Asymptotic behavior of solutions of A 2n^th order nonlinear differential equation

Abstract

In this paper we prove two results. The first is an extension of the result of G. D. Jones [4[:

Every nontrivial solution for

$$\left\{ \begin{gathered} ( - 1)^n u^{(2n)} + f(t,u) = 0,\;\;{in}\;\;( \alpha ,\infty ), \hfill \\ u^{(i)} ({\xi }) = 0,i = 0.1,\;.\;.\;.\;,n - 1, and \xi \in ( \alpha , \infty ),\hfill \\ \end{gathered} \right.$$

must be unbounded, provided $f(t,z)z \geqslant 0$, in $E \times \mathbb{R}$ and for every bounded subset I, f(t, z) is bounded in E × I.

(B) Every bounded solution for $( - 1)^n u^{(2n)} + f(t,u) = 0$, in $\mathbb{R}$, must be constant, provided $f(t,\;z)z \geqslant 0$ in $\mathbb{R} \times \mathbb{R}$ and for every bounded subset I, $f(t,z)$ is bounded in $\mathbb{R} \times I$.