On the Properties of Semigroups Generated by Volterra Integro-Differential Equations with Kernels Representable by Stieltjes Integrals

A Abstract Volterra integro-differential equations with integral operator kernels representable by Stieltjes integrals of the exponential function are studied. The approach is based on the study of one-parameter semigroups for linear evolution equations. A method for reducing the original initial value problem for a model integro-differential equation with operator coefficients in a Hilbert space to the Cauchy problem for a first-order differential equation in an extended function space is presented. The existence of a contraction C-0-semigroup is proved. As a corollary, we establish the well-posed solvability of the resulting Cauchy problem for the first-order differential equation in an extended function space and the initial value problem for the original abstract integro-differential equation and indicate a relation between their solutions.