Inhomogeneous Fractional Evolutionary Equation in the Sectorial Case

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, which is in the sector. 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 with respect to time, containing in the simplest case initial-boundary-value problems for fractional diffusion and diffusion-wave equations.