Asymptotic behaviour of solutions of linear integro-differential equations

Given the linear integro-differential equation (P_o) on a reflexive Banach space, we prove the existence of unbounded solutions with an exponential growth rate for a class of initial-value problems. Since the appearing kernel functions are of convolution type on a semi-axis, abstract Wiener-Hopf techniques, recently developed by Feldman 3,4,5], are used for the construction of the resolving operator associated with the problems under consideration. Applicability of the results is shown to initial boundary-value problems arising in the theory of generalized heat conduction in materials with memory and of viscoelasticity.