Asymptotic Behavior of Solutions of Functional Differential Equations with Finite Delays

In this paper we consider a sufficient condition for W ( t , x ( t )) to approach zero as t →∞, where x ( t ) is a solution of a non-autonomous functional differential equation with finite delays and W ( t , x ) is a so-called Lyapunov function. We shall show that in the applications this provides useful information for asymptotic behavior of the solution x ( t ). For example, we generalize examples given by J. R. Haddock and J. Terjeki ( J. Differential Equations 48 , 1983, 95–122) to the case of non-autonomous systems.