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

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 $\alpha \gt 0$, $a,b \in C[t_{0},\infty)$, $a(t) \geq 0$ ($t t_{0}$), new oscillation criteria and nonoscillation are given in both cases $\int_{t_{0}}^{\infty} a(s) \, ds \infty$ \lt \infty$. One of main results is an analogue Hartman–Wintner theorem. Our generalize Li Yeh's for second order half-linear scalar equations.