A remark on the existence of specific nonoscillatory solutions of half-linear ordinary differential equations

The half-linear ordinary differential equation ${\left({\left|u^{\prime }\right|}^{\alpha }\mathrm{sgn}{u}^{\prime }\right)}^{\prime }=\alpha ({\lambda }^{\alpha +1}+b\left(t\right)){\left|u\right|}^{\alpha }\mathrm{sgn}u,t\ge t_0,$is considered. Here, $\alpha$ and $\lambda$ are positive constants, and $b\left(t\right)$ is a real-valued continuous function on $[t_0,\infty )$ and does not change sign for sufficiently large $t$. It is shown that, under the assumption $0<\alpha \le 1$, if the above equation has a solution $u\left(t\right)$ which behaves like $e^{\lambda t}$ or $e^{-\lambda t}$ as $t\to \infty$, then $b\left(t\right)$ is absolutely integrable on $[t_0,\infty )$.