#### Abstract

This paper is to present a common fixed point theorem for two *R*-weakly commuting self-mappings satisfying nonlinear contractive type condition defined using a Φ-function, defined on fuzzy metric spaces. Some comments on previously published results and some examples are given.

#### 1. Introduction and Preliminaries

The theory of fuzzy sets was introduced by Zadeh in [1]. Definitions of fuzzy metric spaces were given by Kaleva and Seikkala in [2] and by Kramosil and Michálek in [3], and these definitions are accepted by many mathematicians and physicists. The modification of these definitions were given by George and Veeramani in [4].

*Definition 1 (see [5]). *A binary operation is a continuous -norm if is a topological monoid with unit 1 such that whenever and are satisfied, for all .

*Example 2. *.

*Definition 3 (see [1]). *A fuzzy set in is a function with domain in and values in .

*Definition 4 (see [3]). *The 3-tuple is said to be a fuzzy metric space in the sense of Kramosil and Michálek, where is an arbitrary set, is a continuous -norm, and is a fuzzy set defined on set satisfying the following conditions: (FM-1) ,
(FM-2) , for every if and only if ,(FM-3), for all and ,(FM-4), for all and ,(FM-5) is a left continuous function, for all .

A function is called a fuzzy metric.

*Remark 5 (see [4]). *Let be a metric space and and
for . Then, is a fuzzy metric space. The 3-tuple is a fuzzy metric space, also, if we observe a -norm instead of . Taking that in (1), we get

A fuzzy metric given by (2) is called the standard fuzzy metric, and it is induced by a metric .

Grabiec in [6] proved the next lemma.

Lemma 6 (see [6]). *Let be a fuzzy metric space. Then, is a nondecreasing function, for all .*

*Definition 7 (see [4]). *A sequence in a fuzzy metric space is said to converge to if and only if for every and , there exists such that , for every , for every . In that case, write one .

*Definition 8 (see [4]). *A sequence in a fuzzy metric space is said to be a Cauchy sequence if for every and , there exists such that , for all . A fuzzy metric space is said to be complete if every Cauchy sequence is convergent.

The following lemma is essentially given in [6].

Lemma 9. *Let be a fuzzy metric space and and . Then, for every fixed , it holds
*

George and Veeramani in [4] introduced Hausdorff topology in fuzzy metric spaces. This topology is first countable.

*Definition 10 (see [4]). *Let be a fuzzy metric space. Open ball with center in , and radius , for is defined as follows:

Topology in fuzzy metric spaces is defined as follows:

Theorem 11 (see [4]). *Every open ball is an open set. *

Theorem 12 (see [4]). *Every fuzzy metric space is Hausdorff. *

*Definition 13. *Let be a fuzzy metric space and . The closure of the set is the smallest closed set containing , denoted by .

Obviously, having in mind the Hausdorff topology and the definition of converging sequences, we have that the next remark holds.

*Remark 14. * if and only if there exists a sequence in such that .

The facts that will play the important role in the proof of the main result are the following definition and theorem which are given and proved in [7].

*Definition 15 (see [7]). *Let be a fuzzy metric space. A collection is said to have fuzzy diameter zero if for each and each , there exists such that , for all .

Theorem 16 (see [7]). *A fuzzy metric space is complete if and only if every nested sequence of nonempty closed sets with fuzzy diameter zero has nonempty intersection. *

*Remark 17 (see [7]). *The element is unique.

The concept of bounded sets in probabilistic metric spaces is very close to the concept of strong boundness in fuzzy metric spaces.

*Definition 18. *Let be a fuzzy metric space. Let the mappings be defined as

The constant will be called fuzzy diameter of set . If the set will be called -strongly bounded.

Lemma 19. *Let be a fuzzy metric space. A set is an -strongly bounded if and only if for each , there exists such that , for all . *

*Proof. *The proof follows from the definitions of and of nonempty sets.

*Example 20. *Let be a fuzzy metric space induced by a metric on given in Remark 5. is metrically bounded if and only if it is an -strongly bounded.

*Proof. *Let be metrically bounded, that is, , for some and all . Let be arbitrary. We will prove that there exists such that , for all . Let us take such that . For this , it follows that from which it follows that . Finally, from the last inequality, we get that . Since , for all , it follows that , for all . Applying Lemma 19, we get that is an -strongly bounded.

Conversely, if is an -strongly bounded set, then for arbitrary , there exists such that , for all . From this inequality, it follows that , for all , that is, the set is metrically bounded. This completes the proof.

Properties of fuzzy metric spaces and fixed point results in these spaces are obtained by several authors [4, 7–11]. Results presented in [10, 11] consider the fixed point results for mappings satisfying nonlinear contractive type condition, but the class of fuzzy metric spaces is restrictive because the authors added an additional axiom for fuzzy metric spaces. Our result is applicable for arbitrary fuzzy metric space without restrictions. Recently, Miheţ in [12] proved a fixed point theorem for mappings defined on probabilistic metric spaces satisfying nonlinear contractive type condition defined using a -function. We will prove a common fixed point theorem for mappings defined on fuzzy metric spaces satisfying nonlinear contractive type condition defined using -function.

Khan et al. in [13] introduced the concept of altering distance functions that alter the distance between two points in metric spaces.

*Definition 21 (see [13]). *A function is an altering distance function if (i) is monotone increasing and continuous, (ii) if and only if .

Choudhury and Das [14] extended the concept of altering distance functions to Menger PM-spaces.

*Definition 22 (see [14]). *A function is said to be a -function if the following conditions hold(i) if and only if , (ii) is strictly increasing and as , (iii) is left continuous in , (iv) is continuous at 0. The class of all -functions will be denoted by .

Choudhury and Das [14] proved the following result.

Theorem 23 (see [14]). *Let be a complete Menger PM-space, with continuous -norm given by , and let be a continuous self-mapping on such that for every , and all holds
**
where is a -function and . Then, has a unique fixed point. *

The previous result is generalized and improved by Miheţ in [12].

Theorem 24 (see [12]). *Let be a complete Menger PM-space with continuous -norm , and let be a self-mapping on such that for every , and all holds
**
where is a -function and . If there exists such that the orbit of in , is probabilistic bounded, then has a unique fixed point. *

By proving common fixed point results, several authors observed different generalizations of commutativity. The concept of *R*-weakly commuting mappings was introduced by Pant in [15]. Following Pant, the definition of *R*-weakly commutativity for mappings defined on spaces with nondeterministic distances was given in [8, 9, 16].

*Definition 25. *Let be a fuzzy metric space, and let and be self-mappings of . The mappings and will be called *R*-weakly commutings, if there exists some positive real number such that
for all and each .

#### 2. Main Results

Lemma 26. *Let be a fuzzy metric space. Let be a -function and . If for , it holds that
**
for all , then .*

*Proof. *Let us suppose that and . From this condition, we have and, by induction, we have . Taking limit as , we get that , for all , which is a contradiction, that is, .

Theorem 27. *Let be a complete fuzzy metric space. Let and be R-weakly commuting self-mappings on , and let be continuous such that . Let, for all and every ,
**
hold, where is a -function and . If there exists a point and such that the set
**
where is an -strongly bounded set, then the mappings and have a unique common fixed point in .*

*Proof. *First, we will prove that the mapping is continuous. We will prove that for sequence , it follows that as . Let be arbitrary. For arbitrary , there exists such that . Since is continuous, it follows that , that is, for all , it holds that as . Since the function is nondecreasing for all , applying (11), we have
as as , for all , that is, as .

For from , it follows that there exists a point such that . By induction, a sequence can be chosen such that , and the set is an -strongly bounded.

Let us consider nested sequence of nonempty closed sets defined by

We will prove that the family has fuzzy diameter zero.

Let and be arbitrary. We will prove that there exists such that , for all . For arbitrary , from , it follows that is an -strongly bounded set, that is, there exists such that

Since is a -function, it follows that there exists such that . Let and be arbitrary. There exist sequences in such that and .

Since for arbitrary , there exists such that , from (11), we have

Thus, by induction, we get

Since and because is a nondecreasing function, from previous inequalities, it follows that

As and are sequences in from (15) and (18), it follows that, for all , it holds

Taking as and applying Lemma 9, we get that , for all , that is, the family has fuzzy diameter zero.

Applying Theorem 16, we conclude that this family has nonempty intersection, which consists of exactly one point . Since the family has fuzzy diameter zero and , for all , then for each and each there exists such that for all holds

From the last, it follows that for each and each holds

Taking that , we get that for each , it holds that
that is, . From the definition of , we have .

Since and are *R*-weakly commutings, we have that for each it holds that

Taking as from previous inequality, for each , it holds that
that is, it holds that

Let us prove that . From (11), it follows that for every , it holds that

Taking as , we have that holds, for every . Since , it follows that

Applying Lemma 26, we get that is a fixed point of . From (25), it follows that is a common fixed point of and .

Let us prove that is a unique common fixed point. For this purpose, let us suppose that there exists another common fixed point, denoted by . From the starting condition, follow
for every . Therefore, we get that
for every . Finally, applying Lemma 26, it follows that . This completes the proof.

If we take that is identical mapping, in the statement of Theorem 27, since identical mappings commute with and commuting mappings are -weakly commutings, we get the following theorem.

Theorem 28. *Let be a complete fuzzy metric space. Let be a self-mapping on such that for all and every , it holds
**
where is a -function and . If there exists a point and such that the set
**
where is an -strongly bounded set, then the mapping has a unique common fixed point in .*

Since every Menger probabilistic metric space is fuzzy metric space, Theorem 28 is an improvement of main result presented in [12].

*Example 29. *Let be a complete fuzzy metric space induced by the metric on given in Remark 5. Let and .

We will prove that all the conditions of Theorem 27 are satisfied. Because
we conclude that and are not commuting mappings. On the other hand, we have that

Since , for every , we have
that is, for , the condition (9) is satisfied, that is, we conclude that and are *R*-weakly commutings, for .

We will prove that the condition (11) is satisfied too. Since , for all , then we have

Since all the conditions of Theorem 27 are satisfied, we have that and have a unique common fixed point. It is easy to see that this point is .

#### Acknowledgments

This paper was supported by the Ministry of Education and Science of Serbia, Project Grant no. 174032. The authors are very grateful to the referees for their helpful suggestions.