Boundedness for solutions of nonlinear Hill's equations with periodic forcing terms via Moser's twist theorem (Q1122051)

The author considers the Hill's equation with periodic forcing term \[ (*)\quad x''+\beta x^{2n+1}+(a_ 1+\epsilon a(t))x=p(t),\quad n\geq 1, \] where a(t) and p(t) are continuous and 1-periodic functions in \(t\in R\), \(a_ 1\) and \(\beta\) are positive constants and \(\epsilon\) is a small parameter. He proves that there exists an \(\epsilon_ 0\) such that for a ll \(\epsilon \in (0,\epsilon_ 0)\) every solution to the equation (*) is bounded together with its derivative. Under some assumptions there exists quasi-periodic solution or m-periodic solution for every integer \(m\geq 1\).