# American Institute of Mathematical Sciences

July  2016, 36(7): 3961-3991. doi: 10.3934/dcds.2016.36.3961

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

 1 School of Mathematical Sciences, Soochow University, Suzhou 215006 2 Department of Mathematics, Nanjing University, Nanjing 210093, China 3 School of Mathematical Sciences, Ocean University of China, Qingdao 266100

Received  February 2015 Revised  November 2015 Published  March 2016

We introduce a new method for the boundedness problem of semilinear Duffing equations at resonance. In particular, it can be used to study a class of semilinear equations at resonance without 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.
Citation: Zhiguo Wang, Yiqian Wang, Daxiong Piao. A new method for the boundedness of semilinear Duffing equations at resonance. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3961-3991. doi: 10.3934/dcds.2016.36.3961
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