Fundamental Results about the Fractional Integro-Differential Equation Described with Caputo Derivative

Abstract

In this paper, we study the existence and uniqueness of the mild solution of the fractional integro-differential with the nonlocal initial condition described by the Caputo fractional operator. Note that here the order of the Caputo derivative satisfies the condition that $\alpha \in \left(1,2\right)$ . The existence of $\alpha$ -resolvent operator in Banach space and fixed point theorem has been utilized in the proof of the existence of the mild solution. We have established in this paper the Hyers-Ulam stability of the mild solution of the considered fractional integro-differential equation. An illustrative example has been provided to support the main findings of the paper.