Boundedness for sublinear reversible systems with a nonlinear damping and periodic forcing term

In this paper, we are concerned with the sublinear reversible systems with a nonlinear damping and periodic forcing term x ″ + f ( x ) g ( x ′ ) + γ | x | α − 1 x = p ( t ) , where f ( x ) , p ( t ) are odd functions, p ( t ) is smooth 1-periodic function and γ ≠ 0 is a constant. A sufficient and necessary condition for the boundedness of all solutions of the above equation is established. Moreover, we show the existence of Aubry–Mather sets as well.