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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2015, Vol. 32 ›› Issue (4): 568-576.doi: 10.3969/j.issn.1005-3085.2015.04.010

• • 上一篇    下一篇

四阶非线性差分方程周期解的存在性

吴云宗1,   石海平2,   郭承军3   

  1. 1- 广东白云学院基础部，广州 510450
    2- 广东建设职业技术学院现代商务与管理系，广州 510440
    3- 广东工业大学应用数学学院，广州 510006
  • 收稿日期:2012-01-12 接受日期:2014-01-06 出版日期:2015-08-15 发布日期:2015-10-15
  • 基金资助:
    国家自然科学基金 (11401121)；广东省自然科学基金 (S2013010014460)；广东省教育科学“十一五”规划课题 (2010tjk074).

Existence of Periodic Solutions of Fourth-order Nonlinear Difference Equations

WU Yun-zong1,   SHI Hai-ping2,   GUO Cheng-jun3   

  1. 1- Basic Courses Department, Guangdong Baiyun Institute, Guangzhou 510450
    2- Modern Business and Management Department, Guangdong Construction Vocational Technology Institute, Guangzhou 510440
    3- School of Applied Mathematics, Guangdong University of Technology, Guangzhou 510006
  • Received:2012-01-12 Accepted:2014-01-06 Online:2015-08-15 Published:2015-10-15
  • Supported by:
    The National Natural Science Foundation of China (11401121); the Natural Science Foundation of Guangdong Province (S2013010014460); the Guangdong Province Education Science in the Eleventh Five-year Plan (2010tjk074).

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摘要： 本文研究一类四阶非线性差分方程周期解的存在性．首先建立四阶差分方程的变分泛函，然后将四阶差分方程周期解的存在性问题转化成相应泛函临界点的存在性问题加以求解．利用临界点理论和一些空间分解技巧，获得了该方程至少存在一个周期解的充分条件．通过一个实例，说明所得定理在应用中的有效性．所得结果丰富了对四阶差分方程周期解问题的研究．

关键词: 四阶差分方程, 周期解, 鞍点定理, 离散变分理论

Abstract:

In this paper, the existence of periodic solutions to a fourth-order nonlinear difference equation is studied. First, a variational functional corresponding to the difference equations as aforementioned is established. Next, the existence of periodic solutions can be simplified as the existence of critical point of the functional. By using critical point theory and some decomposition techniques of space, the sufficient conditions for the existence of a periodic solution are obtained. The practical effectiveness of our main theorem in view of an example is illustrated. Main results of this paper are helpful to enrich the theory of periodic solutions to fourth-order difference equations.

Key words: fourth-order difference equations, periodic solutions, saddle point theorem, discrete variational theory

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