- Research Article
- Open Access

# Impact of Common Property (E.A.) on Fixed Point Theorems in Fuzzy Metric Spaces

- D Gopal
^{1}Email author, - M Imdad
^{2}and - C Vetro
^{3}

**2011**:297360

https://doi.org/10.1155/2011/297360

© D. Gopal et al. 2011

**Received:**7 November 2010**Accepted:**9 March 2011**Published:**14 March 2011

## Abstract

We observe that the notion of common property (E.A.) relaxes the required containment of range of one mapping into the range of other which is utilized to construct the sequence of joint iterates. As a consequence, a multitude of recent fixed point theorems of the existing literature are sharpened and enriched.

## Keywords

- Closed Subset
- Fixed Point Theorem
- Common Property
- Common Fixed Point
- Semicontinuous Function

## 1. Introduction and Preliminaries

The evolution of fuzzy mathematics solely rests on the notion of fuzzy sets which was introduced by Zadeh [1] in 1965 with a view to represent the vagueness in everyday life. In mathematical programming, the problems are often expressed as optimizing some goal functions equipped with specific constraints suggested by some concrete practical situations. There exist many real-life problems that consider multiple objectives, and generally, it is very difficult to get a feasible solution that brings us to the optimum of all the objective functions. Thus, a feasible method of resolving such problems is the use of fuzzy sets [2]. In fact, the richness of applications has engineered the all round development of fuzzy mathematics. Then, the study of fuzzy metric spaces has been carried out in several ways (e.g., [3, 4]). George and Veeramani [5] modified the concept of fuzzy metric space introduced by Kramosil and Michálek [6] with a view to obtain a Hausdorff topology on fuzzy metric spaces, and this has recently found very fruitful applications in quantum particle physics, particularly in connection with both string and theory (see [7] and references cited therein). In recent years, many authors have proved fixed point and common fixed point theorems in fuzzy metric spaces. To mention a few, we cite [2, 8–15]. As patterned in Jungck [16], a metrical common fixed point theorem generally involves conditions on commutatively, continuity, completeness together with a suitable condition on containment of ranges of involved mappings by an appropriate contraction condition. Thus, research in this domain is aimed at weakening one or more of these conditions. In this paper, we observe that the notion of common property (E.A.) relatively relaxes the required containment of the range of one mapping into the range of other which is utilized to construct the sequence of joint iterates. Consequently, we obtain some common fixed point theorems in fuzzy metric spaces which improve many known earlier results (e.g., [11, 15, 17]).

Before presenting our results, we collect relevant background material as follows.

Definition 1.1 (see [18]).

Let be any set. A fuzzy set in is a function with domain and values in .

Definition 1.2 (see [6]).

A binary operation is a continuous -norm if it satisfies the following conditions:

(i) is associative and commutative,

Definition 1.3 (see [5]).

A triplet is a fuzzy metric space whenever is an arbitrary set, is a continuous -norm, and is a fuzzy set on satisfying, for every and , the following conditions:

Note that can be realized as the measure of nearness between and with respect to . It is known that is nondecreasing for all . Let be a fuzzy metric space. For , the open ball with center and radius is defined by . Now, the collection , , is a neighborhood system for a topology on induced by the fuzzy metric . This topology is Hausdorff and first countable.

Definition 1.4 (see [5]).

A sequence in converges to if and only if for each and each , there exists such that for all .

Remark 1.5 (see [5]).

Let be a metric space. We define for all and for every , then is a fuzzy metric space. The fuzzy metric space is complete if and only if the metric space is complete.

With a view to accommodate a wider class of mappings in the context of common fixed point theorems, Sessa [19] introduced the notion of weakly commuting mappings which was further enlarged by Jungck [20] by defining compatible mappings. After this, there came a host of such definitions which are scattered throughout the recent literature whose survey and illustration (up to 2001) is available in Murthy [21]. Here, we enlist the only those weak commutatively conditions which are relevant to presentation.

Definition 1.6 (see [20]).

whenever is a sequence in such that , for some . Also, the pair is called noncompatible, if there exists a sequence in such that , but either or the limit does not exist.

Definition 1.7 (see [10]).

A pair of self-mappings defined on a fuzzy metric space is said to satisfy the property (E.A.) if there exists a sequence in such that for some .

Clearly, compatible as well as noncompatible pairs satisfy the property (E.A.).

Definition 1.8 (see [10]).

Two pairs of self mappings and defined on a fuzzy metric space are said to share common property (E.A.) if there exist sequences and in such that for some .

For more on properties (E.A.) and common (E.A.), one can consult [22] and [10], respectively.

Definition 1.9.

Two self mappings and on a fuzzy metric space are called weakly compatible if they commute at their point of coincidence; that is, implies .

Definition 1.10 (see [23]).

Two finite families of self mappings and are said to be pairwise commuting if

The following definitions will be utilized to state various results in Section 3.

Definition 1.11 (see [15]).

Definition 1.12 (see [15]).

Definition 1.13.

## 2. Main Results

Now, we state and prove our main theorem as follows.

Theorem 2.1.

Let and be self mappings of a fuzzy metric space such that the mappings and are a generalized fuzzy contraction with respect to mappings and . Suppose that the pairs and share the common property (E.A.) and and are closed subsets of . Then, the pair as well as have a point of coincidence each. Further, and have a unique common fixed point provided that both the pairs and are weakly compatible.

Proof.

Since is a closed subset of , therefore , and henceforth, there exists a point such that .

that is a contradiction yielding thereby . Therefore, is a coincidence point of the pair .

If is a closed subset of , then . Therefore, there exists a point such that .

which is a contradiction as earlier. It follows that which shows that is a point of coincidence of the pair . Since the pair is weakly compatible and , hence .

Finally, using the notion of weak compatibility of the pair together with (1.6), we get . Hence, is a common fixed point of both the pairs and .

Uniqueness of the common fixed point is an easy consequence of condition (1.6).

The following example is utilized to highlight the utility of Theorem 2.1 over earlier relevant results.

Example 2.2.

Then, and satisfy all the conditions of the Theorem 2.1 with , where and have a unique common fixed point which also remains a point of discontinuity.

Moreover, it can be seen that and . Here, it is worth noting that none of the earlier theorems (with rare possible exceptions) can be used in the context of this example as most of earlier theorems require conditions on the containment of range of one mapping into the range of other.

In the foregoing theorem, if we set , , and , then we get the following result which improves and generalizes the result of Jungck [16, Corollary 3.2] in metric space.

Corollary 2.3.

for every , . Suppose that the pairs and share the common property (E.A.) and and are closed subsets of . Then, the pair as well as have a point of coincidence each. Further, and have a unique common fixed point provided that both the pairs and are weakly compatible.

By choosing and suitably, one can deduce corollaries for a pair as well as for two different trios of mappings. For the sake of brevity, we deduce, by setting and , a corollary for a pair of mappings which is an improvement over the result of C. Vetro and P. Vetro [15, Theorem 2].

Corollary 2.4.

Let be a pair of self mappings of a fuzzy metric space such that satisfies the property (E.A.), is a fuzzy contraction with respect to and is a closed subset of . Then, the pair has a point of coincidence, whereas the pair has a unique common fixed point provided that it is weakly compatible.

Now, we know that
fuzzy *k*-contraction with respect to
implies
fuzzy contraction with respect to
. Thus, we get the following corollary which sharpen of [15, Theorem 4 ].

Corollary 2.5.

Let and be self mappings of a fuzzy metric space such that the pair enjoys the property (E.A.), is a fuzzy -contraction with respect to , and is a closed subset of . Then, the pair has a point of coincidence. Further, and have a unique common fixed point provided that the pair is weakly compatible.

## 3. Implicit Functions and Common Fixed Point

We recall the following two implicit functions defined and studied in [14] and [23], respectively.

Firstly, following Singh and Jain [14], let be the set of all real continuous functions , non decreasing in first argument, and satisfying the following conditions:

Example 3.1.

Secondly, following Imdad and Ali [23], let denote the family of all continuous functions satisfying the following conditions:

(i) : for every , with or , we have ,

The following examples of functions are essentially contained in [23].

Example 3.2.

Define
as
*,* where
is a continuous function such that
for
*.*

Example 3.3.

Example 3.4.

Example 3.5.

Example 3.6.

Example 3.7.

Before proving our results, it may be noted that above-mentioned classes of functions and are independent classes as the implicit function , where (belonging to ) does not belongs to as for all , whereas implicit function (belonging to ) does not belongs to as implies instead of .

The following lemma interrelates the property (E.A.) with the common property (E.A.).

Lemma 3.8.

for all and . Suppose that pair (or ) satisfies the property (E.A.), and (or ). If for each , in such that (or ), we have for all , then, the pairs and share the common property (E.A.).

Proof.

If the pair enjoys the property (E.A.), then there exists a sequence in such that for some . Since , hence for each there exists in such that , henceforth . Thus, we have , and .

implying thereby that , which is a contradiction. Hence which shows that the pairs and share the common property (E.A.).

With a view to generalize some fixed point theorems contained in Imdad and Ali [11, 23] we prove the following fixed point theorem which in turn generalizes several previously known results due to Chugh and Kumar [24], Turkoglu et al. [25], Vasuki [18], and some others.

Theorem 3.9.

for all and . Suppose that the pairs and share the common property (E.A.) and and are closed subsets of . Then, the pair as well as have a point of coincidence each. Further, and have a unique common fixed point provided that both the pairs and are weakly compatible.

Proof.

which gives for all , that is, . Hence, . Therefore, is a point of coincidence of the pair .

implying thereby that , for all . Hence , which shows that is a point of coincidence of the pair . Since the pair is weakly compatible and , we deduce that .

that is . Hence, for all and therefore .

Now, using the notion of the weak compatibility of the pair and (3.5), we get . Hence, is a common fixed point of both the pairs and . Uniqueness of is an easy consequence of (3.5).

Example 3.10.

In the setting of Example 2.2, retain the same mappings
and
and define
as
with
*.*

Then, and satisfy all the conditions of Theorem 3.9 and have a unique common fixed point which also remains a point of discontinuity.

Further, we remark that Theorem 2 of Imdad and Ali [23] cannot be used in the context of this example, as the required conditions on containment in respect of ranges of the involved mappings are not satisfied.

Corollary 3.11.

The conclusions of Theorem 3.9 remain true if (3.5) is replaced by one of the following conditions:

(i) , where is a continuous function such that for all .

Proof.

The proof of various corollaries corresponding to contractive conditions (i)–(vi) follows from Theorem 3.9 and Examples 3.2–3.7.

Remark 3.12.

Corollary 3.11 corresponding to condition (i) is a result due to Imdad and Ali [11], whereas Corollary 3.11 corresponding to various conditions presents a sharpened form of Corollary 2 of Imdad and Ali [23]. Similar to this corollary, one can also deduce generalized versions of certain results contained in [17, 18, 24].

The following theorem generalizes a theorem contained in Singh and Jain [14].

Theorem 3.13.

for all , and . Suppose that the pairs and enjoy the common property (E.A.) and and are closed subsets of . Then, the pairs and have a point of coincidence each. Further, and have a unique common fixed point provided that both the pairs and are weakly compatible.

Proof.

The proof of this theorem can be completed on the lines of the proof of Theorem 3.9, hence details are omitted.

Example 3.14.

In the setting of Example 2.2, we define
*,* besides retaining the rest of the example as it stands.

Then, all the conditions of Theorem 3.13 with are satisfied.

Notice that 2 is the unique common fixed point of and , but this example cannot be covered by Theorem 3.1 due to Singh and Jain [14] as and . This example cannot also be covered by Theorem 3.9 of this paper as implies which contradicts .

Now, we state (without proof) the following result.

Theorem 3.15.

Let , and be four finite families of self mappings of a fuzzy metric space such that the mappings and satisfy (3.5). Suppose that the pairs and share the common property (E.A.) and as well as are closed subsets of . Then, the pairs and have a point of coincidence each. Further, provided the pairs of families and commute pairwise, where , and , then and have a unique common fixed point.

Proof.

The proof of this theorem can be completed on the lines of Theorem 3.1 due to Imdad et al. [26], hence details are avoided.

By setting , and in Theorem 3.15, one can deduce the following result for certain iterates of mappings which is a partial generalization of Theorem 3.9.

Corollary 3.16.

Let and be four self mappings of a fuzzy metric space such that and satisfy the condition (3.5). Suppose that the pairs and share the common property (E.A.) and as well as are closed subsets of . Then, the pairs and have a point of coincidence each. Further, and have a unique common fixed point provided that the pairs and commute pairwise.

Remark 3.17.

Results similar to Corollary 3.11 as well as Corollary 3.16 can be outlined in respect of Theorem 3.13, Theorem 3.15, and Corollary 3.16. But due to the repetition, details are avoided.

Now, we conclude this note by deriving the following results of integral type.

Corollary 3.18.

then the pairs and have a point of coincidence each. Further, and have a unique common fixed point provided that both the pairs and are weakly compatible.

Proof.

Being a closed subset of , repeating the same argument, we deduce that there exists a point such that .

Since the pair is weakly compatible and , we deduce that .

that implies . Hence . Similarly, we prove that and so is a common fixed point of and . Uniqueness of is a consequence of condition (3.16).

Corollary 3.19.

Suppose that the pairs and enjoy the common property (E.A.) and and are closed subsets of . Then, the pairs and have a point of coincidence each. Further, and have a unique common fixed point provided that both the pairs and are weakly compatible.

Proof.

The proof is the same of Corollary 3.18, so details are omitted.

## Declarations

### Acknowledgment

C. Vetro is supported by University of Palermo, Local University project R. S. ex 60%. The authors are grateful to Professor Dorel Mihet for going through the manuscript and for useful suggestions.

## Authors’ Affiliations

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