具有普通二分性非线性离散系统的周期解

Periodic solutions for nonlinear discrete systems with ordinary dichotomy

研究了一类具有普通二分性非线性离散系统的周期解.首先指出若齐次线性系统具有普通二分性,则对应非齐次线性系统存在有界的周期解,并给出了该有界周期解的表达式;然后借助这个结论并应用Banach不动点定理,探讨了当线性部分具有普通二分性时,对应非线性系统周期解的存在唯一性,给出了非线性离散系统存在唯一周期解的充分条件;最后通过一个具体例子说明了主要结论在实际问题中的应用.

In this paper, the periodic solutions for nonlinear discrete systems with ordinacy dichotomy were studied. Firstly, it is pointed out that if the homogeneous linear system has ordinary dichotomy, then the nonhomogeneous linear system admits a bounded periodic solution, the expression of the bounded periodic solution is also given. Then, by using the above conclusion and the Banach fixed point theorem, a sufficient condition for the existence and uniqueness of periodic solutions for nonlinear discrete systems is established, if the linear part admired ordinary exponential dichotomy. Finally, an example is given to illustrate the results we obtained.