本文引进了合适的作用-角变量变换并结合新的估计方法,对Duffing方程的Poincare映射应用推广的Aubry-Mather定理,证明了一类半线性Duffing方程的Aubry-Mather集的存在性.改进了已有的半线性Duffing方程的Aubry-Mather集的研究成果,我们的非线性扰动项φ(x)既可是有界函数又可是无界函数.

In this paper,by introducing an appropriate action-angle variable transformation and combining with new estimates,we obtain the existence of Aubry-Mather sets for a class of semilinear Duffing equations via generalized Aubry-Mather theorem on Poincare map of Duffing equations.Our work improves the existing research results for semilinear Duffing equations,and nonlinear perturbation term φ(x) in this paper can be both bounded functions and unbounded functions.