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　d(m+1)V/dt(m+1)　is　bounded　then　the　condition　that　dV/dt　is　negative　definite　can　be　weakened　and　replaced　by　that　dV/dt　less　than　or　equal　to　0　and　-(\dV/dt\　+　\d(2)V/dt(2)\　+　...　+　\d(m)V/dt(m)\　+　\d(m+p)V/dt(m+p)\)　is　negative　definite.　(C)　2004　Elsevier　Inc.　All　rights　reserved.