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Let q(t)$q(t)$ be a continuous 2π-periodic function with 12π∫02πq(t)dt>0$\frac{1}{2\pi}\int_{0}^{2\pi}q(t)\,dt>0$. We propose a new approach to establish the existence of Aubry-Mather sets and quasi-periodic solutions for the following time-periodic parameters semilinear Duffing-type equation: x″+q(t)x+f(t,x)=0,$$x''+q(t) x+f(t,x)=0, $$ where f(t,x)$f(t,x)$ is a continuous function, 2π-periodic in the first argument and continuously differentiable in the second one. Under some assumptions on the functions q and f, we prove that there are infinitely many generalized quasi-periodic solutions via a version of the Aubry-Mather theorem given by Pei. Especially, an advantage of our approach is that it does not require any high smoothness assumptions on the functions q(t)$q(t)$ and f(t,x)$f(t,x)$.

A new approach to the existence of quasiperiodic solutions for a class of semilinear Duffing-type equations with time-periodic parameters