Positive Solutions and Asymptotic Behavior of Delay Differential Equations with Nonlinear Impulses*

Consider the delay differential equation DDE.with nonlinear impulses n ˙x t.q pi t.x tyti.s0, t/tj , tGt0 , is1 x tq.yx t .sI x t .., js1, 2, . . . , ). j j j j where pigC wt0,`.,Rq., tiG0 for is1, 2, . . . , n, and IjgC R, R. for js 1, 2, . . . . The purpose of this paper is to obtain a necessary and sufficient condition for the existence of positive solutions of DDE without impulses and to establish a kind of order persistence of solutions of Eq. ). and criteria of the asymptotic behavior of Eq. )., which can be used to improve and develop some of the known results in the literature.