Inhomogeneous Fractional Evolutionary Equation in the Sectorial Case

Abstract

In this paper, we prove the unique solvability of the Cauchy problem for a linear inhomogeneous equation in a Banach space that is solved with respect to the Gerasimov–Caputo fractional derivative. We assume that the operator acting on the unknown function in the equation generates a set of resolving operators of the corresponding homogeneous equation, which is exponentially bounded and analytic in a sector containing the positive semiaxis. The general form of solutions to the Cauchy problem is obtained. The general results are applied to the study of the unique solvability of a certain class of initial-boundary-value problems for partial differential equations solvable with respect to the Gerasimov–Caputo fractional derivative by time, containing in the simplest case initial-boundary-value problems for fractional diffusion and diffusion-wave equations.