Investigation of Existence and Ulam's type Stability for Coupled Fractal Fractional Differential Equations

Abstract

In this study, we investigate the coupled system of fractal fractional differential equations (FFDEs) from an existence and stability point of view. The area related to coupled FFDEs is significant because its permitting us to analyze and predict the relationships between several variables throughout the real-world phenomena. Coupled systems  have numerous applications in different fields, such as modeling of brain activities and spreads of disease in biology and medicine, modeling of mechanical systems, electrical circuits in engineering, and modeling of population dynamics, predator-prey models in ecology and financial mathematical in economics. Furthermore, the afore mentioned area is significant for environmental science in pollution control, climate change modeling, artificial intelligence, and control systems in technology. We  analysis a coupled system and make more informed choices in a variety of fields through coupled system of FFDEs. Keeping these informative aspects of coupled systems in mind, this article aims to explore the qualitative analysis such as existence, uniqueness, and stability for the solution of underlying coupled systems of FFDEs. The tools of functional analysis (FA) and fixed point theory (FPT) have been applied to deduce our required results. We have used the Banach contraction principle (BCP) and the Krasnoselskii fixed point theorem (KFPT) to demonstrate the conditions for existence and uniqueness of a solutions (EUS). Additionally, the results related to stability have demonstrated by using Ulam's concept. Subsequently, a suitable example is given to illustrates the validity of this work.