American Institute of Mathematical Sciences

June  2017, 10(3): 543-556. doi: 10.3934/dcdss.2017027

Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term

 1 School of Mathematics and Computer Science, Shangrao Normal University, Shangrao 334001, China 2 Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China

* Corresponding author

Received  December 2015 Revised  October 2016 Published  February 2017

Fund Project: The work is supported by NSF of China under 11461056.

In this paper we consider the existence of Aubry-Mather sets and quasi-periodic solutions for a class of second order differential equation with a nonlinear damping term
 $x''+α x^+-β x^-+q(x)f(x')+g(t,x)=p(t),$
where
 $q, f∈ C^1(\mathbb{R}),$
 $g(t,x)∈ C^{0,1}(\mathbf{S}^1× \mathbb{R})$
and
 $p(t)∈ C^0(\mathbf{S}^1)$
,
 $\mathbf{S}^1= \mathbb{R}/2π\mathbb{Z}$
,
 $α$
and
 $β$
are two positive constants satisfying
 $\frac{1}{\sqrt{α}}+\frac{1}{\sqrt{β}}=\frac{2}{ω}$
with
 $ω∈ \mathbb{R}^+$
. Under some assumptions on the parities of
 $f,$
 $g$
and
 $p$
, we obtain the existence of infinitely many generalized quasi-periodic solutions via a result of Chow and Pei from the Aubry-Mather theory of reversible mapping. In particular, an advantage of our approach is that it does not require any high smoothness assumptions on the functions
 $q, f, g$
and
 $p$
.
Citation: Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027
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