Development of the Lyapunov function method in the stability problem for functional-differential equations

In the present paper, we consider the stability problem for delay functional-differential equations with finite delay. We suggest a development of the Lyapunov function method involving the use of scalar comparison equations and limit functions and equations. We prove a localization theorem for the positive limit set of a bounded solution and a theorem on the asymptotic stability of the zero solution. We present examples of sufficient conditions for the asymptotic stability of solutions of systems of the first, second, and arbitrary orders.