Open Access

ARTICLE

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

Ouafaa Bouftouh¹, Samir Kabbaj¹, Thabet Abdeljawad^2,3,*, Aziz Khan²
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.

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Keywords

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

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