Skip to main content

Unified fixed point theorems in fuzzy metric spaces via common limit range property

Abstract

The object of this paper is to utilize the notion of common limit range property to prove unified fixed point theorems for weakly compatible mappings in fuzzy metric spaces satisfying an implicit relation due to Rao et al. (Hacet. J. Math. Stat. 37(2):97-106, 2008). Some illustrative examples are furnished which demonstrate the validity of the hypotheses and degree of utility of our results. As an application to our main result, we prove an integral type fixed point theorem in fuzzy metric space.

MSC:54H25, 47H10.

1 Introduction

The concept of a fuzzy set was introduced by Zadeh [1] in his seminal paper in 1965. In the last two decades, there has been a tremendous development and growth in fuzzy mathematics. In 1975, Kramosil and Michalek [2] introduced the concept of fuzzy metric space, which opened an avenue for further development of analysis in such spaces. Further, George and Veeramani [3] modified the concept of fuzzy metric space introduced by Kramosil and Michalek [2] with a view to obtain a Hausdorff topology on it. Fuzzy set theory has applications in applied sciences such as neural network theory, stability theory, mathematical programming, modeling theory, engineering sciences, medical sciences (medical genetics, nervous system), image processing, control theory, communication, etc.

Mishra et al. [4] extended the notion of compatible mappings to fuzzy metric spaces and proved common fixed point theorems in presence of continuity of at least one of the mappings, completeness of the underlying space and containment of the ranges amongst involved mappings. Further, Singh and Jain [5] weakened the notion of compatibility by using the notion of weakly compatible mappings in fuzzy metric spaces and showed that every pair of compatible mappings is weakly compatible but reverse is not true. Many mathematicians used different contractive conditions on self-mappings and proved several fixed point theorems in fuzzy metric spaces (see [513]). However, the study of common fixed points of non-compatible mappings is also of great interest due to Pant [14]. In 2002, Aamri and Moutawakil [15] defined the notion of property (E.A) for a pair of self-mappings which contains the class of non-compatible mappings. In an interesting paper of Ali and Imdad [16], it was pointed out that property (E.A) allows replacing the completeness requirement of the space with a more natural condition of closedness of the range. Afterward, Liu et al. [17] defined a new property which contains the property (E.A) and proved some common fixed point theorems under hybrid contractive conditions. It was observed that the notion of common property (E.A) relatively relaxes the requirement on containment of the range of one mapping into the range of other which is utilized to construct the sequence of joint iterates. Subsequently, there are a number of results proved for contraction mappings satisfying property (E.A) and common property (E.A) in fuzzy metric spaces (see [1826]). In 2011, Sintunavarat and Kumam [27] coined the idea of ’common limit range property’ (also see [2832]), which relaxes the condition of closedness of the underlying subspace. Recently, Imdad et al. [33] extended the notion of common limit range property to two pairs of self-mappings without any requirement on closedness of the underlying subspaces. Several common fixed point theorems have been proved by many researchers in framework of fuzzy metric spaces via implicit relations (see [5, 18, 23, 34]).

In this paper, by using an implicit relation due to Rao et al. [12], we prove some common fixed point theorems for weakly compatible mappings with common limit range property. Some related results are also derived besides furnishing illustrative examples. We also present an integral type common fixed point theorem in fuzzy metric space. Our results improve, extend and generalize a host of previously known results including the ones contained in Rao et al. [12].

2 Preliminaries

Definition 2.1 [35]

A triangular norm (shortly t-norm) is a binary operation on the unit interval [0,1] such that for all a,b,c,d[0,1] and the following conditions are satisfied:

  1. (1)

    a1=a,

  2. (2)

    ab=ba,

  3. (3)

    abcd whenever ac and bd,

  4. (4)

    (ab)c=a(bc).

Two typical examples of continuous t-norms are ab=min{a,b} and ab=ab.

Definition 2.2 [3]

A 3-tuple (X,M,) is said to be a fuzzy metric space if X is an arbitrary set, is a continuous t-norm and M is a fuzzy set on X 2 ×(0,) satisfying the following conditions: for all x,y,zX, t,s>0,

  1. (1)

    M(x,y,t)>0,

  2. (2)

    M(x,y,t)=1 iff x=y,

  3. (3)

    M(x,y,t)=M(y,x,t),

  4. (4)

    M(x,z,t+s)M(x,y,t)M(y,z,s),

  5. (5)

    M(x,y,):(0,)(0,1] is continuous.

Then M is called a fuzzy metric on X whereas M(x,y,t) denotes the degree of nearness between x and y with respect to t.

Let (X,M,) be a fuzzy metric space. For t>0, the open ball B(x,r,t) with center xX and radius 0<r<1 is defined by

B(x,r,t)= { y X : M ( x , y , t ) > 1 r } .

Now let (X,M,) be a fuzzy metric space and τ the set of all AX with xA if and only if there exist t>0 and 0<r<1 such that B(x,r,t)A. Then τ is a topology on X induced by the fuzzy metric M.

In the following example (see [3]), we know that every metric induces a fuzzy metric.

Example 2.1 Let (X,d) be a metric space. Denote ab=ab (or ab=min{a,b}) for all a,b[0,1] and let M d be fuzzy sets on X 2 ×(0,) defined as follows:

M d (x,y,t)= t ( t + d ( x , y ) ) .

Then (X, M d ,) is a fuzzy metric space and the fuzzy metric M induced by the metric d is often referred to as the standard fuzzy metric.

Lemma 2.1 [36]

Let (X,M,) be a fuzzy metric space. Then M(x,y,t) is non-decreasing for all x,yX.

Definition 2.3 [4]

A pair (A,S) of self-mappings of a fuzzy metric space (X,M,) is said to be compatible iff M(AS x n ,SA x n ,t)1 for all t>0, whenever { x n } is a sequence in X such that A x n ,S x n z for some zX as n.

Definition 2.4 [37]

A pair (A,S) of self-mappings of a non-empty set X is said to be weakly compatible (or coincidentally commuting) if they commute at their coincidence points, that is, if Az=Sz some zX, then ASz=SAz.

Remark 2.1 [37]

Two compatible self-mappings are weakly compatible but the converse is not true. Hence, the notion of weak compatibility is more general than compatibility.

Definition 2.5 [19]

A pair (A,S) of self-mappings of a fuzzy metric space (X,M,) is said to satisfy the property (E.A), if there exists a sequence { x n } in X for some zX such that

lim n A x n = lim n S x n =z,

for some zX.

Note that weak compatibility and property (E.A) are independent to each other (see [[38], Examples 2.1-2.2]).

Remark 2.2 From Definition 2.3, it is inferred that a pair (A,S) of self-mappings of a fuzzy metric space (X,M,) is said to be non-compatible iff there exists at least one sequence { x n } in X such that lim n A x n = lim n S x n =z for some zX, but for some t>0, lim n M(AS x n ,SA x n ,t) is either less than 1 or non-existent.

Therefore, in view of Definition 2.5, a pair of non-compatible mappings of a fuzzy metric space (X,M,) satisfies the property (E.A) but the converse need not be true (see [[38], Remark 4.8]).

Definition 2.6 [19]

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 { x n }, { y n } in X such that

lim n A x n = lim n S x n = lim n B y n = lim n T y n =z,

for some zX.

Definition 2.7 [27]

A pair (A,S) of self-mappings of a fuzzy metric space (X,M,) is said to satisfy the common limit range property with respect to mapping S (briefly, ( CLR S ) property), if there exists a sequence { x n } in X such that

lim n A x n = lim n S x n =z,

where zS(X).

Definition 2.8 [28]

Two pairs (A,S) and (B,T) of self-mappings of a fuzzy metric space (X,M,) are said to satisfy the common limit range property with respect to mappings S and T (briefly, ( CLR ST ) property), if there exist two sequences { x n }, { y n } in X such that

lim n A x n = lim n S x n = lim n B y n = lim n T y n =z,

where zS(X)T(X).

Definition 2.9 [39]

Two families of self-mappings { A i } i = 1 m and { S k } k = 1 n are said to be pairwise commuting if

  1. (1)

    A i A j = A j A i for all i,j{1,2,,m},

  2. (2)

    S k S l = S l S k for all k,l{1,2,,n},

  3. (3)

    A i S k = S k A i for all i{1,2,,m} and k{1,2,,n}.

Lemma 2.2 [4]

Let (X,M,) be a fuzzy metric space with ttt for all t[0,1]. If there exists a constant k(0,1) such that

M(x,y,kt)M(x,y,t),

for all x,yX, then x=y.

3 Implicit function

Following by Rao et al. [12], let Φ 6 denote the set of all continuous functions ϕ: [ 0 , 1 ] 6 R satisfying the conditions:

( ϕ 1 ) ϕ is decreasing in t 2 , t 3 , t 4 , t 5 and t 6 .

( ϕ 2 ) ϕ(u,v,v,v,v,v)0 implies uv for all u,v[0,1].

Example 3.1 ϕ( t 1 , t 2 , t 3 , t 4 , t 5 , t 6 )= t 1 min{ t 2 , t 3 , t 4 , t 5 , t 6 }.

Example 3.2 ϕ( t 1 , t 2 , t 3 , t 4 , t 5 , t 6 )= t 1 2 min{ t i t j :i,j{2,3,4,5,6}}.

Example 3.3 ϕ( t 1 , t 2 , t 3 , t 4 , t 5 , t 6 )= t 1 3 min{ t i t j t k :i,j,k{2,3,4,5,6}}.

4 Results

We begin with the following observation.

Lemma 4.1 Let (X,M,) be a fuzzy metric space with ttt for all t[0,1]. Let A, B, S and T be mappings from X into itself satisfying:

  1. (1)

    The pair (A,S) (or (B,T)) satisfies the ( CLR S ) (or ( CLR T )) property,

  2. (2)

    A(X)T(X) (or B(X)S(X)),

  3. (3)

    T(X) (or S(X)) is a closed subset of X,

  4. (4)

    B( y n ) converges for every sequence { y n } in X whenever T( y n ) converges (or A( x n ) converges for every sequence { x n } in X whenever S( x n ) converges),

  5. (5)

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

    ϕ( M ( A x , B y , k t ) , M ( S x , T y , t ) , M ( A x , S x , t ) , M ( B y , T y , t ) , M ( A x , T y , t ) , M ( B y , S x , t ) )0
    (4.1)

for all x,yX, t>0 and ϕ Φ 6 .

Then the pairs (A,S) and (B,T) share the ( CLR ST ) property.

Proof Since the pair (A,S) enjoys the ( CLR S ) property, there exists a sequence { x n } in X such that

lim n A x n = lim n S x n =z,

where zS(X). By (2), A(X)T(X), and for each sequence { x n } there exists a sequence { y n } in X such that A x n =T y n . Therefore, due to the closedness of T(X),

lim n T y n = lim n A x n =z,

where zS(X)T(X). Thus, we have A x n z, S x n z and T y n z as n. By (4), the sequence {B y n } converges and in all we need to show that B y n z as n. Let, on the contrary that B y n z (≠z) as n. On using inequality (4.1) with x= x n , y= y n , we have

ϕ( M ( A x n , B y n , k t ) , M ( S x n , T y n , t ) , M ( A x n , S x n , t ) , M ( B y n , T y n , t ) , M ( A x n , T y n , t ) , M ( B y n , S x n , t ) )0.

Taking the limit as n, we have

0ϕ( M ( z , z , k t ) , M ( z , z , t ) , M ( z , z , t ) , M ( z , z , t ) , M ( z , z , t ) , M ( z , z , t ) ),

or, equivalently,

0ϕ ( M ( z , z , k t ) , 1 , 1 , M ( z , z , t ) , 1 , M ( z , z , t ) ) .

Since ϕ is decreasing in t 2 ,, t 6 , we get

0ϕ ( M ( z , z , k t ) , M ( z , z , t ) , M ( z , z , t ) , M ( z , z , t ) , M ( z , z , t ) , M ( z , z , t ) ) .

From ( ϕ 2 ), we have M(z, z ,kt)M(z, z ,t). Appealing to Lemma 2.2, we obtain z= z . Therefore, lim n B y n =z and hence the pairs (A,S) and (B,T) satisfy the ( CLR ST ) property. □

Remark 4.1 The converse of Lemma 4.1 is not true in general. For a counter example, we refer to Example 4.1.

Theorem 4.1 Let (X,M,) be a fuzzy metric space with ttt for all t[0,1]. Let A, B, S and T be mappings from X into itself satisfying inequality (4.1). Suppose that the pairs (A,S) and (B,T) enjoy the ( CLR ST ) property. Then the pairs (A,S) and (B,T) have a coincidence point each. Moreover, A,B, S and T have a unique common fixed point provided both the pairs (A,S) and (B,T) are weakly compatible.

Proof Since the pairs (A,S) and (B,T) satisfy the ( CLR ST ) property, there exist two sequences { x n } and { y n } in X such that

lim n A x n = lim n S x n = lim n B y n = lim n T y n =z,

where zS(X)T(X). Since zS(X), there exists a point uX such that Su=z. First we show that Au=Su, by putting x=u and y= y n in (4.1), we get

ϕ( M ( A u , B y n , k t ) , M ( S u , T y n , t ) , M ( A u , S u , t ) , M ( B y n , T y n , t ) , M ( A u , T y n , t ) , M ( B y n , S u , t ) )0,

which on making n, reduces to

ϕ( M ( A u , z , k t ) , M ( z , z , t ) , M ( A u , z , t ) , M ( z , z , t ) , M ( A u , z , t ) , M ( z , z , t ) )0,

and so,

ϕ ( M ( A u , z , k t ) , 1 , M ( A u , z , t ) , 1 , M ( A u , z , t ) , 1 ) 0.

From ( ϕ 1 ) and ( ϕ 2 ), we have M(Au,z,kt)M(Au,z,t). In view of Lemma 2.2, Au=z, and hence Au=Su=z which shows that u is a coincidence point of the pair (A,S).

Also zT(X), there exists a point vX such that Tv=z. Now we assert that Bv=Tv. On using (4.1) with x=u, y=v, we have

ϕ( M ( A u , B v , k t ) , M ( S u , T v , t ) , M ( A u , S u , t ) , M ( B v , T v , t ) , M ( A u , T v , t ) , M ( B v , S u , t ) )0,

which reduces to

ϕ( M ( z , B v , k t ) , M ( z , z , t ) , M ( z , z , t ) , M ( B v , z , t ) , M ( z , z , t ) , M ( B v , z , t ) )0,

or, equivalently,

ϕ ( M ( z , B v , k t ) , 1 , 1 , M ( B v , z , t ) , 1 , M ( B v , z , t ) ) 0.

From ( ϕ 1 ) and ( ϕ 2 ), we have M(z,Bv,kt)M(z,Bv,t). On employing Lemma 2.2, we have z=bv. Hence, Bv=Tv=z which shows that v is a coincidence point of the pair (B,T).

Since the pair (A,S) is weakly compatible and Au=Su, hence Az=ASu=SAu=Sz. Now, we show that z is a common fixed point of the pair (A,S). Putting x=z and y=v in (4.1), we have

ϕ( M ( A z , B v , k t ) , M ( S z , T v , t ) , M ( A z , S z , t ) , M ( B v , T v , t ) , M ( A z , T v , t ) , M ( B v , S z , t ) )0,

and so

ϕ( M ( A z , z , k t ) , M ( A z , z , t ) , M ( A z , A z , t ) , M ( z , z , t ) , M ( A z , z , t ) , M ( z , A z , t ) )0,

or, equivalently,

ϕ ( M ( A z , z , k t ) , M ( A z , z , t ) , 1 , 1 , M ( A z , z , t ) , M ( z , A z , t ) ) 0.

From ( ϕ 1 ) and ( ϕ 2 ), we have M(Az,z,kt)M(Az,z,t). Owing Lemma 2.2, Az=z=Sz, which shows that z is a common fixed point of the pair (A,S).

Also the pair (B,T) is weakly compatible and Bv=Tv, then Bz=BTv=TBv=Tz. On using (4.1) with x=u, y=z, we have

ϕ( M ( A u , B z , k t ) , M ( S u , T z , t ) , M ( A u , S u , t ) , M ( B z , T z , t ) , M ( A u , T z , t ) , M ( B z , S u , t ) )0,

which reduces to

ϕ( M ( z , B z , k t ) , M ( z , B z , t ) , M ( z , z , t ) , M ( B z , B z , t ) , M ( z , B z , t ) , M ( B z , z , t ) )0,

and so

ϕ ( M ( z , B z , k t ) , M ( z , B z , t ) , 1 , 1 , M ( z , B z , t ) , M ( B z , z , t ) ) 0.

From ( ϕ 1 ) and ( ϕ 2 ), we have M(z,Bz,kt)M(z,Bz,t). In view of Lemma 2.2, Bz=z=Tz. Hence, z is a common fixed point of the pair (B,T). Therefore, z is a common fixed point of the mappings A, B, S and T. The uniqueness of common fixed point is an easy consequence of the inequality (4.1) in view of ( ϕ 1 ) and ( ϕ 2 ). □

Remark 4.2 Theorem 4.1 improves the results of Aalam et al. [[18], Theorem 3.1], Rao et al. [[12], Theorem 3.1] and Chauhan and Kumar [[20], Theorem 3.1] as Theorem 4.1 never requires conditions on completeness (or closedness) of the underlying space (or subspace), continuity of the involved mappings and containment amongst range sets of the involved mappings.

Example 4.1 Let (X,M,) be a fuzzy metric space, where X=[1,15), with t-norm defined ab=min{a,b} for all a,b[0,1] and

M(x,y,t)={ t t + x y , if  t > 0 ; 0 , if  t = 0 ,

for all t>0 and x,yX. Define the self-mappings A, B, S and T by

We have A(X)={1,14}[1,13)=T(X) and B(X)={1,5}[1,4)=S(X) which shows that S(X) and T(X) are not closed subsets of X. Consider ϕ: [ 0 , 1 ] 6 R as defined in Examples 3.1-3.3. If we choose two sequences { x n }={3+ 1 n }, { y n }={1} (or { x n }={1}, { y n }={3+ 1 n }), then clearly

lim n A x n = lim n S x n = lim n B y n = lim n T y n =1S(X)T(X).

Hence, both the pairs (A,S) and (B,T) satisfy the ( CLR ST ) property. By a routine calculation, one can verify the inequality (4.1). Thus, all the conditions of Theorem 4.1 are satisfied for some fixed k(0,1) and 1 is a unique common fixed point of the pairs (A,S) and (B,T), which also remains a point of coincidence as well. Also, all the involved mappings are even discontinuous at their unique common fixed point 1.

Theorem 4.2 Let (X,M,) be a fuzzy metric space with ttt for all t[0,1]. Let A, B, S and T be mappings from X into itself. Suppose that the inequality (4.1) and the following hypotheses hold:

  1. (1)

    the pairs (A,S) and (B,T) satisfy the common property (E.A),

  2. (2)

    S(X) and T(X) are closed subsets of X.

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

Proof Since the pairs (A,S) and (B,T) enjoy the common property (E.A), there exist two sequences { x n } and { y n } in X such that

lim n A x n = lim n S x n = lim n B y n = lim n T y n =z,

for some zX. Since S(X) is a closed subset of X, hence lim n S x n =zS(X). Therefore, there exists a point uX such that Su=z. Also, T(X) is a closed subset of X, lim n T y n =zT(X). Therefore, there exists a point vX such that Tv=z. The rest of the proof runs on the lines of the proof of Theorem 4.1. □

Example 4.2 In the setting of Example 4.1, replace the self-mappings A, B, S and T by the following besides retaining the rest:

Then we have A(X)={1,14}[1,13]=T(X) and B(X)={1,5}[1,4]=S(X), hence S(X) and T(X) are closed subsets of X.

Consider two sequences as in Example 4.1, one can see that both the pairs (A,S) and (B,T) enjoy the common property (E.A), that is,

lim n A x n = lim n S x n = lim n B y n = lim n T y n =1X.

Therefore, all the conditions of Theorem 4.2 are satisfied for some fixed k(0,1) and 1 is a unique common fixed point of the pairs (A,S) and (B,T), which also remains a point of coincidence as well.

Theorem 4.3 Let (X,M,) be a fuzzy metric space with ttt for all t[0,1]. Let A, B, S and T be mappings from X into itself satisfying all the hypotheses of Lemma 4.1. Then the pairs (A,S) and (B,T) have a coincidence point each. Moreover, A, B, S and T have a unique common fixed point provided both the pairs (A,S) and (B,T) are weakly compatible.

Proof In view of Lemma 4.1, the pairs (A,S) and (B,T) satisfy the ( CLR ST ) property, there exist two sequences { x n } and { y n } in X such that

lim n A x n = lim n S x n = lim n B y n = lim n T y n =z,

where zS(X)T(X). The rest of the proof can be completed on the lines of the proof of Theorem 4.1. □

Example 4.3 In the setting of Example 4.1, replace the self-mappings A, B, S and T by the following besides retaining the rest:

We obtain A(X)={1,5}[1,13]=T(X) and B(X)={1,4}[1,4]=S(X), which shows that S(X) and T(X) are closed subsets of X. By a routine calculation, one can easily verify the inequality (4.1) for all x,yX and for some fixed k(0,1). Thus, all the conditions of Theorem 4.3 are satisfied and 1 is a unique common fixed point of (A,S) and (B,T), which also remains a point of coincidence as well.

Here, it is worth noting that the majority of earlier proved results Theorem 4.1 cannot be used in the context of this example as Theorem 4.1 never requires any condition on the containment of ranges amongst involved mappings.

Remark 4.3 The conclusions of Lemma 4.1, Theorem 4.1, Theorem 4.2, Corollary 4.1 and Theorem 4.3 remain true if we define ϕ( t 1 , t 2 , t 3 , t 4 , t 5 , t 6 ) as in Examples 3.1-3.3.

Remark 4.4 Theorem 4.1 and Theorem 4.3 (in view of Example 3.1) improve and generalize the results of Mishra et al. [[4], Theorem 1].

By choosing A, B, S and T suitably, we can drive a multitude of common fixed point theorems for a pair or triod of mappings. As a sample, we deduce the following natural result for a pair of self-mappings.

Corollary 4.1 Let (X,M,) be a fuzzy metric space with ttt for all t[0,1]. Let A and S be mappings from X into itself satisfying the following conditions:

  1. (1)

    the pair (A,S) satisfies the ( CLR S ) property,

  2. (2)

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

    ϕ( M ( A x , A y , k t ) , M ( S x , S y , t ) , M ( A x , S x , t ) , M ( A y , S y , t ) , M ( A x , S y , t ) , M ( A y , S x , t ) )0
    (4.2)

for all x,yX, t>0 and ϕ Φ 6 .

Then A and S have a coincidence point. Moreover, if the pair (A,S) is weakly compatible, then A and S have a unique common fixed point.

Remark 4.5 Corollary 4.1 (in view of Example 3.1) generalizes the results of Miheţ [[25], Theorem 3.1].

As an application of Theorem 4.2, we have the following result involving four finite families of self-mappings.

Theorem 4.4 Let (X,M,) be a fuzzy metric space with ttt for all t[0,1]. Let { A i } i = 1 m , { B r } r = 1 n , { S k } k = 1 p and { T g } g = 1 q be four finite families from X into itself such that A= A 1 A 2 A m , B= B 1 B 2 B n , S= S 1 S 2 S p and T= T 1 T 2 T q which satisfy the inequality (4.1). If the pairs (A,S) and (B,T) satisfy the ( CLR ST ) property, then (A,S) and (B,T) have a point of coincidence each.

Moreover, { A i } i = 1 m , { B r } r = 1 n , { S k } k = 1 p and { T g } g = 1 q have a unique common fixed point provided the pairs of families ({ A i },{ S k }) and ({ B r },{ T g }) are commute pairwise, where i{1,2,,m}, k{1,2,,p}, r{1,2,,n} and g{1,2,,q}.

Proof The proof of this theorem is similar to that of Theorem 3.1 contained in Imdad et al. [39], hence details are avoided. □

Now, we indicate that Theorem 4.4 can be utilized to derive common fixed point theorems for any finite number of mappings. As a sample for five mappings, we can derive the following by setting one family of two members while the remaining three of single members.

Corollary 4.2 Let (X,M,) be a fuzzy metric space with ttt for all t[0,1]. Let A, B, R, S and T be mappings from X into itself and satisfy the following conditions:

  1. (1)

    the pairs (A,SR) and (B,T) share the ( CLR ( S R ) ( T ) ) property,

  2. (2)

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

    ϕ( M ( A x , B y , k t ) , M ( S R x , T H y , t ) , M ( A x , S R x , t ) , M ( B y , T H y , t ) , M ( A x , T H y , t ) , M ( B y , S R x , t ) )0
    (4.3)

for all x,yX, t>0 and ϕ Φ 6 .

Then the pairs (A,SR) and (B,T) have a coincidence point each. Moreover, A, B, R, S and T have a unique common fixed point provided the pairs (A,SR) and (B,T) commute pairwise (that is, AS=SA, AR=RA, SR=RS and BT=TB).

Similarly, we can derive a common fixed point theorem for six mappings by setting two families of two members while the rest two of single members.

Corollary 4.3 Let (X,M,) be a fuzzy metric space with ttt for all t[0,1]. Let A, B, R, S, H and T be mappings from X into itself and satisfy the following conditions:

  1. (1)

    the pairs (A,SR) and (B,TH) enjoy the ( CLR ( S R ) ( T H ) ) property,

  2. (2)

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

    ϕ( M ( A x , B y , k t ) , M ( S R x , T H y , t ) , M ( A x , S R x , t ) , M ( B y , T H y , t ) , M ( A x , T H y , t ) , M ( B y , S R x , t ) )0
    (4.4)

for all x,yX, t>0 and ϕ Φ 6 .

Then the pairs (A,SR) and (B,TH) have a coincidence point each. Moreover, A, B, R, S, H and T have a unique common fixed point provided the pairs (A,SR) and (B,TH) commute pairwise (that is, AS=SA, AR=RA, SR=RS, BT=TB, BH=HB and TH=HT).

By setting A 1 = A 2 == A m =A, B 1 = B 2 == B n =B, S 1 = S 2 == S p =S and T 1 = T 2 == T q =T in Theorem 4.4, we deduce the following.

Corollary 4.4 Let (X,M,) be a fuzzy metric space with ttt for all t[0,1]. Let A, B, S and T be mappings from X into itself such that the pairs ( A m , S p ) and ( B n , T q ) share the ( CLR S p , T q ) property. Suppose that there exists a constant k(0,1) such that

ϕ( M ( A m x , B n y , k t ) , M ( S p x , T q y , t ) , M ( A m x , S p x , t ) , M ( B n y , T q y , t ) , M ( A m x , T q y , t ) , M ( B n y , S p x , t ) )0
(4.5)

for all x,yX, t>0, ϕ Φ 6 and m, n, p, q are fixed positive integers.

Then the pairs (A,S) and (B,T) have a point of coincidence each. Further, A, B, S and T have a unique common fixed point provided both the pairs ( A m , S p ) and ( B n , T q ) commute pairwise.

Remark 4.6 Theorem 4.4 and Corollary 4.4 generalize and extend the results of Abbas et al. [[19], Theorem 2.3, Corollary 2.5] to finite families of weakly compatible mappings.

5 Related results for contractive conditions of integral type

A study of contractive conditions of integral type has been initiated by Branciari [40], giving an integral version of the Banach contraction principle. Subsequently, several authors proved common fixed point theorems satisfying contractive conditions of integral type (e.g. [4144]). In this section, we state and prove an integral analogue of Theorem 4.1. First, we need to prove the following lemma in fuzzy metric spaces which is motivated by Altun et al. [45].

Lemma 5.1 Let (X,M,) be a fuzzy metric space with ttt for all t[0,1]. If there exists a constant k(0,1) such that for all x,yX and all t>0,

0 M ( x , y , k t ) φ(s)ds 0 M ( x , y , t ) φ(s)ds,
(5.1)

where φ:[0,)[0,) is a summable non-negative Lebesgue integrable function such that

ϵ 1 φ(s)ds>0,

for each ϵ[0,1), then x=y.

Proof From (5.1), we have

0 M ( x , y , k t ) φ(s)ds 0 M ( x , y , t ) φ(s)ds,

implies

0 M ( x , y , t ) φ(s)ds 0 M ( x , y , k 1 t ) φ(s)ds,

similarly, we can inductively write for nN

0 M ( x , y , t ) φ ( s ) d s 0 M ( x , y , k 1 t ) φ ( s ) d s 0 M ( x , y , k 2 t ) φ ( s ) d s 0 M ( x , y , k n t ) φ ( s ) d s 0 1 φ ( s ) d s ,

as n. Therefore,

0 M ( x , y , t ) φ(s)ds 0 1 φ(s)ds0,

and so

0 M ( x , y , t ) φ(s)ds ( 0 M ( x , y , t ) φ ( s ) d s + M ( x , y , t ) 1 φ ( s ) d s ) 0,

or, equivalently,

M ( x , y , t ) 1 φ(s)ds0,

implying M(x,y,t)1 for all t0. Then we get x=y. □

Remark 5.1 By setting φ(s)=1 for each s0 in (5.1), we have

0 M ( x , y , k t ) ds= [ s ] 0 M ( x , y , k t ) =M(x,y,kt)M(x,y,t)= 0 M ( x , y , t ) ds,

which shows that Lemma 5.1 is a generalization of the Lemma 2 contained in [4].

Theorem 5.1 Let (X,M,) be a fuzzy metric space with ttt for all t[0,1]. Let A, B, S and T be mappings from X into itself. Suppose that there exists a function ϕ Φ 6 satisfying

0 ϕ ( u , v , v , v , v , v ) φ(s)ds0,
(5.2)

where φ:[0,)[0,) is a summable non-negative Lebesgue integrable function such that for each ϵ[0,1)

ϵ 1 φ(s)ds>0,

implies uv for all u,v[0,1]. Suppose that the pairs (A,S) and (B,T) share the ( CLR ST ) property. There exists a constant k(0,1) such that

0 ϕ ( M ( A x , B y , k t ) , M ( S x , T y , t ) , M ( A x , S x , t ) , M ( B y , T y , t ) , M ( A x , T y , t ) , M ( B y , S x , t ) ) φ(s)ds0,
(5.3)

for all x,yX and t>0, 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 both the pairs (A,S) and (B,T) are weakly compatible.

Proof Since the pairs (A,S) and (B,T) share the ( CLR ST ) property, there exist two sequences { x n } and { y n } in X such that

lim n A x n = lim n S x n = lim n B y n = lim n T y n =z,

where zS(X)T(X). Since zS(X), there exists a point uX such that Su=z. Now we assert that Au=Su. On using (5.3) with x=u, y= y n , we have

0 ϕ ( M ( A u , B y n , k t ) , M ( S u , T y n , t ) , M ( A u , S u , t ) , M ( B y n , T y n , t ) , M ( A u , T y n , t ) , M ( B y n , S u , t ) ) φ(s)ds0.

Taking the limit as n, we have

0 0 ϕ ( M ( A u , z , k t ) , M ( z , z , t ) , M ( A u , z , t ) , M ( z , z , t ) , M ( A u , z , t ) , M ( z , z , t ) ) φ(s)ds,

and so

0 0 ϕ ( M ( A u , z , k t ) , 1 , M ( A u , z , t ) , 1 , M ( A u , z , t ) , 1 ) φ(s)ds.

Since ϕ is decreasing in t 2 ,, t 6 , we get

0 0 ϕ ( M ( A u , z , k t ) , M ( z , A u , t ) , M ( A u , z , t ) , M ( z , A u , t ) , M ( A u , z , t ) , M ( z , A u , t ) ) φ(s)ds.

From (5.2), we have M(Au,z,kt)M(Au,z,t). Now appealing to Lemma 4.1, Au=z and hence Au=Su=z which shows that u is a coincidence point of the pair (A,S).

Also zT(X), therefore, there exists a point vX such that Tv=z. Now we show that Bv=Tv. On using (5.3) with x=u, y=v, we have

0 ϕ ( M ( A u , B v , k t ) , M ( S u , T v , t ) , M ( A u , S u , t ) , M ( B v , T v , t ) , M ( A u , T v , t ) , M ( B v , S u , t ) ) φ(s)ds0,

or, equivalently,

0 0 ϕ ( M ( z , B v , k t ) , M ( z , z , t ) , M ( z , z , t ) , M ( B v , z , t ) , M ( z , z , t ) , M ( B v , z , t ) ) φ(s)ds,

and so

0 0 ϕ ( M ( z , B v , k t ) , 1 , 1 , M ( B v , z , t ) , 1 , M ( B v , z , t ) ) φ(s)ds.

Since ϕ is decreasing in t 2 ,, t 6 , we get

0 0 ϕ ( M ( z , B v , k t ) , M ( z , B v , k t ) , M ( z , B v , k t ) , M ( B v , z , t ) , M ( z , B v , k t ) , M ( B v , z , t ) ) φ(s)ds.

From (5.2), we have M(z,Bv,kt)M(z,Bv,t). Owing Lemma 4.1, Bv=z and henceforth Bv=Tv=z. Therefore, v is a coincidence point of the pair (B,T).

Since the pair (A,S) is weakly compatible and Au=Su, therefore, Az=ASu=SAu=Sz. Putting x=z, y=v in (5.3), we get

0 ϕ ( M ( A z , B v , k t ) , M ( S z , T v , t ) , M ( A z , S z , t ) , M ( B v , T v , t ) , M ( A z , T v , t ) , M ( B v , S z , t ) ) φ(s)ds0,

and so

0 0 ϕ ( M ( A z , z , k t ) , M ( z , z , t ) , M ( A z , A z , t ) , M ( z , z , t ) , M ( A z , z , t ) , M ( z , A z , t ) ) φ(s)ds,

or, equivalently,

0 0 ϕ ( M ( A z , z , k t ) , 1 , 1 , 1 , M ( A z , z , t ) , M ( z , A z , t ) ) φ(s)ds.

Since ϕ is decreasing in t 2 ,, t 6 , we obtain

0 0 ϕ ( M ( A z , z , k t ) , M ( A z , z , t ) , M ( A z , z , t ) , M ( A z , z , t ) , M ( A z , z , t ) , M ( z , A z , t ) ) φ(s)ds.

From (5.2), we have M(Az,z,kt)M(Az,z,t). Using Lemma 4.1, we have Az=z=Sz which shows that z is a common fixed point of the pair (A,S).

Also the pair (B,T) is weakly compatible and Bv=Tv, hence Bz=BTv=TBv=Tz. Now we show that z is a common fixed point of the pair (B,T). In order to accomplish this, putting x=u, y=z in (5.3), we have

0 ϕ ( M ( A u , B z , k t ) , M ( S u , T z , t ) , M ( A u , S u , t ) , M ( B z , T z , t ) , M ( A u , T z , t ) , M ( B z , S u , t ) ) φ(s)ds0,

and so

0 0 ϕ ( M ( z , B z , k t ) , M ( z , B z , t ) , M ( z , z , t ) , M ( B z , B z , t ) , M ( z , B z , t ) , M ( B z , z , t ) ) φ(s)ds,

which reduces to

0 0 ϕ ( M ( z , B z , k t ) , M ( z , B z , t ) , 1 , 1 , M ( z , B z , t ) , M ( B z , z , t ) ) φ(s)ds.

Since ϕ is decreasing in t 2 ,, t 6 , we get

0 0 ϕ ( M ( z , B z , k t ) , M ( z , B z , t ) , M ( z , B z , t ) , M ( z , B z , t ) , M ( z , B z , t ) , M ( B z , z , t ) ) φ(s)ds.

From (5.2), we have M(z,Bz,kt)M(z,Bz,t). Appealing to Lemma 4.1, we have Bz=z=Tz which shows that z is a common fixed point of the pair (B,T). Hence z is a common fixed point of mappings A, B, S and T. Uniqueness of common fixed point is an easy consequence of the inequality (5.3). □

Remark 5.2 Results similar to Theorem 4.2, Theorem 4.3, Theorem 4.4, Corollary 4.1, Corollary 4.2 and Corollary 4.3 can be outlined in views of Theorem 5.1, but we do not include the details due to repetition. Earlier proved results can also be described in view of Examples 3.1-3.3 (using Lemma 5.1) which improve the results of Chauhan and Kumar [[20], Theorem 4.1].

Remark 5.3 Theorem 5.1 (in view of Remark 5.2) improves the results of Bhatia [[41], Theorem 3.1], Shao [[43], Theorem 3.2], Murthy et al. [[42], Theorems 2, 3, 5] and extend the result of Sedghi and Shobe [[44], Theorem 2.2].

References

  1. Zadeh LA: Fuzzy sets. Inf. Control 1965, 8: 338–353. 10.1016/S0019-9958(65)90241-X

    MathSciNet  Article  MATH  Google Scholar 

  2. Kramosil I, Michálek J: Fuzzy metric and statistical metric spaces. Kybernetika 1975, 11(5):336–344.

    MathSciNet  MATH  Google Scholar 

  3. George A, Veeramani P: On some result in fuzzy metric space. Fuzzy Sets Syst. 1994, 64: 395–399. 10.1016/0165-0114(94)90162-7

    MathSciNet  Article  MATH  Google Scholar 

  4. Mishra SN, Sharma N, Singh SL: Common fixed points of maps on fuzzy metric spaces. Int. J. Math. Math. Sci. 1994, 17(2):253–258. 10.1155/S0161171294000372

    MathSciNet  Article  MATH  Google Scholar 

  5. Singh B, Jain S: Semicompatibility and fixed point theorems in fuzzy metric space using implicit relation. Int. J. Math. Math. Sci. 2005, 16: 2617–2629.

    MathSciNet  Article  MATH  Google Scholar 

  6. Balasubramaniam P, Muralisankar S, Pant RP: Common fixed points of four mappings in a fuzzy metric space. J. Fuzzy Math. 2002, 10(2):379–384.

    MathSciNet  MATH  Google Scholar 

  7. Beg I, Sedghi S, Shobe N: Fixed point theorems in fuzzy metric spaces. Int. J. Anal. 2013., 2013: Article ID 934145

    Google Scholar 

  8. Chauhan S, Pant BD: Common fixed point theorems in fuzzy metric spaces. Bull. Allahabad Math. Soc. 2012, 27(1):27–43.

    MathSciNet  MATH  Google Scholar 

  9. Cho YJ: Fixed points in fuzzy metric spaces. J. Fuzzy Math. 1997, 5(4):949–962.

    MathSciNet  MATH  Google Scholar 

  10. Imdad M, Ali J: Common fixed point theorems in complete fuzzy metric spaces. Math. Commun. 2006, 11(2):153–163.

    MathSciNet  MATH  Google Scholar 

  11. O’Regan D, Abbas M: Necessary and sufficient conditions for common fixed point theorems in fuzzy metric space. Demonstr. Math. 2009, 42(4):887–900.

    MathSciNet  MATH  Google Scholar 

  12. Rao KPR, Babu GVR, Fisher B: Common fixed point theorems in fuzzy metric spaces under implicit relations. Hacet. J. Math. Stat. 2008, 37(2):97–106.

    MathSciNet  MATH  Google Scholar 

  13. Sharma S: Common fixed point theorems in fuzzy metric spaces. Fuzzy Sets Syst. 2002, 127(3):345–352. 10.1016/S0165-0114(01)00112-9

    Article  MathSciNet  MATH  Google Scholar 

  14. Pant RP: Common fixed points of four mappings. Bull. Calcutta Math. Soc. 1998, 90(4):281–286.

    MathSciNet  MATH  Google Scholar 

  15. Aamri M, El Moutawakil D: Some new common fixed point theorems under strict contractive conditions. J. Math. Anal. Appl. 2002, 270: 181–188. 10.1016/S0022-247X(02)00059-8

    MathSciNet  Article  MATH  Google Scholar 

  16. Ali J, Imdad M: An implicit function implies several contraction conditions. Sarajevo J. Math. 2008, 17(4):269–285.

    MathSciNet  MATH  Google Scholar 

  17. Liu Y, Wu J, Li Z: Common fixed points of single-valued and multivalued maps. Int. J. Math. Math. Sci. 2005, 19: 3045–3055. doi:10.1155/IJMMS.2005.3045

    MathSciNet  Article  MATH  Google Scholar 

  18. Aalam I, Kumar S, Pant BD: A common fixed point theorem in fuzzy metric space. Bull. Math. Anal. Appl. 2010, 2(4):76–82.

    MathSciNet  MATH  Google Scholar 

  19. Abbas M, Altun I, Gopal D: Common fixed point theorems for non compatible mappings in fuzzy metric spaces. Bull. Math. Anal. Appl. 2009, 1(2):47–56.

    MathSciNet  MATH  Google Scholar 

  20. Chauhan S, Kumar S: Coincidence and fixed points in fuzzy metrtc spaces using common property (E.A). Kochi J. Math. 2013, 8: 135–154.

    MathSciNet  MATH  Google Scholar 

  21. Gopal D, Imdad M, Vetro C: Impact of common property (E.A.) on fixed point theorems in fuzzy metric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 297360. doi:10.1155/2011/297360

    Google Scholar 

  22. Imdad M, Ali J, Hasan M: Common fixed point theorems in fuzzy metric spaces employing common property (E.A.). Math. Comput. Model. 2012, 55(3–4):770–778. 10.1016/j.mcm.2011.09.002

    MathSciNet  Article  MATH  Google Scholar 

  23. Kumar S, Fisher B: A common fixed point theorem in fuzzy metric space using property (E.A.) and implicit relation. Thai J. Math. 2010, 8(3):439–446.

    MathSciNet  MATH  Google Scholar 

  24. Kumar S, Chauhan S: Common fixed point theorems using implicit relation and property (E.A) in fuzzy metric spaces. Ann. Fuzzy Math. Inform. 2013, 5(1):107–114.

    MathSciNet  MATH  Google Scholar 

  25. Miheţ D: Fixed point theorems in fuzzy metric spaces using property (E.A). Nonlinear Anal. 2010, 73(7):2184–2188. 10.1016/j.na.2010.05.044

    MathSciNet  Article  MATH  Google Scholar 

  26. Pant V, Pant RP: Fixed points in fuzzy metric space for non-compatible maps. Dōngwú Shùlı̌xué Bào 2007, 33(4):647–655.

    MathSciNet  MATH  Google Scholar 

  27. Sintunavarat W, Kumam P: Common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric spaces. J. Appl. Math. 2011., 2011: Article ID 637958

    Google Scholar 

  28. Chauhan S: Fixed points of weakly compatible mappings in fuzzy metric spaces satisfying common limit in the range property. Indian J. Math. 2012, 54(3):375–397.

    MathSciNet  MATH  Google Scholar 

  29. Chauhan, S, Bhatnagar, S, Radenović, S: Common fixed point theorems for weakly compatible mappings in fuzzy metric spaces. Le Matematiche (2013, in printing)

  30. Chauhan S, Sintunavarat W, Kumam P: Common fixed point theorems for weakly compatible mappings in fuzzy metric spaces using (JCLR) property. Appl. Math. 2012, 3(9):976–982. doi:10.4236/am.2012.39145 10.4236/am.2012.39145

    Article  Google Scholar 

  31. Sintunavarat W, Kumam P: Common fixed points for R -weakly commuting in fuzzy metric spaces. Ann. Univ. Ferrara 2012, 58(2):389–406. 10.1007/s11565-012-0150-z

    MathSciNet  Article  MATH  Google Scholar 

  32. Singh SL, Pant BD, Chauhan S: Fixed point theorems in non-Archimedean Menger PM-spaces. J. Nonlinear Anal. Optim. 2012, 3(2):153–160.

    MathSciNet  Google Scholar 

  33. Imdad M, Pant BD, Chauhan S:Fixed point theorems in Menger spaces using the ( CLR ST ) property and applications. J. Nonlinear Anal. Optim. 2012, 3(2):225–237.

    MathSciNet  Google Scholar 

  34. Imdad M, Ali J: A general fixed point theorem in fuzzy metric spaces via an implicit function. J. Appl. Math. Inform. 2008, 26(3–4):591–603.

    Google Scholar 

  35. Schweizer B, Sklar A North-Holland Series in Probability and Applied Mathematics. In Probabilistic Metric Spaces. North-Holland, New York; 1983.

    Google Scholar 

  36. Grabiec M: Fixed points in fuzzy metric spaces. Fuzzy Sets Syst. 1988, 27(3):385–389. 10.1016/0165-0114(88)90064-4

    MathSciNet  Article  MATH  Google Scholar 

  37. Jungck G: Common fixed points for noncontinuous nonself maps on nonmetric spaces. Far East J. Math. Sci. 1996, 4(2):199–215.

    MathSciNet  MATH  Google Scholar 

  38. Pathak HK, López RR, Verma RK: A common fixed point theorem using implicit relation and property (E.A) in metric spaces. Filomat 2007, 21(2):211–234. 10.2298/FIL0702211P

    MathSciNet  Article  MATH  Google Scholar 

  39. Imdad M, Ali J, Tanveer M: Coincidence and common fixed point theorems for nonlinear contractions in Menger PM spaces. Chaos Solitons Fractals 2009, 42(5):3121–3129. 10.1016/j.chaos.2009.04.017

    MathSciNet  Article  MATH  Google Scholar 

  40. Branciari A: A fixed point theorem for mappings satisfying a general contractive condition of integral type. Int. J. Math. Math. Sci. 2002, 29(9):531–536. 10.1155/S0161171202007524

    MathSciNet  Article  MATH  Google Scholar 

  41. Bhatia SS, Manro S, Kumar S: Fixed point theorem for weakly compatible maps using E.A. property in fuzzy metric spaces satisfying contractive condition of integral type. Int. J. Contemp. Math. Sci. 2010, 5(51):2523–2528.

    MathSciNet  MATH  Google Scholar 

  42. Murthy PP, Kumar S, Taş K: Common fixed points of self maps satisfying an integral type contractive condition in fuzzy metric spaces. Math. Commun. 2010, 15(2):521–537.

    MathSciNet  MATH  Google Scholar 

  43. Shao X, Hu X: Common fixed point under integral type contractive condition in fuzzy metric spaces. Third Internat. Workshop on Adv. Comput. Intel. 2010, 131–134.

    Chapter  Google Scholar 

  44. Sedghi S, Shobe N: Common fixed point theorems for weakly compatible mappings satisfying contractive condition of integral type. J. Adv. Res. Appl. Math. 2011, 3(4):67–78. doi:10.5373/jaram.821.030711 10.5373/jaram.821.030711

    MathSciNet  Article  Google Scholar 

  45. Altun I, Tanveer M, Miheţ D, Imdad M: Common fixed point theorems of integral type in Menger PM spaces. J. Nonlinear Anal. Optim. 2012, 3(1):55–66.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Sunny Chauhan.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

SC, MAK and SK contributed equally. All authors read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Chauhan, S., Khan, M.A. & Kumar, S. Unified fixed point theorems in fuzzy metric spaces via common limit range property. J Inequal Appl 2013, 182 (2013). https://doi.org/10.1186/1029-242X-2013-182

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1029-242X-2013-182

Keywords

  • fuzzy metric space
  • compatible mappings
  • weakly compatible mappings
  • common limit range property
  • fixed point