Approximations of Solutions for an Impulsive Fractional Differential Equation with a Deviated Argument

Abstract

In the present work, we consider an impulsive fractional differential equation with a deviated argument in an arbitrary separable Hilbert space H. We obtain an associated integral equation and then consider a sequence of approximate integral equations. The existence and uniqueness of solutions to every approximate integral equation is obtained by using analytic semigroup and Banach fixed point theorem. Next we demonstrate the convergence of the solutions of the approximate integral equations to the solution of the associated integral equation. We study the Faedo–Galerkin approximation of the solution and establish some convergence results. Finally, we consider an example to show the effectiveness of obtained theory.