Common Coupled Fixed Point Theorems on C ⋆ -Algebra-Valued Partial Metric Spaces

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


Preliminaries
First of all, we recall some basic definitions, notations, and results of C ⋆ -algebra that can be found in [27].An algebra A, together with a conjugate linear involution map a ↦ a ⋆ , is called a ⋆-algebra if ðabÞ ⋆ = b ⋆ a ⋆ and ða ⋆ Þ ⋆ = a for all a, b ∈ A. Moreover, the pair ðA, ⋆Þ is called a unital ⋆-algebra if A contains the identity element 1 A .By a Banach ⋆-algebra, we mean a complete normed unital å-algebra ðA, ⋆Þ 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 ðA, ⋆Þ, 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 σðaÞ ⊂ ℝ + , where σðaÞ = fυ ∈ ℝ : υ1 A − a is noninvertibleg.The set of all positive elements will be denoted by A + .Such elements allow us to define a parial ordering ⪰ on the elements of If a ∈ A is positive, then we write a ⪰ 0 A , where 0 A 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 1 A .Further, A + = fa ∈ A : a ± 0 A g and ða ⋆ aÞ 1/2 =|a | .Now, we recall the definition of C * -algebra-valued partial metric space introduced by Chandok et al. [44].

Main Results
Now, we give our main results.Theorem 7. Let ðΓ, A, ρÞ be a complete C ⋆ -algebra-valued partial metric space.Suppose that the mappings φ : where r ∈ A with ∥r∥<ð1/ ffiffi ffi 2 p Þ.If φðΓ × ΓÞ ⊆ gðΓÞ and gðΓÞ is complete in Γ, then φ and g have a coupled coincidence point and ρðgℵ, gℵÞ = 0 A , ρðgϖ, gϖÞ = 0 A .Moreover, if φ and g are ω-compatible, then they have unique common coupled fixed point in Γ.
Let ðℵ ′ , ϖ ′ Þ be another coupled coincidence point of φ and g.Then, Therefore, φ and g have a coupled point of coincidence ðgv, gvÞ.We know gv = gℵ, then v = gv = φðv, vÞ.Therefore, φ and g have a unique common coupled fixed point ð v, vÞ.
Example 1.Let Γ = R and A = M 2 ðℂÞ, and the map ρ where k ⋗ 0 is a constant.Then, ðΓ, A, ρÞ is a complete C ⋆ -algebra-valued partial metric space.Consider the mappings φ : In this case, ð0, 0Þ is coupled coincidence point of φ and g.Moreover, ð0, 0Þ is a unique common coupled fixed point of φ and g.Corollary 8. Let ðΓ, A, ρÞ be a complete C ⋆ -algebra-valued partial metric space.Suppose that mapping φ : where r ∈ A with ∥r∥<ð1/ ffiffi ffi 2 p Þ.Then, φ has a unique coupled fixed point.
We recall the following lemma of [27].

Lemma 9.
Suppose that A is a unital C ⋆ -algebra with a unit 1 A .
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.

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 E is a measurable, G : where π q : K ⟶ K is the multiplicative operator, which is defined by: Now, we state and prove our result, as follows: Theorem 12. Suppose that (for all ℵ, ϖ ∈ Γ) (S1) There exists a continuous function κ : Subsequently, the integral Equation ( 49) has a unique solution in Γ.

Proof. Define
Journal of Function Spaces Set τ = θI, then τ ∈ A. For any z ∈ K, we have Hence, all the hypotheses of Corollary 8 are verified, and consequently, the integral Equation ( 49) has a unique solution.

Conclusion
In this paper, we proved common coupled fixed point theorems on C * -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 ⋆ -algebravalued metric space and obtain common coupled fixed point results on C ⋆ -algebra-valued bipolar metric spaces.