Nonlinear Analysis 67 (2007) 200–237 www.elsevier.com/locate/na A small twist theorem and boundedness of solutions for semilinear Duffing equations at resonance ✩ Shiwang Ma a,∗ , Jianhong Wu b a School of Mathematical Sciences, Nankai University, Tianjin, 300071, PR China b Laboratory for Industrial and Applied Mathematics, Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada M3J 1P3 Received 17 April 2006; accepted 20 April 2006 Abstract We obtain a new variant of Moser’s small twist theorem and apply this new version to investigate the boundedness of solutions for the following semilinear Duffing equation ¨ x + n 2 x + g(x ) = p(t ), where p is a 2π -periodic smooth function and lim |x |→∞ x -1 g(x ) = 0. We obtain some sharp sufficient conditions for the boundedness of all solutions to the above equation at resonance. Unlike many existing results in the literature where the function g is required to be a bounded function with asymptotic limits, our main results here allow g be unbounded or oscillatory without asymptotic limits. c  2006 Elsevier Ltd. All rights reserved. MSC: 34C11; 34C28; 58F35 Keywords: Semilinear Duffing equation; Resonance; Moser’s small twist theorem; Boundedness 1. Introduction and main results In this paper, we consider the boundedness of solutions to the Duffing equation ¨ x +¯ g(x ) = p(t ), p(t + 2π) ≡ p(t ), (1.0) ✩ Research partially supported by the National Natural Science Foundation of China (SM), by the Natural Sciences and Engineering Research Council of Canada, and by Canada Research Chairs Program (JW). ∗ Corresponding author. E-mail addresses: shiwangm@163.net (S. Ma), wujh@mathstat.yorku.ca (J. Wu). 0362-546X/$ - see front matter c  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2006.04.023