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

Authors: Muhammad Talha Waheed; Saif Ur Rehman; Naeem Jan; Abdu Gumaei; Mabrook Al-Rakhami
Journal: Journal of Function Spaces
 (2021)

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## Abstract

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 (FP-theorems) 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.


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## Body

## 1. 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 FM-space 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 FP-theorems 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 ψ,ϕ-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 FM-spaces. Latterly, Rehman et al. [9] gave out the notion of rational type fuzzy contraction for FP in complete FM-spaces with an application. Some more related FP-results can be found in [10–15].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 [19–22]).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 FP-theorems 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 FP-results 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 FP-findings 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.


## 2. Preliminaries

Definition 1 [32].
A binary operation∗:0,1×0,1⟶0,1 would be a continuous t-norm if ∗ fulfils the following conditions:
(i)
∗ is associative and commutative(ii)
∗ is continuous(iii)
1∗α=α, ∀α∈0,1(iv)
α1∗α2≤α3∗α4 whenever α1≤α3 and α2≤α4, for α1,α2,α3,α4∈0,1Throughout the complete paper,ζ-norm represents a continuous t-norm.Definition 2 [16].
LetE 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≠ϑ(ii)
α1,α2∈R,α1,α2≥0 and a,b∈C, then α1a+α2b∈C(iii)
botha∈C and −a∈C and then a=ϑA partial ordering on a given coneC⊂E is defined by a⪯b⇔b−a∈C. a≺b stands for a⪯b and a≠b, while a≪b stands for b−a∈intC. In this paper, all cones have nonempty interior.Definition 3 [4].
A 3-tupleA,Mc,∗ is said to be a FM-space if A is any set, ∗ is a ζ-norm, and Mc is a fuzzy set on A2×0,∞ satisfying
(i)
Mca,b,ζ>0(ii)
Mca,b,ζ=0⇔a=b(iii)
Mca,b,ζ=Mcb,a,ζ(iv)
Mca,c,ζ∗Mcc,b,s≤Mca,b,ζ+s(v)
Mca,b,.:0,∞⟶0,1is continuous,∀a,b,c∈A and ζ,s>0Definition 4 [23].
A 3-tupleA,Mc,∗ is said to be a FCM-space if C is a cone of E, Ais an arbitrary set,∗is aζ-norm, andMcis a fuzzy set onA2×intCsatisfying(i)
Mca,b,ζ>0(ii)
Mca,b,ζ=0⇔a=b(iii)
Mca,b,ζ=Mcb,a,ζ(iv)
Mca,c,ζ∗Mcc,b,s≤Mca,b,ζ+s(v)
Mca,b,.:intC⟶0,1 is continuous, ∀a,b,c∈A, and ζ,s≫ϑDefinition 5 [23].
Let a 3-tupleA,Mc,∗ be a FCM-space, b1∈A, which is a sequence bjinA(i)
It converges tob if α3∈0,1 and ζ≫ϑ; there is j1∈N such that Mcbj,b,ζ>1−α3, for j≥j1, or we write it as limj⟶∞bj=b1 or bj⟶b1 as j⟶∞(ii)
It is a Cauchy sequence ifα3∈0,1 and ζ≫ϑ; there is j1∈N such that Mcbj,bi,ζ>1−α3, for j,i≥j1(iii)
A,Mc,∗ is complete if every Cauchy sequence is convergent in A(iv)
It is fuzzy cone contractive if∃α∈0,1 and fulfilling(1)1Mcbj,bj+1,ζ−1≤α1Mcbj−1,bj,ζ−1,forζ≫ϑ,j≥1.Lemma 6 [23].
LetA,Mc,∗ be a FCM-space, and let a sequence bj in A converge to a point b∈A⇔Mcbj,b,ζ which converges to 1 as j⟶∞, for ζ≫ϑ.Definition 7 [24].
LetA,Mc,∗ be a FCM-space. The FCM Mc is triangular, if
(2)1Mca,b,ζ−1≤1Mca,c,ζ−1+1Mcc,b,ζ−1,∀a,c,b∈A,ζ≫ϑ.Definition 8 [23].
LetA,Mc,∗ be a FCM-space and Γ:A⟶A. Then, Γ is said to be fuzzy cone contractive if ∃α1∈0,1 such that
(3)1McΓb,Γc,ζ−1≤α11Mcb,c,ζ−1,∀b,c∈A,ζ≫ϑ.Definition 9.
Letb,c be an element in A×A. Then, it is called coupled FP of a mapping Γ:A×A⟶A if
(4)Γb,c=b,Γc,b=c.Now, in the following, we prove some unique couple FP-theorems in FCM-spaces with examples to support our main work. Furthermore, we present an application of Lebesgue integral contractive type mapping to prove a unique coupled FP-theorem in FCM-spaces.


## 3. Main Results

Now, we present our first main result.Theorem 10.
LetΓ:A×A⟶A be a mapping on complete FCM-spaces A,Mc,∗ in which Mc is triangular and satisfies the inequality:
(5)1McΓa,b,Γκ,ϱ,ζ−1≤α11Mca,κ,ζ−1+α21NΓ,a,b,κ,ϱ,ζ−1,where
(6)1NΓ,a,b,κ,ϱ,ζ−1=1Mca,Γa,b,ζ−1+1Mcκ,Γκ,ϱ,ζ−1+1Mca,Γκ,ϱ,ζ−1+1Mcκ,Γa,b,ζ−1,∀a,b,κ,ϱ∈A, ζ≫ϑ, α1∈0,1, and α2≥0 with α1+4α2<1. Then, Γ has a unique couple FP in A.Proof.
Anya0,b0∈A; we define sequences aj and bj in A such that
(7)Γaj,bj=aj+1,Γbj,aj=bj+1,forj≥0.☐Now from (5) for ζ≫ϑ, we have
(8)1Mcaj,aj+1,ζ−1=1McΓaj−1,bj−1,Γaj,bj,ζ−1≤α11Mcaj−1,aj,ζ−1+α21NΓ,aj−1,bj−1,aj,bj,ζ−1,where
(9)1NΓ,aj−1,bj−1,aj,bj,ζ−1=1Mcaj−1,Γaj−1,bj−1,ζ−1+1Mcaj,Γaj,bj,ζ−1+1Mcaj−1,Γaj,bj,ζ−1+1Mcaj,Γaj−1,bj−1,ζ−1=1Mcaj−1,aj,ζ−1+1Mcaj,aj+1,ζ−1+1Mcaj−1,aj+1,ζ−1≤21Mcaj−1,aj,ζ−1+1Mcaj,aj+1,ζ−1.Now from (8) and (9), for ζ≫ϑ,
(10)1Mcaj,aj+1,ζ−1≤α1Mcaj−1,aj,ζ−1+2α21Mcaj−1,aj,ζ−1+1Mcaj,aj+1,ζ−1.We get, after simplification,(11)1Mcaj,aj+1,ζ−1≤λ1Mcaj−1,aj,ζ−1,forζ≫ϑ,where λ=α1+2α2/1−2α2<1. Similarly,
(12)1Mcaj−1,aj,ζ−1≤λ1Mcaj−2,aj−1,ζ−1,forζ≫ϑ.Now, from (11) and (12) and by induction, for ζ≫ϑ,
(13)1Mcaj,aj+1,ζ−1≤λ1Mcaj−1,aj,ζ−1≤λ21Mcaj−2,aj−1,ζ−1≤⋯≤λj1Mca0,a1,ζ−1⟶0,asj⟶∞.It shows that the sequenceaj is a fuzzy cone contractive; therefore,
(14)limj⟶∞Mcaj,aj+1,ζ=1,forζ≫ϑ.Now fori>j and for ζ≫ϑ, we have
(15)1Mcaj,ai,ζ−1≤1Mcaj,aj+1,ζ−1+1Mcaj+1,aj+2,ζ−1+⋯+1Mcai−1,ai,ζ−1≤λj1Mcao,a1,ζ−1+λj+11Mca0,a1,ζ−1+⋯+λi−11Mca0,a1,ζ−1=λj+λj+1+⋯+λi−11Mca0,a1,ζ−1=λj1−λ1Mcao,a1,ζ−1⟶0,asj⟶∞.Hence, the sequenceaj is Cauchy. Now for sequence bj and from (5), for ζ≫ϑ, we have
(16)1Mcbj,bj+1,ζ−1=1McΓbj−1,aj−1,Γbj,aj,ζ−1≤α11Mcbj−1,bj,ζ−1+α21NΓ,bj−1,aj−1,bj,aj,ζ−1,where
(17)1NΓ,bj−1,aj−1,bj,aj,ζ−1=1Mcbj−1,Γbj−1,aj−1,ζ−1+1Mcbj,Γbj,aj,ζ−1+1Mcbj−1,Γbj,aj,ζ−1+1Mcbj,Γbj−1,aj−1,ζ−1=1Mcbj−1,bj,ζ−1+1Mcbj,bj+1,ζ−1+1Mcbj−1,bj+1,ζ−1≤21Mcbj−1,bj,ζ−1+1Mcbj,bj+1,ζ−1.Now from (16) and (17), for ζ≫ϑ,
(18)1Mcbj,bj+1,ζ−1≤α11Mcbj−1,bj,ζ−1+2α21Mcbj−1,bj,ζ−1+1Mcbj,bj+1,ζ−1.We get, after simplification,(19)1Mcbj,bj+1,ζ−1≤λ1Mcbj−1,bj,ζ−1,forζ≫ϑ,where λ=α1+2α2/1−2α2<1. Similarly,
(20)1Mcbj−1,bj,ζ−1≤λ1Mcbj−2,bj−1,ζ−1,forζ≫ϑ.Now, from (19) and (20) and by induction, for ζ≫ϑ,
(21)1Mcbj,bj+1,ζ−1≤λ1Mcbj−1,bj,ζ−1≤λ21Mcbj−2,bj−1,ζ−1≤⋯≤λj1Mcb0,b1,ζ−1⟶0,asj⟶∞It shows that the sequencebj is a fuzzy cone contractive; therefore,
(22)limj⟶∞Mcbj,bj+1,ζ=1,forζ≫ϑ.Now fori>j and for ζ≫ϑ, we have
(23)1Mcbj,bi,ζ−1≤1Mcbj,bj+1,ζ−1+1Mcbj+1,bj+2,ζ−1+⋯+1Mcbi−1,hi,ζ−1≤λj1Mcbo,b1,ζ−1+λj+11Mcb0,b1,ζ−1+⋯+λi−11Mcb0,b1,ζ−1=λj+λj+1+⋯+λi−11Mcb0,b1,ζ−1=λj1−λ1Mcb0,b1,ζ−1⟶0,asj⟶∞.Hence, the sequencebj is Cauchy. Since A is complete and aj,bj are Cauchy sequences in A, so ∃a,b∈A such that aj⟶a and bj⟶b as j⟶∞ or this can be written as limj⟶∞aj=a and limj⟶∞bj=b. Therefore,
(24)limj⟶∞Mcaj,a,ζ=1,limj⟶∞Mcbj,b,ζ=1,forζ≫ϑ.Hence,(25)limj⟶∞aj+1=limj⟶∞Γaj,bj=Γlimj⟶∞aj,limj⟶∞bj⇒Γa,b=a.Similarly,(26)limj⟶∞bj+1=limj⟶∞Γbj,aj=Γlimj⟶∞bj,limj⟶∞aj⇒Γb,a=b.Regarding its uniqueness, supposea1,b1 and b1,a1 are another couple fixed point pairs in A×A such that Γa1,b1=a1 and Γb1,a1=b1. Now, from (5), for ζ≫ϑ, we have
(27)1Mca,a1,ζ−1=1McΓa,b,Γa1,b1,ζ−1≤α11Mca,a1,ζ−1+α21NΓ,a,b,a1,b1,ζ−1,where
(28)1NΓ,a,b,a1,b1,ζ−1=1Mca,Γa,b,ζ−1+1Mca1,Γa1,b1,ζ−1+1Mca,Γa1,b1,ζ−1+1Mca1,Γa,b,ζ−1=1Mca,a,ζ−1+1Mca1,a1,ζ−1+1Mca,a1,ζ−1+1Mca1,a,ζ−1=21Mca,a1,ζ.Now from (27) and for ζ≫ϑ,
(29)1Mca,a1,ζ−1≤α11Mca,a1,ζ−1+2α21Mca,a1,ζ−1=α1+2α21Mca,a1,ζ−1=α1+2α21McΓa,b,Γa1,b1,ζ−1≤α1+2α221Mca,a1,ζ−1≤⋯≤α1+2α2j1Mca,a1,ζ−1⟶0,asj⟶∞,where α1+2α2<1. Hence, we have Mca,a1,ζ=1 for ζ≫ϑ, ⇒a=a1.Similarly, again from (5), for ζ≫ϑ, we have
(30)1Mcb,b1,ζ−1=1McΓb,a,Γb1,a1,ζ−1≤α11Mcb,b1,ζ−1+α21NΓ,b,a,b1,a1,ζ−1,where
(31)1NΓ,b,a,b1,a1,ζ−1=1Mcb,Γb,a,ζ−1+1Mcb1,Γb1,a1,ζ−1+1Mcb,Γb1,a1,ζ−1+1Mcb1,Γb,a,ζ−1=1Mcb,b,ζ−1+1Mcb1,b1,ζ−1+1Mcb,b1,ζ−1+1Mcb1,b,ζ−1=21Mcb,b1,ζ.Now from (30) and for ζ≫ϑ,
(32)1Mcb,b1,ζ−1≤α11Mcb,b1,ζ−1+2α21Mcb,b1,ζ−1=α1+2α21Mcb,b1,ζ−1=α1+2α21McΓb,a,Γb1,a1,ζ−1≤α1+2α221Mcb,b1,ζ−1≤⋯≤α1+2α2j1Mcb,b1,ζ−1⟶0,asj⟶∞.Hence, we haveMcb,b1,ζ=1 for ζ≫ϑ, ⇒b=b1.Corollary 11.
LetΓ:A×A⟶A be a mapping on complete FCM-spaces A,Mc,∗ in which Mc is triangular and satisfies
(33)1McΓa,b,Γκ,ϱ,ζ−1≤α11Mca,κ,ζ−1+α21Mca,Γa,b,ζ−1+1Mcκ,Γκ,ϱ,ζ−1,for all a,b,κ,ϱ∈A, ζ≫ϑ, α1∈0,1, and α2≥0 with α1+2α2<1. Then, Γ has a unique couple FP in A.Corollary 12.
LetΓ:A×A⟶A be a mapping on complete FCM-spaces A,Mc,∗ in which Mc is triangular and satisfies
(34)1McΓa,b,Γκ,ϱ,ζ−1≤α11Mca,κ,ζ−1+α21Mca,Γκ,ϱ,ζ−1+1Mcκ,Γa,b,ζ−1,for all a,b,κ,ϱ∈A, ζ≫ϑ, α1∈0,1, and α2≥0 with α1+2α2<1. Then, Γ has a unique couple FP in A.Example 1.
A=0,∞, ∗ is a ζ-norm, and Mc:A2×0,∞⟶0,1 is defined as
(35)Mca,b,ζ=ζζ+da,b,da,b=a−b,for all a,b∈A and ζ>0. Then, it is easy to verify that Mc is triangular and A,Mc,∗ is a complete FCM-space. We define
(36)Γg,h=a−b12,ifa,b∈0,1,2a+2b−23,ifa,b∈1,∞.
Now from (5), for ζ≫ϑ, we have
(37)1McΓa,b,Γκ,ϱ,ζ−1=1Mca−b/12,κ−ϱ/12,ζ−1=1ζda−b12,κ−ϱ12=112ζa−b−κ+ϱ≤112ζ∣a−κ+a−a−b+κ−κ−ϱ+a−κ−ϱ+κ−a−b∣≤112ζa−κ+112ζa−a−b+κ−κ−ϱ+a−κ−ϱ+κ−a−b=1121Mca,κ,ζ−1+1121Mca,Γa,b,ζ−1+1Mcκ,Γκ,ϱ,ζ−1+1Mca,Γκ,ϱ,ζ−1+1Mcκ,Γa,b,ζ−1=1121Mca,κ,ζ−1+1121NΓ,a,b,κ,ϱ,ζ−1,forζ≫ϑ
It is easy to verify that conditions of Theorem10 are satisfied with α1=α2=1/12. Then, Γ has unique coupled FP for a=2 and b=2.
(38)Γa,b=Γ2,2=22+22−23=2⇒Γ2,2=2.Theorem 13.
LetΓ:A×A⟶A be a mapping in a complete FCM-space A,Mc,∗ in which Mc is triangular and satisfies
(39)1McΓa,b,Γκ,ϱ,ζ−1≤α11Mca,κ,ζ−1+α21Mca,Γa,b,ζ−1+1Mcκ,Γκ,ϱ,ζ−1+α31Mcκ,Γa,b,ζ∗Mcκ,Γκ,ϱ,ζ−1,for all a,b,κ,ϱ∈A, ζ≫ϑα1∈0,1, and α2,α3≥0 with α1+2α2+α3<1. Then, Γ has a unique couple FP in A.Proof.
Anya0,b0∈A, and we define sequence aj by
(40)Γaj,bj=aj+1,Γbj,aj=bj+1,forj≥0.☐Now, from (39), for ζ≫ϑ, we have
(41)1Mcaj,aj+1,ζ−1=1McΓaj−1,bj−1,Γaj,bj,ζ−1≤α11Mcaj−1,aj,ζ−1+α21Mcaj−1,Γaj−1,bj−1,ζ−1+1Mcaj,Γaj,bj,ζ−1+α31Mcaj,Γaj−1,bj−1,ζ∗Mcaj,Γaj,bj,ζ−1=α11Mcaj−1,aj,ζ−1+α21Mcaj−1,aj,ζ−1+1Mcaj,aj+1,ζ−1+α31Mcaj,aj,ζ∗Mcaj,aj+1,ζ−1=α11Mcaj−1,aj,ζ−1+α21Mcaj−1,aj,ζ−1+1Mcaj,aj+1,ζ−1+α31Mcaj,aj+1,ζ−1.We get, after simplification,(42)1Mcaj,aj+1,ζ−1≤δ1Mcaj−1,aj,ζ−1,forζ≫ϑ,whereδ=α1+α2/1−α2−α3<1. Similarly,
(43)1Mcaj−1,aj,ζ−1≤δ1Mcaj−2,aj−1,ζ−1,forζ≫ϑ.Now, from (42) and (43) and by induction, for ζ≫ϑ, we have
(44)1Mcaj,aj+1,ζ−1≤δ1Mcaj−1,aj,ζ−1≤δ21Mcaj−2,aj−1,ζ−1≤⋯≤δj1Mcao,a1,ζ−1⟶0,asj⟶∞.Hence, the sequenceaj is fuzzy cone contractive; therefore,
(45)limj⟶∞Mcaj,aj+1,ζ=1ζ≫ϑ.Now fori>j and for ζ≫ϑ, we have
(46)1Mcaj,ai,ζ−1≤1Mcaj,aj+1,ζ−1+1Mcaj+1,aj+2,ζ−1+⋯+1Mcai−1,ai,ζ−1≤δj1Mcao,a1,ζ−1+δj+11Mcao,a1,ζ−1+⋯+δi−11Mcao,a1,ζ−1=δj+δj+1+⋯+δi−11Mcao,a1,ζ−1=δj1−δ1Mcao,a1,ζ−1⟶0,asj⟶∞.Hence, the sequenceaj is Cauchy. Now for sequence bj, again from (39), for ζ≫ϑ, we have
(47)1Mcbj,bj+1,ζ−1=1McΓbj−1,aj−1,Γbj,aj,ζ−1≤α11Mcbj−1,bj,ζ−1+α21Mcbj−1,Γbj−1,aj−1,ζ−1+1Mcbj,Γbj,aj,ζ−1+α31Mcbj,Γbj−1,aj−1,ζ∗Mcbj,Γbj,aj,ζ−1=α11Mcbj−1,bj,ζ−1+α21Mcbj−1,bj,ζ−1+1Mcbj,bj+1,ζ−1+α31Mcbj,bj,ζ∗Mcbj,bj+1,ζ−1=α11Mchj−1,bj,ζ−1+α21Mcbj−1,bj,ζ−1+1Mcbj,bj+1,ζ−1+α31Mcbj,bj+1,ζ−1.We get, after simplification,(48)1Mcbj,bj+1,ζ−1≤δ1Mcbj−1,bj,ζ−1,forζ≫ϑ,where the value of δ is the same as in (42). Similarly,
(49)1Mcbj−1,bj,ζ−1≤δ1Mcbj−2,bj−1,ζ−1,forζ≫ϑ.Now, from (48) and (49) and by induction, for ζ≫ϑ, we have that
(50)1Mcbj,bj+1,ζ−1≤δ1Mcbj−1,bj,ζ−1≤δ21Mcbj−2,bj−1,ζ−1≤⋯≤δj1Mcb0,b1,ζ−1⟶0,asj⟶∞.Hence, the sequencebj is fuzzy cone contractive; therefore,
(51)limj⟶∞Mcbj,bj+1,ζ=1,forζ≫ϑ.Now fori>j, for ζ≫ϑ, we have
(52)1Mcbj,bi,ζ−1≤1Mcbj,bj+1,ζ−1+1Mcbj+1,bj+2,ζ−1+⋯+1Mcbi−1,bi,ζ−1≤δj1Mcb0,b1,ζ−1+δj+11Mcb0,b1,ζ−1+⋯+δi−11Mcb0,b1,ζ−1=δj+δj+1+⋯+δi−11Mcb0,b1,ζ−1=δj1−δ1Mcb0,b1,ζ−1⟶0,asj⟶∞.Hence, the sequencebj is Cauchy. Since A is complete and aj and bj are Cauchy sequences in A, ∃a,b∈A such that aj⟶a and bj⟶b as j⟶∞, or this can be written as limj⟶∞aj=a and limj⟶∞bj=b. Therefore,
(53)limj⟶∞aj+1=limj⟶∞Γaj,bj=Γlimj⟶∞aj,limj⟶∞bj⇒Γa,b=a.Similarly,(54)limj⟶∞bj+1=limj⟶∞Γbj,aj=Γlimj⟶∞bj,limj⟶∞aj⇒Γb,a=b.Regarding its uniqueness, leta1,b1 and b1,a1 be another couple fixed point pairs in A×A such that Γa1,b1=a1 and Γb1,a1=b1. Now, from (39), for ζ≫ϑ, we have
(55)1Mca,a1,ζ−1=1McΓa,b,Γa1,b1,ζ−1≤α11Mca,a1,ζ−1+α21Mca,Γa,b,ζ−1−1+1Mca1,Γa1,b1,ζ−1+α31Mca1,Γa,b,ζ∗Mca1,Γa1,b1,ζ−1=α1+α31Mca,a1,ζ−1=α1+α31McΓa,b,Γa1,b1,ζ−1≤α1+α321Mca,a1,ζ−1≤⋯≤α1+α3j1Mca,a1,ζ−1⟶0,asj⟶∞.Hence, we get thatMca,a1,ζ=1,⇒a=a1. Similarly, again from (39), for ζ≫ϑ, we have
(56)1Mcb,b1,ζ−1=1McΓb,a,Γb1,a1,ζ−1≤α11Mcb,b1,ζ−1+α21Mcb,Γb,a,ζ−1−1+1Mcb1,Γb1,a1,ζ−1+α31Mcb1,Γb,a,ζ∗Mcb1,Γb1,a1,ζ−1=α1+α31Mcb,b1,ζ−1=α1+α31McΓb,a,Γb1,a1,ζ−1≤α1+α321Mcb,b1,ζ−1≤⋯≤α1+α3j1Mcb,b1,ζ−1⟶0,asj⟶∞.Hence, we get thatMcb,b1,ζ=1 for ζ≫ϑ,⇒b=b1.Corollary 14.
LetΓ:A×A⟶A be a mapping on complete FCM-spaces A,Mc,∗ in which Mc is triangular and satisfies
(57)1McΓa,b,Γκ,ϱ,ζ−1≤α11Mca,κ,ζ−1+α31Mcκ,Γa,b,ζ∗Mcκ,Γκ,ϱ,ζ−1,for all a,b,κ,ϱ∈A, ζ≫ϑ,α1∈0,1, and α3≥0 with α1+α3<1. Then, Γ has a unique couple FP.Example 2.
A=0,∞, ∗ is a ζ-norm, and Mc:A×A×0,∞⟶0,1 is defined as
(58)Mca,b,ζ=ζζ+da,b,da,b=a−b,for all a,b∈A and ζ>0. Then, it is easy to verify that Mc is triangular and A,Mc,∗ is a complete FCM-space. We define
(59)F¨a,b=a−b8,ifa,b∈0,1,2a+2b−33,ifa,b∈1,∞.Now from (39), for ζ≫ϑ, we have
(60)1McΓa,b,Γκ,ϱ,ζ−1=1Mca−b/8,κ−ϱ/8,ζ−1=1ζda−b8,κ−ϱ8=18ζa−b−κ+ϱ≤18ζ∣a−κ∣+∣a−a−b+κ−κ−ϱ∣≤18ζa−κ+18ζa−a−b+κ−κ−ϱ=181Mca,κ,ζ−1+181Mca,Γa,b,ζ−1+1Mcκ,Γκ,ϱ,ζ−1,forζ≫ϑ.It is easy to verify that all the conditions of Theorem13 are satisfied with α1=α2=1/8 and α3=0. Then, Γ has unique coupled FP.
(61)Γa,b=Γ3,3=23+23−33=3⇒Γ3,3=3.


## 4. Application

In this section, we present an application on Lebesgue integral (LI) mapping to support our main work. In 2002, Branciari proved the following result on complete metric space for unique FP (see [33]):Theorem 15.
LetA,d be a complete metric space, α∈0,1, and Γ:A⟶A a mapping such that for each a,b∈A,
(62)∫0dΓa,Γbφsds≤α∫0da,bφsds,where φ:0,∞⟶0,∞ is a Lebesgue integrable mapping which is summable (i.e., with finite integral on each compact subset of 0,∞) and for each τ>0,
(63)∫0τφsds>0.
Then,Γ has a unique FP u∈A such that for any a∈A, limj⟶∞Γja=u.
Now, we are in the position to use the above concept and to prove a unique coupled FP-theorem in FCM-spaces.Theorem 16.
LetΓ:A×A⟶A be a mapping on complete FCM-spaces A,Mc,∗ in which Mc is triangular and satisfies
(64)∫01/McΓa,b,Γκ,ϱ,ζ−1φsds≤α1∫01/Mca,κ,ζ−1φsds+α2∫01/NΓ,a,b,κ,ϱ,ζ−1φsds,where
(65)1NΓ,a,b,κ,ϱ,ζ−1=1Mca,Γa,b,ζ−1+1Mcκ,Γκ,ϱ,ζ−1+1Mca,Γκ,ϱ,ζ−1+1Mcκ,Γa,b,ζ−1,for all a,b,κ,ϱ∈A, ζ≫ϑ, α1∈0,1, and α2≥0 with α1+4α2<1 and φ:0,∞⟶0,∞ is a Lebesgue integrable mapping which is summable (i.e., with finite integral on each compact subset of 0,∞) and for each τ>0,
(66)∫0τφsds>0.
Then,Γ has a unique couple FP in A.Proof.
Anyao,bo∈A; we define sequences aj and bj in A such that
(67)Γaj,bj=aj+1,Γbj,aj=bj+1,forj≥0.☐Now from (64) and from the proof of Theorem 10, for ζ≫ϑ, we have that
(68)∫01/Mcaj,aj+1,ζ−1φsds=∫01/McΓaj−1,bj−1,Γaj,bj,ζ−1φsds≤λ∫01/Mcaj−1,aj,ζ−1φsds,where λ=α1+2α2/1−2α2<1. Similarly, again by using the same arguments, we have
(69)∫01/Mcaj−1,aj,ζ−1φsds≤λ∫01/Mcaj−2,aj−1,ζ−1φsds,forζ≫ϑ.Now, from (68) and (69) and by induction, for ζ≫ϑ, we have
(70)∫01/Mcaj,aj+1,ζ−1φsds≤λ∫01/Mcaj−1,aj,ζ−1φsds≤λ2∫01/Mcaj−2,aj−1,ζ−1φsds≤⋯≤λj∫01/Mca0,a1,ζ−1φsds⟶0,asj⟶∞,which shows that the sequence aj is a fuzzy cone contractive, therefore
(71)limj⟶∞∫01/Mcaj,aj+1,ζ−1φsds=0⇒limj⟶∞1Mcaj,aj+1,ζ−1=0,forζ≫ϑ.Hence, we get that(72)limj⟶∞Mcaj,aj+1,ζ=1,forζ≫ϑ.Now fori>j and for ζ≫ϑ, we have
(73)∫01/Mcaj,ai,ζ−1φsds≤∫01/Mcaj,aj+1,ζ−1φsds+∫01/Mcaj+1,aj+2,ζ−1φsds+⋯+∫01/Mcai−1,ai,ζ−1φsds≤λj∫01/Mcao,a1,ζ−1φsds+λj+1∫01/Mcao,a1,ζ−1φsds+⋯+λi−1∫01/Mca0,a1,ζ−1φsds=λj+λj+1+⋯+λi−1∫01/Mca0,a1,ζ−1φsds=λj1−λ∫01/Mca0,a1,ζ−1φsds⟶0,asj⟶∞.We get that(74)limj⟶∞∫01/Mcaj,ai,ζ−1φsds=0⇒limj⟶∞1Mcaj,ai,ζ−1=0,forζ≫ϑ.Hence proved that the sequenceaj is Cauchy. Now for sequence bj from (64) and from the proof of Theorem 10, for ζ≫ϑ, we have
(75)∫01/Mcbj,bj+1,ζ−1φsds=∫01/McΓbj−1,aj−1,Γbj,aj,ζ−1φsds≤λ∫01/Mcbj−1,bj,ζ−1φsds,where λ=α1+2α2/1−2α2<1. Similarly, again by using the same arguments, we have
(76)∫01/Mcbj−1,bj,ζ−1φsds≤λ∫01/Mcbj−2,aj−1,ζ−1φsds,forζ≫ϑ.Now, from (75) and (76) and by induction, for ζ≫ϑ, we have
(77)∫01/Mcbj,bj+1,ζ−1φsds≤λ∫01/Mcbj−1,bj,ζ−1φsds≤λ2∫01/Mcbj−2,bj−1,ζ−1φsds≤⋯≤λj∫01/Mcb0,b1,ζ−1φsds⟶0,asj⟶∞,which shows that the sequence bj is fuzzy cone contractive; therefore,
(78)limj⟶∞∫01/Mcbj,bj+1,ζ−1φsds=0⇒limj⟶∞1Mcbj,bj+1,ζ−1=0,forζ≫ϑ.Hence, we get that(79)limj⟶∞Mcbj,bj+1,ζ=1,forζ≫ϑ.Now fori>j and for ζ≫ϑ, we have
(80)∫01/Mcbj,bi,ζ−1φsds≤∫01/Mcbj,bj+1,ζ−1φsds+∫01/Mcbj+1,bj+2,ζ−1φsds+⋯+∫01/Mcbi−1,bi,ζ−1φsds≤λj∫01/Mcbo,b1,ζ−1φsds+λj+1∫01/Mcbo,b1,ζ−1φsds+⋯+λi−1∫01/Mcbo,b1,ζ−1φsds=λj+λj+1+⋯+λi−1∫01/Mcbo,b1,ζ−1φsds=λj1−λ∫01/Mcbo,b1,ζ−1φsds⟶0,asj⟶∞.We get that(81)limj⟶∞∫01/Mchj,hi,ζ−1φsds=0⇒limj⟶∞1Mchj,hi,ζ−1=0,forζ≫ϑ.Hence, it was proved that the sequencebj is Cauchy. Since A is complete and aj,bj are Cauchy sequences in A, so ∃a,b∈A such that aj⟶a and bj⟶b as j⟶∞ or this can be written as limj⟶∞aj=a and limj⟶∞bj=b. Therefore,
(82)limj⟶∞Mcaj,a,ζ=1,limj⟶∞Mcbj,b,ζ=1,forζ≫ϑ.Hence,(83)limj⟶∞aj+1=limj⟶∞Γaj,bj=Γlimj⟶∞aj,limj⟶∞bj⇒Γa,b=a.Similarly,(84)limj⟶∞bj+1=limj⟶∞Γbj,aj=Γlimj⟶∞bj,limj⟶∞aj⇒Γb,a=b.Regarding its uniqueness, supposea1,b1 and b1,a1 are another couple fixed point pairs in A×A such that Γa1,b1=a1 and Γb1,a1=b1. Now, from (64) and from the proof of Theorem 10, for ζ≫ϑ, we have that
(85)∫01/Mca,a1,ζ−1φsds=∫01/McΓa,b,Γa1,b1,ζ−1φsds≤α1+2α2∫01/Mca,a1,ζ−1φsds=α1+2α2∫01/McΓa,b,Γa1,b1,ζ−1φsds≤α1+2α22∫01/Mca,a1,ζ−1φsds≤⋯≤α1+2α2j∫01/Mca,a1,ζ−1φsds⟶0,asj⟶∞.Hence, we get thatMca,a1,ζ=1 for ζ≫ϑ; this implies a=a1.Similarly, again from (64)and from the proof of Theorem 10, for ζ≫ϑ, we have that
(86)∫01/Mcb,b1,ζ−1φsds=∫01/McΓb,a,Γb1,a1,t∗−1φsds≤α1+2α2∫01/Mcb,b1,ζ−1φsds=α1+2α2∫01/McΓb,a,Γb1,a1,ζ−1φsds≤α1+2α22∫01/Mcb,b1,ζ−1φsds≤⋯≤α1+2α2j∫01/Mcb,b1,ζ−1φsds⟶0,asj⟶∞.Hence, we get thatMcb,b1,ζ=1 for ζ≫ϑ; this implies b=b1.


## 5. Conclusion

We presented the concept of coupled FP-results in FCM-spaces and prove some unique coupled FP-theorems under the modified contractive type conditions by using “the triangular property of fuzzy cone metric.” We presented examples in support of our result. Further, we presented an application of Lebesgue integral mapping to uplift our main work. With the help of this new concept, one can prove more modified and general contractive type coupled FP-results with different types of integral contractive type of conditions and applications in complete FCM-spaces.



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