An Approach of Lebesgue Integral in Fuzzy Cone Metric Spaces via Unique Coupled Fixed Point Theorems

In the theory of fuzzy fixed point, many authors have been proved different contractive type fixed point results with different types of applications. In this paper, we establish some new fuzzy cone contractive type unique coupled fixed point theorems (FPtheorems) in fuzzy cone metric spaces (FCM-spaces) by using “the triangular property of fuzzy cone metric” and present illustrative examples to support our main work. In addition, we present a Lebesgue integral type mapping application to get the existence result of a unique coupled FP in FCM-spaces to validate our work.


Introduction
The theory of fuzzy sets was introduced by Zadeh [1]. Later on, in 1975, Kramosil and Michalek [2] introduced the concept of fuzzy metric spaces (FM-space); they present some structural properties of FM-space. In 1988, Grabiec [3] used the concept of Kramosil and Michalek [2] and proved two fixed point theorems (FP-theorems) of "Banach and Edelstein contraction mapping theorems on complete and compact FM-spaces, respectively." After that, the idea of FMspace given by Kramosil and Michalek [2] was modified by George and Veeramani [4], and they proved that every metric induces a fuzzy metric and also proved some fundamental properties and Baire's theorem for FM-spaces. In 2002, Gregory and Sapena [5] proved some contractive type FPtheorems in FM-spaces. Roldan et al. [6] presented some new FP-results in FM-spaces, while Jleli et al. [7] proved some results by using cyclic ðψ, ϕÞ-contractions in Kaleva-Seikkala's type fuzzy metric spaces. Kiany and Harandi [8] presented the concept of set-valued fuzzy-contractive type maps and proved some FP and end point results in FMspaces. Latterly, Rehman et al. [9] gave out the notion of rational type fuzzy contraction for FP in complete FMspaces with an application. Some more related FP-results can be found in [10][11][12][13][14][15].
Indeed, Huang and Zhang [16] rediscovered the idea of Banach-valued metric space. Indeed, many mathematicians proposed it; but it becomes popular after Huang and Zhang's study. By adopting the theory that the underlying cone is normal, they demonstrated the convergence properties and some FP-theorems. Rezapour and Hamlbarani [17], in 2008, proved FP-theorems without assuming the cone's normality, while in [18] Karapinar proved some Ćirić-type nonunique FP-theorems on cone metric spaces. After that, many others contributed their ideas to the problem of FP-findings in cone metric spaces. A few of their FP-findings can be found (e.g., see [19][20][21][22]).
In 2015, Oner et al. [23] gave the idea of fuzzy cone metric space (FCM-space), and they also presented some fundamental properties and "a single-valued Banach contraction theorem for FP with the assumption that all the sequences are Cauchy." After that, Rehman and Li [24] settled some generalized fuzzy cone contractive type FP-results neglecting that "all the sequences are Cauchy" in complete FCM-space. And later, Jabeen et al. [25] presented some common FPtheorems for three self-mappings, by taking into consideration the idea of weakly compatible in FCM-spaces with an integral type application. Chen et al. [26], in 2020, gave the idea of coupled fuzzy cone contractive-type mappings. They proved "some coupled FP-theorems in FCM-spaces with non-linear integral type application." Latterly, Rehman and Aydi [27], in 2021, presented the concept of rational type fuzzy cone contraction mappings in FCM-spaces. They used "the triangular property of fuzzy metric" as a fundamental tool and proved some common FP-theorems and give an application.
Guo and Lakshmikantham [28] proved "coupled FPresults for the nonlinear operator with applications". Later, Bhaskar and Lakshmikantham [29] present some coupled FP-theorems in the context of partially ordered metric spaces, and this work is also presented by Lakshmikantham and Ciric [30]. In the year 2010, Sedghi et al. [31] proved some common coupled FP-results for commuting mappings in fuzzy metric spaces.
In this paper, we present some unique coupled FPfindings in FCM-spaces by taking the idea of Guo and Lakshmikantham [28] and Chen et al. [26]. Furthermore, we have also presented an application of the two Lebesgue Integral Equations (LIE) for a common solution to uphold our work. This paper is organized as follows: Section 2 consists of preliminaries. In Section 3, we establish some unique coupled FP-results in FCM-spaces with illustrative examples. In Section 4, we present an application of Lebesgue integral mapping to get the existence result of unique coupled FP in FCM-spaces to hold up our main work. In Section 5, we discuss the conclusion of our work.
Definition 2 [16]. Let E be a real Banach space and ϑ be the zero element of E , and C is a subset of E. Then, C is called a cone if, (i) C is closed and nonempty, and C ≠ fϑg (ii) α 1 , α 2 ∈ R, α 1 , α 2 ≥ 0 and a, b ∈ C, then α 1 a + α 2 b ∈ C (iii) both a ∈ C and −a ∈ C and then a = ϑ A partial ordering on a given cone C ⊂ E is defined by In this paper, all cones have nonempty interior. where Now from (16) and (17), for ζ ≫ ϑ, We get, after simplification, where Now, from (19) and (20) and by induction, for ζ ≫ ϑ, It shows that the sequence fb j g is a fuzzy cone contractive; therefore, Now for i > j and for ζ ≫ ϑ, we have Hence, the sequence fb j g is Cauchy. Since A is complete and fa j g, fb j g are Cauchy sequences in A, so ∃a, b ∈ A such that a j ⟶ a and b j ⟶ b as j ⟶ ∞ or this can be written as lim j⟶∞ a j = a and lim Hence, Similarly, Regarding its uniqueness, suppose ða 1 , b 1 Þ and ðb 1 , a 1 Þ are another couple fixed point pairs in A × A such that Γð a 1 , b 1 Þ = a 1 and Γðb 1 , where 5 Journal of Function Spaces Now from (27) and for ζ ≫ ϑ, where ðα 1 + 2α 2 Þ < 1. Hence, we have M c ða, a 1 , ζÞ = 1 for ζ ≫ ϑ, ⇒a = a 1 . Similarly, again from (5), for ζ ≫ ϑ, we have Now from (30) and for ζ ≫ ϑ, Hence, we have M c ðb, b 1 , ζÞ = 1 for ζ ≫ ϑ, ⇒b = b 1 .
Example 1. A = ð0,∞Þ, * is a ζ-norm, and M c : for all a, b ∈ A and ζ > 0. Then, it is easy to verify that M c is triangular and ðA, M c , * Þ is a complete FCM-space. We define Now from (5), for ζ ≫ ϑ, we have It is easy to verify that conditions of Theorem 10 are satisfied with α 1 = α 2 = 1/12. Then, Γ has unique coupled FP for a = 2 and b = 2.
Proof. Any a 0 , b 0 ∈ A, and we define sequence fa j g by

Journal of Function Spaces
We get, after simplification, where Now, from (42) and (43) and by induction, for ζ ≫ ϑ, we have Hence, the sequence fa j g is fuzzy cone contractive; therefore, Now for i > j and for ζ ≫ ϑ, we have Hence, the sequence fa j g is Cauchy. Now for sequence fb j g, again from (39), for ζ ≫ ϑ, we have We get, after simplification, where the value of δ is the same as in (42). Similarly, Now, from (48) and (49) and by induction, for ζ ≫ ϑ, we have that

Journal of Function Spaces
Hence, the sequence fb j g is fuzzy cone contractive; therefore, Now for i > j, for ζ ≫ ϑ, we have Hence, the sequence fb j g is Cauchy. Since A is complete and fa j g and fb j g are Cauchy sequences in A, ∃a, b ∈ A such that a j ⟶ a and b j ⟶ b as j ⟶ ∞, or this can be written as lim Regarding its uniqueness, let ða 1 , b 1 Þ and ðb 1 , a 1 Þ be another couple fixed point pairs in A × A such that Γða 1 , b 1 Þ = a 1 and Γðb 1 , a 1 Þ = b 1 . Now, from (39), for ζ ≫ ϑ, we have