A constructive comparison technique for determining the asymptotic behaviour of linear functional differential equations with unbounded delay

In this paper, we obtain exact non-exponential rates of growth and decay of solutions of linear functional differential equations with unbounded delay. As a by-product, exponential asymptotic stability is ruled out for asymptotically stable solutions of these equations. Equations with maximum functionals on the right hand side, as well as perturbed equations, are considered. We also give examples showing how the rate of growth or decay of solutions depends on the rate of growth of the unbounded delay. The results established here are obtained by constructing carefully functions which satisfy upper and lower differential inequalities and one aim of the paper is to elucidate this constructive comparison technique.