Representation of solutions of a certain integro-differential equation and applications

In this paper, we consider a second-order integro-differential equation with unbounded operator coefficients in a Hilbert space, which is an abstract hyperbolic equation perturbed by an integral term with a difference-type kernel of a special form. This equation arises in the description of various viscoelastic systems. Using the system of generalized eigenvectors of the operator bundle associated with the equation considered, we construct a $p$-basis in the orthogonal sum of Hilbert spaces. Using this basis, we find a representation of the solution of the equation. We also discuss possible applications to problems of viscoelasticity.