Abstract

In this study, we introduce fuzzy triple controlled metric space that generalizes certain fuzzy metric spaces, like fuzzy rectangular metric space, fuzzy rectangular -metric space, fuzzy -metric space, and extended fuzzy -metric space. We use three noncomparable functions as follows: . We prove Banach fixed point theorem in the settings of fuzzy triple controlled metric space that generalizes Banach fixed point theorem for aforementioned spaces. An example is presented to support our main results. We also apply our technique to the uniqueness for the solution of an integral equation.

1. Introduction and Preliminaries

Banach contraction mapping principle (BCMP) [1] has many applications in various scientific fields ([2–6]). BCMP was proved in 1922 and has been investigated by many researchers in different ways ([7–11]). In 1965, Zadeh [12] defined a fuzzy set that generalizes the definition of a crisp set by associating all elements with membership values between the interval . Since then, the fuzzy set theory has been used extensively in mathematics ([13–18]) and many other areas ([19–21]). The definition of fuzzy metric space was given by Kramosil and Michálek [22] in 1975. The fuzzy version of BCMP was proved by Grabiec [23]. He also extended the Edelstein theorem to fuzzy metric spaces. George and Veeramani [24] introduced the Hausdorff topology in fuzzy metric space and modified the definition of fuzzy metric space given in [22]. In 1989, Bakhtin [25] introduced the notion of a -metric space that generalizes the definition of classical metric space. Branciari [26] introduced the rectangular metric space and proved some fixed point results. Roshan et al. [27] introduced the notion of -rectangular metric space that generalizes the definition of a rectangular metric space.

In [28], Nădăban utilized fuzzy sets in -metric spaces and introduced the notion of a fuzzy -metric space. He also discussed the topological properties of this new space. Kamran et al. [29] further generalized the definition of [25] by introducing the idea of extended -metric space while in [30], Mehmood et al. applied the fuzzy sets to the definition in [29] by introducing the notion of an extended fuzzy -metric space and proved BCMP on this space. In [31], authors defined fuzzy version of rectangular -metric space while in [32], Asim et al. introduced the concept of extended rectangular -metric space and proved related fixed point theorem. Recently, Saleem et al. [33] introduced the notion of fuzzy double controlled metric space and proved BCMP on such space. Abdeljawad et al. [34] modified the definition of controlled metric type space defined in [35] by giving the idea of a double controlled metric type space.

Definition 1 ([36]). Let be a binary operation, then is said to be continuous triangular norm (in short, continuous -norm), if for all the following conditions are satisfied:
;
;
is continuous;
for every ;
whenever .

Definition 2 ([28]). Let be a nonempty set, a real number, a continuous -norm, and be a fuzzy set on . Then, is said to be a fuzzy -metric on if for all , satisfies the following:
for ;
for all iff ;
;
for all ;
is left continuous and .

The quadruple is called fuzzy -metric space.

Definition 3 ([30]). Let be a nonempty set,, a continuous -norm, and is a fuzzy set on . Then, is extended fuzzy -metric if for all , the following conditions are satisfied:
for ;
for all iff ;
;
for all ;
is left continuous.

Then, is known as an extended fuzzy -metric space.

Definition 4 ([37]). Let be a nonempty set, a continuous -norm, and is a fuzzy set on . Then, is called a fuzzy rectangular metric if for any and all distinct points , the following conditions are satisfied:
for ;
for all iff ;
;
for all ;
is left continuous, and .
Then, is called a fuzzy rectangular metric space.

Definition 5 ([31]). Let be a nonempty set, a real number, a continuous -norm, and be a fuzzy set on . Then, is called fuzzy rectangular -metric if for any and all distinct points , the following conditions are satisfied:
;
for all iff ;
;
for all ;
is left continuous, and .
Then, is known as fuzzy rectangular -metric space.

Definition 6 ([34]). Let be two noncomparable functions, if for all satisfies the following:
iff ;
;
.
Then, is known as double controlled metric type space.

2. Main Results

We first give the definition of a fuzzy triple controlled metric space as follows.

Definition 7. Let be three noncomparable functions, a continuous -norm, and is a fuzzy set on . Then, is called fuzzy triple controlled metric if for any and all distinct , the following conditions are satisfied:
;
for all iff ;
;
, for all ;
is continuous.

Then, is called a fuzzy triple controlled metric space.

Remark 8. (i)Taking in (4), then we get the definition of fuzzy rectangular metric space [37].(ii)Taking in (4), then our definition reduces to the definition of fuzzy rectangular -metric space [31].(iii)Taking and in (4), then our definition reduces to the definition of fuzzy double controlled metric space defined in [33].(iv)Taking , and in (4), then our definition reduces to the definition of extended fuzzy -metric space defined in [30].(v)Taking , and in (4), then our definition reduces to the definition of fuzzy -metric space defined in [28].

The following example justifies Definition 7.

Example 1. Consider and let be three noncomparable functions defined as and . Then, defined by with product -norm, that is, , is a fuzzy triple controlled metric space.
Now, Axioms (1) to (3) and (5) can easily be verified, and we check only (4).
Let then either and or and . We prove for and . The proof for and is similar. Now, Clearly, Working like same steps, remaining cases can easily be proved. Hence, is a fuzzy triple controlled metric space.

Remark 9. (i) In Example 1, is not a fuzzy triple controlled metric space, if we use minimum -norm, i.e., instead of product -norm.
Next, we define the convergence of a sequence as well as Cauchy sequence in the context of fuzzy triple controlled metric space.

Definition 10. Consider a fuzzy triple controlled metric space . Then, a sequence in is said to be (1)convergent, if for all there exists in such that(2)Cauchy iff for all and for each there exists such that is called complete fuzzy triple controlled metric space, if every Cauchy sequence in converges to some point in

Definition 11. Let be a fuzzy triple controlled metric space. Then, an open ball , with center , radius , and , is given by and the corresponding topology is defined as Next example shows that a fuzzy triple controlled metric space is not Hausdorff.

Example 2. Consider the fuzzy triple controlled metric space as given in Example 1. Then, the open ball centered at , radius , and is given by Now, Thus, . Now, consider the open ball with centered at , radius , and . Then, Thus, and so Hence, fuzzy triple controlled metric space is not Hausdorff.

Theorem 12. Let be three noncomparable functions and be a complete fuzzy triple controlled metric space such that Let be a given mapping such that Then, has a unique fixed point.

Proof. Choose an arbitrary point in . If , then we are done. If then . Continuing in this way, we have and so on, So, we have iterative sequence . Applying (14) successively, we get

Now, writing and for any sequence , using rectangular property, we have the following cases:

Case 1. When i.e., is odd, then Applying (17) on right hand side, we deduce

Case 2. When , i.e., is even, then Now applying (17), we deduce Using (13) for each case, we obtain which shows that is a Cauchy sequence in and converges to some (as is complete), i.e.,

Now, we show that is the fixed point of . From (14), which shows that , i.e., is a fixed point of . To prove is unique, we assume is also a fixed point of , i.e., . Then, which implies that , so the fixed point of is unique.

Example 3. Let and be defined as for all . Let be defined by and . Then, is a complete fuzzy triple controlled metric space. Let be given by . Then, for all where Since all the conditions of Theorem 12 are satisfied, therefore has a unique fixed point which is .

Theorem 12 generalizes Theorem 2.1 of [31] as follows.

Corollary 13. Let be a complete fuzzy rectangular -metric space with such that Let be a mapping such that Then, has a unique fixed point.

Remark 14. If we choose then our main result reduces to the Banach contraction principle for the fuzzy rectangular metric space defined in [37].

3. Application

Consider the integral equation where .

Define , and Also, for all ,

Then, is a complete fuzzy triple controlled metric space with product -norm, where is the space of all real valued continuous functions defined on

Theorem 15. Consider an integral operator defined on as follows: where satisfies the following condition.
There exists such that for all and for all , we have where Then, the integral equation (29) has a unique solution.

Proof. Let . Note that Thus, for all and consequently, all the conditions of Theorem 12 are satisfied. Therefore, the integral equation (29) has a unique solution.

4. Conclusion

In this article, the concept of fuzzy triple controlled metric space is given which generalizes fuzzy rectangular -metric space, rectangular fuzzy metric space. We have established Banach fixed point theorem in this space. An example is also presented that illustrates our main result. Our newly defined results can be used for further investigation in many existing results in the literature.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare to have no competing interests.

Authors’ Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Acknowledgments

Authors are thankful to the editor and anonymous referees for their valuable comments and suggestions.