Abstract

In this article, we introduce the notions of extended -rectangular and controlled rectangular fuzzy metric-like spaces that generalize many fuzzy metric spaces in the literature. We give examples to justify our newly defined fuzzy metric-like spaces and prove that these spaces are not Hausdorff. We use fuzzy contraction and prove Banach fixed point theorems in these spaces. As an application, we utilize our main results to solve the uniqueness of the solution of a differential equation occurring in the dynamic market equilibrium.

1. Introduction and Preliminaries

In 1965, Zadeh [1] introduced the concept of a fuzzy set that generalizes the concept of an ordinary set or crisp set. Many authors have used fuzzy sets in different branches of mathematics extensively. For example, Kaleva [2] gave the idea of fuzzy differential equations, Buckley and Feuring [3] introduced the concept of fuzzy partial differential equations, and Puri and Ralescu [4] introduced the differentials of fuzzy functions. Fuzzy metric space is one of the most studied topic in fuzzy set theory. The definition of fuzzy metric space was given by Kramosil and Michálek [5] in 1975 which was later modified by George and Veeramani [6]. In 1983, Grabiec [7] established and proved the fuzzy version of the Banach contraction principle. Many researchers have used and extended this version in many fuzzy metric spaces (see [8–15] and references therein). Branciari [16] generalized the definition of classical metric space by introducing rectangular or Branciari metric space and proved some fixed point results. As a generalization of a Branciari metric space, authors in [17] gave the notion of -rectangular metric space.

Hitzler and Seda [18] introduced the idea of dislocated topology in which the self-distance between the points may not be zero. Amini-Harandi [19] introduced the definition of a metric-like space and proved related results. The notion of -metric-like space was introduced by [20] as a generalization of metric-like space. Mlaiki et al. [21] generalized the definition of a rectangular metric space by defining rectangular metric-like space. The concept of a fuzzy metric-like space was introduced by Shukla and Abbas [14] as a generalization of [6] and proved related fixed point results. They generalized the definition of a fuzzy metric space in the sense that the self-distance may not be equal to one. The concept of fuzzy -metric and fuzzy quasi--metric space was introduced by Nadaban [22]. The concepts of fuzzy double controlled metric space and fuzzy triple controlled metric space were given by Saleem et al. [13] and Furqan et al. [23], respectively. They also proved that these spaces are not Hausdorff.

Definition 1 (see [19]). Let be a nonempty set, a mapping is called metric-like if, for all satisfies the following: The pair is called a metric-like space.

Example 1 (see [19]). Let , and is given by if and otherwise. Then, is a metric-like space.

Definition 2 (see [20]). Let be a nonempty set and ; a function is called -metric-like if satisfies the following: The pair is called -metric-like space.

Example 2 (see [20]). Let and be defined as . Then, is a -metric-like space with .

Definition 3 (see [24]). Let be a nonempty set and be a function; then, is said to be a rectangular metric-like space if it satisfies the following: The pair is called rectangular metric-like space.

Definition 4 (see [25]). A binary operation on , where , is called a continuous triangular norm, if for all the following conditions are satisfied:         for every  

2. Extended Fuzzy -Rectangular Metric-Like Space

In this section, we introduce the definition of an extended fuzzy -rectangular metric-like space.

Definition 5. Let be a function; is a fuzzy set on . Then, is called an extended fuzzy -rectangular metric-like with -norm , if for all and all distinct ; satisfies the following: Then, is called an extended fuzzy -rectangular metric-like space.

Remark 6. (i)By taking , then extended fuzzy -rectangular metric-like space reduces to fuzzy -rectangular metric-like space.(ii)By taking , then extended fuzzy -rectangular metric-like space reduces to fuzzy rectangular metric-like space.

The following example justifies Definition 5.

Example 3. Let . If we define by for all in and by Then, , given by is an extended fuzzy -rectangular metric-like on provided that is a minimum -norm, that is, for all

Clearly,

Note that the first three axioms (–) clearly hold. To prove the axiom , we discuss the following cases:

Case 1. Let . Then, we have Also, Note that Clearly, That is, Similarly, we have Hence,

Case 2. Let and Then, Now, Clearly, implies that Similarly, we obtain that Hence,

Case 3. If and then we have Also, Now, Clearly, implies that Similarly, Hence, Thus, in each case, we have for all distinct and . Hence, is an extended fuzzy -rectangular metric-like space.

Definition 7. Let be an extended fuzzy -rectangular metric-like space and be a sequence in , then is (1)a convergent sequence, if there exists in such that(2)a Cauchy sequence, if for all and for An extended fuzzy -rectangular metric-like space is said to be complete, if every Cauchy sequence converges to some .

Definition 8. Let be an extended fuzzy -rectangular metric metric-like space. A mapping is said to be a fuzzy contractive mapping if

Theorem 9. Let be a complete extended fuzzy b-rectangular metric-like space with and be a fuzzy contractive mapping such that Then, has a unique fixed point.

Proof. Let be an arbitrary point in . If , then is the required fixed point. If , then there exists such that . If , then is the required fixed point. Continuing in this way, we have the sequence in such that . Let and using inequality (30), we have so we have Now, so we have Continuing in this way, we have we have Working in the same way, we also can prove that

Let be a sequence in ; then, we have the following cases.

Case 1. If is odd (say) for , then Using inequality (37), we have Taking limit , we have

Case 2. If that is, is even, then Using inequality (37) and (38), we have Taking limit , we have Thus, is a Cauchy sequence and converges to (since is complete). Now, we have to prove is the fixed point of . Consider so, Now, Applying limit on the right-hand side, we have , which shows is the fixed point of . For uniqueness, let be an other fixed point of such that and consider, which is a contradiction; thus, is the only fixed point of .

3. Fuzzy Controlled Rectangular Metric-Like Space

In this section, we will give the definition of fuzzy controlled rectangular metric like space.

Definition 10. Let be a function and is a fuzzy set on . Then, is called fuzzy controlled rectangular metric-like with as continuous -norm; if for any and all distinct , the following conditions are satisfied: Then, is called a fuzzy controlled rectangular metric-like space.

Remark 11. (i)If we take , then a fuzzy controlled rectangular metric-like space reduces to extended fuzzy -rectangular metric-like space(ii)If we take , then a fuzzy controlled rectangular metric-like space reduces to fuzzy -rectangular metric-like space(iii)If we take , then a fuzzy controlled rectangular metric-like space reduces to fuzzy rectangular metric-like space

Following example justifies the definition 10.

Example 4. Let , and . Define as Then, is a fuzzy controlled rectangular metric-like space.
We will prove as – and are easy to prove.
Now, so we have Thus Hence, is a fuzzy controlled rectangular metric-like space.

Example 5. Consider and let be a function defined as , and . Define a rectangular metric-like space by Now define as follows: Then, is a fuzzy controlled rectangular metric-like space with product -norm.

Definition 12. Let be a fuzzy controlled rectangular metric-like space and be a sequence in ; then, the sequence is called a convergent sequence if

Definition 13. A sequence in is called a Cauchy sequence if for If every Cauchy sequence converges in , then is known as a complete fuzzy controlled rectangular metric-like space.

Definition 14. Let be a fuzzy controlled rectangular metric-like space. Then, an open ball , with center and radius , is given by and the corresponding topology is defined as We show that a fuzzy controlled rectangular metric-like space need not be Hausdorff in the following example.

Example 6. Consider the fuzzy controlled rectangular metric-like space as in Example 5. Now, define an open ball with center radius , and as Let , then , so .
Let , then , so .
Let , then , so .
Let , then , so .
Hence, Now, define an open ball with center radius , and as Let , then , so .
Let , then , so .
Let , then , so .
Let , then , so .
Hence, Now, . Thus, a fuzzy controlled rectangular metric-like space is not Hausdorff.

Remark 15. In light of Remark 11, extended fuzzy rectangular -metric-like, fuzzy rectangular -metric-like, and a fuzzy rectangular metric-like spaces are also not Hausdorff.

The following is the Banach fixed point theorem in the settings of a fuzzy controlled rectangular metric-like space.

Theorem 16. Let , be a function, and be a complete fuzzy controlled rectangular metric-like space such that Assume further that be self-mapping such that for all , Then, has a unique fixed point.

Proof. Let be an arbitrary point in ; then, we have the iterative sequence . Now, hence, In the same way, we can prove that

Now writing and for any sequence , we have the cases below.

Case 1. When is odd, (say), then Applying (66), we deduce

Case 2. When is even, (say), then Now applying (66) and (67) on the right-hand side, we deduce Using (63) for each case, we obtain which shows is a Cauchy sequence in , as is complete, so there exists such that Now, to prove is a fixed point of . From (64), as which shows is a fixed point of . For uniqueness, we assume has as an other fixed point, then as Hence , so has a unique fixed point.

Example 7. Let and a mapping given by Define by Note that is a complete fuzzy controlled rectangular metric-like space and Thus, all the conditions of Theorem 16 are satisfied. Moreover, is a unique fixed point of .

4. Application to Dynamic Market Equilibrium

Due to its vast applications in many fields, the fixed point theory has been used to prove the uniqueness of the solutions of many problems. This section is devoted to prove the unique solution of a differential equation appearing in dynamic market equilibrium.

Let and denote the supply and demand of a certain item, respectively, affected by price and trends. The economist wants to know the current price of the item which is falling or rising at a decreasing or increasing rate. Let and denote, respectively, the first and second derivatives of the price , and assume

where , and are constants. In equilibrium, ; hence, we have which implies

Putting , and in (81), we have

On dividing by , we have the following initial value problem:

with . Note that equation (83) is equivalent to the following integral equation:

where is Green’s function defined by

We will prove the uniqueness of the solution of the following integral equation:

Denote , the space of all real-valued continuous functions defined over the interval . Now, define a complete fuzzy controlled metric-like space with product -norm as with controlled functions , and .

Theorem 17. Consider the operator Assume the following conditions hold: where Then, integral equation (86) has a unique solution.

Proof. Let and and consider By the application of Theorem 16, problem (86) has a unique solution.

5. Conclusion

In this article, the concepts of extended -rectangular and fuzzy controlled rectangular metric-like spaces are given that extend numerous fuzzy metric-like spaces. We proved that these classes of fuzzy metric-like spaces are not Hausdorff, and we have given examples to support our main results and definitions; also, we proved the Banach fixed point theorem in these spaces by using different types of contractions. We apply our results to prove the uniqueness of the solution of an integral equation appearing in dynamic market equilibrium.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no competing interests.