On Littlewood’s boundedness problem for sublinear Duffing equations

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