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In this paper, we consider the Littlewood's problem for the second order differential equation x ¨ + x 2 n + 1 + q (t) x l = p (t) , l < n , where the functions p (t) and q (t) do not need to be periodic. Using the theory of non-periodic twist mappings which is developed by Kunze and Ortega in [3–7] , we obtained longtime closeness results on the bounded and unbounded solutions of the equation. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we prove the boundedness of all solutions for the periodic equation , where ω satisfies the Diophantine condition, is bounded, and is bounded for . [Copyright &y& Elsevier]

The authors study the Lagrangian stability for the sublinear Duffing equations ẍ + e(t)∣x∣α−1x = p(t) with 0 < α < 1, where e and p are real analytic quasi-periodic functions with frequency ω. It is proved that if the mean value of e is positive and the frequency ω satisfies Diophantine condition, then every solution of the equation is bounded. [ABSTRACT FROM AUTHOR]