Boundedness of solutions for Duffing equation with low regularity in time

Abstract

It is shown that all solutions are bounded for Duffing equation $\ddot{x}+x^{2n+1}+{\sum }_{j=0}^{2n}{P}_j(t)x^j=0,$ provided that for each $n+1\le j\le 2n$, $P_j(t)\in C^{\gamma }(\mathbb{T})$ with $\gamma >1-\frac{1}{n}$ and for each $0\le j\le n$, $P_j\in L({\mathbb{T}}^1)$.