Regular Article
Positive Solutions and Asymptotic Behavior of Delay Differential Equations with Nonlinear Impulses,☆☆

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Abstract

Consider the delay differential equation (DDE) with nonlinear impulsesẋ(t)+i=1npi(t)x(tτi)=0,ttj,tt0,x(t+j)x(tj)=Ij(x(tj)),j=1,2,…,(∗)wherepi  C([t0, ∞), R+), τi  0 fori = 1, 2,…,n, andIj  C(R, R) forj = 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.

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This work was partially supported by the National Natural Science Foundation of China.

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Communicated by William, F. Ames