扰动的时滞微分方程的多重周期解与Hamilton系统

Multiple Periodic Solutions of a Class of Differential Delay Equations Via Hamiltonian Systems

讨论了一类带周期扰动项的时滞微分方程x′(t)=-[f(x(t-1))+f(x(t-2))+…+f(x(t-(n-1)))]+ε2g(t,ε)具有给定周期的多重周期解的存在性,其中n为正奇数,函数g关于变量t是1-周期的.运用渐近凸哈密顿系统的一些结果证明了此类方程在周期扰动下多重周期解的存在性,且所得周期解的最小重数与当g恒为零时系统的周期解的最小重数是一致的.

In this paper, we study a class of differential delay equations with periodic perturbation x＇（t） = - [f（x（t- 1）） ＋f（x（t-2）） ＋…＋f（x（t- （n-）））] ＋ ε^2g（t,ε）. We get the existence of prescribed multiple periodic solutions of the equations by using some results of convex asymptotically linear Hamiltonian systems. And the smallest multiplicity of the periodic solutions agrees with the multiple number of the system with g = 0.