Abstract
Let \(q(t)\) be a continuous 2π-periodic function with \(\frac{1}{2\pi}\int_{0}^{2\pi}q(t)\,dt>0\). 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: $$x''+q(t) x+f(t,x)=0, $$ where \(f(t,x)\) 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 \(q(t)\) and \(f(t,x)\).
Highlights
The goal of this paper is to study the existence of Aubry-Mather sets and quasi-periodic solutions to the following time-periodic parameters semilinear Duffing-type equation:x + q(t)x + f (t, x) =, ( . )where q(t) is continuous and π -periodic function in the time t, f (t, x) is a continuous function, π -periodic in the first argument and continuously differentiable in the second one
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, where 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
Summary
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, where 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) and f (t, x). 1 Introduction The goal of this paper is to study the existence of Aubry-Mather sets and quasi-periodic solutions to the following time-periodic parameters semilinear Duffing-type equation: x + q(t)x + f (t, x) = ,
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