Boundedness of solutions in a class of Duffing equations with oscillating potentials

In this paper, we prove that the second order differential equation d2xdt2+x2n+1(1+ccosx)+p(t)=0 with p(t+1)=p(t)p(t+1)=p(t) smooth and |c|<1|c|<1, possesses the Lagrangian stability despite the fact that the monotone twist condition is violated.