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KEY WORDS: Lyapunov Functions, almost-periodic systems, asymptotic stability. For the autonomous system & = f(x), f(0) = 0, the following result strengthening the Lyapunov theorem on asymptotic stability is known. A sufficient condition for asymptotic stability of the zero equflibril,m position is the existence of a positive definite Lyapunov function v(x) such that ~ < 0 and the level surfaces v = const > 0 do not contain entire trajectories [1]. In this paper, we show that this result can be extended (with obvious modifications of the statement) to nonautonomous systems, assllrn~ng that the Lyapunov function v(x) and its partial derivatives are ~]most-periodic in t.Earlier [2], we have shown that in the special case in which an almost-periodic system is linear and v = (G(t)x, z) with ~.lmost-periodic G and G, a certain condition stated below is sufficient for exponential stability (here and in the sequel (x, y> = ~ xiyi).

We establish a sufficient condition for asymptotic stability for the systems under consideration in which the condition on the difference derivative by the system is weakened in comparison with the Lyapunov conditionν < 0. We also obtain applications to the analysis of stability of equilibria of dynamical systems with infinite-dimensional phase space.

For a bounded linear operator on Hilbert space we define a sequence of so-called minimal vectors in connection with invariant subspaces and show that this presents a new approach to invariant subspaces. In particular , we show that for any compact operator K some weak limit of the sequence of minimal vectors is noncyclic for all operators commuting with K and that for any normal operator N , the norm limit of the sequence of minimal vectors is noncyclic for all operators commuting with N. Thus, we give a new and more constructive proof of existence of invariant subspaces. The sequence of minimal vectors does not seem to converge in norm for an arbitrary bounded linear operator. We will prove that if T belongs to a certain class C of operators, then the sequence of such vectors converges in norm, and that if T belongs to a subclass of C, then the norm limit is cyclic.

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