共振條件下一類方程無界解和周期解的共存性

黃國安,劉期懷

(1.桂林航天工業高等專科學校計算機系,廣西桂林 541004;2.廣西師范大學數學科學院,廣西桂林 541004;3.蘇州大學數學科學學院,江蘇蘇州 215006)

共振條件下一類方程無界解和周期解的共存性

黃國安1,2,劉期懷3

(1.桂林航天工業高等專科學校計算機系,廣西桂林 541004;2.廣西師范大學數學科學院,廣西桂林 541004;3.蘇州大學數學科學學院,江蘇蘇州 215006)

討論了在共振條件下一類具有等時位勢的方程無界解和周期解的共存性.利用Poincar′e映射軌道的性質,給出了無界解的存在性條件.在此條件下,由Poincar′e-Bohl定理,得到了方程的一個周期解,進而說明共振條件下這類方程無界解和周期解的是可以共存的.最后,給出了一個無界解和周期解共存的具有等時位勢的方程實例.

等時位勢；無界解；周期解；共振

1 引言

共振條件下的無界解和周期解的共存性引起了許多數學研究者的興趣,并已獲得了豐富的結果[1-3].本文考慮一類具有等時位勢的方程

這里f(x)∈C(R,R),V(x)為2π/n?等時位勢(n∈N),?(t,x)∈C(R×R,R)關于t是2π-周期且有界的.稱V(x)為T-等時位勢,即x00+V0(x)=0的所有解為T-周期的.

當V0(x)=ax+?bx?,?(t,x)=?g(x)+p(t)時,在非共振條件下,利用反轉系統的Moser扭轉定理,文[4]給出了方程所有解有界f(x),g(x)應滿足的一些條件.對于一般的?(t,x),隨后作者又給出了非共振條件下該方程有界解和周期解共存的條件[5].因此,一個自然的問題是,在共振條件下,對于方程(1)的解是否滿足相應的有界解和周期解共存性？本文參照文[1-2,6]中的討論方法并對該問題給出了答案.

2 一類平面映射無界軌道的存在性

為了證明我們的主要結果,先給出一類平面映射軌道的性質.

給定σ＞0,令

3 主要結果及其證明

對任意的ρ＞0,記φ(·,ρ)為方程x00+V0(x)=0滿足φ(0,ρ)=ρ,φt(0,ρ)=0的解,ψ(t)為方程x00+ax+?bx?=0滿足x(0)=1,x0(0)=0的解,這里x+=max{x,0},x?=max{?x,0}, 則ψ(t)為2π/n-周期的且

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Coexistence of unbounded and periodic solutions to a class of isochronous oscillators at resonance

HUANG Guo-an1,2,LIU Qi-huai3
(1.Department of Computer Guilin College of Aerospace Technology,Guilin541004,China; 2.Department of Mathematics Guangxi Normal University,Guilin541004,China; 3.Department of Mathematics Suzhou University,Suzhou215006,China)

This paper studies the coexistence of unbounded and periodic solutions to a class of isochronous oscillators at resonance,obtaining the conditions of existence of unbounded solutions by the dynamics of Ponica′e mapping.Under these conditions,we can get a periodic solution using Ponica′e-Bohl theorem.Thus,it implies the coexistence of unbounded and periodic solutions.Finially,an example for the equations which has the coexistence of unbounded and periodic solutions is given.

isochronous potential,unbounded solutions,periodic solutions,resonance

O175.2

A

1008-5513(2009)03-0603-07

2008-09-26.

黃國安(1971-),講師,在讀碩士,研究方向：常微分方程與金融工程.

2000MSC:34C11,34C25