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Oscillation and nonoscillation for second order linear impulsive differential equations. (English) Zbl 0895.34031
The author establishes an oscillation theorem for linear impulsive differential equations (*) $u''=-p(t)u$, $t\ge 0$, where $p(t)$ is an impulsive function defined by $p(t)= \sum^\infty_{n=1} a_n \delta (t-t_n)$ with $a_n>0$ for all $n\in \bbfN$ and $0\le t_0<t_1 <t_2< \cdots <t_n<\dots$, $t_n\to\infty$ as $n\to\infty$. Next, this result is applied to derive sufficient conditions for the nonoscillation and oscillation of (*) in each one of the particular cases corresponding to $t_n=t_0+ \lambda^{n-1}T$, $\lambda>1$, $T>0$, and $t_n= t_0+nT$.

34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
34A37Differential equations with impulses
Full Text: DOI
[1] C. Huang, Oscillation and nonoscillation for second order linear differential equations · Zbl 0880.34034
[2] Hartman, P.: Ordinary differential equations. (1982) · Zbl 0476.34002
[3] Yan, J.: Oscillation theorems for second order linear differential equations with damping. Proc. amer. Math. soc. 98, 276-282 (1986) · Zbl 0622.34027