Localization of limit sets of solutions of nonautonomous delay equations by nonmonotone functionals

We present new generalizations of the Barbashin-Krasovskii theorem which also apply to delay equations with unbounded right-hand side. These generalizations are based on information on the localization of the limit sets of solutions, which is obtained with the use of two classes of not necessarily monotone Lyapunov functionals. The classes of functionals to be used contain sign-definite Lyapunov-Krasovskii functionals as well as Lyapunov-Razumikhin functions.