Geodesic
Asymptotic behavior of solutions of a $2n^{th}$ order nonlinear differential equation
Czechoslovak Mathematical Journal, Tome 52 (2002) no. 3, pp. 665-672 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $\mathbb R \times I$.
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 $\mathbb R \times I$.
Classification : 34C10, 34C11, 34D05
Keywords: asymptotic behavior; higher order differential equation 
@article{CMJ_2002_52_3_a18,
     author = {Lin, C. S.},
     title = {Asymptotic behavior of solutions of a $2n^{th}$ order nonlinear differential equation},
     journal = {Czechoslovak Mathematical Journal},
     pages = {665--672},
     year = {2002},
     volume = {52},
     number = {3},
     mrnumber = {1923270},
     zbl = {1023.34032},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2002_52_3_a18/}
}
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Lin, C. S. Asymptotic behavior of solutions of a $2n^{th}$ order nonlinear differential equation. Czechoslovak Mathematical Journal, Tome 52 (2002) no. 3, pp. 665-672. http://geodesic.mathdoc.fr/item/CMJ_2002_52_3_a18/