On Littlewood’s boundedness problem for sublinear Duffing equations

Liu, Bin

doi:10.1090/S0002-9947-00-02770-7

On Littlewood’s boundedness problem for sublinear Duffing equations
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Trans. Amer. Math. Soc. 353 (2001), 1567-1585 Request permission

Abstract:

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 \[ x^{\prime \prime } + g(x) = e(t), \] where the 1-periodic function $e(t)$ is a smooth function and $g(x)$ satisfies sublinearity: \[ \operatorname {sign}(x)\cdot g(x)\to +\infty ,\quad g(x)/x\to 0 \quad \operatorname {as} |x|\to +\infty . \]

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