A new method for the boundedness of semilinear Duffing equations at resonance

Abstract

We introduce a new method for the boundedness problem ofsemilinear Duffing equations at resonance. In particular, it canbe used to study a class of semilinear equations at resonancewithout the polynomial-like growth condition. As an application,we prove the boundedness of all the solutions for the equation$\ddot{x}+n^2x+g(x)+\psi(x)=p(t)$ under the Lazer-Leach condition on $g$ and $p$, where $n\in \mathbb{N^+}$, $p(t)$ and $\psi(x)$ are periodic and $g(x)$ is bounded.