#### Abstract

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

#### 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 [321] and references therein).

In 2014, Ma et al. [22] introduced the notion of a -algebra-valued metric space and proved fixed point theorem. In 2015, Batul and Kamran [23] proved fixed theorems on -algebra-valued metric space. In 2016, Alsulami et al. [24] proved fixed point theorems on -algebra-valued metric space. In 2016, Cao and Xin [25] proved common coupled fixed point theorems in -algebra-valued metric spaces. The details on -algebra are available in [2629]. 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 [3143]. In 2019, Chandok et al. [44] proved fixed point theorems on -algebra-valued partial metric space. In this paper, we prove common coupled fixed point theorems on -algebra-valued partial metric space.

#### 2. Preliminaries

First of all, we recall some basic definitions, notations, and results of -algebra that can be found in [27]. An algebra , together with a conjugate linear involution map , is called a -algebra if and for all . Moreover, the pair is called a unital -algebra if contains the identity element . By a Banach -algebra, we mean a complete normed unital -algebra such that the norm on is submultiplicative and satisfies for all . Further, if for all , we have in a Banach -algebra , then is known as a -algebra. A positive element of is an element such that and its spectrum , where = . The set of all positive elements will be denoted by . Such elements allow us to define a parial ordering on the elements of . That is,

If is positive, then we write , where is the zero element of . Each positive element of a -algebra has a unique positive square root. From now on, by , we mean a unital -algebra with identity element . Further, and =.

Now, we recall the definition of -algebra-valued partial metric space introduced by Chandok et al. [44].

Definition 1. Let be a nonvoid set and the mapping are defined, with the following properties:
(A1) for all and if and only if
(A2)
(A3) for all
(A4) for all

Then, is said to be a -algebra-valued partial metric on , and is said to be a -algebra-valued partial metric space.

Definition 2. A sequence in is called convergent (with respect to ) to a point , if for given , such that .

Definition 3. A sequence in is called Cauchy (with respect to ), if exists, and it is finite.

Definition 4. The triplet is called complete -algebra-valued partial metric space if every Cauchy sequence in is convergent to some point in such that

Definition 5 (see [18]). Let be a nonvoid set. An element is said to be (1)A couple fixed point of the mapping if and (2)A coupled coincidence point of the mapping and if and . In this case, is said to be coupled point of coincidence(3)A common coupled fixed point of the mapping and if and

Note that Definition 5 (3) reduces to Definition 5 (1) if the mapping is the identity mapping.

Definition 6 (see [18]). The mappings and is said to be -compatible if whenever and .

#### 3. Main Results

Now, we give our main results.

Theorem 7. Let be a complete -algebra-valued partial metric space. Suppose that the mappings and such that where with . If and is complete in , then and have a coupled coincidence point and , . Moreover, if and are -compatible, then they have unique common coupled fixed point in .

Proof. Let , then , and . One can obtain two sequences and by continuing this process such that , and . Then, Similarly, Let Using (4) and (5), we have Let , then implies (Theorem 2.2.5 in [27]). Therefore, for each , If , then and have a coupled coincidence point . Now, letting , then for each , Consequently, which implies that Since we have which is together with and Therefore, and are Cauchy sequences in . Since is complete, such that and , and Now, we show that and . For this, As , we get , and hence, . Similarly, . Therefore, and have a coupled coincidence point .
Let be another coupled coincidence point of and . Then, Consequently, which implies that Since , then Hence, we get and Similarly, we can prove that and Then, and have a unique coupled point of coincidence . Moreover, set , then Since and are -compatible, Therefore, and have a coupled point of coincidence . We know , then . Therefore, and have a unique common coupled fixed point .

Example 1. Let and , and the map is defined by where is a constant. Then, is a complete -algebra-valued partial metric space. Consider the mappings with and with . Set with and , then and . Clearly, and are -compatible. Moreover, one can verify that satisfies the contractive condition

In this case, is coupled coincidence point of and . Moreover, is a unique common coupled fixed point of and .

Corollary 8. Let be a complete -algebra-valued partial metric space. Suppose that mapping such that where with . Then, has a unique coupled fixed point.

We recall the following lemma of [27].

Lemma 9. Suppose that is a unital -algebra with a unit . (1)If with then is invertible(2)If and , then (3)If and then deduces , where

Theorem 10. Let is a complete -algebra-valued partial metric space. Suppose that the mappings and such that where with . If and is complete in , then and have a coupled coincidence point and , . Moreover, if and are -compatible, then they have unique common coupled fixed point in .

Proof. Similar to Theorem 7, construct two sequences and in such that and . Then, by applying (25), we have Since with , we have is invertible and Therefore, Then, Since, Therefore, and are Cauchy sequences in . By the completeness of , such that and , and Since, which implies that Then, or equivalently . Similarly, one can obtain . Let be another coupled coincidence point of and , then and which implies that and . Similarly, we have and . Hence, and have a unique coupled point of coincidence . Moreover, we can show that and have a unique common coupled fixed point.

Theorem 11. Let be a complete -algebra-valued partial metric space. Suppose that mappings and such that where with . If and is complete in , then and have a coupled coincidence point and , . Moreover, if and are -compatible, then they have unique common coupled fixed point in .

Proof. Following similar process given in Theorem 7, we construct two sequences and in such that and . From (36), we have which implies that Because of the symmetry in (36), which implies that From (38) and (40), we obtain Since with then which together with Lemma 9 (3), we obtain Let , then . The same argument in Theorem 10 tells that is a Cauchy sequence in . Similarly, we can derive that is also a Cauchy sequence in . By the completeness of , such that and , and Now, we show that and For this, which implies that By the continuity of the metric and the norm, we obtain Since ; therefore, . Thus, . Similarly, . Hence, is a coupled coincidence point of and . The same reasoning that Theorem 10 tells us that and have unique common coupled fixed point in .

In 2015, Ma and Jiang [45] proved fixed point theorems in -algebra-valued -metric spaces with an application of Fredholm integral equations. In 2016, Xin et al. [46] proved common fixed point theorems in -algebra-valued metric spaces with an application of Fredholm integral equations. In 2020, Mlaiki et al. [47] proved fixed point results on -algebra valued partial -metric spaces with an application of Fredholm integral equations. In 2021, Tomar et al. [48] proved fixed point theorems in -algebra valued partial metric space with an application of Fredholm integral equations.

#### 4. Application

As an application of Corollary 8, we find an existence and uniqueness result for a type of following system of Fredholm integral equations: where is a measurable, and . Let , , and . Define by (for all and ): where is the multiplicative operator, which is defined by:

Now, we state and prove our result, as follows:

Theorem 12. Suppose that (for all )
(S1) There exists a continuous function and , such that for all .
(S2) .

Subsequently, the integral Equation (49) has a unique solution in .

Proof. Define by: Set , then . For any , we have Hence, all the hypotheses of Corollary 8 are verified, and consequently, the integral Equation (49) has a unique solution.

#### 5. Conclusion

In this paper, we proved common coupled fixed point theorems on -algebra-valued partial metric space using -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 -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 -algebra-valued bipolar metric space instead of -algebra-valued metric space and obtain common coupled fixed point results on -algebra-valued bipolar metric spaces.

#### Data Availability

No data were used to support the study.

#### Conflicts of Interest

The authors declare that there is not any competing interest regarding the publication of this manuscript.

#### Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

#### Acknowledgments

The fourth author (YUG) would like to acknowledge that this publication was made possible by a grant from Carnegie Corporation of New York. The statements made and views expressed are solely the responsibility of the author.