Boundedness of semilinear Duffing equations at resonance with oscillating nonlinearities

Abstract

In this paper, we prove the boundedness of all the solutions for the equation $\ddot{x}+n^{2}x+g(x)+{\psi }^{\prime }(x)=p(t)$ with the Lazer-Leach condition on $g$ and $p$, where $n\in {\mathbb{N}}^{+}$, $p(t)$ and ${\psi }^{\prime }(x)$ are periodic and $g(x)$ is bounded. For the critical situation that $|{\int }_0^{2\pi }p(t)e^{int}dt|=2|g(+\infty )-g(-\infty )|$, we also prove a sufficient and necessary condition for the boundedness if ${\psi }^{\prime }(x)\equiv 0$.

arXiv comment42 pages