于躍華, 賈仁偉, 黃祖達(dá)

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一類二階多時(shí)滯泛函微分方程多個(gè)周期解的存在性

于躍華, 賈仁偉, 黃祖達(dá)

(湖南文理學(xué)院 數(shù)學(xué)與計(jì)算科學(xué)學(xué)院,湖南 常德, 415000)

泛函微分方程; 重合度; 多個(gè)周期解

泛函微分方程周期解的存在性問(wèn)題一直是人們廣泛研究的課題. 近年來(lái), Mawhin的重合度理論延拓定理已成為研究周期解存在性問(wèn)題的強(qiáng)有力工具[1-5], 文獻(xiàn)[6]利用這種方法研究了另一類二階非線性泛函微分方程多個(gè)周期解的存在性, 但在證明過(guò)程中還存在一些不夠完善的地方, 本文考慮如下一類多時(shí)滯二階泛函微分方程:

為了行文方便, 引入Mawhin的重合度理論[7], 設(shè),為Banach空間,: dom為指標(biāo)為零的Fredholm算子,為連續(xù)投影, 滿足: Im=Ker, Im=Ker,. 定義的廣義逆為因?yàn)橥瑯?gòu), 記同構(gòu)映射為

則方程(1)至少有兩個(gè)-周期解.

于是由式(3)、(4)知:

從而可得:

由式(5)和(6)可得:

或者

由此得:

從而有:

于是:

即:

[1] Lu S, Ge W. Periodic solutions for a kind of second order differential equation with multiple deviating argument[J]. Appl Math Comput, 2003, 146: 195-209.

[2] Huang C, He Y, Huang L, et al. New result on the periodic solutions for a kind of Rayleigh equation with two deviating argu- ments[J]. J Math & Comput Modell, 2007, 46: 604-611.

[3] Peng L, Liu B, Zhou Q, et al. Periodic solutions for a kind of Rayleigh equation with two deviating arguments[J]. J Franklin Inst, 2006, 7: 676-687 .

[4] S?rma A, Tun? C, ?zlem S. Existence and uniqueness of periodic solutions for a kind of Rayleigh equation with nitely many deviating arguments[J]. Nonlinear Analysis, 2010, 73(2): 358-366.

[5] Burton T A. Stability and Periodic Solutions of Ordinary and Functional Differential Equations[M]. Orlando: Academic Press, 1985.

[6] 田德生, 曾憲武. 一類二階泛函微分方程多個(gè)周期解的存在性[J]. 數(shù)學(xué)物理學(xué)報(bào), 2010, 30A(2): 525-530.

[7] Gains R E, Mauhin JL. Coincidence Degree and Nonlinear Differential Equations[M]. Berlin: Springer, 1977.

On existence of serveral periodic solution for a class of second order multi-delay functional diferential equations

YU Yue-hua, JIA Ren-wei, HUANG Zu-da

(Department of Maths and Computing, Hnnan University of Arts and Science, Changde 415000, China)

Functional differential equations; mult-periodic solutions; coincidence degree

10.3969/j.issn.1672-6146.2012.03.001

O 175.6

1672-6146(2012)03-0001-05

2012-08-30

湖南省教育廳資助課題2010[243], 湖南文理學(xué)院芙蓉學(xué)院2010-2011重點(diǎn)課題資助

于躍華(1971-), 女, 副教授, 研究方向?yàn)槲⒎址匠? E-mail: 2424676032@qq.com

(責(zé)任編校: 劉曉霞)