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On Littlewood's boundedness problem for sublinear Duffing equations

Author: Bin Liu
Journal: Trans. Amer. Math. Soc. 353 (2001), 1567-1585
MSC (1991): Primary 34C15, 58F27
Published electronically: December 18, 2000
MathSciNet review: 1806727
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In this paper, we are concerned with the boundedness of all the solutions and the existence of quasi-periodic solutions for second order differential equations

\begin{displaymath}x^{\prime\prime} + g(x) = e(t), \end{displaymath}

where the 1-periodic function $e(t)$ is a smooth function and $g(x)$satisfies sublinearity:

\begin{displaymath}{sign}(x)\cdot g(x)\to+\infty,\quad g(x)/x\to 0 \quad {as}\,\,\, \vert x\vert\to+\infty. \end{displaymath}

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Additional Information

Bin Liu
Affiliation: School of Mathematical Sciences, Peking University, Beijing 100871, P.R.China

Keywords: Boundedness of solutions, quasi-periodic solutions, Moser's small twist theorem
Received by editor(s): January 16, 1997
Published electronically: December 18, 2000
Additional Notes: Supported by NNSF of China
Article copyright: © Copyright 2000 American Mathematical Society