Solvability of an Initial-value Problem of Non-linear Implicit Fractal Differential Equation

Abstract

In this paper we study the initial-value problem of the fractal differential equation𝑑𝑥(𝑡)/𝑑𝑡^𝛽 = 𝑓( 𝑡, 𝑑𝑥(𝑡)/𝑑𝑡^𝛼 ), 𝑎.𝑒. 𝑡∈(0,𝑇], 𝑥(0)=𝑥_0.We discuss the existence of at least one solution 𝑥∈AC[0,T]. The Uniqueness of the solution will be proved. The continuous dependence on the initial data 𝑥0 and on the function 𝑓 will be analysed. Also, the Hyers - Ulam stability of the problem will be established. Finally, some examples will be given to verify our results.2020 MSC. 34A12, 34A34, 45G05.Key words. Fractal derivative, functional equation, existence of solution, continuous dependence, Hyers - Ulam stability.Implicit fractal differential equations represent a fascinating area of study that combines fractal geometry and differential equations. These equations involve fractal-like structures and demonstrate intricate and self-similar patterns. They are characterized by their non-linearity and often exhibit complex behaviours, such as chaos and self-replication (see [13, 14]). Differential equations and fractal differential equations have applications in various fields, including physics, biology, and finance, and have garnered significant interest due to their ability to model complex systems with remarkable precision.