzzy metric space with X= [−1,1]and for all x,y∈X , define M(x,y,t) =e−|x−y|t if t>0and M(x,y,0) =0. Define self mappings A,B,S and T on X by Ax=x

3,Bx=

−x

3,Sx=x,T x=−x for all x∈X .

Then with sequences{xn=1_n}and{yn=−_n1}in X , one can easily verify that

lim

n→+∞Axn=n→lim+∞Sxn=n→lim+∞Byn=n→lim+∞Tyn=S0=0.

This shows that the pairs (A,S) and (B,T ) share the common limit in the range of S property.

Definition 2.13([8]). Two families of self-mappings{Ai}and{Sj}are said to be pairwise commuting if:

1. AiAj=AjAi, i,j∈ {1,2, . . . ,m},

2. SiSj=SjSi, i,j∈ {1,2, . . . ,n},

3. AiSj=SjAi, i∈ {1,2, . . . ,m}, j∈ {1,2, . . . ,n}.

Lemma 2.2([16]). Let(X,M,∗)be a fuzzy metric space, where∗is a continuous t-norm. If there exists a constant

k∈(0,1)such that M(x,y,kt)≥M(x,y,t), for all x,y∈X and t>0, then x=y.

3 Main Results

Lemma 3.1. Let A,B,S and T be self mappings of a fuzzy metric space(X,M,∗)satisfying the followings:

(i) the pair (A,S) (or (B,T )) satisfies the common limit in the range of S property (or T property); (ii) there exists a constant k∈(0,1)such that

(M(Ax,By,kt))2≥min((M(Sx,Ty,t))2,M(Sx,Ax,t).M(Ty,By,t),M(Sx,By,2t). (3.1)

M(Ty,Ax,t),M(Ty,Ax,t),M(Sx,By,2t).M(Ty,By,t))

for any x,y∈X and t>0;

(iii) A(X)⊆T(X)( or B(X)⊆S(X)).

Then the pairs (A,S) and (B,T ) share the common limit in the range of S property.

Proof. Suppose that the pair (A,S) satisfies the common limit in the range ofSproperty, then there exists a sequence {xn} inX such thatlimn→∞Axn=limn→∞Sxn=Szfor some pointz∈X. SinceA(X)⊆T(X), therefore for each {xn}, there exist{yn}inX such thatAxn=Tyn. Thuslimn→∞Tyn=Sz.Hence we havelimn→∞Axn=limn→∞Sxn=

limn→∞Tyn=Sz.Now, we assert thatlimn→∞Byn=Sz.From (3.1), we get

(M(Axn,Byn,kt))2≥min((M(Sxn,Tyn,t))2,M(Sxn,Axn,t).M(Tyn,Byn,t),M(Sxn,Byn,2t).

M(Tyn,Axn,t),M(Tyn,Axn,t),M(Sxn,Byn,2t).M(Tyn,Byn,t)).

Taking limit asn→∞, we get

(M(Sz,limn→∞Byn,kt))2≥min((M(Sz,Sz,t))2,M(Sz,Sz,t).M(Sz,limn→∞Byn,t),

M(Sz,limn→∞Byn,2t).M(Sz,Sz,t),M(Sz,Sz,t),

M(Sz,limn→∞Byn,2t).M(Sz,limn→∞Byn,t))

(M(Sz,limn→∞Byn,kt))2≥min(1,M(Sz,limn→∞Byn,t),M(Sz,limn→∞Byn,2t),

1,M(Sz,limn→∞Byn,2t).M(Sz,limn→∞Byn,t))

(M(Sz,limn→∞Byn,kt))2≥(M(Sz,limn→∞Byn,t))2

by Lemma 2.2, we have therefore,limn→∞Byn=Sz.Then the pairs (A,S) and (B,T) share the common limit in the

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Theorem 3.1. Let A,B,S and T be self mappings of a fuzzy metric space(X,M,∗)satisfying inequality (3.1). Suppose

that

(i) the pair (A,S) (or (B,T )) satisfies the common limit in the range of S property (or T property); (ii) A(X)⊆T(X)( or B(X)⊆S(X)).

Then the pairs (A,S) and (B,T ) have a point of coincidence each.

Moreover, A,B,S and T have a unique common fixed point provided that both the pairs (A,S) and (B,T ) are weakly compatible.

Proof. In view of Lemma 3.1, the pairs (A,S) and (B,T) share the common limit in the range ofS property, that is there exists two sequences{xn}and{yn}inXsuch that

limn→∞Axn=limn→∞Sxn=limn→∞Tyn=limn→∞Byn=Szfor somez∈X.

Firstly, we assert thatAz=Sz. By (3.1), we have

(M(Az,Byn,kt))2≥min((M(Sz,Tyn,t))2,M(Sz,Az,t).M(Tyn,Byn,t),M(Sz,Byn,2t).

M(Tyn,Az,t),M(Tyn,Az,t),M(Sz,Byn,2t).M(Tyn,Byn,t))

Proceeding limit asn→∞, we get

(M(Az,Sz,kt))2≥min((M(Sz,Sz,t))2,M(Sz,Az,t).M(Sz,Sz,t),M(Sz,Sz,2t).M(Sz,Az,t),

M(Sz,Az,t),M(Sz,Sz,2t).M(Sz,Sz,t))

(M(Az,Sz,kt))2≥min(1,M(Sz,Az,t),M(Sz,Az,t),1)

(M(Az,Sz,kt))2≥(M(Az,Sz,t))2

By Lemma 2.2, we haveAz=Sz.

Since,A(X)⊆T(X), there existv∈Xsuch thatAz=T v. Secondly, we assert thatBv=T v. By (3.1), we get

(M(Az,Bv,kt))2≥min((M(Sz,T v,t))2,M(Sz,Az,t).M(T v,Bv,t),M(Sz,Bv,2t).M(T v,Az,t),

M(T v,Az,t),M(Sz,Bv,2t).M(T v,Bv,t))

(M(T v,Bv,kt))2≥min((M(Sz,Sz,t))2,M(Sz,Sz,t).M(T v,Bv,t),M(T v,Bv,2t).M(T v,T v,t),

M(T v,T v,t),M(T v,Bv,2t).M(T v,Bv,t))

(M(T v,Bv,kt))2≥(M(T v,Bv,t))2.

From Lemma 2.2, we haveT v=Bv=Az=Sz.

Since the pairs (A,S) and (B,T) are weakly compatible andAz=Sz andT v=Bv, therefore,ASz=SAz=AAz= SSz,BT v=T Bv=T T v=BBv.

Finally, we assert thatAAz=Az.Again by (3.1), we have

(M(AAz,Bv,kt))2≥min((M(SAz,T v,t))2,M(SAz,AAz,t).M(T v,Bv,t),M(SAz,Bv,2t).

M(T v,AAz,t),M(T v,AAz,t),M(SAz,Bv,2t).M(T v,Bv,t))

(M(AAz,Az,kt))2≥min((M(AAz,Az,t))2,M(AAz,AAz,t).M(Az,Az,t),M(AAz,Az,2t).

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(M(AAz,Az,kt))2≥min((M(AAz,Az,t))2,1,M(AAz,Az,2t).M(Az,AAz,t),

M(Az,AAz,t),M(AAz,Az,2t))

(M(AAz,Az,kt))2≥(M(AAz,Az,t))2

which again by Lemma 2.2 gives,AAz=Az=SAzwhich gives, Az is common fixed point of AandS. Similarly, one can easily prove thatBBv=Bv=T Bv, that isBvis common fixed point ofBandT. AsAz=Bv, thereforeAz

is common fixed point ofA,B,SandT. The uniqueness of common fixed point is an easy consequence of inequality (3.1).

Our result (Theorem 3.1) improve and extend the results of Cho et al. [4], Pathak et al.[20] and Imdad et. al. [10] besides several known results.

By choosingA,B,SandT suitably, one can derive corollaries involving two or three mappings.

Corollary 3.1. Let A and S be self mappings of a fuzzy metric space(X,M,∗)satisfying:

(i) the pair (A,S) satisfies the common limit in the range of S property; (ii) A(X)⊆S(X);

(iii) there exists a constant k∈(0,1)such that

(M(Ax,Ay,kt))2≥min((M(Sx,Sy,t))2,M(Sx,Ax,t).M(Sy,Ay,t),M(Sx,Ay,2t).

M(Sy,Ax,t),M(Sy,Ax,t),M(Sx,Ay,2t).M(Sy,Ay,t))

for any x,y∈X and t>0.

Then A and S have a point of coincidence. Moreover, A and S have a unique common fixed point provided that A and S are weakly compatible.

Proof. TakingB=AandT=Sin Theorem 3.1.

Corollary 3.2. Let A,B,S and T be self mappings of a fuzzy metric space(X,M,∗)satisfying inequality (3.1). Suppose

that

(i) the pairs (A,S) and (B,T ) satisfies the common limit in the range of S property. (ii) A(X)⊆T(X)( or B(X)⊆S(X)).

Then the pairs (A,S) and (B,T ) have a point of coincidence each. Moreover, A,B,S and T have a unique common fixed point provided that both the pairs (A,S) and (B,T ) are weakly compatible.

Proof. Proof easily follows on same lines of Theorem 3.1 using Lemma 3.1

Theorem 3.2. Let A,B,S and T be self mappings of a fuzzy metric space(X,M,∗)satisfying inequality (3.1). Suppose

that

(i) the pair (A,S) (or (B,T )) satisfies property (E.A.) and S(X)is a closed subspace of X ; (ii)A(X)⊆T(X)( or B(X)⊆S(X)).

Proof. Suppose pair(A,S)satisfy property (E.A.), there exist a sequence{xn}inXsuch thatlimn→∞Axn=limn→∞Sxn=

pfor somep∈X. It follows fromS(X)is a closed subspace ofXthat p=Szfor somez∈X and then the pair (A,S) satisfies the common limit in the range ofSproperty. By Theorem 3.1, we getA,B,SandT have a unique common fixed point.

that

(i) the pairs (A,S) and (B,T ) satisfies common property (E.A.) and S(X)is a closed subspace of X ; (ii) A(X)⊆T(X)( or B(X)⊆S(X)).

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Proof. Since the pairs (A,S) and (B,T) satisfies common property (E.A.), there exists two sequences{xn}and{yn}

inX such thatlimn→∞Axn=limn→∞Sxn=limn→∞Tyn=limn→∞Byn=pfor some p∈X. It follows fromS(X)is a

closed subspace ofX that p=Szfor somez∈X and then the pairs (A,S) and (B,T) share the common limit in the range ofSproperty. By Theorem 3.1, we getA,B,SandT have a unique common fixed point.

Since the pair of non compatible mappings implies to the pair satisfying property (E.A.), we get the following corol-lary.

that

(i) the pair (A,S) (or (B,T )) is non compatible mappings and S(X)is a closed subspace of X ; (ii) A(X)⊆T(X)( or B(X)⊆S(X)).

Then the pairs(A,S) and (B,T ) have a point of coincidence each. Moreover, A,B,S and T have a unique common fixed point provided that both the pairs (A,S) and (B,T ) are weakly compatible.

Proof. Since the pair (A,S) is non compatible mappings, we getA andS satisfy property (E.A.). Therefore, by Theorem 3.2, we getA,B,SandT have a unique common fixed point inX.

Remark 3.1. The property (E.A.) (common property (E.A.)) is an essential tool to claim the existence of common

fixed points of some mappings. However these properties require the condition of closedness of S(X). Note that Theorem 3.1 weakens the condition of closed subspace of S(X). Therefore it is most interesting to used common limit in the range of S property as another auxiliary tool to claim the existence of a common fixed point. However, all the main results in this paper are some of the choices for claim that the existence of common fixed point in fuzzy metric spaces. Our result may be the motivation to other authors for extending and improving these results to suitable tools for these problems.

As an application of Theorems 3.1, we prove a common fixed point theorem for four finite families of mappings on fuzzy metric spaces. While proving our result, we utilize Definition 2.13 which is a natural extension of commutativity condition to two finite families.

Theorem 3.3. Let {A1,A2, . . . ,Am},{B1,B2, . . . ,Bn},{S1,S2, . . . ,Sp}and {T1,T2, . . . ,Tq}be four finite families of

self-mappings of a fuzzy metric space(X,M,∗)such that A=A1A2· · ·Am, B=B1B2· · ·Bn, S=S1S2· · ·Spand T=

T1T2· · ·Tqsatisfy the conditions of Theorem 3.1. Then

(a) the pairs(A,S)and(B,T)have a point of coincidence each;

(b){Ai},{Bj},{Sk}and{Tr}have a unique common fixed point provided that the pairs of families({Ai},{Sk})and

({Bj},{Tr})commute pairwise, for all i=1, . . . ,m, j=1, . . . ,n, k=1, . . . ,p and r=1, . . . ,q.

Proof. Since the pairs of families({Ai},{Sk})and({Bj},{Tr})commute pairwise, we first show thatAS=SA. In

fact, we have

AS = (A1A2· · ·Am)(S1S2· · ·Sp)

= (A1A2· · ·Am−1)(AmS1S2· · ·Sp)

= (A1A2· · ·Am−1)(S1S2· · ·SpAm)

= (A1A2· · ·Am−2)(Am−1S1S2· · ·SpAm)

= (A1A2· · ·Am−2)(S1S2· · ·SpAm−1Am)

= . . .

= A1(S1S2· · ·SpA2· · ·Am)

= (S1S2· · ·Sp)(A1A2· · ·Am) =SA.

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Now, we need to prove thatzremains the fixed point of all the component mappings. To this aim, consider

A(Aiz) = ((A1A2· · ·Am)Ai)z= (A1A2· · ·Am−1)(AmAi)z

= (A1A2· · ·Am−1)(AiAm)z= (A1A2· · ·Am−2)(Am−1AiAm)z

= (A1A2· · ·Am−2)(AiAm−1Am)z=· · ·=A1(AiA2· · ·Am)z

= (A1Ai)(A2· · ·Am)z= (AiA1)(A2· · ·Am)z

= Ai(A1A2· · ·Am)z=AiAz=Aiz.

Similarly, one can prove thatA(Skz) =Sk(Az) =Skz,S(Skz) =Sk(Sz) =Skz,S(Aiz) =Ai(Sz) =Aiz,B(Bjz) =B j(Bz) =

Bjz,B(Trz) =Tr(Bz) =Trz,T(Trz) =Tr(T z) =TrzandT(Bjz) =Bj(T z) =Bjz, which show that (for alli,j,kand

r)AizandSkzare other fixed points of the pair(A,S)whereasBjzandTrzare other fixed points of the pair(B,T). SinceA,B,SandT have a unique common fixed point, then we getz=Aiz=Skz=Bjz=Trz, for alli=1, . . . ,m,

j=1, . . . ,n,k=1, . . . ,pandr=1, . . . ,q. Thuszis the unique common fixed point of{Ai},{Bj},{Sk}and{Tr}.

Now we prove common fixed point theorem using lower semi continuous function,ψ:[0,1]→[0,1]such thatψ(t)>t

for allt∈(0,1)along withψ(0) =0 andψ(1) =1, for integral mappings satisfying contractive conditions.

Theorem 3.4. Let A,B,S and T be self mappings of a fuzzy metric space(X,M,∗)satisfying the conditions (i) and

(ii) of Theorem 3.1 and for all x,y∈X,t>0

∫ M(Ax,By,t)+min{M(Sx,By,t),M(Ax,Ty,t)}

2

0 φ(s)ds≥ψ(

∫ min{M(Sx,Ty,t),M(Sx,Ax,t),M(Ty,By,t),M(Sx,By,t),M(Ty,Ax,t)}

0 φ(s)ds) (3.2)

whereφ :R+→R+ is a Lebesgue integrable function which is summable and satisfies0<∫₀εφ(s)ds<1 for all

0<ε<1and∫₀1φ(s)ds=1.

Proof. Suppose that the pair (A,S) satisfies the common limit in the range ofSproperty, then there exists a sequence {xn} inX such thatlimn→∞Axn=limn→∞Sxn=Szfor some pointz∈X. SinceA(X)⊆T(X), therefore for each {xn}, there exist{yn}inX such thatAxn=Tyn. Thuslimn→∞Tyn=Sz.Hence we havelimn→∞Axn=limn→∞Sxn=

limn→∞Tyn=Sz.Now, we assert thatlimn→∞Byn=Sz.Suppose not, then applying inequality (3.2), we get

∫ M(Axn,Byn,t)+min{M(Sxn,Byn,t),M(Axn,Tyn,t)}

2

0

φ(s)ds≥ψ(

∫ min{M(Sxn,Tyn,t),M(Sxn,Axn,t),M(Tyn,Byn,t),M(Sxn,Byn,t),M(Tyn,Axn,t)}

0

φ(s)ds)

takingn→∞, we have

∫ M(Sz,limn→∞Byn,t)

0 φ(s)ds≥ψ(

∫ min{M(Sz,Sz,t),M(Sz,Sz,t),M(Sz,limn→∞Byn,t),M(Sz,limn→∞Byn,t),M(Sz,Sz,t)}

0 φ(s)ds)

=ψ(

∫ min{1,1,M(Sz,limn→∞Byn,t),M(Sz,limn→∞Byn,t),1}

0 φ(s)ds)

=ψ(

∫ min{M(Sz,limn→∞Byn,t)}

0

φ(s)ds)

>

∫ min{M(Sz,limn→∞Byn,t)}

0

φ(s)ds

which is a contradiction, therefore,limn→∞Byn=Sz.Then the pairs (A,S) and (B,T) share the common limit in the

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limn→∞Tyn=limn→∞Byn=Szfor somez∈X.

Firstly, we assert thatAz=Sz. Suppose not, then by (3.2), we have

∫ M(Az,Byn,t)+min{M(Sz,Byn,t),M(Az,Tyn,t)}

2

0 φ(s)ds≥ψ(

∫ min{M(Sz,Tyn,t),M(Sz,Az,t),M(Tyn,Byn,t),M(Sz,Byn,t),M(Tyn,Az,t)}

0 φ(s)ds)

takingn→∞, we have

∫ M(Az,Sz,t)

0 φ(s)ds≥ψ(

∫ min{M(Sz,Sz,t),M(Sz,Az,t),M(Sz,Sz,t),M(Sz,Sz,t),M(Sz,Az,t)}

0 φ(s)ds)

=ψ(

∫ min{1,M(Sz,Az,t),1,1,M(Sz,Az,t)}

0

φ(s)ds)

=ψ(

∫ M(Sz,Az,t)

0

φ(s)ds)

>

∫ minM(Sz,Az,t)

0

φ(s)ds

which is contradiction and therefore,Az=Sz. Since,A(X)⊆T(X), there existv∈X such thatAz=T v. Secondly, we assert thatBv=T v. Suppose not, then by (3.2), we get

∫ M(Az,Bv,t)+min{M(Sz,Bv,t),M(Az,T v,t)}

2

0

φ(s)ds≥ψ(

∫ min{M(Sz,T v,t),M(Sz,Az,t),M(T v,Bv,t),M(Sz,Bv,t),M(T v,Az,t)}

0

φ(s)ds)

∫ M(T v,Bv,t)

0

φ(s)ds≥ψ(

∫ min{M(T v,T v,t),M(T v,T v,t),M(T v,Bv,t),M(T v,Bv,t),M(T v,T v,t)}

0

φ(s)ds)

=ψ(

∫ min{1,1,M(T v,Bv,t),M(T v,Bv,t),1}

0 φ(s)ds)

=ψ(

∫ M(T v,Bv,t)

0 φ(s)ds)

>

∫ M(T v,Bv,t)

0 φ(s)ds

a contradiction and thereforeT v=Bv=Az=Sz. Since the pairs (A,S) and (B,T) are weakly compatible andAz=Sz

andT v=Bv, therefore,ASz=SAz=AAz=SSz,BT v=T Bv=T T v=BBv.Finally, we assert thatAAz=Az. Suppose not, then again by (3.2), we have

∫ M(AAz,Bv,t)+min{M(SAz,Bv,t),M(AAz,T v,t)}

2

0 φ(s)ds≥ψ(

∫ min{M(SAz,T v,t),M(SAz,AAz,t),M(T v,Bv,t),M(SAz,Bv,t),M(T v,AAz,t)}

0 φ(s)ds)

∫ M(AAz,Az,t)

0 φ(s)ds≥ψ(

∫ min{M(AAz,Bv,t),M(AAz,AAz,t),M(Bv,Bv,t),M(AAz,Bv,t),M(Bv,AAz,t)}

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=ψ(

∫ min{M(AAz,Az,t),1,1,M(AAz,Az,t),M(Az,AAz,t)}

0

φ(s)ds)

=ψ(

∫ M(AAz,Az,t)

0

φ(s)ds)

>

∫ M(AAz,Az,t)

0

φ(s)ds

which is contradiction and thereforeAAz=Az=SAzwhich gives, Az is common fixed point ofAandS. Similarly, one can easily prove thatBBv=Bv=T Bv, that isBvis common fixed point of B and T. AsAz=Bv, therefore,Az

is common fixed point ofA,B,SandT. The uniqueness of common fixed point is an easy consequence of inequality (3.2).

Remark 3.2. Notice that results similar to Theorems 3.2, 3.3 and Corollaries 3.1, 3.2, 3.3, 3.4 can also be outlined

in respect to Theorem 3.4, but we may omit the details with a view to avoid any repetition.

Example 3.1. Let(X,M,∗)be a fuzzy metric space where X = [0, 2) with a t- norm defined by a∗b = min{a,b}.

Define M(x,y,t) = _t+_|_xt₋_y_| if t>0 and M(x,y,0) =0. Define A,B,S and T by A(X) =B(X) =1 and Sx=1 if x∈Q,Sx=2₃ otherwise and T x=1if x∈Q and T x=1₃ otherwise. Clearly, the pairs (A,S) and (B,T ) satisfies all conditions of Theorem 3.1 and shares the common limit in the range of S property and A(X)⊆T(X). A,B,S and T have a unique common fixed point x=1.

Remark 3.3. One can obtain the similar results in setting of probabilistic metric spaces.

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