Existence and Uniqueness of Solutions For Cauchy-Type Problems Involving Mth-Level Fractional Derivatives

Abstract

This paper investigates the existence and uniqueness of global solutions in the space of weighted continuous functions for Cauchy-type problems involving fractional differential equations with mth-level fractional derivatives. Using the Banach fixed-point theorem and the step method, we establish our results within an appropriate functional space, demonstrating the equivalence between the given problem and a corresponding Volterra integral equation. As a particular case, we examine the Cauchy-type problem for fractional differential equations with second-level fractional derivatives. Key properties and fundamental results related to this type of fractional calculus are discussed. Additionally, we derive significant Cauchy-type problems from both mth- and second-level fractional derivatives, which have been extensively studied in Riemann–Liouville, Caputo, and Hilfer fractional derivatives. Finally, we analyze the stability of the solutions to the weighted Cauchy-type problem.