Geodesic
A note on the existence of solutions with prescribed asymptotic behavior for half-linear ordinary differential equations
Mathematica Bohemica, Tome 149 (2024) no. 3, pp. 317-336
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The half-linear differential equation $$ (|u'|^{\alpha }{\rm sgn} u')' = \alpha (\lambda ^{\alpha + 1} + b(t))|u|^{\alpha }{\rm sgn} u, \quad t \geq t_{0}, $$ 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 $).
The half-linear differential equation $$ (|u'|^{\alpha }{\rm sgn} u')' = \alpha (\lambda ^{\alpha + 1} + b(t))|u|^{\alpha }{\rm sgn} u, \quad t \geq t_{0}, $$ 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 $).
DOI : 10.21136/MB.2023.0158-22
Classification : 34C11, 34D05, 34D10
Keywords: half-linear differential equation; nonoscillatory solution; asymptotic form 
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Naito, Manabu. A note on the existence of solutions with prescribed asymptotic behavior for half-linear ordinary differential equations. Mathematica Bohemica, Tome 149 (2024) no. 3, pp. 317-336. doi: 10.21136/MB.2023.0158-22

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