Approximations of solutions to a retarded type fractional differential equation with a deviated argument

Abstract

In the present work, we are concerned with approximations of solutions to a retarded type fractional differential equation with a deviated argument in a separable Hilbert space~$H$. We consider an integral equation associated with a given problem and then consider a sequence of approximate integral equations. We prove the existence, uniqueness and convergence to each of the approximate integral equations by using analytic semigroup theory and the fixed point method. We also prove that the limiting function satisfies the associated integral equation. Finally, we consider Faedo-Galerkin approximations of solutions and prove some convergence results.