Poincaré-Perron problem for half-linear ordinary differential equations

This paper deals with the Poincaré-Perron problem for the second order half-linear ordinary differential equation $$ (\Phi_{\alpha}(u'))' + (a_{1} + f_{1}(t)) \Phi_{\alpha}(u') + (a_{2} + f_{2}(t))\Phi_{\alpha}(u) = 0, $$ where $\Phi_{\alpha}(u) = |u|^{\alpha}\mathrm{sgn}\, u$ with $\alpha > 0$, and $a_{1}$ and $a_{2}$ are real constants, and $f_{1}(t)$ and $f_{2}(t)$ are real-valued continuous functions on $[t_{0},\infty)$. It is assumed that $\alpha|\lambda|^{\alpha + 1} + a_{1}\Phi_{\alpha}(\lambda) + a_{2} = 0$ has distinct real roots $\lambda_{1}$ and $\lambda_{2}$. Then it is possible to obtain a necessary and sufficient condition for the existence of nonoscillatory solutions $u_{i}(t)$ which satisfy $u_{i}'(t)/u_{i}(t) \to \lambda_{i}$ as $t \to \infty$ ($i = 1, 2$). It is also shown that, in this case, any nontrivial solution $u(t)$ satisfies $u'(t)/u(t) \to \lambda_{1}$ or $\lambda_{2}$ as $t \to \infty$. The classical results for linear equations ($\alpha = 1$) by Hartman-Wintner and the recent results by Řehák, Naito-Usami, Luey-Usami are unified and generalized to the above equation.