Perturbed linear equations with Hilfer fractional derivative

Abstract

We study issues on unique solvability for the Cauchy–type problem for linear equations in a Banach space with the Hilfer fractional derivative, with an operator generating a strongly continuous resolving family, and with a bounded perturbation operator, either constant or time–dependent. It is shown that the addition of a constant bounded operator does not move us outside the class of generators of strongly continuous resolving families of operators. Based on the technique of the proof of this result, we show the existence of a unique solution to the Cauchy–type problem for the equation with a nonautonomous perturbation operator. We employ abstract results in the study of the unique solvability of an initial boundary value problem with periodic conditions in the spatial variable for the equation with the Hilfer fractional derivative in time and with the Laplace operator perturbed by a nonautonomous linear integral operator.