We study a class of equations in Banach spaces with a Riemann–Liouville-type integro-differential operator with an operator-valued convolution kernel. The properties of \(k\)-resolving operators of such equations are studied and the class \(\mathcal A_{m,K,\chi}\) of linear closed operators is defined such that the belonging to this class is necessary and, in the case of commutation of the operator with the convolution kernel, is sufficient for the existence of analytic in the sector \(k\)-resolving families of operators of the equation under study. Under certain additional conditions on the convolution kernel, we prove theorems on the unique solvability of the nonhomogeneous linear equation of the class under consideration if the nonhomogeneity is continuous in the norm of the graph of the operator from the equation or Hölder continuous. We obtain the theorem on sufficient conditions on an additive perturbation of an operator of the class \(\mathcal A_{m,K,\chi}\) in order that the perturbed operator also belong to such a class. Abstract results are used in the study of initial-boundary value problems for a system of partial differential equations with several fractional Riemann–Liouville derivatives of different orders with respect to time and for an equation with a fractional Prabhakar derivative with respect to time.

The paper investigates integro-differential equations in Banach spaces with operators, which are a composition of convolution and differentiation operators. Depending on the order of action of these two operators, we talk about integro-differential operators of the Riemann—Liouville type, when the convolution operator acts first, and integro-differential operators of the Gerasimov type otherwise. Special cases of the operators under consideration are the fractional derivatives of Riemann—Liouville and Gerasimov, respectively. The classes of integro-differential operators under study also include those in which the convolution has an integral kernel without singularities. The conditions of the unique solvability of the Cauchy type problem for a linear integro-differential equation of the Riemann—Liouville type and the Cauchy problem for a linear integrodifferential equation of the Gerasimov type with a bounded operator at the unknown function are obtained. These results are used in the study of similar equations with a degenerate operator at an integro-differential operator under the condition of relative boundedness of the pair of operators from the equation. Abstract results are applied to the study of initial boundary value problems for partial differential equations with an integro-differential operator, the convolution in which is given by the Mittag-Leffler function multiplied by a power function.