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We study the boundedness of all solutions for the following differential equation x′′+f(x)x′+(B+εe(t))|x|α-1x=p(t), where f(x),  p(t) are odd functions, e(t) is an even function, e(t),  p(t) are smooth 1-periodic functions, B is a nonzero constant, and ε 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.