二阶差分方程组周期解的多重性

Multiplicity of Periodic Solutions for the Second Order Nonlinear Difference System

众多领域将其重要问题的数学模型转化为非线性差分方程来研究,针对一类带参数的非线性二阶差分方程组(P)λ,研究了其周期解的存在性和多重性,得到了该方程组的变分结构,证明了问题(P)λ的解等价于泛函J在Banach空间E上的临界点.利用临界点理论、分歧方法和Morse理论证明了问题(P)λ在一定的假设条件下至少存在三个不同的非平凡周期解.所得结论完善了非线性差分方程组的研究结果,对非线性离散问题周期解的研究有一定的指导意义.

As many mathematic models of important research in various fields have turned to nonlinear difference system in recent years,the existence and the multiplicity of periodic solutions for the second order nonlinear difference system（P）λwith the parameter were studied.The variational framework,which was corresponding to a class of the above problem was obtained,and proved the desired solutions were equivalent to the critical points of the functional Jin Banach space E.It is proved that the problem（P）λhas at least three distinct nontrivial periodic solutions under suitable conditions by critical point theory,bifurcation method and Morse theory.The conclusions enrich research results of the nonlinear difference system and direct the study of the periodic solutions for the nonlinear discrete problem.