Explicit estimates and limit formulae for the solutions of linear delay functional differential systems with nonnegative Volterra type operators

In this paper we discuss the asymptotic properties of solutions of special inhomogeneous linear delay functional differential systems generated by nonnegative Volterra type operators. In a recent paper of us a variation of constants formula has been given for such systems, and this is particularly suited to handling the studied problems. First, we investigate the boundedness of the solutions. Then some estimates are given for the upper and lower limits of the solutions. By using this, we study the existence of the limit of the solutions, and obtain a formula for the limit. The applicability of our results is illustrated by studying the dependence of limits of the solutions on the choice of inhomogeneities, synchronisation, and Lotka-Volterra type delay functional differential systems.