In 1965, Zadeh [1] investigated the concept of a fuzzy set in his seminal paper. In the last two decades there has been a tremendous development and growth in fuzzy mathematics. The concept of fuzzy metric space was introduced by Kramosil and Michalek [2] in 1975, 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 Hausdoroff topology which has very important applications in quantum particle physics, particularly in connection with both string and theory (see, [4] and references mentioned therein). Fuzzy set theory also 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.

In 2002, Aamri and El-Moutawakil [5] defined the notion of (E.A) property for self mappings which contained the class of non-compatible mappings in metric spaces. It was pointed out that (E.A) property allows replacing the completeness requirement of the space with a more natural condition of closedness of the range as well as relaxes the complexness of the whole space, continuity of one or more mappings and 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 (E.A) property in fuzzy metric spaces (see [6-11]). Most recently, Sintunavarat and Kumam [12] defined the notion of “common limit in the range” property (or (CLR) property) in fuzzy metric spaces and improved the results of Mihet [10]. In [12], it is observed that the notion of (CLR) property never requires the condition of the closedness of the subspace while (E.A) property requires this condition for the existence of the fixed point (also see [13]). Many authors have proved common fixed point theorems in fuzzy metric spaces for different contractive conditions. For details, we refer to [14-25].

The aim of this paper is to introduce the notion of the joint common limit in the range of mappings property called (JCLR) property and prove a common fixed point theorem for a pair of weakly compatible mappings using (JCLR) property in fuzzy metric space. As an application to our main result, we present a common fixed point theorem for two finite families of self mappings in fuzzy metric space using the notion of pairwise commuting due to Imdad et al. [15]. Our results improve and generalize the results of Cho et al. [26], Abbas et al. [7] and Kumar [8].

Definition 2.1 [27] A binary operation

is a continuous t-norm if it satisfies the following conditions: 

1) is associative and commutative

2) is continuous

3) for all

4) whenever and for all.

Examples of continuous t-norms are and.

Definition 2.2 [3] A 3-tuple 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 satisfying the following conditions: For all, , 

1)

2) if and only if

3)

4)5) is continuous.

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

Let be a fuzzy metric space. For, the open ball with center and radius is defined by

Now let be a fuzzy metric space and the set of all with if and only if there exist and such that. Then 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 be a metric space. Denote (or) for all and let be fuzzy sets on defined as follows:

.

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

Definition 2.3 Let be a fuzzy metric space. M is said to be continuous on if

whenever a sequence in converge to a point, i.e.,

and

Lemma 2.1 [28] Let be a fuzzy metric space. Then is non-decreasing for all.

Lemma 2.2 [29] Let be a fuzzy metric space. If there exists such that

for all and, then.

Definition 2.4 [30] Two self mappings f and g of a non-empty set X are said to be weakly compatible (or coincidentally commuting) if they commute at their coincidence points, i.e. if some , then.

Remark 2.1 [30] Two compatible self mappings are weakly compatible, but the converse is not true. Therefore the concept of weak compatibility is more general than that of compatibility.

Definition 2.5 [7] A pair of self mappings f and g of a fuzzy metric space are said to satisfy the (E.A) property, if there exists a sequence in X for some such that

Remark 2.2 It is noted that weak compatibility and (E.A) property are independent to each other (see [31], Example 2.1, Example 2.2).

In 2011, Sintunavarat and Kumam [12] defined the notion of “common limit in the range” property in fuzzy metric space as follows:

Definition 2.6 A pair of self mappings of a fuzzy metric space is said to satisfy the “common limit in the range of g” property (shortly, (CLRg) property) if there exists a sequence in X such that

for some.

Now, we show examples of self mappings f and g which are satisfying the (CLRg) property.

Example 2.2 Let be a fuzzy metric space with and

for all. Define self mappings f and g on X by and for all. Let a sequence in X, we have

which shows that f and g satisfy the (CLRg) property.

Example 2.3 The conclusion of Example 2.2 remains true if the self mappings f and g is defined on X by

and for all. Let a sequence in X. Since

therefore f and g satisfy the (CLRg) property.

The following definition is on the lines due to Imdad et al. [32].

Definition 2.7 [32] Two families of self mappings

and are said to be pairwise commuting if 

1) for all

2) for all

3) for all and .

Throughout this paper, is considered to be a fuzzy metric space with condition

for all.

In this section, we first introduce the notion of “the joint common limit in the range property” of two pairs of self mappings.

Definition 3.1 Let be a fuzzy metric space and. The pair and are said to satisfy the “joint common limit in the range of b and g” property (shortly, (JCLRbg) property) if there exists a sequence and in X such that

for some.

Remark 3.1 If, and in (1), then we get the definition of (CLRg).

Throughout this section, denotes the set of all continuous and increasing functions in any coordinate and for all.

Following are examples of some function:

1) for some.

2) for some.

3) for some and for all t-norm such that.

Now, we state and prove main results in this paper.

Theorem 3.1 Let be a fuzzy metric space, where is a continuous t-norm and f, g, a and b be mappings from X into itself. Further, let the pair and are weakly compatible and there exists a constant such that

holds for all, , and. If and satisfy the (JCLRbg) property, then f, g, a and b have a unique common fixed point in X.

Proof. Since the pairs and satisfy the (JCLRbg) property, there exists a sequence and in X such that

for some.

Now we assert that. Using (2), with, , for, we get

Taking the limit as, we have

Since is increasing in each of its coordinate and for all, we get . By Lemma 2.2, we have.

Next we show that. Using (2), with, , for, we get

Taking the limit as, we have

Since is increasing in each of its coordinate and for all, we get

. By Lemma 2.2, we have.

Now, we assume that. Since the pair is weakly compatible, and then. It follows from is weakly compatible, and hence .

We show that. To prove this, using (2) with, , for, we get

and so

Since is increasing in each of its coordinate and for all, , which implies that. Hence.

Next, we show that. To prove this, using (2) with, , for, we get

and so

Since is increasing in each of its coordinate and for all, , which implies that. Hence. Therefore, we conclude that this implies f, g, a and b have common fixed point that is a point z.

For uniqueness of common fixed point, we let w be another common fixed point of the mappings f, g, a and b. On using (2) with, , for, we have

and then

Since is increasing in each of its coordinate and for all, , which implies that. Therefore f, g, a and b have a unique a common fixed point.

Remark 3.2 From the result, it is asserted that (JCLRgb) property never requires any condition closedness of the subspace, continuity of one or more mappings and containment of ranges amongst involved mappings.

Remark 3.3 Theorem 3.1 improves and generalizes the results of Abbas et al. ([7], Theorem 2.1) and Kumar ([8], Theorem 2.3) without any requirement of containment amongst range sets of the involved mappings and closedness of the underlying subspace.

Remark 3.4 Since the condition of t-norm with for all is replaced by arbitrary continuous t-norm, Theorem 3.1 also improves the result of Cho et al. ([26], Theorem 3.1) without any requirement of completeness of the whole space, continuity of one or more mappings and containment of ranges amongst involved mappings.

Corollary 3.1 Let be a fuzzy metric space, where is a continuous t-norm and f, g, a and b be mappings from X into itself. Further, let the pair and are weakly compatible and there exists a constant such that

holds for all, , and such that. If and satisfy the (JCLRbg) property, then f, g, a and b have a unique common fixed point in X.

Proof. By Theorem 3.1, if we define

then the result follows.

Remark 3.5 Corollary 3.1 improves the result of Cho et al. ([26], Corollary 3.4) without any requirement of completeness of the whole space, continuity of one or more mappings and containment of ranges amongst involved mappings while the condition of t-norm for all is replaced by arbitrary continuous t-norm.

Corollary 3.2 Let be a fuzzy metric space, where is a continuous t-norm and f and g be mappings from X into itself. Further, let the pair is weakly compatible and there exists a constant such that

holds for all, , and. If satisfies the (CLRg) property, then f and g have a unique common fixed point in X.

Proof. Take and in Theorem 3.1, then we get the result.

Our next theorem is proved for a pair of weakly compatible mappings in fuzzy metric space using (E.A) property under additional condition closedness of the subspace.

Theorem 3.2 Let be a fuzzy metric space, where is a continuous t-norm. Further, let the pair of self mappings is weakly compatible satisfying inequality (4) of Corollary 3.2. If f and g satisfy the (E.A) property and the range of g is a closed subspace of X, then f and g have a unique common fixed point in X.

Proof. Since the pair satisfies the (E.A) property, there exists a sequence in X such that

for some. It follows from being a closed subspace of X that there exists in which. Therefore f and g satisfy the (CLRg) property. From Corollary 3.2, the result follows.

In what follows, we present some illustrative examples which demonstrate the validity of the hypotheses and degree of utility of our results.

Example 3.1 Let with the metric d defined by and for each define

for all. Clearly be a fuzzy metric space with t-norm defined by for all

. Consider a function defined by. Then we have

. Define the self mappings f and g on X by

and

Taking or, it is clear that the pair satisfies the (CLRg) property since

It is noted that. Thus, all the conditions of Corollary 3.2 are satisfied for a fixed constant and 2 is a unique common fixed point of the pair. Also, all the involved mappings are even discontinuous at their unique common fixed point 2. Here, it may be pointed out that is not a closed subspace of X.

Example 3.2 In the setting of Example 3.1, replace the mapping g by the following, besides retaining the rest:

Taking or, it is clear that the pair satisfies the (E.A) property since

It is noted that. Thus, all the conditions of Theorem 3.2 are satisfied and 2 is a unique common fixed point of the mappings f and g. Notice that all the involved mappings are even discontinuous at their unique common fixed point 2. Here, it is worth noting that is a closed subspace of X.

Now, we utilize Definition 2.7 which is a natural extension of commutativity condition to two finite families of self mappings. Our next theorem extends Corollary 3.2 in the following sense:

Theorem 3.3 Let and be two finite families of self mappings in fuzzy metric space, where is a continuous t-norm such that and which satisfy the inequalities (4) of Corollary 3.2. If the pair shares (CLRg) property, then f and g have a unique point of coincidence.

Moreover, and have a unique common fixed point provided the pair of families

commutes pairwise, where and .

Proof. The proof of this theorem can be completed on the lines of Theorem 3.1 contained in Imdad et al. [15], hence details are avoided.

Putting and

in Theorem 3.3, we get the following result:

Corollary 3.3 Let f and g be two self mappings of a fuzzy metric space, where is a continuous t-norm. Further, let the pair shares (CLRg) property. Then there exists a constant such that

holds for all, , , and m and n are fixed positive integers, then f and g have a unique common fixed point provided the pair commutes pairwise.

Remark 3.6 Theorem 3.2, Theorem 3.3 and Corollary 3.3 can also be outlined in respect of Corollary 3.1.

Remark 3.7 Using Example 2.2, we can obtain several fixed point theorems in fuzzy metric spaces in respect of Theorems 3.2 and 3.3 and Corollaries 3.2, 3.1 and 3.3.

The authors would like to express their sincere thanks to Professor Mujahid Abbas for his paper [18]. The second author would like to thank the Research Professional Development Project under the Science Achievement Scholarship of Thailand (SAST). This study was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission under the Computational Science and Engineering Research Cluster (CSEC Grant No. 55000613).