Oscillations of second order nonlinear ODEs with impulses
Four oscillation theorems and six corollaries are proved for the semilinear impulse problem \[ \begin{cases} -y''= f(t,y)\quad\text{in }I= [t_0,\infty),\;t\neq t_k,\;k=1,2,\dots\\ y(t^+_0)= y_0,\;y'(t^+_0)= y_0'\\ y(t^+_k)= g_k(y(t_k)),\;q'(t^+_k)= h_k(y'(t_k)),\;k=1,2,\dots\end{cases}\tag{1} \] for \(f\in C(I\times\mathbb{R},\mathbb{R})\), \(sf(t,s)>0\) for all \(s\neq 0\), \(g_k, h_k\in C(\mathbb{R},\mathbb{R})\), \(g_k(s)/s\), \(h_k(s)/s\) are bounded above and below by positive constants for all \(s\neq 0\), and \(0\leq t_0<t_1<\cdots< t_k\to+\infty\) as \(k\to\infty\). The results provide various sufficient conditions for all solutions \(y\) of (1) to be oscillatory at \(\infty\). Examples are included for linear, superlinear, and sublinear differential equations, indicating that insertion of impulses can transform nonoscillatory problems into oscillatory ones.
scientific article; zbMATH DE number 1031640
scientific article