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

Authors:

C. S. Lin

Affiliation:

(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

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 $$f(t,;z)z geqslant 0$$

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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