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

Abstract

We consider the nonlinear Hill's equation with periodic forcing term x′' + βx 2 n + 1 + ( a 1 + εa( t)) x = p( t), n ⩾ 1, where a( t) and p( t) are continuous and 1-periodic, a 1 and β are positive constants, and ε is a small parameter. The purpose of this paper is to prove that all solutions of the above-mentioned equation are bounded for t ϵ R and that there are infinitely many quasi-periodic solutions and an infinity of periodic solutions of minimal period m, for each positive integer m.