# Common Coupled Fixed Point Theorems onC⋆-Algebra-Valued Partial Metric Spaces

Authors: Gunaseelan Mani; Arul Joseph Gnanaprakasam; Precious B. Khumalo; Yaé U. Gaba
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, we prove common coupled fixed point theorems on completeC⋆-algebra-valued partial metric spaces. An example and application to support our result are presented.


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## Body

## 1. Introduction

In 1987, Guo and Lakshmikantham [1] introduced the concept of a coupled fixed point. In 2006, Bhaskar and Lakshmikantham [2] introduced the concept of a mixed monotone property for the first time and investigated some coupled fixed point theorems for mappings. As a result, many authors obtained many coupled fixed point and coupled coincidence theorems (see [3–21] and references therein).In 2014, Ma et al. [22] introduced the notion of a C⋆-algebra-valued metric space and proved fixed point theorem. In 2015, Batul and Kamran [23] proved fixed theorems on C⋆-algebra-valued metric space. In 2016, Alsulami et al. [24] proved fixed point theorems on C∗-algebra-valued metric space. In 2016, Cao and Xin [25] proved common coupled fixed point theorems in C∗-algebra-valued metric spaces. The details on C⋆-algebra are available in [26–29]. In 2011, Aydi et al. [30] proved coupled fixed point theorems on ordered partial metric space. The details on partial metric space are available in [31–43]. In 2019, Chandok et al. [44] proved fixed point theorems on C∗-algebra-valued partial metric space. In this paper, we prove common coupled fixed point theorems on C∗-algebra-valued partial metric space.


## 2. Preliminaries

First of all, we recall some basic definitions, notations, and results ofC⋆-algebra that can be found in [27]. An algebra A, together with a conjugate linear involution map a↦a⋆, is called a ⋆-algebra if ab⋆=b⋆a⋆ and a⋆⋆=a for all a,b∈A. Moreover, the pair A,⋆ is called a unital ⋆-algebra if A contains the identity element 1A. By a Banach ⋆-algebra, we mean a complete normed unital å-algebra A,⋆ such that the norm on A is submultiplicative and satisfies ∥a⋆∥=∥a∥ for all a∈A. Further, if for all a∈A, we have ∥a⋆a∥=∥a∥2 in a Banach ⋆-algebra A,⋆, then A is known as a C⋆-algebra. A positive element of A is an element a∈A such that a=a⋆ and its spectrum σa⊂ℝ+, where σa = υ∈ℝ:υ1A−aisnoninvertible. The set of all positive elements will be denoted by A+. Such elements allow us to define a parial ordering ⪰ on the elements of A. That is,
(1)b⪰aifandonlyifb−a∈A+.Ifa∈A is positive, then we write a⪰0A, where 0A is the zero element of A. Each positive element a of a C⋆-algebra A has a unique positive square root. From now on, by A, we mean a unital C⋆-algebra with identity element 1A. Further, A+=a∈A:a±0A and a⋆a1/2 =∣a∣.Now, we recall the definition ofC∗-algebra-valued partial metric space introduced by Chandok et al. [44].Definition 1.
LetΓ be a nonvoid set and the mapping ρ:Γ×Γ⟶A are defined, with the following properties:
(A1) 0A⪯ρℵ,ϖ for all ℵ,ϖ∈Γ and ρℵ,ℵ=ρϖ,ϖ=ρℵ,ϖ if and only if ℵ=ϖ
(A2) ρℵ,ℵ⪯ρℵ,ϖ
(A3) ρℵ,ϖ=ρϖ,ℵ for all ℵ,ϖ∈Γ
(A4) ρℵ,ϖ⪯ρℵ,γ+ργ,ϖ−ργ,γ for all ℵ,ϖ,γ∈ΓThen,ρ is said to be a C⋆-algebra-valued partial metric on Γ, and Γ,A,ρ is said to be a C⋆-algebra-valued partial metric space.Definition 2.
A sequenceℵα in Γ,A,ρ is called convergent (with respect to A) to a point ℵ∈Γ, if for given ε>0, ∃k∈ℕ such that ρℵα,ℵ−ρℵ,ℵ<ε,∀α>k.Definition 3.
A sequenceℵα in Γ,A,ρ is called Cauchy (with respect to A), if limα⟶∞ρℵα,ℵm exists, and it is finite.Definition 4.
The tripletΓ,A,ρ is called complete C∗-algebra-valued partial metric space if every Cauchy sequence in Γ is convergent to some point ℵ in Γ such that
(2)limα⟶∞ρℵα,ℵm=limα⟶∞ρℵα,ℵ=ρℵ,ℵ.Definition 5 (see [18]).
LetΓ be a nonvoid set. An element ℵ,ϖ∈Γ×Γ is said to be
(1)
A couple fixed point of the mappingφ:Γ×Γ⟶Γ if φℵ,ϖ=ℵ and φϖ,ℵ=ϖ(2)
A coupled coincidence point of the mappingφ:Γ×Γ⟶Γ and g:Γ⟶Γ if φℵ,ϖ=gℵ and φϖ,ℵ=gϖ. In this case, gℵ,gϖ is said to be coupled point of coincidence(3)
A common coupled fixed point of the mappingφ:Γ×Γ⟶Γ and g:Γ⟶Γ if φℵ,ϖ=gℵ=ℵ and φϖ,ℵ=gϖ=ϖNote that Definition5 (3) reduces to Definition 5 (1) if the mapping g is the identity mapping.Definition 6 (see [18]).
The mappingsφ:Γ×Γ⟶Γ and g:Γ⟶Γ is said to be ω-compatible if gφℵ,ϖ=φgℵ,gϖ whenever gℵ=φℵ,ϖ and gϖ=φϖ,ℵ.


## 3. Main Results

Now, we give our main results.Theorem 7.
LetΓ,A,ρ be a complete C⋆-algebra-valued partial metric space. Suppose that the mappings φ:Γ×Γ⟶Γ and g:Γ⟶Γ such that
(3)ρφℵ,ϖ,φi,v⪯ρgℵ,gir+r⋆ρgϖ,gvr,foranyℵ,ϖ,i,v∈Γ,where r∈A with ∥r∥<1/2. If φΓ×Γ⊆gΓ and gΓ is complete in Γ, then φ and g have a coupled coincidence point and ρgℵ,gℵ=0A, ρgϖ,gϖ=0A. Moreover, if φ and g are ω-compatible, then they have unique common coupled fixed point in Γ.Proof.
Letℵ0,ϖ0∈Γ, then gℵ1=φℵ0,ϖ0, and gϖ1=φϖ0,ℵ0. One can obtain two sequences ℵα and ϖα by continuing this process such that gℵα+1=φℵα,ϖα, and gϖα+1=φϖα,ℵα. Then,
(4)ρgℵα,gℵα+1=ρφℵα−1,ϖα−1,φℵα,ϖα⪯r⋆ρgℵα−1,gℵαr+r⋆ρgϖα−1,gϖαr⪯r⋆ρgℵα−1,gℵα+ρgϖα−1,gϖαr.
Similarly,(5)ρgϖα,gϖα+1=ρφϖα−1,ℵα−1,φϖα,ℵα⪯r⋆ρgϖα−1,gϖαr+r⋆ρgℵα−1,gℵαr⪯r⋆ρgϖα−1,gϖα+ρgℵα−1,gℵαr.
Let(6)Iα=ρgℵα,gℵα+1+ρgϖα,gϖα+1.
Using (4) and (5), we have
(7)Iα=ρgℵα,gℵα+1+ρgϖα,gϖα+1⪯r⋆ρgℵα−1,gℵα+ρgϖα−1,gϖαr+r⋆ρgϖα−1,gϖα+ρgℵα−1,gℵαr⪯2r⋆ρgℵα−1,gℵα+ρgϖα−1,gϖα2r⪯2r⋆Iα−12r.
Lets,t∈Aþ, then s⪯t implies r⋆sr⪯r⋆tr (Theorem 2.2.5 in [27]). Therefore, for each α∈ℕ,
(8)0A⪯Iα⪯2r⋆Iα−12r⪯⋯⪯2r⋆αI02rα.
IfI0=0A, then φ and g have a coupled coincidence point ℵ0,ϖ0. Now, letting 0A⪯I0, then for each α,℘∈ℕ,
(9)ρgℵα+℘,gℵα⪯ρgℵα+℘,gℵα+℘−1+ρgℵα+℘−1,gℵα+℘−2−ρgℵα+℘−1,gℵα+℘−1+⋯+ρgℵα+2,gℵα+1+ρgℵα+1,gℵα−ρgℵα+1,gℵα+1⪯ρgℵα+℘,gℵα+℘−1+ρgℵα+℘−1,gℵα+℘−2+⋯+ρgℵα+2,gℵα+1+ρgℵα+1,gℵα,ρgϖα+℘,gϖα⪯ρgϖα+℘,gϖα+℘−1+ρgϖα+℘−1,gϖα+℘−2−ρgϖα+℘−1,gϖα+℘−1+⋯+ρgϖα+2,gϖα+1+ρgϖα+1,gϖα−ρgϖα+1,gϖα+1⪯ρgϖα+℘,gϖα+℘−1+ρgϖα+℘−1,gϖα+℘−2+⋯+ρgϖα+2,gϖα+1+ρgϖα+1,gϖα.
Consequently,(10)ρgℵα+℘,gℵα+ρgϖα+℘,gϖα⪯Iα+℘−1+Iα+wp−2+⋯+Iα⪯∑k=αα+℘−12r⋆kI02rk,which implies that
(11)ρgℵα+℘,gℵα+ρgϖα+℘,gϖα⪯∑k=αα+℘−12r2kI0⪯∑k=α∞2r2kI0=2r2α1−2r2I0.
Since∥r∥⋖1/√2, we have
(12)ρgℵα+℘,gℵα+ρgϖα+℘,gϖα⪯2r2α1−2r2I0⟶0,which is together with
(13)ρgℵα+℘,gℵα⪯ρgℵα+℘,gℵα+ρgϖα+℘,gϖα,and
(14)ρgϖα+℘,gϖα⪯ρgℵα+℘,gℵα+ρgϖα+℘,gϖα.
Therefore,gℵα and gϖα are Cauchy sequences in gΓ. Since gϖα is complete, ∃ℵ,ϖ∈Γ such that limα⪯∞gℵα=gℵ and
(15)ρgℵ,gℵ=limn⟶∞ρgℵα,gℵ=limn⟶∞ρgℵα,gℵα=0A,
limα⪯∞gϖα=gϖ, and
(16)ρgϖ,gϖ=limn⟶∞ρgϖα,gϖ=limn⟶∞ρgϖα,gϖα=0A.
Now, we show thatφℵ,ϖ=gℵ and φϖ,ℵ=gϖ. For this,
(17)ρφℵ,ϖ,gℵ⪯ρφℵ,ϖ,gℵα+1+ρgℵα+1,gℵ⪯ρφℵ,ϖ,φℵα,ϖα+ρgℵα+1,gℵ⪯r⋆ρgℵα,gℵr+r⋆ρgϖα,gϖr+ρgℵα+1,gℵ.
Asα⟶∞, we get ρφℵ,ϖ,gℵ=0A, and hence, φℵ,ϖ=gℵ. Similarly, φϖ,ℵ=gϖ. Therefore, φ and g have a coupled coincidence point ℵ,ϖ.
Letℵ′,ϖ′ be another coupled coincidence point of φ and g. Then,
(18)ρgℵ,gℵ′=ρφℵ,ϖ,φℵ′,ϖ′⪯r⋆ρgℵ,gℵ′r+r⋆ρgϖ,gϖ′r,ρgϖ,gϖ′=ρφϖ,ℵ,φϖ′,ℵ′⪯r⋆ρgϖ,gϖ′r+r⋆ρgℵ,gℵ′r.
Consequently,(19)ρgℵ,gℵ′+ρgϖ,gϖ′⪯2r⋆ρgℵ,gℵ′+ρgϖ,gϖ′2r,which implies that
(20)ρgℵ,gℵ′+ρgϖ,gϖ′⪯2r2ρgℵ,gℵ′+ρgϖ,gϖ′.
Since2r<1, then ρgℵ,gℵ′+ρgϖ,gϖ′=0. Hence, we get gℵ=gℵ′ and gϖ=gϖ′. Similarly, we can prove that gℵ=gϖ′ and gϖ=gℵ′. Then, φ and g have a unique coupled point of coincidence gℵ,gℵ. Moreover, set v=gℵ, then v=gℵ=φℵ,ℵ. Since φ and g are ω-compatible,
(21)gv=ggℵ=gφℵ,ℵ=φgℵ,gℵ=φv,v.
Therefore,φ and g have a coupled point of coincidence gv,gv. We know gv=gℵ, then v=gv=φv,v. Therefore, φ and g have a unique common coupled fixed point v,v.Example 1.
LetΓ=R and A=M2ℂ, and the map ρ:Γ×Γ⟶A is defined by
(22)ρℵ,ϖ=ℵ−ϖ00kℵ−ϖ+maxℵ,ϖ00kmaxℵ,ϖ,where k⋗0 is a constant. Then, Γ,A,ρ is a complete C⋆-algebra-valued partial metric space. Consider the mappings φ:Γ×Γ⟶Γ with φℵ,ϖ=ℵ+ϖ/2 and g:Γ⟶Γ with gℵ=2ℵ. Set λ∈ℂ with ∣λ∣<1/2, and r=λ00λ, then r∈A and ∥r∥∞=∣λ∣. Clearly, φ and g are ω-compatible. Moreover, one can verify that φ satisfies the contractive condition
(23)ρφℵ,ϖ,φu,v⪯r⋆φℵ,ur+r⋆φϖ,vr,foranyℵ,ϖ,u,v∈Γ.In this case,0,0 is coupled coincidence point of φ and g. Moreover, 0,0 is a unique common coupled fixed point of φ and g.Corollary 8.
LetΓ,A,ρ be a complete C⋆-algebra-valued partial metric space. Suppose that mapping φ:Γ×Γ⟶Γ such that
(24)ρφℵ,ϖ,φu,v⪯r⋆ρℵ,ur+r⋆ρϖ,vr,foranyℵ,ϖ,u,v∈Γ,where r∈A with ∥r∥<1/2. Then, φ has a unique coupled fixed point.We recall the following lemma of [27].Lemma 9.
Suppose thatA is a unital C⋆-algebra with a unit 1A.
(1)
Ifr∈A+ with ∥r∥<1/2, then 1A−r is invertible(2)
Ifr,s∈A+ and rs=sr, then 0A⪯rs(3)
Ifr,s∈Aþ and t∈A+′ then r⪯s deduces tr⪯ts, where A+′=A+∩A′Theorem 10.
LetΓ,A,ρ is a complete C⋆-algebra-valued partial metric space. Suppose that the mappings φ:Γ×Γ⟶Γ and g:Γ⟶Γ such that
(25)ρφℵ,ϖ,φu,v⪯rρφℵ,ϖ,gℵ+sρφu,v,gu,foranyℵ,ϖ,u,v∈Γ,where r,s∈A+′ with ∥r∥+∥s∥<1. If φΓ×Γ⊆gΓ and gΓ is complete in Γ, then φ and g have a coupled coincidence point and ρgℵ,gℵ=0A, ρgϖ,gϖ=0A. Moreover, if φ and g are ω-compatible, then they have unique common coupled fixed point in Γ.Proof.
Similar to Theorem7, construct two sequences ℵα and ϖα in Γ such that gℵα+1=φℵα,ϖα and gϖα+1=φϖα,ℵα. Then, by applying (25), we have
(26)1A−sρgℵα,gℵα+1⪯rρgℵα,gℵα−1,1A−sρgϖα,gϖα+1⪯rρgϖα,gϖα−1.
Sincer,s∈A+′ with ∥r∥+∥s∥<1, we have 1A−s is invertible and 1A−s−1r∈A+′. Therefore,
(27)ρgℵα,gℵα+1⪯1A−s−1rρgℵα,gℵα−1,ρgϖα,gϖα+1⪯1A−s−1rρgϖα,gϖα−1.
Then,(28)ρgℵα,gℵα+1⪯1A−s−1rρgℵα,gℵα−1,ρgϖα,gϖα+1⪯1A−s−1rρgϖα,gϖα−1.
Since,(29)1A−s−1r⪯1A−s−1r⪯∑k=0∞skr=r1−s.⋖1.
Therefore,gℵα and gϖα are Cauchy sequences in gΓ. By the completeness of gΓ, ∃ℵ,ϖ∈Γ such that limα°∞gℵα=gℵ and
(30)ρgℵ,gℵ=limn⟶∞ρgℵα,gℵ=limn⟶∞ρgℵα,gℵα=0A,
limα⟶∞gϖα=gϖ, and
(31)ρgϖ,gϖ=limn⟶∞ρgϖα,gϖ=limn⟶∞ρgϖα,gϖα=0A.
Since,(32)ρφℵ,ϖ,gℵ⪯ρgℵα+1,φℵ,ϖ+ρgℵα+1,gℵ=ρφℵα,ϖα,φℵ,ϖ+ρgℵα+1,gℵ⪯rρφℵα,ϖα,gℵα+sρφℵ,ϖ,gℵ+ρgℵα+1,gℵ⪯rρgℵα+1,gℵα+sρφℵ,ϖ,gℵ+ρgℵα+1,gℵ,which implies that
(33)ρφℵ,ϖ,gℵ⪯1−s−1rρgℵα+1,gℵα+1−s−1ρgℵα+1,gℵα.
Then,ρφℵ,ϖ,gℵ=0A or equivalently φℵ,ϖ=gℵ. Similarly, one can obtain φϖ,ℵ=gϖ. Let ℵ′,ϖ′ be another coupled coincidence point of φ and g, then
(34)ρgℵ′,gℵ⪯ρφℵ′,ϖ′,φℵ,ϖ⪯rρφℵ′,ϖ′,gℵ′+sρφℵ,ϖ,gℵ=rρgℵ′,gℵ′+sρgℵ,gℵ=0A,and
(35)ρgϖ′,gϖ⪯ρφϖ′,ℵ′,φϖ,ℵ⪯rρφϖ′,ℵ′,gϖ′+sρφϖ,ℵ,gϖ=rρgϖ′,gϖ′+sρgϖ,gϖ=0A,which implies that gℵ′=gℵ and gϖ′=gϖ. Similarly, we have gℵ′=gϖ and gϖ′=gℵ. Hence, φ and g have a unique coupled point of coincidence gℵ,gℵ. Moreover, we can show that φ and g have a unique common coupled fixed point.Theorem 11.
LetΓ,A,ρ be a complete C⋆-algebra-valued partial metric space. Suppose that mappings φ:Γ×Γ⟶Γ and g:Γ⟶Γ such that
(36)ρφℵ,ϖ,φu,v⪯rρφℵ,ϖ,gu+sρφu,v,gℵ,foranyℵ,ϖ,u,v∈Γ,where r,s∈A+′ with ∥r∥+∥s∥<1. If φΓ×Γ⊆gΓ and gΓ is complete in Γ, then φ and g have a coupled coincidence point and ρgℵ,gℵ=0A, ρgϖ,gϖ=0A. Moreover, if φ and g are ω-compatible, then they have unique common coupled fixed point in Γ.Proof.
Following similar process given in Theorem7, we construct two sequences ℵα and ϖα in Γ such that gℵα+1=φℵα,ϖα and gϖα+1=φϖα,ℵα. From (36), we have
(37)ρgℵα,gℵα+1=ρφℵα−1,ϖα−1,φℵα,ϖα⪯rρφℵα−1,ϖα−1,gℵα+sρφℵα,ϖα,gℵα−1⪯rρgℵα,gℵα+sρgℵα+1,gℵα−1⪯rρgℵα+1,gℵα+sρgℵα+1,gℵα+sρgℵα,gℵα−1−sρρgℵα,gℵα⪯rρgℵα+1,gℵα+sρgℵα+1,gℵα+sρgℵα,gℵα−1,which implies that
(38)1A−r+sρgℵα,gℵα+1⪯sρgℵα,gℵα−1.
Because of the symmetry in (36),
(39)ρgℵα+1,gℵα=ρφℵα,ϖα,φℵα−1,ϖα−1⪯rρφℵα,ϖα,gℵα−1+sρφℵα−1,ϖα−1,gℵα⪯rρgℵα+1,gℵα−1+sρgℵα,gℵα⪯rρgℵα+1,gℵα+rρgℵα,gℵα−1−rρgℵα,gℵα+sρgℵα+1,gℵα⪯rρgℵα+1,gℵα+rρgℵα,gℵα−1+sρgℵα+1,gℵα,which implies that
(40)1A−r+sρgℵα,gℵα+1⪯rρgℵα,gℵα−1.
From (38) and (40), we obtain
(41)1A−r+sρgℵα,gℵα+1⪯r+s2ρgℵα,gℵα−1.
Sincer,s∈A+′ with ∥r+s∥⪯∥r∥+∥s∥⋖1, then 1A−r+s−1∈A+′, which together with Lemma 9 (3), we obtain
(42)ρgℵα,gℵα+1⪯1A−r+s−1r+s2ρgℵα,gℵα−1.
Lete=1A−r+s−1r+s/2, then ∥e∥=∥1A−r+s−1r+s/2∥<1. The same argument in Theorem 10 tells that gℵα is a Cauchy sequence in gΓ. Similarly, we can derive that gϖα is also a Cauchy sequence in gΓ. By the completeness of gΓ, ∃ℵ,ϖ∈Γ such that limα⪯∞gℵα=gℵ and
(43)ρgℵ,gℵ=limn⟶∞ρgℵα,gℵ=limn⟶∞ρgℵα,gℵα=0A,
limα⪯∞gϖα=gϖ, and
(44)ρgϖ,gϖ=limn⟶∞ρgϖα,gϖ=limn⟶∞ρgϖα,gϖα=0A.
Now, we show thatφℵ,ϖ=gℵ and φϖ,ℵ=gϖ. For this,
(45)ρφℵ,ϖ,gℵ⪯ρgℵα+1,φℵ,ϖ+ρgℵα+1,gℵ=ρφℵα,ϖα,φℵ,ϖ+ρgℵα+1,gℵ⪯rρφℵα,ϖα,gℵ+sρφℵ,ϖ,gℵα+ρgℵα+1,gℵ⪯rρgℵα+1,gℵ+sρφℵ,ϖ,gℵα+ρgℵα+1,gℵ,which implies that
(46)ρφℵ,ϖ,gℵ⪯rρgℵα+1,gℵ+sρφℵ,ϖ,gℵα+ρgℵα+1,gℵ.
By the continuity of the metric and the norm, we obtain(47)ρφℵ,ϖ,gℵ⪯sρφℵ,ϖ,gℵ.
Since∥s∥<1; therefore, ∥ρφℵ,ϖ,gℵ∥=0. Thus, φℵ,ϖ=gℵ. Similarly, φϖ,ℵ=gϖ. Hence, ℵ,ϖ is a coupled coincidence point of φ and g. The same reasoning that Theorem 10 tells us that φ and g have unique common coupled fixed point in Γ.In 2015, Ma and Jiang [45] proved fixed point theorems in C⋆-algebra-valued b-metric spaces with an application of Fredholm integral equations. In 2016, Xin et al. [46] proved common fixed point theorems in C∗-algebra-valued metric spaces with an application of Fredholm integral equations. In 2020, Mlaiki et al. [47] proved fixed point results on C⋆-algebra valued partial b-metric spaces with an application of Fredholm integral equations. In 2021, Tomar et al. [48] proved fixed point theorems in C⋆-algebra valued partial metric space with an application of Fredholm integral equations.


## 4. Application

As an application of Corollary8, we find an existence and uniqueness result for a type of following system of Fredholm integral equations:
(48)ℵμ=∫EGμ,p,ℵp,ϖpdp+δμ,μ,p∈E,(49)ϖμ=∫EGμ,p,ϖp,ℵpdp+δμ,μ,p∈E,where E is a measurable, G:E×E×ℝ×ℝ⟶ℝ and δ∈L∞E. Let Γ=L∞E, K=L2E, and LK=A. Define ρ:Γ×Γ⟶A by (for all δ,θ,I∈Γ and τ=θ<1):
(50)ρδ,θ=π∣δ−θ∣+I,where πq:K⟶K is the multiplicative operator, which is defined by:
(51)πqψ=q.ψ.Now, we state and prove our result, as follows:Theorem 12.
Suppose that (for allℵ,ϖ∈Γ)
(S1) There exists a continuous function κ:E×E⟶ℝ and θ∈0,1, such that
(52)∣Gμ,p,ℵp,ϖp−Gμ,p,up,up∣≤θ∣κμ,p∣∣ℵp−up∣+∣ϖp−vp∣+I−θ−1I,for all μ,p∈E.
(S2) supμ∈E∫E∣κμ,p∣dp≤1.Subsequently, the integral Equation (49) has a unique solution in Γ.Proof.
Defineφ:Γ×Γ⟶Γ by:
(53)φℵ,ϖμ=∫EGμ,p,ℵp,ϖpdp+δμ,∀μ,p∈E,
Setτ=θI, then τ∈A. For any z∈K, we have
(54)ρφℵ,ϖ,φu,v=supz=1π∣φℵ,ϖ−φu,v∣+Iz,z=supz=1∫E∣φℵ,ϖ−φu,v∣+Izμzμ¯dμ≤supz=1∫E∫E∣Gμ,p,ℵp,ϖp−Gμ,p,ℵp,ϖp∣dpzμ2dμ+supz=1∫E∫Edpzμ2dμI≤supz=1∫E∫Eθ∣κμ,p∣∣ℵp−up∣+∣ϖp−vp∣+I−θ−1Idpzμ2dμ+I≤θsupz=1∫E∫E∣κμ,p∣dpzμ2dμℵ−u∞+ϖ−v∞≤θsupμ∈E∫E∣κμ,p∣dpsupz=1∫Ezμ2dμℵ−u∞+ϖ−v∞≤θℵ−u∞+ϖ−v∞=τρℵ,u+ρϖ,v.
Hence, all the hypotheses of Corollary8 are verified, and consequently, the integral Equation (49) has a unique solution.


## 5. Conclusion

In this paper, we proved common coupled fixed point theorems onC∗-algebra-valued partial metric space using ω-compatible mappings. An illustrative example is provided that shows the validity of the hypothesis and the degree of usefulness of our findings. Moreover, we introduced an application to show that the useful of C⋆-algebra-valued metric space to study the existence and uniqueness of system of Fredholm integral equations. Recently, Mutlu et al. [49] proved coupled fixed point theorems on bipolar metric spaces. It is an interesting open problem to study the C⋆-algebra-valued bipolar metric space instead of C⋆-algebra-valued metric space and obtain common coupled fixed point results on C⋆-algebra-valued bipolar metric spaces.



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*Source: 3336095-2022-01-24.xml*