Nonoscillation and oscillation of second-order impulsive differential equations with periodic coefficients (Q427589)

The authors consider a second-order half-linear system of the form NEWLINE\[NEWLINE\begin{gathered} (\phi_\alpha(x'))'+ p(t)\phi_\alpha(x')+ q(t) \phi_\alpha(x)= 0,\quad t\neq\theta_i,\\ \Delta\phi_\alpha(x')+ \beta_i\phi_\alpha(x)= 0,\quad t=\theta_i,\end{gathered}\tag{1.1}NEWLINE\]NEWLINE where \(p\), \(q\) are \(\omega\)-periodical functions and \(\alpha> 1\).NEWLINENEWLINE For \(\alpha= 2\) the equation (1.1) is said to be an impulse Hill equation. The authors find sufficiently conditions (1.1) to be nonoscillatory or oscillatory.