Oscillation and Nonoscillation for Two-dimensional Nonlinear Systems of Ordinary Differential Equations

Abstract

For the two-dimensional nonlinear system \[ u' = a(t) |v|^{1/\alpha} \operatorname{sgn}v, \quad v' = - b(t) |u|^{\alpha} \operatorname{sgn}u \] with $\alpha \gt 0$, $a,b \in C[t_{0},\infty)$, $a(t) \geq 0$ ($t \geq t_{0}$), new oscillation criteria and nonoscillation criteria are given in both cases $\int_{t_{0}}^{\infty} a(s) \, ds = \infty$ and $\int_{t_{0}}^{\infty} a(s) \, ds \lt \infty$. One of the main results is an analogue of the Hartman–Wintner oscillation theorem. Our results generalize Li and Yeh's results for second order half-linear scalar equations.