Boundedness of solutions for asymmetric Duffing equations

In this paper, we are concerned with the boundedness of all the solutions and the existence of quasiperiodic solutions for some Duffing equations x¨+g(x)=e(t), where e(t) is of period 1, and g : R → R possesses the characters: g(x) is superlinear when x ⩾ d0, d0 is a positive constant and g(x) is semilinear when x ⩽ 0.