Semigroups of operators generated by integro-differential equations with kernels representable by Stieltjes integrals

Abstract Volterra integro-differential equations with kernels of integral operators representable by Stieltjes integrals are investigated. The presented results are based on the approach related to the study of one-parameter semigroups for linear evolution equations. We present the method of reduction of 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. The existence of the contractive $C_0$-semigroup is proved. An estimate for the exponential decay of the semigroup is obtained.