# Some Rational Coupled Fuzzy Cone Contraction Theorems in Fuzzy Cone Metric Spaces with an Application

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

---

## Abstract

In this paper, we establish the new concept of rational coupled fuzzy cone contraction mapping in fuzzy cone metric spaces and prove some unique rational-type coupled fixed-point theorems in the framework of fuzzy cone metric spaces by using “the triangular property of fuzzy cone metric.” To ensure the existence of our results, we present some illustrative unique coupled fixed-point examples. Furthermore, we present an application of a Lebesgue integral-type contraction mapping in fuzzy cone metric spaces and to prove a unique coupled fixed-point theorem.


---

## Body

## 1. Introduction

In 1965, the theory of fuzzy sets was introduced by Zadeh [1]. Kramosil and Michalek [2] introduced the notion of FMS by using continuous t-norm with fuzzy sets. Afterward, Grabiec [3] established the completeness property of the FMS and proved a “Fuzzy Banach Contraction Principle for a unique fixed point (FP) in complete FMS.” Since then, many contributed to this theory concerning FP results (e.g., see [4–6]). Later on, in 1994, George and Veeramani [7] modified the concept of FMS introduced by Kramosil and Michalek [2], and they presented the topological properties and proved Baire’s theorem on complete FMS. In 2002, some contractive-type FP theorems were proved by Gregory and Sapena [8] on complete FMS by using the concept of [2, 7]. Some related FP concepts in FMSs can be found in [9–12]. Recently, the rational-type fuzzy contraction concept in FMS is given by Rehman et al. [13], and they proved some FP results with an application.Jaggi [14] proved the rational-type FP result for a contractive condition. However, Harjani et al. [15] modified the concept of Jaggi [14] and proved a generalized result in “partially ordered metric space.” In 2011, Luong and Thuan [16] proved generalized rational weak contraction results in “partially ordered metric space,” which is a generalization of the result of [14]. In [17], Guo and Lakshmikantham presented the concept of coupled FP with applications by using the nonlinear operator. Later on, Bhaskar [18] and Lakshmikantham [19] proved coupled FP results in “partially ordered metric space.” In [20], Sedghi et al. used commuting mappings and established some common coupled FP theorems in FMSs.In 2007, the notion of cone metric space (CMS) was introduced by Huang and Zhang [21]. They proved some basic convergence properties and FP theorems on CMS. In 2008, Abbas et al. [22] proved some common FP theorems without continuity for noncommuting mappings on CMS. After that, many others contributed their ideas to the problem of FP results in CMS. Some of their FP contributions can be found in [23–25].Oner et al. [26] introduced the concept of fuzzy cone metric space (FCMS) and proved a “fuzzy cone Banach contraction theorem” for FP in complete FCMSs in which they assumed that the “fuzzy cone contractive fc−contractive sequences are Cauchy.” In [27], Rehman and Li proved some FP theorems in FCMSs without the assumption of “fc−contractive sequences are Cauchy” by using the “triangular property of FCM.” Some more FP findings in the said space can be found in [28–31]. Recently, Chen et al. [32] and Rehman and Aydi [33] established some coupled FP and common FP results, respectively, in FCMs with integral types of applications. Waheed et al. [34] proved some coupled FP theorems in FCMSs depending on another function with an application to Volterra integral equations.In this paper, we prove some rational-type unique coupled FP theorems in FCMSs under the rational typefc−contractive conditions with supportive examples. In addition, to verify the validity of our work, we present an application of a Lebesgue integral-type contraction condition theorem to support our work. The layout of this paper is as follows: Section 2 consists of some basic preliminary concepts. In Section 3, we define the rational coupled fc−contractive mapping in FCMS and prove some unique rational coupled FP results in complete FCMSs with suitable examples. Section 4 deals with the application of Lebesgue integral-type contraction mapping to get the existence result of unique coupled FP theorems in complete FCMSs.


## 2. Preliminaries

In this section, we recall some basic definitions and lemmas related to our main results. Throughout the complete paper,ℕ represents a set of natural numbers and ⊺-norm represents a continuous t-norm as defined in [35].Definition 1.
An operation∗:0,12⟶0,1 would be a ⊺-norm if ∗ fulfils the following conditions:(1)
∗ is associative, commutative, and continuous(2)
1∗κ1=κ1, ∀κ1∈0,1(3)
κ1∗κ2≤κ3∗κ4 whenever κ1≤κ3 and κ2≤κ4,forκ1,κ2,κ3,κ4∈0,1Definition 2.
LetE be a real Banach space, 0∈E. Then, a subset C⊂E is called a cone:(1)
IfC≠∅, closed, and C≠0(2)
Ifκ1,κ2≥0 and g,z∈C, then κ1g+κ2z∈C(3)
If−z,z∈C, then z=0.
A partial ordering is defined on a given coneC⊂E by g≤z⇔z−g∈C. g≺z stands for g≤z and g≠z, while g≪z stands for z−g∈intC. In this paper, all cones have a nonempty interior.Definition 3.
A 3-tupleG,Mr,∗ is said to be a FMS if G is any set, ∗ is a ⊺-norm, and Mr is a fuzzy set on G×G×0,∞ which satisfies the following:(1)
Mrg1,g2,⊺>0(2)
Mrg1,g2,⊺=0⇔g1=g2(3)
Mrg1,g2,⊺=Mrg2,g1,⊺(4)
Mrg1,g3,⊺∗Mrg3,g2,s≤Mrg1,g2,⊺+s(5)
Mrg1,g2,.:0,∞⟶0,1 is continuous; ∀g1,g2,g3∈G and s,⊺>0Definition 4.
A 3-tupleG,Mr,∗ is said to be a FCMS if C is a cone of E, G is an arbitrary set, ∗ is a ⊺-norm, and Mr is a fuzzy set on G×G×intC which satisfies the following:(1)
Mrg1,g2,⊺>0(2)
Mrg1,g2,⊺=0⇔g1=g2(3)
Mrg1,g2,⊺=Mrg2,g1,⊺(4)
Mrg1,g3,⊺∗Mrg3,g2,s≤Mrg1,g2,⊺+s(5)
Mrg1,g2,.:intC⟶0,1 is continuous; ∀g1,g2,g3∈G, and s,⊺≫0Definition 5.
Let a 3-tupleG,Mr,∗ be a FCMS and ϱ∈G and a sequence gJ in G(1)
Converges toϱ if γ∈0,1 and ⊺≫0 and there is J1∈ℕ∋MrgJ,ϱ,⊺>1−γ, for J≥J1. We may write this limJ⟶∞gJ=ϱ or gJ⟶ϱ as J⟶∞.(2)
Is a Cauchy sequence ifγ∈0,1 and ⊺≫0 and there is J1∈ℕ∋MrgJ,gℓ,⊺>1−γ, for J,ℓ≥J1.(3)
G,Mr,∗ is complete if every Cauchy sequence is convergent in G.(4)
Isfc−contractive if ∃η∈0,1, satisfying 1/MrgJ,gJ+1,⊺−1≤η1/MrgJ−1,gJ,⊺−1,for⊺≫0,J≥1.Lemma 1.
LetG,Mr,∗ be a FCMS, and let a sequence gJ in G converge to a point ϱ∈G if MrgJ,ϱ,⊺⟶1 as J⟶∞, for ⊺≫0.Definition 6.
LetG,Mr,∗ be a FCMS. The FCM Mr is triangular, if 1/Mrg,h,⊺−1≤1/Mrg,z,⊺−1+1/Mrz,h,⊺−1,∀g,h,z∈G,⊺≫0.Γ.Definition 7 (see [26]).
LetG,Mr,∗ be a FCMS and Γ:G⟶G. Then, Γ is known as a fc−contractive if ∃η∈0,1 such that(1)1MrΓg,Γh,⊺−1≤η1Mrg,h,⊺−1,∀g,h∈G,⊺≫0.Definition 8.
An elementg,h∈G×G is known as a coupled FP of a function F¨:G×G⟶G if(2)Γg,h=g,andΓh,g=h.
Furthermore, we shall study some unique coupled FP results in FCMSs under the rational coupledfc−contraction conditions with examples. Also, we present an application of the Lebesgue integral-type rational coupled fc−contraction mapping to get a unique rational coupled FP result in FCMSs.


## 3. Main Results

In this section, we shall present our main results with illustrative examples.Definition 9.
LetG,Mr,∗ be a FCMS. A mapping Γ:G×G⟶G is called a rational coupled fc−contraction if ∃η1∈0,1 and η2≥0 such tha(3)1MrΓg,h,Γξ,κ,⊺−1≤η11Mrg,ξ,⊺−1+η2Mrg,ξ,⊺Mrg,Γg,h,⊺∗Mrξ,Γg,h,2⊺−1,∀g,h,ξ,κ∈Gand⊺≫0.Theorem 1.
Assume thatG,Mr,∗ be a complete FCMS in which Mr is triangular and a mapping Γ:G×G⟶G is a rational coupled fc−contraction satisfying (3). Then, Γ has a unique coupled FP in G.Proof.
Let anyg0,h0∈G; we define sequences gJ and hJ in G such that(4)ΓgJ,hJ=gJ+1,andΓhJ,gJ=hJ+1,forJ≥0.
Now, from (3) and (4), for ⊺≫0,(5)1MrgJ,gJ+1,⊺−1=1MrΓgJ−1,hJ−1,ΓgJ,hJ,t−1,≤η11MrgJ−1,gJ,⊺−1+η2MrgJ−1,gJ,⊺MrgJ−1,ΓgJ−1,hJ−1,⊺∗MrgJ,ΓgJ−1,hJ−1,2⊺−1=η11MrgJ−1,gJ,⊺−1+η2MrgJ−1,gJ,⊺MrgJ−1,gJ,⊺∗MrgJ,gJ,2⊺−1.
This implies that(6)1MrgJ,gJ+1,⊺−1≤η11MrgJ−1,gJ,⊺−1,for⊺≫0.
Similarly,(7)1MrgJ−1,gJ,⊺−1≤η11MrgJ−2,gJ−1,⊺−1,for⊺≫0.
Now, from (6) and (7) and by induction, for ⊺≫0, we have that(8)1MrgJ,gJ+1,⊺−1≤η11MrgJ−1,gJ,⊺−1,≤η121MrgJ−2,gJ−1,⊺−1≤⋯≤η1J1Mrg0,g1,⊺−1⟶0,asJ⟶∞.
This showsgJ is a fc−contractive sequence; therefore,(9)limJ⟶∞MrgJ,gJ+1,⊺=1,for⊺≫0.
Now, forℓ>J and for ⊺≫0,(10)1MrgJ,gℓ,⊺−1≤1MrgJ,gJ+1,⊺−1+1MrgJ+1,gJ+2,⊺−1+⋯+1Mrgℓ−1,gℓ,⊺−1≤η1J1Mrgo,g1,⊺−1+η1J+11Mrg0,g1,⊺−1+⋯+η1ℓ−11Mrg0,g1,⊺−1=η1J+η1J+1+⋯+η1ℓ−11Mrg0,g1,⊺−1=η1J1−η11Mrgo,g1,⊺−1⟶0,asJ⟶∞.
Hence,gJ is a Cauchy sequence. Since, by the completeness of G,Mr,∗, ∃g∈G, so that(11)limJ⟶∞MrgJ,g,⊺=1,for⊺≫0.
Now, for sequencehJ and from (3) and (4), for ⊺≫0,(12)1MrhJ,hJ+1,⊺−1=1MrΓhJ−1,gJ−1,ΓhJ,gJ,⊺−1,≤η11MrhJ−1,hJ,⊺−1+η2MrhJ−1,hJ,⊺MrhJ−1,ΓhJ−1,gJ−1,⊺∗MrhJ,ΓhJ−1,gJ−1,2⊺−1=η11MrhJ−1,hJ,⊺−1+η2MrhJ−1,hJ,⊺MrhJ−1,hJ,⊺∗MrhJ,hJ,2⊺−1.
This implies that(13)1MrhJ,hJ+1,⊺−1≤η11MrhJ−1,hJ,⊺−1,for⊺≫0.
Similarly,(14)1MrhJ−1,hJ,⊺−1≤η11MrhJ−2,hJ−1,⊺−1,for⊺≫0.
Now, from (13) and (14) and by induction, for ⊺≫0, we have that(15)1MrhJ,hJ+1,⊺−1≤η11MrhJ−1,hJ,⊺−1,≤η121MrhJ−2,hJ−1,⊺−1≤⋯≤η1J1Mrh0,h1,⊺−1⟶0,asJ⟶∞.
This showshJ is a fc−contractive sequence; therefore,(16)limJ⟶∞MrhJ,hJ+1,⊺=1,for⊺≫0.
Now, forℓ>J and for ⊺≫0, we have(17)1MrhJ,hℓ,⊺−1≤1MrhJ,hJ+1,⊺−1+1MrhJ+1,hJ+2,⊺−1+⋯+1Mrhℓ−1,hℓ,⊺−1,≤η1J1Mrh0,h1,⊺−1+η1J+11Mrh0,h1,⊺−1+⋯+η1ℓ−11Mrh0,h1,⊺−1=η1J+η1J+1+⋯+η1ℓ−11Mrh0,h1,⊺−1=η1J1−η11Mrh0,h1,⊺−1⟶0,asJ⟶∞.
Hence,hJ is a Cauchy sequence. Since, by the completeness of G,Mr,∗, ∃g∈G so that(18)limJ⟶∞MrhJ,h,⊺=1,for⊺≫0.
Now, we shall prove thatΓg,h=g. Since Mr is triangular and by the view of (3), (9), and (11), for ⊺≫0,(19)1Mrg,Γg,h,⊺−1≤1Mrg,ΓgJ,hJ,⊺−1+1MrΓgJ,hJ,Γg,h,⊺−1,≤1Mrg,ΓgJ,hJ,⊺−1+η11Mrgℓ,g,⊺−1+η2MrgJ,g,⊺MrgJ,ΓgJ,hJ,⊺∗Mrg,ΓgJ,hJ,2⊺−1=1Mrg,gJ+1,⊺−1+η11MrgJ,g,⊺−1+η2MrgJ,g,⊺MrgJ,gJ+1,⊺∗Mrg,gJ+1,2⊺−1⟶0,asJ⟶∞.
Hence,Mrg,Γg,h,⊺=1⇒g=Γg,h for ⊺≫0. Next, we have to prove that h=Γh,g; therefore, by the triangular property of Mr and by the view of (3), (16), and (18), for ⊺≫0,(20)1Mrh,Γh,g,⊺−1≤1Mrh,ΓhJ,gJ,⊺−1+1MrΓhJ,gJ,Γh,g,⊺−1≤1Mrh,ΓhJ,gJ,⊺−1+η11MrhJ,h,⊺−1+η2MrhJ,h,⊺MrhJ,ΓhJ,gJ,⊺∗Mrh,ΓhJ,gJ,2⊺−1=1Mrh,hJ+1,⊺−1+η11MrhJ,h,⊺−1+η2MrhJ,h,⊺MrhJ,hJ+1,⊺∗Mrh,hJ+1,2⊺−1⟶0,asJ⟶∞.
Hence,Mrg,Γh,g,⊺=1⇒Γh,g=h, for ⊺≫0.
Uniqueness: supposeg1,h1 and h1,g1 are other coupled fixed-point pairs in G×G such that Γg1,h1=g1 and Γh1,g1=h1. Now, from (3) and by using Definition 4 (4), for ⊺≫0,(21)1Mrg,g1,⊺−1=1MrΓg,h,Γg1,h1,⊺−1≤η11Mrg,g1,⊺−1+η2Mrg,g1,⊺Mrg,Γg,h,⊺∗Mrg1,Γg,h,2⊺−1=η11Mrg,g1,⊺−1+η2Mrg,g1,⊺Mrg,g,⊺∗Mrg1,g,2⊺−1=η11Mrg,g1,⊺−1=η11MrΓg,h,Γg1,h1,⊺−1≤η121Mrg,g1,⊺−1≤⋯≤η1J1Mrg,g1,⊺−1⟶0,asJ⟶∞.
Hence, we get thatMrg,g1,⊺=1⇒g=g1 for ⊺≫0. Similarly, again from (3) and by using Definition 4 (4), for ⊺≫0,(22)1Mrh,h1,⊺−1=1MrΓh,g,Γh1,g1,⊺−1,≤η11Mrh,h1,⊺−1+η2Mrh,h1,⊺Mrh,Γh,g,⊺∗Mrh1,Γh,g,2⊺−1=η11Mrh,h1,⊺−1+η2Mrh,h1,⊺Mrh,h1,⊺∗Mrh1,h,2⊺−1=η11Mrh,h1,⊺−1=η11MrΓh,g,Γh1,g1,⊺−1≤η121Mrh,h1,⊺−1≤⋯≤η1J1Mrh,h1,⊺−1⟶0,asJ⟶∞.
Hence, we get thatMrh,h1,⊺=1⇒h=h1 for ⊺≫0.Corollary 1.
LetΓ:G×G⟶G be a mapping on a complete FCMS G,Mr,∗ in which Mr is triangular and Γ satisfies the inequality(23)1MrΓg,h,Γξ,κ,⊺−1≤η11Mrg,ξ,⊺−1,∀g,h,ξ,κ∈G,⊺≫0, and η1∈0,1. Then, Γ has a unique coupled FP in G.Example 1.
LetG=0,∞; ∗ is a ⊺-norm, and Mr:G×G×0,∞⟶0,1 is defined as(24)Mrg,h,⊺=⊺⊺+g−h,∀g,h∈Gand⊺>0.
Then, one can easily prove the triangular property of FCM from the above example andG,Mr,∗ is a complete FCMS. We define a mapping Γ:G×G⟶G by(25)Γg,h=3g8,g,h∈0,1,g+5h4−76,g,h∈1,∞.
Then, we have(26)1MrΓg,h,Γξ,κ,⊺−1=1Mr3g/8,3ξ/8,⊺−1,=38⊺g−ξ=381Mrg,ξ,⊺−1.∀g,h,ξ,κ∈Gand⊺≫0. Hence, we proved that Γ:G×G⟶G is a rational coupled fc−contractive. Now, by using Definition 4 (4), for ⊺≫0,(27)Mrg,ξ,⊺Mrg,Γg,h,⊺∗Mrξ,Γg,h,2⊺−1≤Mrg,ξ,⊺Mrg,Γg,h,⊺∗Mrξ,g,⊺∗Mrg,Γg,h,⊺−1,=1Mrg,Γg,h,⊺∗Mrg,Γg,h,⊺−1=1Mrg,Γg,h,⊺2−1=5g64⊺25g+16⊺.∀g,h,ξ,κ∈G. Hence, from the above, we conclude that all the conditions of Theorem 1 are satisfied with η1=3/8, η2∈0,8/13, and Γg,h=Γ7/3,7/3=7/3∈0,∞.Theorem 2.
LetΓ:G×G⟶G be a mapping on a complete FCMS G,Mr,∗ in which Mr is triangular and Γ satisfies the inequality(28)1MrΓg,h,Γξ,κ,⊺−1≤η11Mrg,ξ,⊺−1+η2Mrg,ξ,⊺∗Mrξ,Γξ,κ,⊺Mrg,Γg,h,⊺∗Mrg,Γξ,κ,2⊺−1,+η3Mrg,Γg,h,⊺Mrg,Γξ,κ,2⊺−1+Mrξ,Γξ,κ,⊺Mrg,Γξ,κ,2⊺−1+η41Mrg,Γg,h,⊺−1+1Mrξ,Γξ,κ,⊺−1,∀g,h,ξ,κ∈G, ⊺≫0, η1∈0,1, and η2,η3,η4≥0 with η1+η2+2η3+2η4<1. Then, Γ has a unique coupled FP in G.Proof.
Let anyg0,h0∈G; we define sequences gJ and hJ in G such that(29)ΓgJ,hJ=gJ+1,andΓhJ,gJ=hJ+1,forJ≥0.
Now, from (28) and (29), for ⊺≫0,(30)1MrgJ,gJ+1,⊺−1=1MrΓgJ−1,hJ−1,ΓgJ,hJ,⊺−1,≤η11MrgJ−1,gJ,⊺−1+η2MrgJ−1,gJ,⊺∗MrgJ,ΓgJ,hJ,⊺MrgJ−1,ΓgJ−1,hJ−1,⊺∗MrgJ−1,ΓgJ,hJ,2⊺−1+η3MrgJ−1,ΓgJ−1,hJ−1,⊺MrgJ−1,ΓgJ,hJ,2⊺−1+MrgJ,ΓgJ,hJ,⊺MrgJ−1,ΓgJ,hJ,2⊺−1+η41MrgJ−1,ΓgJ−1,hJ−1,⊺−1+1MrgJ,ΓgJ,hJ,⊺−1=η11MrgJ−1,gJ,⊺−1+η2MrgJ−1,gJ,⊺∗MrgJ,gJ+1,⊺MrgJ−1,gJ,⊺∗MrgJ−1,gJ+1,2⊺−1+η3MrgJ−1,gJ,⊺MrgJ−1,gJ+1,2⊺−1+MrgJ,gJ+1,⊺MrgJ−1,gJ+1,2⊺−1+η41MrgJ−1,gJ,⊺−1+1MrgJ,gJ+1,⊺−1.
Now, by Definition4 (4), for ⊺≫0,(31)1MrgJ,gJ+1,⊺−1≤η11MrgJ−1,gJ,⊺−1+η2MrgJ−1,gJ,⊺∗MrgJ,gJ+1,⊺MrgJ−1,gJ,⊺∗MrgJ−1,gJ,⊺∗MrgJ,gJ+1,⊺−1+η3MrgJ−1,gJ,⊺MrgJ−1,gJ,⊺∗MrgJ,gJ+1,⊺−1+MrgJ,gJ+1,⊺MrgJ−1,gJ,⊺∗MrgJ,gJ+1,⊺−1+η41MrgJ−1,gJ,⊺−1+1MrgJ,gJ+1,⊺−1.
After simplification, we get that(32)1MrgJ,gJ+1,⊺−1≤ρ1MrgJ−1,gJ,⊺−1,for⊺≫0.where ρ=η1+η2+η3+η4/1−η3−η4<1. Similarly, again, by using (28) and Definition 4 (4), we get that(33)1MrgJ−1,gJ,⊺−1≤ρ1MrgJ−2,gJ−1,⊺−1,for⊺≫0.where ρ=η1+η2+η3+η4/1−η3−η4<1. Now, from (32) and (33) and by induction, for ⊺≫0, we have that(34)1MrgJ,gJ+1,⊺−1≤ρ1MrgJ−1,gJ,⊺−1,≤ρ21MrgJ−2,gJ−1,⊺−1≤⋯≤ρJ1Mrg0,g1,⊺−1⟶0,asJ⟶∞.
This showsgJ is a fc−contractive sequence; therefore,(35)limℓ⟶∞MrgJ,gJ+1,⊺=1,for⊺≫0.
Now, forℓ>J and for ⊺≫0, we have(36)1MrgJ,gℓ,⊺−1≤1MrgJ,gJ+1,⊺−1+1MrgJ+1,gJ+2,⊺−1+⋯+1Mrgℓ−1,gℓ,⊺−1,≤ρJ1Mrgo,g1,⊺−1+ρJ+11Mrg0,g1,⊺−1+⋯+ρℓ−11Mrg0,g1,⊺−1=ρJ+ρJ+1+⋯+ρℓ−11Mrg0,g1,⊺−1=ρJ1−ρ1Mrgo,g1,⊺−1⟶0,asJ⟶∞.
Hence,gJ is a Cauchy sequence. Since, by the completeness of G,Mr,∗, ∃g∈G so that(37)limJ⟶∞MgJ,g,⊺=1,for⊺≫0.
Now, for sequencehJ, from (28) and (29), for ⊺≫0,(38)1MrhJ,hJ+1,⊺−1=1MrΓhJ−1,gJ−1,ΓhJ,gJ,⊺−1≤η11MrhJ−1,hJ,⊺−1+η2MrhJ−1,hJ,⊺∗MrhJ,ΓhJ,gJ,⊺MrhJ−1,ΓhJ−1,gJ−1,⊺∗MrhJ−1,ΓhJ,gJ,2⊺−1+η3MrhJ−1,ΓhJ−1,gJ−1,⊺MrhJ−1,ΓhJ,gJ,2⊺−1+MrhJ,ΓhJ,gJ,⊺MrhJ−1,ΓhJ,gJ,2⊺−1+η41MrhJ−1,ΓhJ−1,gJ−1,⊺−1+1MrhJ,ΓhJ,gJ,⊺−1=η11MrhJ−1,hJ,⊺−1+η2MrhJ−1,hJ,⊺∗MrhJ,hJ+1,⊺MrhJ−1,hJ,⊺∗MrhJ−1,hJ+1,2⊺−1+η3MrhJ−1,hJ,⊺MrhJ−1,hJ+1,2⊺−1+MrhJ,hJ+1,⊺MrhJ−1,hJ+1,2⊺−1+η41MrhJ−1,hJ,⊺−1+1MrhJ,hJ+1,⊺−1.
Now, again by Definition4 (4), for ⊺≫0,(39)1MrhJ,hJ+1,⊺−1≤η11MrhJ−1,hJ,⊺−1+η2MrhJ−1,hJ,⊺∗MrhJ,hJ+1,⊺MrhJ−1,hJ,⊺∗MrhJ−1,hJ,⊺∗MrhJ,hJ+1,⊺−1+η3MrhJ−1,hJ,⊺MrhJ−1,hJ,⊺∗MrhJ,hJ+1,⊺−1+MrhJ,hJ+1,⊺MrhJ−1,hJ,⊺∗MrhJ,hJ+1,⊺−1+η41MrhJ−1,hJ,⊺−1+1MrhJ,hJ+1,⊺−1.
After simplification, we get that(40)1MrhJ,hJ+1,⊺−1≤ρ1MrhJ−1,hJ,⊺−1,for⊺≫0,where the value of ρ is same as in (32). Similarly, again by using (28) and Definition 4 (4), for ⊺≫0, we get that(41)1MrhJ−1,hJ,⊺−1≤ρ1MrhJ−2,hJ−1,⊺−1,for⊺≫0.
Now, from (40) and (41) and by induction, for ⊺≫0, we have that(42)1MrhJ,hJ+1,⊺−1≤ρ1MrhJ−1,hJ,⊺−1,≤ρ21MrhJ−2,hJ−1,⊺−1≤⋯≤ρJ1Mrh0,h1,⊺−1⟶0,asJ⟶∞.
This showshJ is a fc−contractive sequence; therefore,(43)limJ⟶∞MrhJ,hJ+1,⊺=1,for⊺≫0.
Now, forℓ>J and for ⊺≫0,(44)1MrhJ,hJ,⊺−1≤1MrhJ,hJ+1,⊺−1+1MrhJ+1,hJ+2,⊺−1+⋯+1Mrhℓ−1,hℓ,⊺−1,≤ρJ1Mrho,h1,⊺−1+ρJ+11Mrh0,h1,⊺−1+⋯+ρℓ−11Mrh0,h1,⊺−1=ρJ+ρJ+1+⋯+ρℓ−11Mrh0,h1,⊺−1=ρJ1−ρ1Mrho,h1,⊺−1⟶0,asJ⟶∞.
Hence,hJ is a Cauchy sequence. Since, by the completeness of G,Mr,∗, ∃h∈G so that(45)limJ⟶∞MrhJ,h,⊺=1,for⊺≫0.
Now, we shall prove thatΓg,h=g. Since Mr is triangular,(46)1Mrg,Γg,h,⊺−1=1Mrg,gJ+1,⊺−1+1MrgJ+1,Γg,h,t−1,for⊺≫0.
Now, by the view of (28), (35), and (37), and by using Definition 4 (4), for ⊺≫0,(47)1MrgJ+1,Γg,h,⊺−1=1MrΓgJ,hJ,Γg,h,⊺−1,≤η11MrgJ,g,⊺−1+η2MrgJ,g,⊺∗Mrg,Γg,h,⊺MrgJ,ΓgJ,hJ,⊺∗MrgJ,Γg,h,2⊺−1+η3MrgJ,ΓgJ,hJ,⊺MrgJ,Γg,h,2⊺−1+Mrg,Γg,h,⊺MrgJ,Γg,h,2⊺−1+η41MrgJ,ΓgJ,hJ,⊺−1+1Mrg,Γg,h,⊺−1≤η11MrgJ,g,⊺−1+η2MrgJ,g,⊺∗Mrg,Γg,h,⊺MrgJ,gJ+1,⊺∗MrgJ,g,⊺∗Mrg,Γg,h,⊺−1+η3MrgJ,gJ+1,⊺MrgJ,g,⊺∗Mrg,Γg,h,⊺−1+Mrg,Γg,h,⊺MrgJ,g,⊺∗Mrg,Γg,h,⊺−1+η41MrgJ,gJ+1,⊺−1+1Mrg,Γg,h,⊺−1⟶η3+η41Mrg,Γg,h,⊺−1,asJ⟶∞.
Hence,(48)limsupJ⟶∞1MrgJ+1,Γg,h,⊺−1≤η3+η41Mrg,Γg,h,⊺−1,for⊺≫0.
Now, from (35), (46), and (48), for ⊺≫0,(49)1Mrg,Γg,h,⊺−1≤η3+η41Mrg,Γg,h,⊺−1,⇒1−η3+η41Mrg,Γg,h,⊺−1≤0,for⊺≫0,which is a contradiction. As 1−η3+η4≠0, we get that Mrg,Γg,h,⊺=1⇒Γg,h=g for ⊺≫0. Next, we prove that Γh,g=h. Now, again from the triangularity of M,(50)1Mrh,Γh,g,⊺−1≤1Mrh,hJ+1,⊺−1+1MrhJ+1,Γh,g,⊺−1,for⊺≫0.
Now, by the view of (28), (43), and (45), and by using Definition 4 (4), for ⊺≫0,(51)1MrhJ+1,Γh,g,⊺−1=1MrΓhJ,gJ,Γh,g,⊺−1,≤η11MrhJ,h,⊺−1+η2MrhJ,h,⊺∗Mrh,Γh,g,⊺MrhJ,ΓhJ,gJ,⊺∗MrhJ,Γh,g,2⊺−1+η3MrhJ,ΓhJ,gJ,⊺MrhJ,Γh,g,2⊺−1+Mrh,Γh,g,⊺MrhJ,Γh,g,2⊺−1+η41MrhJ,ΓhJ,gJ,⊺−1+1Mrh,Γh,g,⊺−1≤η11MrhJ,h,⊺−1+η2MrhJ,h,⊺∗Mrh,Γh,g,⊺MrhJ,hJ+1,⊺∗MrhJ,h,⊺∗Mrh,Γh,g,⊺−1+η3MrhJ,hJ+1,⊺MrhJ,h,⊺∗Mrh,Γh,g,⊺−1+Mrh,Γh,g,⊺MrhJ,h,⊺∗Mrh,Γh,g,⊺−1+η41MrhJ,hJ+1,⊺−1+1Mrh,Γh,g,⊺−1⟶η3+η41Mrh,Γh,g,⊺−1,asJ⟶∞.
Hence,(52)limsupJ⟶∞1MrhJ+1,Γh,g,⊺−1≤η3+η41Mrh,Γh,g,⊺−1,for⊺≫0.
Now, from (43), (50), and (52), for ⊺≫0,(53)1Mrh,Γh,g,⊺−1≤η3+η41Mrh,Γh,g,⊺−1,⇒1−η3+η41Mrh,Γh,g,⊺−1≤0,for⊺≫0,which is a contradiction. As 1−η3+η4≠0, we get that Mrh,Γh,g,⊺=1⇒Γh,g=h for ⊺≫0.
Uniqueness: supposeg1,h1 and h1,g1 are other coupled FP pairs in G×G such that Γg1,h1=g1 and Γh1,g1=h1. Now, from (28) and by using Definition 4 (4), for ⊺≫0,(54)1Mrg,g1,⊺−1=1MrΓg,h,Γg1,h1,⊺−1,≤η11Mrg,g1,⊺−1+η2Mrg,g1,⊺∗Mrg1,Γg1,h1,⊺Mrg,Γg,h,⊺∗Mrg,Γg1,h1,2⊺−1+η3Mrg,Γg,h,⊺Mrg,Γg1,h1,2⊺−1+Mrg1,Γg1,h1,⊺Mrg,Γg1,h1,2⊺−1+η41Mrg,Γg,h,⊺−1+1Mrg1,Γg1,h1,⊺−1=η11Mrg,g1,⊺−1+η2Mrg,g1,⊺∗Mrg1,g1,⊺Mrg,g,⊺∗Mrg,g1,2⊺−1+η3Mrg,g,⊺Mrg,g1,2⊺−1+Mrg1,g1,⊺Mrg,g1,2⊺−1+η41Mrg,g,⊺−1+1Mrg1,g1,⊺−1=η11Mrg,g1,⊺−1+η2Mrg,g1,⊺Mrg,g1,2⊺−1+η31Mrg,g1,2⊺−1+1Mrg,g1,2⊺−1≤η11Mrg,g1,⊺−1+η2Mrg,g1,⊺Mrg,g1,⊺∗Mrg1,g1,⊺−1+η31Mrg,g1,⊺∗Mrg1,g1,⊺−1+1Mrg,g1,⊺∗Mrg1,g1,⊺−1=η1+2η31Mrg,g1,⊺−1=η1+2η31MrΓg,h,Γg1,h1,⊺−1≤η1+2η321Mrg,g1,⊺−1≤⋯≤η1+2η3J1Mrg,g1,⊺−1⟶0,asJ⟶∞.
Hence,Mrg,g1,⊺=1⇒g=g1 for ⊺≫0. Next, we shall show that h=h1, again from (28), and by using Definition 4 (4), for ⊺≫0, we have(55)1Mrh,h1,⊺−1=1MrΓh,g,Γh1,g1,⊺−1≤η11Mrh,h1,⊺−1+η2Mrh,h1,⊺∗Mrh1,Γh1,g1,⊺Mrh,Γh,g,⊺∗Mrh,Γh1,g1,2⊺−1+η3Mrh,Γh,g,⊺Mrh,Γh1,g1,2⊺−1+Mrh1,Γh1,g1,⊺Mrh,Γh1,g1,2⊺−1+η41Mrh,Γh,g,⊺−1+1Mrh1,Γh1,g1,⊺−1=η11Mrh,h1,⊺−1+η2Mrh,h1,⊺∗Mrh1,h1,⊺Mrh,h1,⊺∗Mrh,h1,2⊺−1+η3Mrh,h,⊺Mrh,h1,2⊺−1+Mrh1,h1,⊺Mrh,h1,2⊺−1+η41Mrh,h,⊺−1+1Mrh1,h1,⊺−1=η11Mrh,h1,⊺−1+η2Mrh,h1,⊺Mrh,h1,2⊺−1+η31Mrh,h1,2⊺−1+1Mrh,h1,2⊺−1≤η11Mrh,h1,⊺−1+η2Mrh,h1,⊺Mrh,h1,⊺∗Mrh1,h1,⊺−1+η31Mrh,h1,⊺∗Mrh1,h1,⊺−1+1Mrh,h1,⊺∗Mrh1,h1,⊺−1=η1+2η31Mrh,h1,⊺−1=η1+2η31MrΓh,g,Γh1,g1,⊺−1≤η1+2η321Mrh,h1,⊺−1≤⋯≤η1+2η3J1Mrh,h1,⊺−1⟶0,asJ⟶∞.
Hence,Mrh,h1,⊺=1⇒h=h1 for ⊺≫0.Corollary 2.
LetΓ:G×G⟶G be a mapping on a complete FCMS G,Mr,∗ in which Mr is triangular and Γ satisfies the inequality(56)1MrΓg,h,Γξ,κ,⊺−1≤η11Mrg,ξ,⊺−1+η2Mrg,ξ,⊺∗Mrξ,Γξ,κ,⊺Mrg,Γg,h,⊺∗Mrg,Γξ,κ,2⊺−1+η41Mrg,Γg,h,⊺−1+1Mrξ,Γξ,κ,⊺−1,∀g,h,ξ,κ∈G,⊺≫0, η1∈0,1, and η2,η4≥0 with η1+η2+2η4<1. Then, Γ has a unique coupled FP in G.Corollary 3.
LetΓ:G×G⟶G be a mapping on a complete FCMS G,Mr,∗ in which Mr is triangular and Γ satisfies the inequality(57)1MrΓg,h,Γξ,κ,⊺−1≤η11Mrg,ξ,⊺−1+η3Mrg,Γg,h,⊺Mrg,Γξ,κ,2⊺−1+Mrξ,Γξ,κ,⊺Mrg,Γξ,κ,2⊺−1+η41Mrg,Γg,h,⊺−1+1Mrξ,Γξ,κ,⊺−1,∀g,h,ξ,κ∈G,⊺≫0, η1∈0,1, and η3,η4≥0 with η1+2η3+2η4<1. Then, Γ has a unique coupled FP in G.Corollary 4.
LetΓ:G×G⟶G be a mapping on a complete FCMS G,Mr,∗ in which Mr is triangular and Γ satisfies the inequality(58)1MrΓg,h,Γξ,κ,⊺−1≤η11Mrg,ξ,⊺−1+η41Mrg,Γg,h,⊺−1+1Mrξ,Γξ,κ,⊺−1,∀g,h,ξ,κ∈G,⊺≫0, η1∈0,1, and η4≥0 with η1+2η4<1. Then, Γ has a unique coupled FP in G.Example 2.
From Example1, we define a FM Mr:G×G×0,∞⟶0,1 by(59)Mrg,h,⊺=⊺⊺+g−h/3,∀g,h∈Gand⊺>0.
Then, one can easily prove the triangular property of FCM from the above example andG,Mr,∗ is a complete FCMS. We define a mapping Γ:G×G⟶G by(60)Γg,h=2g5,g,h∈0,1,2g+hh+1,g,h∈1,∞.
Then,(61)1MrΓg,h,Γξ,κ,⊺−1=251Mrg,ξ,⊺−1,∀g,h,ξ,κ∈Gand⊺≫0. Hence, we proved that a mapping Γ is a coupled fc−contractive. Now, By using Definition 4 (4) to simplify the η2 rational term of (28), for ⊺≫0,(62)Mrg,ξ,⊺∗Mrξ,Γξ,κ,⊺Mrg,Γg,h,⊺∗Mrg,Γξ,κ,2⊺−1≤Mrg,ξ,⊺∗Mrξ,Γξ,κ,⊺Mrg,Γg,h,⊺∗Mrg,ξ,⊺∗Mrξ,Γξ,κ,⊺−1=1Mrg,Γg,h,⊺=g5⊺,∀g,h,ξ,κ∈G. Again, by using Definition 4 (4) and (61) to simplify the η3 rational term of (28), for ⊺≫0, we have(63)Mrg,Γg,h,⊺Mrg,Γξ,κ,2⊺−1+Mrξ,Γξ,κ,⊺Mrg,Γξ,κ,2⊺−1≤Mrg,Γg,h,⊺Mrg,Γg,h,⊺∗MrΓg,h,Γξ,κ,⊺−1+Mrξ,Γξ,κ,⊺Mrg,ξ,⊺∗Mrξ,Γξ,κ,⊺−1=1MrΓg,h,Γξ,κ,⊺−1+1Mrg,ξ,⊺−1=751Mrg,ξ,⊺−1=715⊺g−ξ,∀g,h,ξ,κ∈G. After simple routine calculation, we can get the η4 term result of (28) as follows:(64)1Mrg,Γg,h,⊺−1+1Mrξ,Γξ,κ,⊺−1=15⊺g+ξ,for⊺≫0.
Hence, from the above, we conclude that all the conditions of the Theorem2 are satisfied with η1=2/5,η2=1/6, η3=η4=1/15, and Γg,h=Γ4,4=4∈0,∞.


## 4. Application

In this section, we present an application on Lebesgue integral-type contraction mapping to support our main work. In 2002, Branciari proved the following result on complete metric space for a unique FP (see [36]).Theorem 3.
LetG,d be a complete metric space, η1∈0,1, and Γ:G⟶G be a mapping such that for each g,h∈G,(65)∫0dΓg,Γhφτdτ≤η1∫0dg,hφτdτ,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 κ>0,(66)∫0κφsds>0.
Then,Γ has a unique FP u∈G such that for any g∈G, limJ⟶∞ΓJg=u.”
Now, we are in the position to use the above concept and to prove a unique coupled FP theorem in complete FCMSs.Theorem 4.
LetΓ:G×G⟶G be a mapping on a complete FCMS G,Mr,∗ in which Mr is triangular and satisfies(67)∫01/MrΓg,h,Γξ,κ,⊺−1φτdτ≤η1∫01/Mrg,ξ,⊺−1φτdτ+η2∫0Mrg,ξ,⊺/Mrg,Γg,h,⊺∗Mrξ,Γg,h,2⊺−1φτdτ,for all g,h,ξ,κ∈G, ⊺≫0, η1∈0,1, and η2≥0, 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 κ>0,(68)∫0κφτdτ>0.
Then,Γ has a unique coupled FP in G.Proof.
Let anygo,ho∈G; we define sequences gJ and hJ in G such that(69)ΓgJ,hJ=gJ+1,andΓhJ,gJ=hJ+1,forJ≥0.
Now, from (67) and from the proof of Theorem 1, for ⊺≫0,(70)∫01/MrgJ,gJ+1,⊺−1φτdτ=∫01/MrΓgJ−1,hJ−1,ΓgJ,hJ,⊺−1φτdτ≤η1∫01/MrgJ−1,gJ,⊺−1φτdτ.
Similarly, again by using the arguments, we have(71)∫01/MrgJ−1,gJ,⊺−1φτdτ≤η1∫01/MrgJ−2,gJ−1,⊺−1φτdτ,for⊺≫0.
Now, from (70) and (71) and by induction, for ⊺≫0, we have(72)∫01/MrgJ,gJ+1,⊺−1φτdτ≤η1∫01/MrgJ−1,gJ,⊺−1φτdτ,≤η12∫01/MrgJ−2,gJ−1,⊺−1φτdτ≤⋯≤η1J∫01/Mrg0,g1,⊺−1φτdτ⟶0,asJ⟶∞.
This shows thatgJ is a fc−contractive sequence, and therefore,(73)limJ⟶∞∫01/MrgJ,gJ+1,⊺−1φτdτ=0⇒limJ⟶∞1MrgJ,gJ+1,⊺−1=0,for⊺≫0.
Hence, we get that(74)limJ⟶∞MrgJ,gJ+1,⊺=1,for⊺≫0.
Now, forℓ>J and for ⊺≫0,(75)∫01/MrgJ,gℓ,⊺−1φτdτ≤∫01/MrgJ,gJ+1,⊺−1φτdτ+∫01/MrgJ+1,gJ+2,⊺−1φτdτ+⋯+∫01/Mrgℓ−1,gℓ,⊺−1φτdτ,≤η1J∫01/Mrg0,g1,⊺−1φτdτ+η1J+1∫01/Mrg0,g1,⊺−1φτdτ+⋯+η1ℓ−1∫01/Mrg0,g1,⊺−1φτdτ=η−1J+η1J+1+⋯+η1ℓ−1∫01/Mrg0,g1,⊺−1φτdτ=η1J1−η1∫01/Mrg0,g1,⊺−1φτdτ⟶0,asJ⟶∞.
We get that(76)limJ⟶∞∫01/MrgJ,gℓ,⊺−1φτdτ=0⇒limJ⟶∞1MrgJ,gℓ,⊺−1=0,for⊺≫0.
Hence,gJ is a Cauchy sequence. Since, by the completeness of G,Mr,∗, ∃g∈G so that(77)limJ⟶∞MrgJ,g,⊺=1,for⊺≫0.
Now, for sequencehJ from (67) and from the proof of Theorem 1, for ⊺≫0, we have(78)∫01/MrhJ,hJ+1,⊺−1φτdτ=∫01/MrΓhJ−1,gJ−1,ΓhJ,gJ,t−1φτdτ,≤η1∫01/MrhJ−1,hJ,⊺−1φτdτ.
Similarly, again by using the same arguments, we have(79)∫01/MrhJ−1,hJ,⊺−1φτdτ≤η1∫01/MrhJ−2,gJ−1,⊺−1φτdτ,for⊺≫0.
Now, from (78) and (79) and by induction, for ⊺≫0, we have(80)∫01/MrhJ,hJ+1,⊺−1φτdτ≤η1∫01/MrhJ−1,hJ,⊺−1φτdτ,≤η12∫01/MrhJ−2,gJ−1,⊺−1φτdτ≤⋯≤η1J∫01/Mrh0,h1,⊺−1φτdτ⟶0,asJ⟶∞.
This shows thathJ is a fc−contractive sequence, and therefore,(81)limJ⟶∞∫01/MrhJ,hJ+1,⊺−1φτdτ=0⇒limJ⟶∞1MrhJ,hJ+1,⊺−1=0,for⊺≫0.
Hence, we get that(82)limJ⟶∞MrhJ,hJ+1,⊺=1,for⊺≫0.
Now, forℓ>J and for ⊺≫0, we have(83)∫01/MrhJ,hℓ,⊺−1φτdτ≤∫01/MrhJ,hJ+1,⊺−1φτdτ+∫01/MrhJ+1,hJ+2,⊺−1φτdτ+⋯+∫01/Mrhℓ−1,hℓ,⊺−1φτdτ,≤η2J∫01/Mrho,h1,⊺−1φτdτ+η1J+1∫01/Mrho,h1,⊺−1φτdτ+⋯+η1ℓ−1∫01/Mrho,h1,⊺−1φτdτ=η1J+η1J+1+⋯+η1ℓ−1∫01/Mrho,h1,⊺−1φτdτ=η1J1−η1∫01/Mrho,h1,⊺−1φτdτ⟶0,asJ⟶∞.
We get that(84)limJ⟶∞∫01/MrhJ,hℓ,⊺−1φτdτ=0⇒limJ⟶∞1MrhJ,hℓ,⊺−1=0,for⊺≫0.
Hence,hJ is a Cauchy sequence. Since, by the completeness of G,Mr,∗, ∃h∈G so that(85)limJ⟶∞MrhJ,h,⊺=1,for⊺≫0.
Now, we proveΓg,h=g. Then, from (67), (74), and (77), for ⊺≫0,(86)∫01/Mrg,Γg,h,⊺−1≤∫01/Mrg,gJ+1,⊺−1+∫01/MrΓgJ,hJ,Γg,h,⊺−1,≤∫01/Mrg,gJ+1,⊺−1+η1∫01/MrgJ,g,⊺−1+η2∫0MrgJ,g,⊺/MrgJ,ΓgJ,hJ,⊺∗Mrg,ΓgJ,hJ,2⊺−1=∫01/Mrg,gJ+1,⊺−1+η1∫01/MrgJ,g,⊺−1+η2∫0MrgJ,g,⊺/MrgJ,gJ+1,⊺∗Mrg,gJ+1,2⊺−1⟶0,asJ⟶∞.
Hence,Mrg,Γg,h,⊺=1⇒Γg,h=g for ⊺≫0. Next, we shall prove that Γh,g=h; again from (67), (82), and (85), for ⊺≫0,(87)∫01/Mrh,Γh,g,⊺−1≤∫01/Mrh,hJ+1,⊺−1+∫01/MrΓhJ,gJ,Γh,g,⊺−1,≤∫01/Mrh,hJ+1,⊺−1+η1∫01/MrhJ,h,⊺−1+η2∫0MrhJ,h,⊺/MrhJ,ΓhJ,gJ,⊺∗Mrh,ΓhJ,gJ,2⊺−1=∫01/Mrh,hJ+1,⊺−1+η1∫01/MrhJ,h,⊺−1+η2∫0MrhJ,h,⊺/MrhJ,hJ+1,⊺∗Mrh,hJ+1,2⊺−1⟶0,asJ⟶∞.
Hence,Mrh,Γh,g,⊺=1 which implies Γh,g=h for ⊺≫0.
Uniqueness: supposeg1,h1 and h1,g1 are other coupled fixed-point pairs in G×G such that Γg1,h1=g1 and Γh1,g1=h1. Now, from (67) and from the proof of Theorem 1, for ⊺≫0,(88)∫01/Mrg,g1,⊺−1φτdτ=∫01/MrΓg,h,Γg1,h1,⊺−1φτdτ,≤η1∫01/Mrg,g1,⊺−1φτdτ=η1∫01/MrΓg,h,Γg1,h1,⊺−1φτdτ≤η12∫01/Mrg,g1,⊺−1φτdτ≤⋯≤η1J∫01/Mrg,g1,⊺−1φτdτ⟶0,asJ⟶∞.
Hence, we get thatMrg,g1,⊺=1⇒g=g1 for ⊺≫0. Next, we have to prove h=h1, and now, by using (67) and from the proof of Theorem 1, for ⊺≫0, we have that(89)∫01/Mrh,h1,⊺−1φτdτ=∫01/MrΓh,g,Γh1,g1,⊺−1φτdτ,≤η1∫01/Mrh,h1,⊺−1φτdτ=η1∫01/MrΓh,g,Γh1,g1,⊺−1φτdτ≤η12∫01/Mrh,h1,⊺−1φτdτ≤⋯≤η1J∫01/Mrh,h1,⊺−1φτdτ⟶0,asJ⟶∞.
Hence, we get thatMrh,h1,⊺=1⇒h=h1 for ⊺≫0.


## 5. Conclusion

We established the new concept of rational coupled fc-contraction mapping in FCMSs and proved some unique rational coupled FP theorems in FCMSs under the rational coupled fc-contraction conditions by using the “triangular property of fuzzy cone metric” with the help of some suitable examples to unify our work. In the last section, we presented an application of the Lebesgue integral-type coupled contraction theorem for unique rational coupled FP in complete FCMSs. By using this concept, one can prove more rational coupled-type fc-contraction results in complete FCMSs with different integral types of application to prove unique coupled FP results.



---

*Source: 4764441-2021-08-18.xml*