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 
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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/

[1] M. Biernacki: Sur l’equation differentielle $y^{(4)} + A(x)y = 0$. Ann. Univ. Mariae Curie-Skłodowska 6 (1952), 65–78. | MR

[2] S. P. Hastings and A. C. Lazer: On the asymptotic behavior of solutions of the differential equation $y^{(4)} = p(x)y$. Czechoslovak Math.  J. 18(93) (1968), 224–229. | MR

[3] G. D. Jones: Asymptotic behavior of solutions of a fourth order linear differential equation. Czechoslovak Math. J. 38(113) (1988), 578–584. | MR | Zbl

[4] G. D. Jones: Oscillatory solutions of a fourth order linear differential equation. Lecture notes in pure and apllied Math. Vol 127, 1991, pp. 261–266. | MR

[5] M. K. Kwong and A. Zettl: Norm Inequalities for Derivatives and Differences. Lecture notes in Mathematics, 1536. Springer-Verlag, Berlin, 1992. | MR

[6] M.  Švec: Sur le comportement asymtotique des intégrales de l’équation differentielle $y^{(4)} + Q(x)y = 0$. Czechoslovak Math. J. 8(83) (1958), 230–245. | MR