Boundedness for solutions of nonlinear periodic differential equations via Moser's twist theorem

Abstract

We consider the nonlinear periodic differential equation $$\frac{{d^2 x}}{{dt^2 }} + \beta x^{2n + 1} + a(t)x = p(t), n \geqslant 1,$$ wherea(t) andp(t) are continuous and 1-periodic, β is a positive constant. The purpose of this paper is to prove that all solutions of the above-mentioned equation are bounded fort∈R and there are infinitely many quasi-periodic solutions and an infinity of periodic solutions of minimal periodm, for each positive integerm.