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Oscillations of second-order nonlinear impulsive ordinary differential equations. (English) Zbl 1042.34063

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 \(r\in C({\mathbb R}, (0,\infty))\), \(f\in C({\mathbb R}\times {\mathbb R}, {\mathbb R})\) 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 \(q\in C({\mathbb R}, [0,\infty))\), \(x\phi( x)>0\) 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 \(g_k, h_k\in C({\mathbb R},{\mathbb R})\) 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:
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.

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34A37 Ordinary differential equations with impulses
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