A note on the existence of solutions with prescribed asymptotic behavior for half-linear ordinary differential equations
Manabu Naito
Received November 25, 2022. Published online July 13, 2023.
Abstract: The half-linear differential equation $
(|u'|^{\alpha}{\rm sgn} u')' = \alpha(\lambda^{\alpha+ 1} + b(t))|u|^{\alpha}{\rm sgn} u$, $t \geq t_0$,
is considered, where $\alpha$ and $\lambda$ are positive constants and $b(t)$ is a real-valued continuous function on $[t_0,\infty)$. It is proved that, under a mild integral smallness condition of $b(t)$ which is weaker than the absolutely integrable condition of $b(t)$, the above equation has a nonoscillatory solution $u_0(t)$ such that $u_0(t) \sim{\rm e}^{- \lambda t}$ and $u_0'(t) \sim- \lambda{\rm e}^{- \lambda t}$ ($t \to\infty$), and a nonoscillatory solution $u_1(t)$ such that $u_1(t) \sim{\rm e}^{\lambda t}$ and $u_1'(t) \sim\lambda{\rm e}^{\lambda t}$ ($t \to\infty$).
Keywords: half-linear differential equation; nonoscillatory solution; asymptotic form
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Affiliations: Manabu Naito, Department of Mathematics, Faculty of Science, Ehime University, Matsuyama 790-8577, Japan, e-mail: jpywm078@yahoo.co.jp
