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

Authors: Gunaseelan Mani; Arul Joseph Gnanaprakasam; Absar Ul Haq; Fahd Jarad; Imran Abbas Baloch
Journal: Journal of Function Spaces
 (2021)

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## 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.


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## Body

## 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 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 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 ψ-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]).
LetΠ be a nonvoid set. A 3-triple Ω,Γ,∗ is called an FM-space if Γ is a fuzzy set on Ω2×0,∞ and ∗ is a continuous η-norm justifying the hypotheses below:
(1)
Γμ,σ,η>0(2)
Γμ,σ,η=1 iff μ=σ(3)
Γμ,σ,η=Γσ,μ,η(4)
Γμ,ϑ,η+ζ≥Γμ,σ,η∗Γσ,ϑ,ζ(5)
Γμ,σ,.:0,∞⟶0,1 is continuous
for allμ,σ,ϑ∈Ω and η,ζ>0.Lemma 2. (see [21]).
LetΠ,Γ,∗ be an FM-space. If for all μ,σ∈Π and η>0.
(1)Γμ,σ,kη≥Γμ,σ,η,where k∈0,1, then μ=σ.Definition 3. (see [20]).
LetΠ and Ω be two nonvoid sets. A 4-tuple Π,Ω,Γb,∗ is said to be an FBM-space, where ∗ is continuous η-norm and Γb is a fuzzy set on Π×Ω×0,∞, fulfilling the subsequent assumptions:
(1)
Γbμ,σ,η>0 for all μ,σ∈Π×Ω(2)
Γbμ,σ,η=1 iff μ=σ for μ∈Π and σ∈Ω(3)
Γbμ,σ,η=Γbσ,μ,η for all μ,σ∈Π∩Ω(4)
Γbμ1,σ2,η+ζ+r≥Γbμ1,σ1,η∗Γbμ2,σ1,ζ∗Γbμ2,σ2,r for all μ1,μ2∈Π and σ1,σ2∈Ω(5)
Γbμ,σ,.:0,∞⟶0,1 is left continuous(6)
Γbμ,σ,. is nondecreasing for all μ∈Π and σ∈Ω, for all η,ζ,r>0Remark 4. (see [20]).
In an FBM-spaceΠ,Ω,Γb,∗, if Π=Ω, then Π,Γb,∗ is an FM-space.Lemma 5. (see [20]).
LetΠ,Ω,Γb,∗ be an FBM-space so that
(2)Γbμ,σ,kη≥Γbμ,σ,η,
forμ∈Π,σ∈Ω and k∈0,1. Then, μ=σ.Definition 6. (see [20]).
LetΠ,Ω,Γb,∗ 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 σα on the set Π is called a left sequence, and a sequence σα on Ω is called a right sequence. In an FBM-space, a left or a right sequence is called simply a sequence. A sequence σα is said to be convergent to a point σ, iff σα is a left sequence, σ is a right point, and limα⟶∞Γbσα,σ,η=1. A bisequence σα,μα on Π,Ω,Γb,∗ is a sequence on the set Π×Ω. If the sequence σα and μα are convergent, then the bisequence σα,μα is said to be convergent, and if σα and μα converge to a common point, then σα,μα is called biconvergent. A bisequence σα,μα is a Cauchy bisequence, if limα,β⟶∞Γbσα,μβ,η=1. 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]).
LetΠ,Ω,Γb,∗be an FBM-space, and ifμ∈Π∩Ωis a limit of a sequence, then it is a unique limit of the sequence.Definition 9.
A pointμ∈Π∩Ω is said to be common fixed point for the mappings Λ,Θ on μ∈Π∩Ω such that μ=Λμ=Θμ.


## 3. Main Results

Now, we present the first main theorem.Theorem 10.
LetΠ,Ω,Γb,∗ be a complete FBM-space such that
(3)limη⟶∞Γbμ,σ,η=1forallμ∈Π,σ∈Ω.
LetΛ,Θ:Π∪Ω⟶Π∪Ω be two mappings satisfying
(1)
ΛΠ⊆Π, ΘΠ⊆Π, and ΛΩ⊆Ω, ΘΩ⊆Ω(2)
ΓbΛμ,Θσ,kη≥Γbμ,σ,η for all μ∈Π,σ∈Ω and η>0, where 0<k<1
Then,Λ and Θ have a unique common fixed point.Proof.
Fixμ0∈Π and σ0∈Ω and assume thatΛμ2α=μ2α+1, Θμ2α+1=μ2α+2, Λσ2α=σ2α+1, andΘσ2α+1=σ2α+2 for all α∈ℕ∪0. Then, we get μα,σα as a bisequence on the FBM-space Π,Ω,Γb,∗. Now, we have
(4)Γbμ1,σ1,η=ΓbΛμ0,Θσ0,η≥Γbμ0,σ0,ηk,
∀
η
>
0 and α∈ℕ. By induction, we obtain
(5)Γbμ2α+1,σ2α+1,η=ΓbΛμ2α,Θσ2α,η≥Γbμ0,σ0,ηk2α+1,Γbμ2α+1,σ2α+2,η=ΓbΛμ2α,Θσ2α+1,η≥Γbμ0,σ1,ηk2α+1,
for allη>0 and α∈ℕ.Lettingα<β, for α,β∈ℕ. Then, from the definition of the FBM-space, we get
(6)Γbμα,σβ,η≥Γbμα,σα,η3∗Γbμα+1,σα,η3∗Γbμα+1,σβ,η3,⋮≥Γbμα,σα,η3∗Γbμα+1,σα,η3∗⋯∗Γbμβ−1,σβ−1,η3β−1∗Γbμβ,σβ−1,η3β−1∗Γbμβ,σβ,η3β−1.Therefore,(7)Γbμα,σβ,η≥Γbμ0,σ0,η3kα∗Γbμ0,σ1,η3kα+1∗⋯,⋯∗Γbμ0,σ0,η3β−1kβ+1.From (3), as α,β⟶∞, we get
(8)Γbμα,σβ,η≥1∀η>0.Thus, bisequenceμα,σα is a Cauchy bisequence. Since Π,Ω,Γb,∗ is a complete FBM-space. By Lemma 7, bisequence μα,σα is a biconvergent sequence. Therefore, μα⟶u and σα⟶u, where u∈Π∩Ω. By Lemma 8, both sequences μα and σα have a unique limit. From the triangular property of fuzzy bipolar metric spaces, we have
(9)ΓbΛu,u,η≥ΓbΛu,σα+1,η3∗Γbμα+1,σα+1,η3∗Γbμα+1,u,η3=ΓbΛu,Θσα,η3∗Γbμα+1,σα+1,η3∗Γbμα+1,u,η3≥Γbu,σα,η3∗Γbμα+1,σα+1,η3∗Γbμα+1,u,η3,for allα∈ℕ and η>0 and as α⟶∞,
(10)ΓbΛu,u,η⟶1∗1∗1=1.From Definition3 condition (2), Λu=u. Again,
(11)Γbu,Θu,η=ΓbΛu,Θu,η≥Γbu,u,ηk=1.Therefore,Θu=u. Hence, u is a common fixed point of Λ and Θ.Letv∈Π∩Ω be another fixed point of Λ and Θ. Then,
(12)Γbu,v,η=ΓbΛu,Θv,η≥Γbu,v,ηk,for0<k<1 and ∀η>0. By Lemma 5, we have u=v.The following example supports the above theorem.Example 11.
LetΠ=0,2 and Ω=0∪ℕ−1,2. Define Γb=η/η+μ−σ for all η>0, μ∈Π, and σ∈Ω. Clearly, Π,Ω,Γb,∗ is a complete FBM-space, where ∗ is a continuous η-norm defined as p∗q=pq.LetΛ,Θ:Π∪Ω⟶Π∪Ω be mappings defined by
(13)Λμ=2−μ,ifμ∈0,2,2,ifμ∈ℕ−1,2,Θμ=μ,ifμ∈0,2,2,ifμ∈ℕ−1,2,for allμ∈Π∪Ω. Now, suppose that k=1/2, then for all η>0, we discuss the following cases:Case 1.
Ifμ∈0,2 and σ∈ℕ−1,2, then
(14)ΓbΛμ,Θσ,kη=Γb2−μ,2,kη=kηkη+2−μ−2≥ηη+μ−σ=Γbμ,σ,η.Case 2.
Ifμ∈ℕ−1,2 and σ∈0,2, then
(15)ΓbΛμ,Θσ,kη=Γb2,σ,kη=kηkη+2−σ≥ηη+μ−σ=Γbμ,σ,η.Therefore, the conditions 1 and 2 of Theorem10 are fulfilled byΛandΘ. By Theorem 10, Λ and Θ have a unique common fixed point, i.e., μ=1.The second result of this part is as follows.Theorem 12.
LetΠ,Ω,Γb,∗ be a complete FBM-space such that
(16)limη⟶∞Γbμ,σ,η=1forallμ∈Π,σ∈Ω.LetΛ:Π∪Ω⟶Π∪Ω be two mappings satisfying
(1)
ΛΠ⊆Ω, ΛΩ⊆Π, and ΘΠ⊆Ω, ΘΩ⊆Π(2)
ΓbΛσ,Θμ,kη≥Γbμ,σ,η for all μ∈Π,σ∈Ω, and η>0, where 0<k<1Then,Λ and Θ have a unique common fixed point.Proof.
Fixμ0∈Π and σ0∈Ω and assume that Λμ2α=σ2α, Θμ2α+1=σ2α+1, Λσ2α=μ2α+1, and Θσ2α+1=μ2α+2 for all α∈ℕ∪0. Then, we get μα,σα as a bisequence on the FBM-space Π,Ω,Γb,∗. Now, we have
(17)Γbμ1,σ0,η=ΓbΛσ0,Θμ0,η≥Γbμ0,σ0,ηk,
∀
η
>
0 and α∈ℕ. By induction, we get
(18)Γbμ2α+1,σ2α+1,η=ΓbΛσ2α,Θμ2α+1,η≥Γbμ0,σ0,ηk4α+1,Γbμ2α+1,σ2α,η=ΓbΛσ2α,Θμ2α,η≥Γbμ0,σ0,ηk4α,
for allη>0 and α∈ℕ. Letting α<β, for α,β∈ℕ. Then, from the definition of the fuzzy bipolar metric space, we get
(19)Γbμα,σβ,η≥Γbμα,σα,η3∗Γbμα+1,σα,η3∗Γbμα+1,σβ,η3,⋮≥Γbμα,σα,η3∗Γbμα+1,σα,η3∗⋯∗Γbμβ−1,σβ−1,η3β−1∗Γbμβ,σβ−1,η3β−1∗Γbμβ,σβ,η3β−1.
Therefore,(20)Γbμα,σβ,η≥Γbμ0,σ0,η3k2α+1∗Γbμ0,σ0,η3k2α∗⋯,⋯∗Γbμ0,σ0,η3β−1k2β+1.
From (16), as α,β⟶∞, we get
(21)Γbμα,σβ,η≥1∀η>0.
Thus, bisequenceμα,σα is a Cauchy bisequence, sinceΠ,Ω,Γb,⋆is a complete FBM-space. By Lemma 7, bisequence μα,σα is a biconvergent sequence. Therefore, μα⟶u and σα⟶u, where u∈Π∩Ω. By Lemma 8, both sequences μα and σα have a unique limit. From the triangular property of fuzzy bipolar metric spaces, we have
(22)ΓbΛu,u,η≥ΓbΛu,σα+1,η3∗Γbμα+1,σα+1,η3∗Γbμα+1,u,η3=ΓbΛu,Θμα+1,η3∗ΓbΛσα,Θμα+1,η3∗Γbμα+1,u,η3≥Γbu,μα+1,η3k∗Γbσα,μα+1,η3k∗Γbμα+1,u,η3,
for allα∈ℕ and η>0 and as α⟶∞,
(23)ΓbΛu,u,η⟶1∗1∗1=1.
From Definition3 condition (2), Λu=u. Again,
(24)Γbu,Θu,η=ΓbΛu,Θu,η≥Γbu,u,ηk=1.
Therefore,Θu=u. Hence, u is common fixed point of Λ and Θ. Let v∈Π∩Ω be a another fixed point of Λ and Θ. Then,
(25)Γbu,v,η=ΓbΛu,Θv,η≥Γbu,v,ηk,
for0<k<1 and ∀η>0. By Lemma 5, we have u=v.To support the above theorem, we present the following example.Example 13.
LetΠ=0,1,2,7 and Ω=0,1/3,1/2,3 and define a continuous η-norm as r∗ζ=minr,ζ. Define Γbμ,σ,η=exp−μ−σ/η for all η>0,μ∈Π, and σ∈Ω. Then, Π,Ω,Γb,∗ is a complete FBM-space. Suppose we define a mapping Λ,Θ:Π∪Ω⟶Π∪Ω by
(26)Λμ=13,ifμ∈7,2,0,ifμ∈0,13,12,1,3,Θμ=12,ifμ∈7,2,0,ifμ∈0,13,12,1,3.
Now, suppose thatk=1/2, then for all η>0, we obtain the following cases.Case 1.
Letμ∈7,2 and σ∈0,1/3,1/2,1,3, then
(27)ΓbΛμ,Θσ,kη=Γb13,0,kη=kηkη+1/3≥ηη+μ−σ=Γbμ,σ,η.Case 2.
Letμ∈0,1/3,1/2,1,3 and σ∈7,2, then
(28)ΓbΛμ,Θσ,kη=Γb0,12,kη=kηkη+1/2≥ηη+μ−σ=Γbμ,σ,η.Therefore, the conditions 1 and 2 of Theorem12 were also satisfied by Λ and Θ. Based on Theorem 12, we get Λ and Θ that have a unique common fixed point, i.e., μ=0.


## 4. Supportive Application

In this section, we apply Theorem10 to discuss the existence and uniqueness solution to the following nonlinear integral equations:
(29)μγ=bγ+∫E1∪E2G1γ,ζ,μζdζ,γ∈E1∪E2,μγ=bγ+∫E1∪E2G2γ,ζ,μζdζ,γ∈E1∪E2,where E1∪E2 is a Lebesgue measurable set with mE1∪E2<∞. Let Π=L∞E1 and Ω=L∞E2 be two normed linear spaces. Define Γb:Π×Ω×0,∞⟶0,1 by
(30)Γbμ,σ,η=e−supγ∈E1∪E2μγ−σγ/η,for all μ∈Π,σ∈Ω. Clearly, Π,Ω,Γb,∗ is a complete FBM-space.System (29) will be considered under the following hypotheses:
(i)
G1,G2:E12∪E22×0,∞⟶0,∞ and b∈L∞E1∪L∞E2(ii)
There is a continuous functionθ:E12∪E22⟶0,∞ and k∈0,1 such that G1γ,ζ,μζ−G2γ,ζ,σζ≤kθγ,ζμγ−σγ, for γ,ζ∈E12∪E22(iii)
supγ∈E1∪E2∫E1∪E2θγ,ζdζ≤1Theorem 14.
Under hypotheses (i)–(iii), System (29) has a unique common solution in L∞E1∪L∞E2.Proof.
Define the mappingsΛ,Θ:L∞E1∪L∞E2⟶L∞E1∪L∞E2 by
(31)Λμγ=bγ+∫E1∪E2G1γ,ζ,μζdζ,γ∈E1∪E2,Θμγ=bγ+∫E1∪E2G2γ,ζ,μζdζ,γ∈E1∪E2.
Now, we have(32)ΓbΛμγ,Θσγ,kη=e−supγ∈E1∪E2Λμγ−Θσγ/kη=e−supγ∈E1∪E2bγ+∫E1∪E2G1γ,ζ,μζdζ−bγ−∫E1∪E2G2γ,ζ,σζdζ/kη=e−supγ∈E1∪E2bγ+∫E1∪E2G1γ,ζ,μζdζ−bγ+∫E1∪E2G2γ,ζ,σζdζ/kη≥e−supγ∈E1∪E2∫E1∪E2G1γ,ζ,μζ−G2γ,ζ,σζdζ/kη≥e−supγ∈E1∪E2∫E1∪E2kθγ,ζμγ−σγdζ/kη≥e−supγ∈E1∪E2∫E1∪E2kθγ,ζμγ−σγdζ/kη≥e−supγ∈E1∪E2μγ−σγ/η=Γbμ,σ,η.
Hence, all hypotheses of Theorem10 are fulfilled, and consequently, the system (29) has a unique common solution.Example 15.
LetE1=0,1, E2=1,2, Π=L∞E1, and Ω=L∞E2. Now, consider the following nonlinear integral equations as
(33)Λμγ=γ+∫E1∪E2γ+ζ+μζ81+μζdζ,Θμγ=γ+∫E1∪E2γ+ζ+μζ81+μζdζ,
for allγ∈E1∪E2. Then clearly, the above equation is in the form of the following equation:
(34)Λμγ=bγ+∫E1∪E2G1γ,ζ,μζdζ,γ∈E1∪E2,Θμγ=bγ+∫E1∪E2G2γ,ζ,μζdζ,γ∈E1∪E2,
wherebγ=γ and
(35)G1γ,ζ,μζ=γ+ζ+μζ81+μζ,G2γ,ζ,μζ=γ+ζ+μζ81+μζ.
That is, (33) is a particular case of system (29). Now, we have
(36)G1γ,ζ,μζ−G2γ,ζ,σζ=μζ81+μζ−σζ81+σζ=18μζ−σζ1+μζ1+σζ≤18μζ−σζ.
Consider a continuous functionθ:E12∪E22⟶[0,∞ defined by θγ,ζ=1/4∀γ,ζ∈E12∪E22. Then, we obtain
(37)supγ∈E1∪E2∫E1∪E2θγ,ζdζ≤1.
Therefore, all the conditions of Theorem14 are satisfied. Hence, system (33) has a unique common solution in L∞E1∪L∞E2.


## 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.



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