On the asymptotic stability of equilibrium

The classical criterion of asymptotic stability of the zero solution of equations x′=f(t,x) is that there exists a positive definite function V which has infinitesimal upper bound such that dVdt is negative definite. In this paper we prove that if dm+1Vdtm+1 is bounded then the condition that dVdt is negative definite can be weakened and replaced by that dVdt⩽0 and −(|dVdt|+|d2Vdt2|+⋯+|dmVdtm|+|dm+pVdtm+p|) is negative definite.