Correct solvability and exponential stability for solutions of Volterra integro-differential equations

Abstract integro-differential equations that are operator models of viscoelasticity problems are studied. The kernels of the integral operators can be specified as sums of decreasing exponentials or sums of Rabotnov functions with positive coefficients, which are widely used in viscoelasticity theory. A method is described whereby the original initial value problem for a model integro-differential equation with operator coefficients in a Hilbert space is reduced to the Cauchy problem for a first-order differential equation. Exponential stability of solutions is established under known assumptions on the kernels of the integral operators. The results are used to establish the correct solvability of the original initial value problem for a Volterra integro-differential equation with corresponding solution estimates.