Manabu Naito

Oscillation and Nonoscillation for Certain Nonlinear Systems of Ordinary Differential Equations

The two-dimensional nonlinear system
\[ u' = a(t)|v|^{1/\alpha}\mathrm{sgn}\,v, \quad v' = - b(t)|u|^{\alpha}\mathrm{sgn}\,u \eqno{(1.1)} \]
is considered under the assumptions that $\alpha > 0$, $a, b \in C[t_{0},\infty)$, $a(t) \geq 0$, $a(t) \not\equiv 0$ $(t \geq t_{0})$. It is shown that, under certain conditions on $a(t)$ and $b(t)$, if system (1.1) is nonoscillatory, then the integral averages with the weight $a(t)$ of the functions $|c(t)|$ and $|c(t)|^{(\alpha + 1)/\alpha}$ tend to $0$ as $t\to\infty$. Here,
c(t) = \lim_{\tau \to \infty} \bigg(\int\limits_{t}^{\tau}a(s)\,ds\bigg)^{-1}\int\limits_{t}^{\tau}a(s)\bigg(\int\limits_{t}^{s}b(r)\,dr\bigg)\,ds, \;\; t \geq t_{0}.
Using this result, we can establish many kinds of Hartman-Wintner type oscillation criteria for (1.1).

Mathematics Subject Classification: 34C10

Key words and phrases: Half-linear system, oscillation, nonoscillation, Hartman-Wintner theorem