Analysis of impulsive Caputo fractional integro-differential equations with delay

The main focus of this manuscript is to study an impulsive fractional integro-differential equation with delay and Caputo fractional derivative. The existence solution of such a class of fractional differential equations is discussed for linear and nonlinear case with the help of direct integral method. Moreover, Banach's fixed point theorem and Schaefer's fixed point theorem are use to discuss the uniqueness and at least one solution of the said fractional differential equations, respectively. Some hypothesis and inequalities are utilize to present four different types of Hyers–Ulam stability of the mentioned impulsive integro-differential equation. Example is provide for the illustration of main results.