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Zbl 1042.34063
He, Zhimin; Ge, Weigao
Oscillations of second-order nonlinear impulsive ordinary differential equations. (English)
[J] J. Comput. Appl. Math. 158, No. 2, 397-406 (2003). ISSN 0377-0427

The authors study the second-order impulsive ordinary differential equation $$\left(r(t)\bigl(x'(t)\bigr)^\sigma \right)'+f(t,x(t))=0, \qquad t\geq t_0, \ t\neq t_k, \ k=1,2,\dots \eqno(1)$$ where $r\in C({\Bbb R}, (0,\infty))$, $f\in C({\Bbb R}\times {\Bbb R}, {\Bbb R})$ and $f$ satisfies the sign condition $xf(t,x)>0$ for all $x\neq 0$ and the growth condition $\frac{f(t,x)}{\phi( x)}\geq q(t)$ for all $x\neq 0$ where $q\in C({\Bbb R}, [0,\infty))$, $x\phi( x)>0$ for $x\neq 0$ and $\phi'(x)\geq 0$. The impulses $$x(t_k^+)=g_k(x(t_k)), \qquad x'(t_k^+)=h_k(x'(t_k)), \qquad k=1,2,\dots$$ are supposed to satisfy $g_k, h_k\in C({\Bbb R},{\Bbb R})$ and $$ \overline{a}_k\leq \frac {g_k(x)}x\leq a_k,\qquad \overline{b}_k\leq \frac {h_k(x)}x\leq b_k,\qquad k=1,2,\dots,$$ where $a_k$, $\overline{a}_k$, $b_k$ and $\overline{b}_k$ are positive real numbers. Sufficient conditions for the oscillation of equation (1) are established. One of the typical results is the following theorem: \par Assume that $$\lim_{t\to\infty}\int_{t_0}^t\left(\frac 1{r(s)}\right)^{1/\sigma} \prod_{t_o<t_k<s}\frac {\overline b_k}{a_k}\,ds=+\infty$$ and there exists a positive integer $k_0$ such that $\overline a_k\geq 1$ for $k\geq k_0$. If $$ \int_\varepsilon^\infty\frac{du}{(\phi(u))^{1/\sigma}}<+\infty, \qquad \int_{-\varepsilon}^{-\infty} \frac{du}{(\phi(u))^{1/\sigma}}<+\infty, $$ hold for some $\varepsilon>0$ and $$ \sum_{k=0}^{\infty}\int_{t_{k}^+}^{t_{k+1}} \left(\frac 1{r(s)}\right)^{1/\sigma} \left( \lim_{t\to+\infty}\int_s^t\prod_{s<t_k<u} \left(\frac 1{b_k}\right)^\sigma q(u)\,du \right)^{1/\sigma}\,ds=+\infty, $$ then every solution of equation (1) is oscillatory.
[Robert Mař\'ik (Brno)]

MSC 2000:: *34C10 Qualitative theory of oscillations of ODE: Zeros, etc.
34A37 Differential equations with impulses

Keywords: impulsive differential equation; oscillation; second-order; differential equation

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