Boundedness of Solutions for a Class of Sublinear Reversible Oscillators with Periodic Forcing

Abstract

We study the boundedness of all solutions for the following differential equation $x^{\prime \prime }+f(x)x^{\prime }+(B+\epsilon e(t))|x{|}^{\alpha -\mathrm{1}}x=p(t),$ where $f(x),\hspace{0.17em}\hspace{0.17em}p(t)$ are odd functions, $e(t)$ is an even function, $e(t),\hspace{0.17em}\hspace{0.17em}p(t)$ are smooth $\mathrm{1}$-periodic functions, $B$ is a nonzero constant, and $\epsilon$ is a small parameter. A sufficient and necessary condition for the boundedness of all solutions of the above equation is established. Moreover, the existence of Aubry-Mather sets is obtained as well.