Boundedness of solutions for semilinear Duffing equations (Q1265090)

The paper concerns the semilinear Duffing equation \(x''(t)+\lambda^2x(t)+\phi(x(t))=e(t)\). Here, \(\phi(x)=o(x)\) as \(| x|\to+\infty\) and \(e\) is \(2\pi\)-periodic. For this equation, the main result states that all solutions are bounded and some solutions are quasi-periodic. The assumptions on \(e\) do not require any smallness condition [cf. \textit{R. Ortega}, J. Lond. Math. Soc., II. Ser. 53, No. 2, 325-342 (1996; Zbl 0860.34017)].