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# Quasi Controlled -Metric Spaces over -Algebras with an Application to Stochastic Integral Equations

1 Department of Mathematics, Laboratory of Partial Differential Equations, Algebra and Spectral Geometry, Faculty of Sciences, Ibn Tofail University, Kenitra, BP 133, Morocco

2 Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, Riyadh, 11586, Saudi Arabia

3 Department of Medical Research, China Medical University, Taichung, 40402, Taiwan

* Corresponding Author: Thabet Abdeljawad. Email:

(This article belongs to this Special Issue: Computational Aspects of Nonlinear Operator and Fixed Point Theory with Applications)

*Computer Modeling in Engineering & Sciences* **2023**, *135*(3), 2649-2663. https://doi.org/10.32604/cmes.2023.023496

**Received** 28 April 2022; **Accepted** 16 August 2022; **Issue published** 23 November 2022

## Abstract

Generally, the field of fixed point theory has attracted the attention of researchers in different fields of science and engineering due to its use in proving the existence and uniqueness of solutions of real-world dynamic models.*C*-algebra is being continually used to explain a physical system in quantum field theory and statistical mechanics and has subsequently become an important area of research. The concept of a

^{*}*C*-algebra-valued metric space was introduced in 2014 to generalize the concept of metric space. In fact, It is a generalization by replacing the set of real numbers with a

^{*}*C*-algebra. After that, this line of research continued, where several fixed point results have been obtained in the framework of

^{*}*C*-algebra valued metric, as well as (more general)

^{*}*C*-algebra-valued b-metric spaces and

^{*}*C*-algebra-valued extended b-metric spaces. Very recently, based on the concept and properties of

^{*}*C*-algebras, we have studied the quasi-case of such spaces to give a more general notion of relaxing the triangular inequality in the asymmetric case. In this paper, we first introduce the concept of

^{*}*C*-algebra-valued quasi-controlled -metric spaces and prove some fixed point theorems that remain valid in this setting. To support our main results, we also furnish some examples which demonstrate the utility of our main result. Finally, as an application, we use our results to prove the existence and uniqueness of the solution to a nonlinear stochastic integral equation.

^{*}## Keywords

*C*

^{*}-algebra-valued quasi controlled K-metric spaces; left convergence; right convergence; fixed point; contraction

One of the most relevant theories marking the passage from classical to modern analysis is the fixed point theory which was implemented by Banach [1]. Several mathematicians have created diverse generalizations of Banach fixed point theory. Wilson, on the other hand, introduced the quasi-metric space that is one of the abstractions of the metric spaces [2]. This theory, however, does not include the commutative condition. Numerous mathematicians have adopted this concept to demonstrate some fixed point outcomes, see [3].

The b-metric spaces concept was first set up by Bakhtin [4] and Czerwik [5]. Besides, numerous authors obtained a lot of fixed point results. For example, see [6–10]. The extended b-metric spaces idea was elaborated by Kamran et al. [11] and generalized by Abdeljawad et al. [12] by imposing the control or the double control of the s-relaxed inequality by one or two functions. Mudasir et al. [13] stated new results in the context of dislocated b-metric spaces and presented an application related to electrical engineering and extended the notion of Kannan maps in view of the F-contraction in this framework, see [14].

In [15,16], Ma et al. introduced

In this work, we introduce the notion of

Throughout this paper,

Note that

To prove our main results, it will be useful to introduce the following lemma.

Lemma 2.1. [20] Suppose that

1. if

2. if

3. for all

4.

Definition 2.1. [17] Let

1.

2.

3.

The triplet

In this section, by omitting the symmetry condition, we introduce the notion of

Definition 3.1. A

1.

2.

Remark 3.1. In particular, by taking

Example 3.1. Let

Define a

Given the

Then,

Example 3.2. Let

Let the

Example 3.3. Consider

We take

Thus,

Next, we introduce some topological concepts on

Definition 3.2. Let

Example 3.4. Let us define a

with the

Then, it is evident that

The open ball B is given by

if

if

Remark 3.2. We can also define the closed ball by

Definition 3.3. Let

1.

2.

3.

Definition 3.4. Let

1.

2.

3.

4. If every Cauchy sequence

Example 3.5. Take

Let

Then,

Example 3.6. Let

Let us define the

The condition (i) of Definition 3.1 is clearly satisfied by

Therefore,

This prove that

We deduce

and

We conclude that the sequence

We will fix the notion of a continuous metric in the context presented in this paper since in the literature during the proof of the results in fixed point certain problems arise due to the possible discontinuity of the b-metric with respect to the topology it generates.

Definition 3.5. Let

Lemma 3.1. Let

Proof. Fix

As

Our main result runs as follows.

Theorem 3.1. Let

where

Proof. Let the sequence

Now we prove that

Since

Thus, the above inequality implies

Letting

Remains to see that

Therefore,

so that

Then, we get a contradiction, as a result

Dynamic programming is a powerful technique for solving some complex problems in computer sciences. We illustrate Theorem 3.2 by studying the existence and uniqueness of the solutions of the functional equation presented in the following example.

Example 3.7. Let X and Y be Banach spaces.

such that

for all

It is easy to get

Therefore, the Eq. (1) possesses unique bounded solution on S.

Example 3.8. Let

Let the

We define a mapping

It is easy to get

Definition 3.6. Let

Definition 3.7. Let

with

Theorem 3.2. Let

Proof. Similar to Theorem 3.1, we prove that

We find

and this completes the proof.

By applying the previous results and involving the

where

1.

2.

3.

4.

Let

We consider

We now claim

where

Assume now the function

where

Define the operator

Moreover, under the conditions

Hence,

We prove the existence of solutions to problem 4 utilising our deduced fixed point theorems. Now, let

Similar to the Example 6, one can easily verify the completeness of

Since

Example 4.1. Let

Note that for all

Assume that

Now let

We see that

The results obtained are supported by non-trivial examples and complement and extend some of the most recent results from the literature. We have made a contribution by establishing some basic fixed-point problems considering a

Future study is to investigate the sufficient conditions to guarantee the existence of a unique positive definite solution of the nonlinear matrix equations in the setting of

Acknowledgement: The authors Thabet Abdeljawad and Aziz Khan would like to thank Prince Sultan University for the support through the TAS research lab.

Funding Statement: The article is financially supported by Prince Sultan University.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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## Cite This Article

Bouftouh, O., Kabbaj, S., Abdeljawad, T., Khan, A. (2023). Quasi Controlled -Metric Spaces over -Algebras with an Application to Stochastic Integral Equations.*CMES-Computer Modeling in Engineering & Sciences, 135(3)*, 2649–2663.