Linearized oscillation theory for a nonlinear delay impulsive equation

摘要

For a scalar nonlinear impulsive delay differential equationẏ(t)+∑k=1mrk(t)fk[y(hk(t))]=0,t≠τj,y(τj)=Ij(y(τj−))with rk(t)⩾0,hk(t)⩽t, limj→∞τj=∞, such an auxiliary linear impulsive delay differential equationẋ(t)+∑k=1mrk(t)ak(t)x(hk(t))=0,x(τj)=bjx(τj−)is constructed that oscillation (nonoscillation) of the nonlinear equation can be deduced from the corresponding properties of the linear equation. Coefficients rk(t) and delays are not assumed to be continuous. Explicit oscillation and nonoscillation conditions are established for some nonlinear impulsive models of population dynamics, such as the impulsive logistic equation and the impulsive generalized Lasota–Wazewska equation which describes the survival of red blood cells. It is noted that unlike nonimpulsive delay logistic equations a solution of a delay impulsive logistic equation may become negative.