Asymptotic behaviour of linear evolutionary integral equations

The asymptotic behaviour of bounded solutions of evolutionary integral equa tions in a Banach space X (u)over dot(t) = integral (infinity)(0) A(0)(tau)(u)over dot(t-tau) d tau integral (infinity)(0) dA(1)(tau )u(t-tau) + f(t), t epsilon R, on the real line and of (v)over dot(t) = (integral (t)(0) A(t-tau )v(tau) d tau) + g(t), t epsilon R+, on the half-line are studied. Assuming that the inhomogeneity f (resp. g) b elongs to a given homogeneous subspace of BUC(R; X) (resp. BUC(R+;X)) it is shown that given bounded solutions u (resp. v) belong also to E provided t he spectra of these equations are countable. The results are applied to an equation of scalar type which is of importance in applications :like viscoe lasticity.