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　dV/dt　is　negative　definite.　In　this　paper　we　prove　that　if　dm+1V/dtm+1　is　bounded　then　the　condition　that　dV/dt　is　negative　definite　can　be　weakened　and　replaced　by　that　dV/dt　≤　0　and　-　(　dV/dt　+　d2V/dt2　+　⋯　+　dmV/dtm　+　dm+p　V/dtm+p　)　is　negative　definite.　©　2004　Elsevier　Inc.　All　rights　reserved.