The least possible impulse for oscillating all nontrivial solutions of second-order nonoscillatory differential equations (Q2114409)

The author considers the impulsive differential equation \[ \begin{aligned} &x''+c(t)\,x=0\,,\quad t\neq \theta_k>t_0\;\\ & \Delta x'(\theta_k)+b_k\,x(\theta_k)=0\,, \end{aligned} \] where \(c\) is a continuous real-valued function on \([t_0,\infty]\), \(\{ \theta_k\}\) is a sequence satisfying \(\theta_i<\theta_{i+1}\) for \(i\in\mathbb{N}\) and \(\lim_{k\to\infty}\theta_k=\infty\), \(\Delta\) is the difference operator \(\Delta z(\theta_k)=z(\theta_k^+)-z(\theta_k^-)\), and \(\{b_k\}\) is a sequence of positive real numbers.\par The aim of the paper is to calculate the least amount of impulse required to change the nonoscillatory character of the solutions of the second order linear differential equation \(x''+c(t)\,x=0\,\) to oscillatory.