Integro-Differential Equations of Gerasimov Type with Sectorial Operators

The issues of existence and uniqueness of a solution to the Cauchy problem are studied for a linear equation in a Banach space with a closed operator at the unknown function that is resolved with respect to a first-order integro-differential operator of Gerasimov type. The properties of resolving families of operators of the homogeneous equations are investigated. It is shown that sectoriality, i.e., belonging to the class of operators $$\mathcal{A}_{K}$$ introduced here, is a necessary and sufficient condition for the existence of an analytical resolving family of operators in a sector. A theorem on the perturbation of operators of the class $$\mathcal{A}_{K}$$ is obtained, and two versions of the theorem on the existence and uniqueness of a solution to a linear inhomogeneous equation are proved. Abstract results are used to study initial–boundary value problems for an equation with the Prabhakar time derivative and for a system of partial differential equations with Gerasimov–Caputo time derivatives of different orders.