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

The purpose of this manuscript is to obtain some ﬁ xed point results under mild contractive conditions in fuzzy bipolar metric spaces. Our results generalize and extend many of the previous ﬁ ndings 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.


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][4][5][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 t-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 contractivetype 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 ψ-contractions in 2021. The bipolar metric space has been studied by many authors, and important results have been obtained [15][16][17][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.

Basic Facts
This part is devoted to present some basic definitions, lemmas, and propositions of FBM-spaces as follows.
Definition 6. (see [20]). Let ðΠ, Ω, Γ b , * Þ be an FBM-space. A point σ ∈ Π ∪ Ω is called a left point if σ ∈ Π, a right point if σ ∈ Ω, and a central point if it is both a left and a right point. Similarly, a sequence fσ α g on the set Π is called a left sequence, and a sequence fσ α g on Ω is called a right sequence. In an FBM-space, a left or a right sequence is called simply a sequence. A sequence fσ α g is said to be convergent to a point σ, iff fσ α g is a left sequence, σ is a right point, and lim α⟶∞ Γ b ðσ α , σ, ηÞ = 1. A bisequence ðfσ α g, f μ α gÞ on ðΠ, Ω, Γ b , * Þ is a sequence on the set Π × Ω. If the sequence fσ α g and fμ α g are convergent, then the bisequence ðfσ α g, fμ α gÞ is said to be convergent, and if fσ α g and fμ α g converge to a common point, then ðfσ α g, fμ α gÞ is called biconvergent. A bisequence ðfσ α g, fμ α gÞ is a Cauchy bisequence, if lim α,β⟶∞ Γ b ðσ α , μ β , ηÞ = 1. An FBMspace is called complete, if every Cauchy bisequence is convergent, hence biconvergent.

Main Results
Now, we present the first main theorem.
Then, Λ and Θ have a unique common fixed point.
The following example supports the above theorem.
The second result of this part is as follows.
To support the above theorem, we present the following example.
Journal of Function Spaces define a mapping Λ, Θ : Now, suppose that k = 1/2, then for all η > 0, we obtain the following cases.

Supportive Application
In this section, we apply Theorem 10 to discuss the existence and uniqueness solution to the following nonlinear integral equations: where for all μ ∈ Π,σ ∈ Ω. Clearly, ðΠ, Ω, Γ b , * Þ is a complete FBMspace.
System (29) will be considered under the following hypotheses: Under hypotheses (i)-(iii), System (29) has a unique common solution in Proof. Define the mappings Λ, Θ : Now, we have Hence, all hypotheses of Theorem 10 are fulfilled, and consequently, the system (29) has a unique common solution.

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.