A weaker criterion of asymptotic stability

The classical criterion of asymptotic stability for differential equations requires the existence of a Liapunov function V with negative definite . Successive efforts have been made to weaken the negative definiteness of to semi-negative definiteness. Recently, it was given an interesting result that, under the boundedness of , the negative definiteness can be weakened to that together with that is negative definite. Unfortunately, its basic lemma is proved to be false by a counter example and cannot support this interesting result. In this paper we re-establish the weak criterion for asymptotic stability with less requirements.