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Sucient conditions are given for the asymptotic stability and uniform asymptotic stability of the zero solution of the nonautonomous FDE's whose right-hand sides can be unbounded functions of the time. The theo- rems are based upon Lyapunov{Krasovski functionals whose derivatives with respect to the equations are negative semidenite and can vanish at long in- tervals. The functionals and their derivatives are estimated by either x(t), the norm of the instantaneous value of the solutions orkxtk2 ,t heL2-norm of the phase segment of the solutions. Examples are given to show that the conditions are sharp, and the main theorems with the two dierent types of estimates are independent and improve earlier results. The theorems are applied to linear and nonlinear retarded FDE's with one delay and with distributed delays.

On the asymptotic stability for nonautonomous functional differential equations by Lyapunov functionals