A generalization of Barbashin–Krasovski theorem

The classical criterion of asymptotic stability of the zero solution of equations x = f (t,x) is that there exists a function V (t,x), a( x ) V (t,x) b( x ) for some a, b ∈ K, such that V˙ (t,x) −c( x ) for some c ∈ K. In this paper we prove that if f (t,x) is bounded, V˙ (t,x) is uniformly continuous and bounded, then the condition that V˙ (t,x) −c( x ) can be weakened and replaced by V˙ (t,x) 0 and {(t, x): x = 0, V˙¯ (t,x) = 0} contains no complete trajectory of x = f¯(t,x), t ∈ [−T,T ], where V¯ (t,x) = limk→∞V (t +tk, x), f¯(t,x) = limk→∞f (t +tk, x) uniformly for (t, x) ∈ [−T,T ] × BH . © 2006 Elsevier Inc. All rights reserved