A qualitative study on generalized Caputo fractional integro-differential equations

Abstract

The aim of this article is to discuss the uniqueness and Ulam–Hyers stability of solutions for a nonlinear fractional integro-differential equation involving a generalized Caputo fractional operator. The used fractional operator is generated by iterating a local integral of the form $(I_{a}^{\rho }f)(t)=\int _{a}^{t}f(s)s^{\rho -1}\,ds$ ( I a ρ f ) ( t ) = ∫ a t f ( s ) s ρ − 1 d s . Our reported results are obtained in the Banach space of absolutely continuous functions that rely on Babenko’s technique and Banach’s fixed point theorem. Besides, our main findings are illustrated by some examples.

Existence and Ulam–Hyers Stability of a Fractional-Order Coupled System in the Frame of Generalized Hilfer Derivatives

“…Moreover, Li and Zada in [28] provided connections between the stability of U-H and uniform exponential over Banach space. These types of stability have been very wellinvestigated for FDEs, see [29][30][31][32][33][34]. The existence and stability of solutions of the following ϑ-Hilfer type FDE:…”