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

摘要

Let q ( t ) $q(t)$ be a continuous 2π-periodic function with 1 2 π ∫ 0 2 π q ( t ) d t 0 $rac{1}{2pi}int_{0}^{2pi}q(t),dt0$ . 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 ) $f(t,x)$ .