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Asymptotic behavior of solutions of A 2nth order nonlinear differential equation

Authors:

C S Lin

Institution:

(1) Department of Mathematics, Hsing Wu College No 11-2, Fen-liao Rd., Lin-kou, Taipei, 224, Taiwan, R.O.C.

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 $\mathbb{R} \times I$ .

Keywords:

asymptotic behavior higher order differential equation

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