四阶泛函微分方程周期解的存在性和全局吸引性

The existence and global attractivity of periodic solutions for fourth order functional differential equations

研究四阶泛函微分方程x(4)(t)+a1x(t)+a2x″(t)+a3x′(t)+a4x(t)+g(t,x(t-τ))=p(t)的周期解问题.首先将该方程转变为四维的拟线性微分方程(组),得到该方程存在唯一周期解的充分条件;然后通过选取适当的李雅普诺夫函数,推导方程任一解的全局吸引性,并进一步得到方程周期解的全局吸引性.最后,通过实例证实了本文结果的正确性.

We consider the periodic solution for the following fourth order functional differential equation x（4）（t）＋a1x″′（t）＋a2x″（t）＋a3x′（t）＋a4x（t）＋g（t,x（t-τ））=p（t）. Firstly we obtain the sufficient condi- tions for the existence of uniqu periodic solution of this equation by converting the equation into a four-dimensi- nal quasi-linear differential equation. Then, by selecting an appropriate Lyapunov function, the global attrac- tivity of an arbitraty solution for this equation is derived, and furthermore, the global attractivity of a periodic solution for this equation is given. Finally, an example is provided to verify the correctness of the results in this paper.