Comparison theorems for second-order differential equations with deviating arguments (Q792528)

The author generalizes the classical Sturm comparison theorem to two nonlinear systems of second order differential equations with deviating arguments \[ (1)\quad x''(t)+p(t)f(x(\tau(t)))=0,\quad(2)\quad x''(t)+q(t)g(x(\theta(t)))=0, \] where p(t), q(t) are continuous and nonnegative on the semi-axis \(t\geq t_ 0\), f(z), g(z) are continuous for \(z\in {\mathbb{R}}\), strictly increasing for \(| z| \geq z_ 0\geq 0\) and satisfying sgn f(z)\(=sgn g(z)=sgn z\), and \(\tau\) (t), \(\theta\) (t) are continuous and satisfying \(\lim_{t\to +\infty}\tau(t)=\lim_{t\to +\infty}\theta(t)=+\infty.\) The main theorem is as follows: If q(t)\(\geq p(t)\), \(\theta(t)\geq \tau(t), | g(z)| \geq | f(z)|\) for \(t\geq t_ 1\geq t_ 0\), \(| z| \geq z_ 1\geq z_ 0\), then the fact that all solutions of (1) are oscillating implies that all solutions of (2) are oscillating.