A note on the existence of solutions with prescribed asymptotic behavior for half-linear ordinary differential equations

Naito, Manabu. "A note on the existence of solutions with prescribed asymptotic behavior for half-linear ordinary differential equations." Mathematica Bohemica 149.3 (2024): 317-336. <http://eudml.org/doc/299377>.

@article{Naito2024,
abstract = {The half-linear differential equation \[ (|u^\{\prime \}|^\{\alpha \}\{\rm sgn\} u^\{\prime \})^\{\prime \} = \alpha (\lambda ^\{\alpha + 1\} + b(t))|u|^\{\alpha \}\{\rm sgn\} u, \quad t \ge t\_\{0\}, \] 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 $u_\{0\}^\{\prime \}(t) \sim - \lambda \{\rm e\}^\{- \lambda t\}$ ($t \rightarrow \infty $), and a nonoscillatory solution $u_\{1\}(t)$ such that $u_\{1\}(t) \sim \{\rm e\}^\{\lambda t\}$ and $u_\{1\}^\{\prime \}(t) \sim \lambda \{\rm e\}^\{\lambda t\}$ ($t \rightarrow \infty $).},
author = {Naito, Manabu},
journal = {Mathematica Bohemica},
keywords = {half-linear differential equation; nonoscillatory solution; asymptotic form},
language = {eng},
number = {3},
pages = {317-336},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A note on the existence of solutions with prescribed asymptotic behavior for half-linear ordinary differential equations},
url = {http://eudml.org/doc/299377},
volume = {149},
year = {2024},
}

TY - JOUR
AU - Naito, Manabu
TI - A note on the existence of solutions with prescribed asymptotic behavior for half-linear ordinary differential equations
JO - Mathematica Bohemica
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 149
IS - 3
SP - 317
EP - 336
AB - The half-linear differential equation \[ (|u^{\prime }|^{\alpha }{\rm sgn} u^{\prime })^{\prime } = \alpha (\lambda ^{\alpha + 1} + b(t))|u|^{\alpha }{\rm sgn} u, \quad t \ge t_{0}, \] 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 $u_{0}^{\prime }(t) \sim - \lambda {\rm e}^{- \lambda t}$ ($t \rightarrow \infty $), and a nonoscillatory solution $u_{1}(t)$ such that $u_{1}(t) \sim {\rm e}^{\lambda t}$ and $u_{1}^{\prime }(t) \sim \lambda {\rm e}^{\lambda t}$ ($t \rightarrow \infty $).
LA - eng
KW - half-linear differential equation; nonoscillatory solution; asymptotic form
UR - http://eudml.org/doc/299377
ER -