Asymptotic Behavior Theorems for Non-autonomous Functional Differential Equations via Lyapunov-Razumikhin Method

In this paper we are concerned with non-autonomous functional differential equations with finite delays and we present sufficient conditions for every bounded solution to approach a closed set Ω which contains zero. As the application, we generalize Cooke and Huang′s theorem (Theorems 6 and 9, "Delay and Differential Equations," pp. 65-77, World Scientific, 1992). Furthermore, we also extend the result proved by Ladas, Sficas, and Stavroulakis (Theorem 3, Proc. Amer. Math. Soc.88 (1983), 247-253).