Abstract

This paper consists of main two sections. In the first section, we prove a common fixed point theorem in modified intuitionistic fuzzy metric space by combining the ideas of pointwise -weak commutativity and reciprocal continuity of mappings satisfying contractive conditions. In the second section, we prove common fixed point theorems in modified intuitionistic fuzzy metric space from the class of compatible continuous mappings to noncompatible and discontinuous mappings. Lastly, as an application, we prove fixed point theorems using weakly reciprocally continuous noncompatible self-mappings on modified intuitionistic fuzzy metric space satisfying some implicit relations.

1. Introduction and Preliminaries

Recently, Saadati et al. [1] introduced the modified intuitionistic fuzzy metric space and proved some fixed point theorems for compatible and weakly compatible maps. Consequently, in this modified setting of intuitionistic fuzzy metric space, Jain et al. [2] discussed the notion of the compatibility of type ; Sedghi et al. [3] proved some common fixed point theorems for weakly compatible maps using contractive conditions of integral type. The paper [1] is the inspiration of a large number of papers [4–7] that employ the use of modified intuitionistic fuzzy metric space and its applications.

In this paper, we prove some new common fixed point theorems in modified intuitionistic fuzzy metric spaces. While proving our results, we utilize the idea of compatibility due to Jungck [8] together with weakly reciprocal continuity due to Pant et al. [16]. Consequently, our results improve and sharpen many known common fixed point theorems available in the existing literature of modified intuitionistic fuzzy fixed point theory.

Firstly, we recall the following notions that will be used in the sequel.

Lemma 1 (see [10]). Consider the set and the operation defined by for every , in . Then, is a complete lattice.
One denotes its units by and .

Definition 2 (see [11]). A triangular norm ( -norm) on is a mapping satisfying the following conditions:(1) for all in ,(2) for all , in , (3) for all , , in ,(4)if for all in , and imply .

Definition 3 (see [10, 11]). A continuous -norm on is called continuous -representable if and only if there exist a continuous -norm * and a continuous -conorm  on such that for all .

Definition 4 (see [1]). Let are fuzzy sets from such that for all in and . The 3-tuple is said to be a modified intuitionistic fuzzy metric space if is an arbitrary nonempty set, is a continuous -representable, and is a mapping satisfying the following conditions for every in and :(a) ,(b) if and only if ,(c) ,(d) ,(e) is continuous.
In this case, is called a modified intuitionistic fuzzy metric. Here, .

Remark 5 (see [12]). In a modified intuitionistic fuzzy metric space , is nondecreasing, and is nonincreasing for all , in . Hence, is nondecreasing with respect to for all , in .

Definition 6 (see [1]). A sequence in a modified intuitionistic fuzzy metric space is called a Cauchy sequence if for each and , there exists such that for each and for all .

Definition 7 (see [1]). A sequence in a modified intuitionistic fuzzy metric space is said to be convergent to in , denoted by if for all .

Definition 8 (see [1]). A modified intuitionistic fuzzy metric space is said to be complete if and only if every Cauchy sequence is convergent to a point of it.

Definition 9 (see [1, 13]). A pair of self-mappings of modified intuitionistic fuzzy metric space is said to be compatible if whenever is a sequence in such that for some in .

Definition 10 (see [13]). Two self-mappings and are called noncompatible if there exists at least one sequence such that for some in but either or the limit does not exist for all in .

Definition 11 (see [14, 15]). A pair of self mappings of a modified intuitionistic fuzzy metric space is said to be -weakly commuting at a point in if for some .

Definition 12 (see [14, 15]). The two self-maps and of a modified intuitionistic fuzzy metric space are called pointwise -weakly commuting on if given in , there exists such that .

Definition 13 (see [15]). The two self-maps and of a modified intuitionistic fuzzy metric space are called -weakly commuting of type if there exists some such that for all in . Similarly, two self-mappings and of a modified intuitionistic fuzzy metric space are called -weakly commuting of type if there exists some such that for all in .
It is obvious that pointwise -weakly commuting maps commute at their coincidence points and pointwise -weak commutativity is equivalent to commutativity at coincidence points. It may be noted that both compatible and noncompatible mappings can be R-weakly commuting of type or , but the converse needs not be true.

Definition 14 (see [2]). Two self-mappings and of a modified intuitionistic fuzzy metric space are called -weakly commuting of type if there exists some such that for all in .

In 1999, Pant [9] introduced a new continuity condition, known as reciprocal continuity as follows.

Definition 15 (see [9]). Two self-mappings and are called reciprocally continuous if and , whenever is a sequence such that for some in .
If and are both continuous, then they are obviously reciprocally continuous, but the converse is not true.

Recently, Pant et al. [16] generalized the notion of reciprocal continuity to weak reciprocal continuity as follows.

Definition 16 (see [16]). Two self-mappings and are called weakly reciprocally continuous if or whenever is a sequence such that for some in .

If and are reciprocally continuous, then they are obviously weak reciprocally continuous, but the converse is not true. Now, with an application of weak reciprocal continuity, we prove common fixed point theorems under contractive conditions that extend the scope of the study of common fixed point theorems from the class of compatible continuous mappings to a wider class of mappings which also includes noncompatible mappings.

2. Lemmas

The proof of our result is based upon the following lemmas.

Lemma 17 (see [2]). Let be a modified intuitionistic fuzzy metric space and for all , and if for a number , Then .

Lemma 18 (see [2]). Let be a modified intuitionistic fuzzy metric space and a sequence in . If there exist a number such that then is a Cauchy sequence in .

3. Main Results

3.1. Section I: Pointwise -weakly Commuting Pairs and Fixed Point

Lemma 19. Let be a modified intuitionistic fuzzy metric space, and let and be pairs of self-mappings on satisfying for all , and . Then, the continuity of one of the mappings in compatible pair or on implies their reciprocal continuity.

Proof. First, assume that and are compatible and is continuous. We show that and are reciprocally continuous. Let be a sequence such that and for some in as . Since is continuous, we have and as and since is compatible, we have for all , That is, as . By (4), for each , there exists in such that . Thus, we have , , , and as whenever .
Now, we claim that as .
Suppose not, then, by (5), taking , Taking , we have By Lemma 17, we have .
Claim that . Again, by (5), taking , Taking , we get By Lemma 17, .
Therefore, , as .
Hence, and are reciprocally continuous on . If the pair is assumed to be compatible and is continuous, the proof is similar.

Theorem 20. Let be a complete modified intuitionistic fuzzy metric space. Further, let and be pointwise -weakly commuting pairs of self-mappings of satisfying (4), (5). If one of the mappings in compatible pair or is continuous, then , , , and have a unique common fixed point in .

Proof. Let . By (4), we define the sequences and in such that for all , We show that is a Cauchy sequence in . By (5) taking , we have Taking , we have Similarly, Therefore, for any and , we have Hence, by Lemma 18, is a Cauchy sequence in . Since is complete, converges to in . Its subsequences , , , and also converge to .
Now, suppose that is a compatible pair and is continuous. Then, by Lemma 17, and are reciprocally continuous, then , as . As, , is a compatible pair. This implies that Hence, .
Since , there exists a point in such that .
By (5), taking , Thus, by Lemma 17, we have .
Thus, .
Since and are pointwise -weakly commuting mappings, there exists , such that Therefore, and .
Similarly, since and are pointwise -weakly commuting mappings, we have .
Again, by (5), taking , By Lemma 17, we have . Hence is a common fixed point of and . Similarly, by (5), is a common fixed point of and . Hence, is a common fixed point of , , , and .

For Uniqueness. Suppose that   is another common fixed point of , , , and . Then, by (5), taking , By Lemma 17, .

Thus, the uniqueness follows.

Corollary 21. Let be a complete modified intuitionistic fuzzy metric space. Further, let and be reciprocally continuous mappings on satisfying for all , and , then pair and has a unique common fixed point.

Example 22. Let and for each , define Then, is complete modified intuitionistic fuzzy metric space. Let , and be self-mappings of defined as Then, , and satisfy all the conditions of the above theorem with and have a unique common fixed point .

3.2. Section II: Weak Reciprocal Continuity and Fixed Point Theorem

Theorem 23. Let and be weakly reciprocally continuous self-mappings of a complete modified intuitionistic fuzzy metric space satisfying the following conditions: If and are either compatible or -weakly commuting of type or -weakly commuting of type or -weakly commuting of type , then and have a unique common fixed point.

Proof. Let be any point in . Then, as , there exist a sequence of points such that .
Also, define a sequence in as Now, we show that is a Cauchy sequence in . For proving this, by (25), we have Then, by Lemma 18, is a Cauchy sequence in . As is complete, there exists a point in such that . Therefore, by (26), we have .
Suppose that and are compatible mappings. Now, by weak reciprocal continuity of and , it implies that or . Let , then the compatibility of and gives This gives Hence, . By (26), we get .
Therefore, by (25), we get Taking , we get Hence, by Lemma 17, we get . Again, the compatibility of and implies commutativity at a coincidence point. Hence . Using (25), we obtain That is, . Hence, and is a common fixed point of and .
Next, suppose that . Then, implies that for some and, therefore, .
The compatibility of and implies that . By the virtue of (26), this gives Using (25), we get Taking , we have By Lemma 17, we get . The compatibility of and yields . Finally, using (25), we obtain That is, . Hence and is a common fixed point of and .
Now, suppose that and are -weakly commuting of type . Now, weak reciprocal continuity of and implies that or . Let us first assume that . Then, -weak commutativity of type of and yields This gives . Also, using (25), we get Taking , we have Hence, by Lemma 17, we get . Again, by using -weak commutativity of type ,
This yields . Therefore, . Using (25), we get That is, . Hence, and is a common fixed point of and .
Similarly, we prove if .
Suppose that and are -weakly commuting of type . Again, as done above, we can easily prove that is a common fixed point of and .
Finally, suppose that and are -weakly commuting of type . Weak reciprocal continuity of and implies that or . Let us assume that . Then, -weak commutativity of type of and yields This gives .
Using (24) and (26), we have as ; this gives as . Also, using (25) we get Taking , we have By Lemma 17, we get . Again, by using -weak commutativity of type , This yields .
Therefore, . Using (25), we get By Lemma 17, we get . Hence, and is a common fixed point of and .
Similarly, we prove that if .

Uniqueness of the common fixed point theorem follows easily in each of the three cases by using (5).

We now give an example to illustrate Theorem 23.

Example 24 (see [16]). Let be modified intuitionistic fuzzy metric space, where , as defined in Example 22.
Define by Let be a sequence in such that either or for each .
Then, clearly, and satisfy all the conditions of Theorem 23 and have a unique common fixed point at .

Theorem 25. Let and be weakly reciprocally continuous noncompatible self-mappings of a modified intuitionistic fuzzy metric space satisfying (24) and for all whenever for all and .
If and are -weakly commuting of type or -weakly commuting of type or -weakly commuting of type , then and have common fixed point.

Proof. Since and are noncompatible maps, there exists a sequence in such that and for some in as but either or the limit does not exist. Since , for each there exists , in such that . Thus, and as . By the virtue of this and using (48), we obtain This gives as . Therefore, we have .
Suppose that and are -weakly commuting of type . Then, weak reciprocal continuity of and implies that or . Similarly, or . Let us first assume that . Then, -weak commutativity of type of and yields This gives . Using (48), we get This implies that . Again, by the virtue of -weak commutativity of type , This yields and . If , then by using (49), we get which is a contradiction. Hence, and is a common fixed point of and .
Similarly, we can prove that if , then again is a common fixed point of and . Proof is similar if and are -weakly commuting of type or .

We now give an example to illustrate Theorem 25.

Example 26 (see [16]). Let be modified intuitionistic fuzzy metric space as defined in Example 22 and let be a sequence in as in Example 22.
Define by Let be a sequence in such that either or for each . Then, clearly, and satisfy all the conditions of Theorem 25 and have a common fixed point at .

Theorem 27. Let and be weakly reciprocally continuous noncompatible self-mappings of a modified intuitionistic fuzzy metric space satisfying the conditions (24) and whenever for all and .
If and are -weakly commuting of type or -weakly commuting of type or -weakly commuting of type , then and have common fixed point.

Proof. Since and are noncompatible maps, there exists a sequence in such that and for some in as but either or the limit does not exist.
Since , for each , there exists in such that . Thus, , and as . By the virtue of this and (56), we obtain . Therefore, we have .
Suppose that and are -weakly commuting of type . Then, weak reciprocal continuity of and implies that or . Similarly, or . Let us first assume that . Then, -weak commutativity of type of and yields This gives . Using (56), we get Taking , we have This implies that . Again, by the virtue of -weak commutativity of type , This yields and . If , then by using (57), we get which is a contradiction. Hence and is a common fixed point of and .
Similarly, we can prove that if , then, again is a common fixed point of and . Proof is similar if and are -weakly commuting of type or -weakly commuting of type .

We now give an example to illustrate Theorem 27.

Example 28 (see [16]). Let be modified intuitionistic fuzzy metric space as defined in Example 22 and let be a sequence in as in Example 22.
Define as follows: Then, and satisfy all the conditions of Theorem 27 and have two common fixed points at and .

Remark 29. Theorems 23, 25, and 27 generalize the result of Pant et al. [16] for a pair of mappings in modified intuitionistic fuzzy metric space.

4. Fixed Point and Implicit Relation

Let denote the class of those functions such that is continuous and .

There are examples of :(1) ,(2) ,(3) .

Now, we prove our results using this implicit relation.

Theorem 30. Let and be weakly reciprocally continuous noncompatible self-mappings of a modified intuitionistic fuzzy metric space satisfying the conditions: whenever for all , and for some , where is a Lebesgue integrable mapping which is summable nonnegative and such that for each .
If and are -weakly commuting of type or -weakly commuting of type or -weakly commuting of type , then and have common fixed point.

Proof. Since and are noncompatible maps, there exists a sequence in such that and for some in as but either or the limit does not exist. Since , for each , there exists in such that . Thus, and as . By the virtue of this and using (66), we obtain Taking , we have which implies that as . Therefore, we have .
Suppose that and are -weakly commuting of type . Then, weak reciprocal continuity of and implies that or . Similarly, or . Let us first assume that . Then -weak commutativity of type of and yields This gives . Using (66), we get Taking , we have This implies that . Again, by the virtue of -weak commutativity of type , This yields and . If , then by using (67), we get which is a contradiction. Hence, and is a common fixed point of and .
Similarly, we can prove if , then, again, is a common fixed point of and . Proof is similar if and are -weakly commuting of type or .

If we take as , then we get the following corollaries.

Corollary 31. Let and be weakly reciprocally continuous noncompatible self-mappings of a modified intuitionistic fuzzy metric space satisfying the conditions (65), (66) and whenever for all , , where is a Lebesgue integrable mapping which is summable nonnegative and such that for each .
If and are -weakly commuting of type or -weakly commuting of type or -weakly commuting of type , then and have common fixed point.

Corollary 32. Let and be weakly reciprocally continuous noncompatible self-mappings of a modified intuitionistic fuzzy metric space satisfying the conditions (65), (66) and whenever for all , , where is a Lebesgue integrable mapping which is summable nonnegative and such that for each .
If and are -weakly commuting of type or -weakly commuting of type or -weakly commuting of type , then and have common fixed point.

Corollary 33. Let and be weakly reciprocally continuous noncompatible self-mappings of a modified intuitionistic fuzzy metric space satisfying the conditions (65), (66), and whenever for all , , where is a Lebesgue integrable mapping which is summable nonnegative and such that for each .
If and are -weakly commuting of type or -weakly commuting of type or -weakly commuting of type , then and have common fixed point.
Let denote the class of those functions such that is continuous and .
There are examples of :(1) ,(2) .

Now, we prove our main results.

Theorem 34. Let and be weakly reciprocally continuous noncompatible self-mappings of a modified intuitionistic fuzzy metric space satisfying the conditions (65), (66), and whenever for all , and for some where is a Lebesgue integrable mapping which is summable nonnegative and such that for each .
If and are -weakly commuting of type or -weakly commuting of type or -weakly commuting of type , then and have common fixed point.

Proof. Since and are noncompatible maps, there exists a sequence in such that and for some in as but either or the limit does not exist. Since , for each , there exists in such that . Thus and as . By the virtue of this and using (78), we obtain Taking , we have which implies that, as . Therefore, we have .
Suppose that and are -weakly commuting of type . Then, weak reciprocal continuity of and implies that or . Similarly, or . Let us first assume that . Then -weak commutativity of type of and yields This gives . Using (78), we get Taking , we have This implies that . Again, by the virtue of -weak commutativity of type , . This yields and . If , then by using (78), we get which is a contradiction. Hence, and is a common fixed point of and .
Similarly, we can prove that if , then, again, is a common fixed point of and . Proof is similar if and are -weakly commuting of type or .

If we take as , then we get the following corollaries.

Corollary 35. Let and be weakly reciprocally continuous noncompatible self-mappings of a modified intuitionistic fuzzy metric space satisfying the conditions (65), (66), and whenever for all , , where is a Lebesgue integrable mapping which is summable nonnegative and such that for each .
If and are -weakly commuting of type or -weakly commuting of type or -weakly commuting of type , then and have common fixed point.

Corollary 36. Let and be weakly reciprocally continuous noncompatible self-mappings of a modified intuitionistic fuzzy metric space satisfying the conditions (65), (66) and whenever for all , and for some , where is a Lebesgue integrable mapping which is summable nonnegative and such that for each .
If and are -weakly commuting of type or -weakly commuting of type or -weakly commuting of type , then and have common fixed point.