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Second-order quasilinear oscillation with impulses. (English) Zbl 1063.34004

The author considers the second-order impulsive differential equation \[ \biggl( r(t)\bigl| x'(t)\bigr|^{\alpha-1} x'(t)\biggr)'+ f \bigl(t, x(t)\bigr)=0, \quad t\geq t_0,\;t\neq t_k,\;k=1,2,3,\dots \]
\[ x(t_k^+)= g_k \bigl(x(t_k)\bigr),\;x'(t^+_k)=h_k\bigl(x'(t_k)\bigr),\;k=1,2,3, \dots \]
\[ x(t^+_0)=x_0,\;x'(t^+_0)= x'(t_0), \]
\[ 0\leq t_0<t_1 <\cdots<t_k< \dots,\;\lim t_k=\infty\;(k\to\infty). \] Sufficiently conditions for every solution to be oscillatory are derived.

MSC:

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