Solving an Integral Equation by Using Fixed Point Approach in Fuzzy Bipolar Metric Spaces

Abstract

The purpose of this manuscript is to obtain some fixed point results under mild contractive conditions in fuzzy bipolar metric spaces. Our results generalize and extend many of the previous findings in the same approach. Moreover, two examples to support our theorems are obtained. Finally, to examine and strengthen the theoretical results, the existence and uniqueness of the solution to a nonlinear integral equation was studied as a kind of applications.

1. Introduction

The notion of the continuous triangular norm was introduced in 1960 by Schweizer and Sklar in their paper [1]. The concept of fuzzy set theory was initiated by Zadeh [2] in 1965. Some references to a fuzzy logic-based education system can be found in [3–6]. The other direction of the fuzzy set is the fuzzy metric theory. The idea of fuzzy metric space (FM-space) was presented by Kramosil and Michalek [7]. With the help of continuous -norm property, they obtained some pivotal fixed point results under the mild contractive conditions in the mentioned space. Many authors worked in this direction; they either modified the definition of FM-spaces [8] or extended the well-known fixed point theorem of Banach to fuzzy metric spaces [9]. Moreover, Gregori and Sapena [5, 10] obtained some contractive-type fixed point theorems in FM-spaces. Recently, in 2020, Li et al. [11] showed some strongly coupled fixed point theorems by using cyclic contractive-type mappings in complete FM-spaces. In 2019, Beloul and Tomar [12] proved integral-type common fixed point theorems in modified intuitionistic fuzzy metric spaces. Prasad et al. [13] presented coincidence theorems via contractive mappings in ordered non-Archimedean fuzzy metric spaces. Again Prasad [14] analyzed coincidence points of relational -contractions in 2021. The bipolar metric space has been studied by many authors, and important results have been obtained [15–18].

Recently, FM-space was extended and generalized to fuzzy bipolar metric space (FBM-space) by Mutlu and Gurdal [19]. They gave new concepts for measurement of the distance between the elements of two different sets. Bartwal et al. [20] introduced the notion of fuzzy bipolar metric space and obtained some fixed point results under mild conditions.

A continuation of this approach, in this manuscript, we shall obtain some fixed point theorems via contractive-type mappings in FBM-spaces. Our results generalize, unify, and extend the results of Bartwal et al. [20] and many other papers in this direction. Also, two examples are given to support our theorems. Ultimately, the existence and uniqueness solution to an integral equation in the sense of Lebesgue measurable functions are obtained as an application.

2. Basic Facts

This part is devoted to present some basic definitions, lemmas, and propositions of FBM-spaces as follows.

Definition 1. (see [8]). Let be a nonvoid set. A 3-triple is called an FM-space if is a fuzzy set on and is a continuous -norm justifying the hypotheses below: (1)(2) iff (3)(4)(5) is continuousfor all and .

Lemma 2. (see [21]). Let be an FM-space. If for all and . where , then .

Definition 3. (see [20]). Let and be two nonvoid sets. A -tuple is said to be an FBM-space, where is continuous -norm and is a fuzzy set on , fulfilling the subsequent assumptions: (1) for all (2) iff for and (3) for all (4) for all and (5) is left continuous(6) is nondecreasing for all and , for all

Remark 4. (see [20]). In an FBM-space , if , then is an FM-space.

Lemma 5. (see [20]). Let be an FBM-space so that for and . Then, .

Definition 6. (see [20]). Let be an FBM-space. A point is called a left point if , a right point if , and a central point if it is both a left and a right point. Similarly, a sequence on the set is called a left sequence, and a sequence on is called a right sequence. In an FBM-space, a left or a right sequence is called simply a sequence. A sequence is said to be convergent to a point , iff is a left sequence, is a right point, and . A bisequence on is a sequence on the set . If the sequence and are convergent, then the bisequence is said to be convergent, and if and converge to a common point, then is called biconvergent. A bisequence is a Cauchy bisequence, if . An FBM-space is called complete, if every Cauchy bisequence is convergent, hence biconvergent.

Lemma 7. (see [20]). In an FBM-space, every convergent Cauchy bisequence is biconvergent.

Lemma 8. (see [20]). Letbe an FBM-space, and ifis a limit of a sequence, then it is a unique limit of the sequence.

Definition 9. A point is said to be common fixed point for the mappings on such that .

3. Main Results

Now, we present the first main theorem.

Theorem 10. Let be a complete FBM-space such that Let be two mappings satisfying (1), , and , (2) for all and , where Then, and have a unique common fixed point.

Proof. Fix and and assume that, , , and for all . Then, we get as a bisequence on the FBM-space . Now, we have and . By induction, we obtain for all and .

Letting , for . Then, from the definition of the FBM-space, we get

Therefore,

From (3), as , we get

Thus, bisequence is a Cauchy bisequence. Since is a complete FBM-space. By Lemma 7, bisequence is a biconvergent sequence. Therefore, and , where . By Lemma 8, both sequences and have a unique limit. From the triangular property of fuzzy bipolar metric spaces, we have

for all and and as ,

From Definition 3 condition (2), . Again,

Therefore, . Hence, is a common fixed point of and .

Let be another fixed point of and . Then,

for and . By Lemma 5, we have .

The following example supports the above theorem.

Example 11. Let and . Define for all , , and . Clearly, is a complete FBM-space, where is a continuous -norm defined as .

Let be mappings defined by

for all . Now, suppose that , then for all , we discuss the following cases:

Case 1. If and , then

Case 2. If and , then

Therefore, the conditions 1 and 2 of Theorem 10 are fulfilled byand. By Theorem 10, and have a unique common fixed point, i.e., .

The second result of this part is as follows.

Theorem 12. Let be a complete FBM-space such that

Let be two mappings satisfying (1), , and , (2) for all , and , where

Then, and have a unique common fixed point.

Proof. Fix and and assume that , , , and for all . Then, we get as a bisequence on the FBM-space . Now, we have and . By induction, we get for all and . Letting , for . Then, from the definition of the fuzzy bipolar metric space, we get Therefore, From (16), as , we get Thus, bisequence is a Cauchy bisequence, sinceis a complete FBM-space. By Lemma 7, bisequence is a biconvergent sequence. Therefore, and , where . By Lemma 8, both sequences and have a unique limit. From the triangular property of fuzzy bipolar metric spaces, we have for all and and as , From Definition 3 condition (2), . Again, Therefore, . Hence, is common fixed point of and . Let be a another fixed point of and . Then, for and . By Lemma 5, we have .

To support the above theorem, we present the following example.

Example 13. Let and and define a continuous -norm as . Define for all , and . Then, is a complete FBM-space. Suppose we define a mapping by Now, suppose that , then for all , we obtain the following cases.

Case 1. Let and , then

Case 2. Let and , then

Therefore, the conditions 1 and 2 of Theorem 12 were also satisfied by and . Based on Theorem 12, we get and that have a unique common fixed point, i.e., .

4. Supportive Application

In this section, we apply Theorem 10 to discuss the existence and uniqueness solution to the following nonlinear integral equations: where is a Lebesgue measurable set with . Let and be two normed linear spaces. Define by for all . Clearly, is a complete FBM-space.

System (29) will be considered under the following hypotheses: (i) and (ii)There is a continuous function and such that for (iii)

Theorem 14. Under hypotheses (i)–(iii), System (29) has a unique common solution in .

Proof. Define the mappings by Now, we have Hence, all hypotheses of Theorem 10 are fulfilled, and consequently, the system (29) has a unique common solution.

Example 15. Let , , , and . Now, consider the following nonlinear integral equations as for all . Then clearly, the above equation is in the form of the following equation: where and That is, (33) is a particular case of system (29). Now, we have Consider a continuous function defined by . Then, we obtain Therefore, all the conditions of Theorem 14 are satisfied. Hence, system (33) has a unique common solution in .

5. Conclusion

First of all, we proved common fixed point theorems on fuzzy bipolar metric space with an application. On the basis of the ideas of this paper along with the literature present on FBM-spaces, we encourage the interested researcher to explore more interesting results for these spaces.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

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