Research Article | Open Access

# Unified Fixed Point Theorems via Common Limit Range Property in Modified Intuitionistic Fuzzy Metric Spaces

**Academic Editor:**E. Karapınar

#### Abstract

The purpose of this paper is to emphasize the role of “common limit range property” to ascertain the existence of common fixed points in modified intuitionistic fuzzy metric spaces enjoying an implicit function utilized in Tanveer et al. (2012) and Imdad et al. (2012). As an application to our main result, we derive a fixed point theorem for finite families of self-mappings. We also give some examples which demonstrate the validity of the hypotheses and degree of generality of our main results. Our results improve and extend several previously known fixed point theorems of the existing literature.

#### 1. Introduction

The fruitful and productive idea of fuzzy set was initiated by Zadeh [1]. In an attempt to generalize the idea of fuzzy set, Atanassov [2] introduced the notion of intuitionistic fuzzy set. Thereafter, Çoker [3] defined a topology on intuitionistic fuzzy sets, while Mondal and Samanta [4] introduced the idea of intuitionistic gradation of openness. Park [5] introduced the notion of intuitionistic fuzzy metric space (abbreviated by IFMS in the sequel) as a generalization of fuzzy metric space, especially the one due to George and Veeramani [6]. In recent years, many authors proved a multitude of fixed point theorems in IFMS (e.g., see [7–15]).

Later on, Gregori et al. [16] showed that the topology induced by fuzzy metric coincides with the topology induced by intuitionistic fuzzy metric. In an attempt to remove this shortcoming, Saadati and Park [17] proposed the idea of modified IFMS wherein the notions of continuous t-norm and continuous t-conorm are employed besides adopting the notion of compatible mappings (essentially due to Jungck [18]). Jain et al. [19] proved some unique common fixed point theorems for four self-mappings satisfying a new contractive condition in modified IFMS through compatibility of type . Saadati and Park [17] extended the notion of weak compatibility (due to Jungck and Rhoades [20]) to modified IFMS. However, the study of common fixed points of noncompatible mappings due to Pant [21] is also equally natural. Tanveer et al. [22] and Imdad et al. [23] utilized the notions of the property (E.A) (due to Aamri and Moutawakil [24]) and the common property (E.A) (due to Liu et al. [25]) to prove some interesting results in modified intuitionistic fuzzy metric spaces. One may notice that the property (E.A) does require the closedness of certain underlying subspaces to ascertain the existence of common fixed point. Sintunavarat and Kumam [26] coined the idea of “common limit range property” which never requires the closedness of any underlying subspace for the existence of common fixed points (also see [27]). Most recently, Chauhan et al. [28, 29] and Sintunavarat et al. [14] proved some interesting fixed point results for mappings defined on modified IFMS via common limit range property. Imdad et al. [30] extended the notion of common limit range property to two pairs of self-mappings and proved some fixed point results in Menger and metric spaces. We cite some recent papers (e.g., [31–37]) which demonstrate the superiority of common limit range property over the property (E.A) in various settings.

In this paper, utilizing an implicit function due to Tanveer et al. [22] (also Imdad et al. [23]), we prove some common fixed point theorems for two pairs of weakly compatible mappings in modified IFMS employing the common limit range property. In process, many known results (especially those contained in Imdad et al. [23]) are enriched and improved. Some related results are also derived besides furnishing illustrative examples.

#### 2. Preliminaries

Lemma 1 (see [38]). *Consider the set and operation defined by
** and , for every . Then is a complete lattice.*

*Definition 2 (see [2]). *An intuitionistic fuzzy set in a universe is an object , where, for all , and are, respectively, called the membership degree and the nonmembership degree of which also satisfy .

For every , if such that , then it is easy to see that

We denote its units by and . Classically, a triangular norm on is defined as an increasing, commutative, associative mapping satisfying , for all . A triangular conorm is defined as an increasing, commutative, associative mapping satisfying , for all . Using the lattice , these definitions can be easily extended.

*Definition 3 (see [39]). *A triangular norm (t-norm) on is a mapping satisfying the following conditions for all : (1), (2), (3), (4) and .

*Definition 4 (see [38, 39]). *A continuous t-norm on is called continuous t-representable if and only if there exist a continuous t-norm and a continuous t-conorm on such that, for all , ,
Now, we define a sequence recursively by and
for and .

*Definition 5 (see [38, 39]). *A negator on is any decreasing mapping satisfying and . If , for all , then is called an involutive negator. A negator on is a decreasing mapping satisfying and . Notice that stands for standard negator on defined by (for all ) .

*Definition 6 (see [17]). *Let , be fuzzy sets from to such that for all and . The 3-tuple is said to be a modified IFMS if is an arbitrary nonempty set, is a continuous t-representable, and is an intuitionistic fuzzy set from satisfying the following conditions (for every and ): (1), (2) if and only if , (3), (4), (5) is continuous.In this case, is called a modified intuitionistic fuzzy metric. Here,

*Remark 7. *In an intuitionistic fuzzy metric space , is nondecreasing and is nonincreasing for all . Hence is nondecreasing function for all .

*Example 8 (see [17]). *Let be a metric space. Define for all and , and let and be fuzzy sets on defined as follows:
for all . Then is a modified IFMS.

*Example 9 (see [17]). *Let . Define for all and , and let and be fuzzy sets on . Then is defined as (for all and ) follows:
Then is a modified IFMS.

*Definition 10 (see [17]). *Let be a modified IFMS. For , define the open ball with center and radius as

A subset is called open if for each there exist and such that . If denotes the family of all open subsets of , then is called the topology induced by intuitionistic fuzzy metric . Notice that this topology is Hausdorff (see [5], Remark 3.3, Theorem 3.5).

*Definition 11 (see [17]). *A sequence in a modified IFMS is called a Cauchy sequence if for each and there exists such that
and for each , where is a standard negator. The sequence is said to be convergent to in the modified IFMS and is generally denoted by if whenever for every . A modified IFMS is said to be complete if and only if every Cauchy sequence is convergent.

Lemma 12 (see [17]). *Let be an intuitionistic fuzzy metric. Then, for any , is nondecreasing with respect to in , for all .*

*Definition 13 (see [17]). *Let be a modified IFMS. Then is said to be continuous on , if
whenever a sequence in converges to a point ; that is,

Lemma 14 (see [17]). *Let be a modified IFMS. Then, is continuous function on .*

*Definition 15. *Let and be two mappings from a modified IFM space into itself. Then this pair of mappings is said to be (1)commuting if , for all ;(2)weakly commuting [17] if
for all and ;(3)compatible [17] if
for all whenever is a sequence in such that
(4)noncompatible [22] if there exists at least one sequence in such that
but or nonexistent for at least one .

*Definition 16 (see [40]). *Two families of self-mappings and are said to be pairwise commuting if (1), for all ; (2), for all ; (3), for all and .

*Definition 17 (see [12]). *Let and be two mappings from a modified IFMS into itself. Then this pair of mappings is said to satisfy the property (E.A) if there exists a sequence in such that for all
for some .

*Definition 18 (see [22]). *Two pairs and of self-mappings of a modified IFMS are said to satisfy the common property (E.A) if there exist sequences and in such that
for some and .

*Definition 19 (see [14]). *A pair of self-mappings of a modified IFMS is said to satisfy the common limit range property with respect to , denoted by , if there exists a sequence in such that for all
where .

Thus, one can infer that a pair satisfying the property (E.A) along with closedness of the subspace always enjoys the property with respect to the mapping (see [14, 29]).

Now, we extend common limit range property for two pairs of self-mappings in the framework of modified IFMS as follows.

*Definition 20. *Two pairs and of self-mappings of a modified IFMS are said to satisfy the common limit range property with respect to mappings and , denoted by , if there exist two sequences and in such that
where and .

By setting and in Definition 20 implies Definition 19 (due to Sintunavarat et al. [14]), whereas Definition 20 implies Definition 18, but the converse implications are not true in general. The following example substantiates this fact.

*Example 21. *Let be a modified IFMS, where and for all and . Define four self-mappings , , , and on as

If we choose two sequences as and ), then the pairs and enjoy the common property (E.A) for all :
where . Here it is noticed that . Therefore, the pairs and do not satisfy the common limit range property with respect to mappings and .

In view of Example 21, the following proposition is predictable.

Proposition 22. *If the pairs and share the common property (E.A) and as well as are closed subsets of , then the pairs also enjoy the property.*

#### 3. Implicit Relations

On the lines of Imdad et al. [23], we adopt an implicit function which covers a multitude of contraction conditions in one go as exhibited by demonstrative examples.

Let be the set of all upper continuous functions , satisfying the following conditions (for all , where , , and ):, for all ;, for all ;, for all .

*Example 23. *Define as
where .

*Example 24. *Define as
where , , and .

*Example 25. *Define as
where .

*Example 26. *Define as
where .

*Example 27. *Define as
where and is a continuous function such that for all .

*Example 28. *Define as
where .

*Example 29. *Define as
where is a continuous function such that for all .

*Example 30. *Define as
where .

*Example 31. *Define as
where and .

*Example 32. *Define as
where and .

*Example 33. *Define as
where , , , and .

On the lines of Tanveer et al. [22], let be the set of all continuous functions , satisfying (for all , where , , and ) the following: for all , or implies that ; implies that .

*Example 34. *Define . Then .

*Example 35. *Define . Then .

Here, it can be pointed out that the abovementioned classes of functions, namely, and , are independent to each other as the implicit function (where and ) does not belong to as , for all , while the implicit function (where ) does not belong to as implies instead of .

For an extensive collection of implicit relations on different settings, we refer to [41–45].

#### 4. Results

Before proving our main results, we observe the following.

Lemma 36. *Let , , , and be self-mappings of a modified IFMS . Suppose that *(1)*the pair satisfies the property (or satisfies the property), *(2)* (or ), *(3)* (or ) is a closed subset of , *(4)* converges for every sequence in whenever converges (or converges for every sequence in whenever converges), *(5)*for all and **
Then the pairs and share the property.*

*Proof. *If the pair enjoys the property with respect to mapping , then there exists a sequence in such that
where . Since , for each sequence , there exists a sequence in such that . Therefore, due to closedness of ,
where . Thus, in all, we have , , and as . Moreover, in view of (4), converges. Now, we show that as . On using inequality (33) with , , we have

Let, on contrary, as . Then, on making , we get
or, equivalently,
which is a contradiction to . Hence , that is, , which shows that as . Hence both the pairs and share the property. This concludes the proof.

*Remark 37. *In general, the converse of Lemma 36 is not true. For a counter example, one can see Example 42.

Now, we state and prove our first main result as follows.

Theorem 38. *Let , , , and be self-mappings of a modified IFMS satisfying inequality (33) of Lemma 36. If the pairs and share the property, then and have a coincidence point each. Moreover, , , , and have a unique common fixed point provided both the pairs and are weakly compatible.*

*Proof. *Since the pairs and satisfy the property, there exist two sequences and in such that
where . Since , there exists a point such that . We show that . If not, then using inequality (33) with , , we get

which, on making , reduces to

so that

a contradiction to . Hence ; that is, . Therefore, is a coincidence point of the pair .

Also ; there exists a point such that . We assert that . If not, then using inequality (33) with , , we get

so that

or

a contradiction to . Hence , and so , which shows that is a coincidence point of the pair .

Since the pair is weakly compatible and , hence . Now, we show that is a common fixed point of the pair . Suppose that ; using inequality (33) with , , we have

so that

or

a contradiction to yielding thereby which shows that is a common fixed point of the pair .

Also the pair is weakly compatible, and ; therefore, . Suppose that ; then using inequality (33) with , , we have

so that

or

a contradiction to . Therefore, which shows that is a common fixed point of the pair . Hence is a common fixed point of both the pairs and . Uniqueness of common fixed point is an easy consequence of inequality (33) (owing to condition ). This completes the proof.

*Remark 39. *Theorem 38 improves the corresponding results contained in Imdad et al. [23] as closedness of the underlying subspaces is not required.

Now, we present an example which demonstrates the validity of the hypotheses and degree of generality of our main result over comparable ones from the existing literature.

*Example 40. *Let be a modified IFMS, wherein , for all and with for all and . Define four self-mappings , , , and by
Define an implicit function by
where is a continuous function such that (that is, ) for all and . Hence (53) implies

for all and . With two sequences and (or , ), the pairs and satisfy the property:
where . Also, and . By a routine calculation, one can easily verify the inequality (54) for all . Thus all the conditions of Theorem 38 are satisfied, and 5 is a unique common fixed point of the pairs and , which also remains a point of coincidence as well. Here, one may notice that all the involved mappings are discontinuous even at their unique common fixed point 5.

Notice that the subspaces and are not closed subspaces of ; therefore, the main result contained in Imdad et al. [23] can not be used in the context of this example which establishes the genuineness of our extension.

In the proof of our next theorem, Lemma 36 is utilized.

Theorem 41. *Let , , , and be self-mappings of a modified IFMS satisfying all the hypotheses of Lemma 36. Then , , , and have a unique common fixed point provided both the pairs and are weakly compatible.*

*Proof. *In view of Lemma 36, the pairs and share the property so that there exist two sequences and in such that
where . The rest of the proof can be completed on the lines of the proof of Theorem 38. This completes the proof.

The following example demonstrates the utility of Theorem 41 over Theorem 38.

*Example 42. *In the setting of Example 40, replace the self-mappings , , , and by the following, besides retaining the rest:
Then, like the earlier example, it is easy to see that both the pairs and satisfy the property. Consider an implicit function described by Example 40. Also, and . The pairs and commute at 5 which is also their common coincidence point. Thus all the conditions of Theorems 41 are satisfied, and 5 is a unique common fixed point of the involved mappings , , , and .

Here, it can be pointed out that Theorem 38 is not applicable to this example as both , are closed subsets of which demonstrates the situational utility of Theorem 41 over Theorem 38.

In view of the earlier demonstrative examples, one can outline the following corollary.

Corollary 43. *The conclusions of Lemma 36, Theorem 38, and Theorem 41 remain true if inequality (33) is replaced by one of the following contraction conditions. For all and ,**
where ,**
where , and ,*