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Keywords:
asymptotic behavior; higher order differential equation
asymptotic behavior; higher order differential equation
Summary:
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 $\mathbb R \times I$.
References:
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[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 0226110
[3] G. D. Jones: Asymptotic behavior of solutions of a fourth order linear differential equation. Czechoslovak Math. J. 38(113) (1988), 578–584. MR 0962901 | Zbl 0672.34052
[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 1096762
[5] M. K. Kwong and A. Zettl: Norm Inequalities for Derivatives and Differences. Lecture notes in Mathematics, 1536. Springer-Verlag, Berlin, 1992. MR 1223546
[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 0101355
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