Asymptotic behavior of solutions of a 2n^{\rm th} order nonlinear differential equation

Author: Lin C.S.¹

Source: Czechoslovak Mathematical Journal, Volume 52, Number 3, September 2002 , pp. 665-672(8)

Publisher: Springer

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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 \cases{(-)^nu^{(2n)}+f(t,u)=0, \enspace {\rm in} \enspace (\alpha, \infty),\cr\cr u^{(i)}(\xi)=0,\quad i=0,1,\ldots,n-1, \quad {\rm and}\quad \xi \in (\alpha, \infty),} must be unbounded, provided f(t,\thinspace z)z\geq 0,, in E \times\Bbb R and for every bounded subset I, f(t, z) is bounded in E \times I.

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

Keywords: asymptotic behavior; higher order differential equation

Language: English

Document Type: Research article

Affiliations: 1: Department of Mathematics, Hsing Wu College No 11-2, Fen-liao Rd., Lin-kou, Taipei 224, Taiwan, R.O.C., t10035@mail.hwc.edu.tw

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