Boundedness of solutions for Duffing equation with low regularity in time

It is shown that all solutions are bounded for Duffing equation $\ddot{x}+ x^{2n+1}+\sum_{j=0}^{2n}P_{j}(t)x^{j}=0,$providedthatforeach$n+1\le j\le 2n$,$P_j(t)\in C^{\gamma}(\mathbb T)$with$\gamma>1-\frac1n$andforeach$0\le j\le n$,$P_j\in L(\mathbb T^1)$.