Solving an integral equation via C (cid:2) -algebra-valued partial b -metrics

In this paper, we prove some common coupled ﬁxed-point theorems on complete C (cid:2) -algebra-valued partial b -metric spaces. Some of the well-known facts in the literature are generalized and expanded by the results shown. An example to illustrate our ﬁndings is presented. We also explore some of the applications of our key results.


Preliminaries
First of all, we recall some basic definitions, notations, and results of C -algebra that can be found in [27]. Let H be a unital algebra. An involution on H is a conjugate-linear map r → r on H such that r = r and (rs) = s r for any r, s ∈ H . The pair (H , ) is called a -algebra. A -algebra H together with a complete submultiplicative norm such that r = r , is said to be a Banach -algebra. Furthermore, a C -algebra is a Banach -algebra with r r = r 2 , for all r ∈ H . An element r in H is self-adjoint, or hermitian, if r = r * . Let H sa be the set of all self-adjoint elements in H , and define the spectrum of r ∈ H to be the set σ (r) = {λ ∈ C : λIr is not invertible}. An element r of a C -algebra © The Author(s) 2022. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.  ([7, 24]) Let ϒ be a nonempty set. Suppose that the mapping ρ : ϒ × ϒ → H is defined, with the following properties: Then ρ is said to be a C -algebra-valued metric on ϒ, and (ϒ, H , ρ) is said to be a Calgebra-valued metric space.
The following definition was introduced by Ma and Jiang [28].

Definition 2.2
Let ϒ be a nonempty set and s ∈ H such that s I. Suppose that the mapping ρ : ϒ × ϒ → H is defined, with the following properties: Then ρ is said to be a C -algebra-valued b-metric on ϒ, and (ϒ, H , ρ) is said to be a C -algebra-valued b-metric space. Now, we recall the definition of a C * -algebra-valued partial b-metric space introduced by Mlaiki et al [26].

Definition 2.3
Let ϒ be a nonempty set and s ∈ H such that s I. Suppose that the mapping ρ : ϒ × ϒ → H is defined, with the following properties: Then ρ is said to be a C -algebra-valued partial b-metric on ϒ, and (ϒ, H , ρ) is said to be a C -algebra-valued partial b-metric space.
where k ≥ 0 and for all ℵ, ∈ ϒ. Then, (ϒ, H , ρ) is a C -algebra-valued partial b-metric space. However, it is easy to see that (ϒ, H , ρ) is not a C * -algebra-valued b-metric space. To substantiate the claim, for any nonzero element ℵ ∈ ϒ, we have . (2) a coupled coincidence point of the mapping T : ϒ × ϒ → ϒ and g : ϒ → ϒ if T (ℵ, ) = gℵ and T ( , ℵ) = g . In this case (gℵ, g ) is said to be a coupled point of coincidence.
Note that Definition 2.7(3) reduces to Definition 2.7(1) if the mapping g is the identity mapping. In this paper, we prove coupled fixed point theorems on C * -algebra-valued partial bmetric space.

Main results
In this section we shall prove some common coupled fixed point theorems for different contractive mappings in the setting of C -algebra-valued partial b-metric spaces. Now we give our main results. condition: where r ∈ H with r < 1 √ 2 and s √ 2r 2 < 1. If T (ϒ × ϒ) ⊆ g(ϒ) and g(ϒ) is complete in ϒ, then T and g have a coupled coincidence point and ρ(gℵ, gℵ) = 0 H and ρ(g , g ) = 0 H . Moreover, if T and g are ω-compatible, then they have a unique common coupled fixed point in ϒ.

Corollary 3.3 Let
where r ∈ H with r < 1 √ 2 and s √ 2r 2 < 1. Then T has a unique coupled fixed point.
Before going to another theorem, we recall the following lemma of [27].
It follows from the fact