A weaker criterion of asymptotic stability

The classical criterion of asymptotic stability for differential equations requires the existence of a Liapunov function VV with negative definite dV/dt. Successive efforts have been made to weaken the negative definiteness of dV/dt to semi-negative definiteness. Recently, it was given an interesting result that, under the boundedness of dm+1V/dtm+1, the negative definiteness can be weakened to that dV/dt≤0 together with that −(|dV/dt|+|d2V/dt2|+⋯+|dmV/dtm|+|dm+pV/dtm+p|) 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.