A note on the asymptotic behavior of nonoscillatory solutions of half-linear ordinary differential equations

The asymptotic behavior of nonoscillatory solutions of the half-linear differential equation \[ (p(t)|x'|^{\alpha}\operatorname{sgn}x')'+q(t)|x|^{\alpha}\operatorname{sgn}x=0,\qquad t\geq t_{0}, \] is discussed. It is assumed that $P(t)\equiv\int_{t_{0}}^{t}p(s)^{-1/\alpha}ds$ $(t\geq t_{0})$ diverges to $\infty$ as $t\to\infty$, and that $Q(t)\equiv\int_{t}^{\infty}q(s)ds$ $(t\geq t_{0})$ exists and is finite. It is shown that, under certain conditions on $P(t)$ and $Q(t)$, if a nonoscillatory solution $x(t)$ of the above equation satisfies the asymptotic property of the type $\displaystyle p(t)^{1/\alpha}P(t)[x'(t)/x(t)]\to\lambda\neq 0$ $(t\to\infty)$, then $\displaystyle x(t)\sim cP(t)^{\lambda}$ and $\displaystyle x'(t)\sim c\lambda p(t)^{-1/\alpha}P(t)^{\lambda-1}$ $(t\to\infty)$, where $c$ is a nonzero constant.