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In this paper, we prove a common fixed point theorem for a pair of weakly compatible mappings in fuzzy metric space using the joint common limit in the range property of mappings called (JCLR) property. An example is also furnished which demonstrates the validity of main result. We also extend our main result to two finite families of self mappings. Our results improve and generalize results of Cho et al. [Y. J. Cho, S. Sedghi and N. Shobe, “Generalized fixed point theorems for compatible mappings with some types in fuzzy metric spaces,” Chaos, Solitons & Fractals, Vol. 39, No. 5, 2009, pp. 2233-2244.] and several known results existing in the literature.

In 1965, Zadeh [

In 2002, Aamri and El-Moutawakil [

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. [

Definition 2.1 [

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 [

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 [

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 [

Lemma 2.2 [

for all and, then.

Definition 2.4 [

Remark 2.1 [

Definition 2.5 [

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

In 2011, Sintunavarat and Kumam [

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. [

Definition 2.7 [

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. ([

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. ([

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. ([

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. [

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 [