Asymptotic behavior of solutions of a $2n^{th}$ order nonlinear differential equation

Lin, C. S.. "Asymptotic behavior of solutions of a $2n^{th}$ order nonlinear differential equation." Czechoslovak Mathematical Journal 52.3 (2002): 665-672. <http://eudml.org/doc/30733>.

@article{Lin2002,
abstract = {In this paper we prove two results. The first is an extension of the result of G. D. Jones [4]: (A) Every nontrivial solution for \[ \left\rbrace \begin\{array\}\{ll\}(-1)^n u^\{(2n)\} + f(t,u) = 0,\hspace\{5.0pt\}\text\{in\} \hspace\{5.0pt\}(\alpha , \infty ), u^\{(i)\}(\xi ) = 0, \quad i = 0,1,\dots , n-1, \hspace\{5.0pt\} \text\{and\} \hspace\{5.0pt\}\xi \in (\alpha , \infty ), \end\{array\}\right.\] must be unbounded, provided $f(t,z)z\ge 0$, in $E \times \mathbb \{R\}$ and for every bounded subset $I$, $f(t,z)$ is bounded in $E \times I$. (B) Every bounded solution for $(-1)^n u^\{(2n)\} + f(t,u) = 0$, in $\mathbb \{R\}$, must be constant, provided $f(t,z)z\ge 0$ in $\mathbb \{R\} \times \mathbb \{R\}$ and for every bounded subset $I$, $f(t,z)$ is bounded in $\mathbb \{R\} \times I$.},
author = {Lin, C. S.},
journal = {Czechoslovak Mathematical Journal},
keywords = {asymptotic behavior; higher order differential equation; asymptotic behavior; higher-order differential equation},
language = {eng},
number = {3},
pages = {665-672},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Asymptotic behavior of solutions of a $2n^\{th\}$ order nonlinear differential equation},
url = {http://eudml.org/doc/30733},
volume = {52},
year = {2002},
}

TY - JOUR
AU - Lin, C. S.
TI - Asymptotic behavior of solutions of a $2n^{th}$ order nonlinear differential equation
JO - Czechoslovak Mathematical Journal
PY - 2002
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 52
IS - 3
SP - 665
EP - 672
AB - In this paper we prove two results. The first is an extension of the result of G. D. Jones [4]: (A) Every nontrivial solution for \[ \left\rbrace \begin{array}{ll}(-1)^n u^{(2n)} + f(t,u) = 0,\hspace{5.0pt}\text{in} \hspace{5.0pt}(\alpha , \infty ), u^{(i)}(\xi ) = 0, \quad i = 0,1,\dots , n-1, \hspace{5.0pt} \text{and} \hspace{5.0pt}\xi \in (\alpha , \infty ), \end{array}\right.\] must be unbounded, provided $f(t,z)z\ge 0$, in $E \times \mathbb {R}$ and for every bounded subset $I$, $f(t,z)$ is bounded in $E \times I$. (B) Every bounded solution for $(-1)^n u^{(2n)} + f(t,u) = 0$, in $\mathbb {R}$, must be constant, provided $f(t,z)z\ge 0$ in $\mathbb {R} \times \mathbb {R}$ and for every bounded subset $I$, $f(t,z)$ is bounded in $\mathbb {R} \times I$.
LA - eng
KW - asymptotic behavior; higher order differential equation; asymptotic behavior; higher-order differential equation
UR - http://eudml.org/doc/30733
ER -