Boundedness for the general semilinear Duffing equations via the twist theorem

In this paper, we prove the boundedness of all solutions for the periodic equation x″+ω2x+ϕ(x)=Gx(x,t)+p(t), where ω satisfies the Diophantine condition, ϕ(x)∈C19 is bounded, p(t)∈C23 and is bounded for 0⩽i+j⩽21.