Boundedness of Solutions for a Class of Semilinear Oscillators

We are concerned with the boundedness for the equation x″ + f(x, x′) + ω²x = p(t), where p is quasi-periodic function. Since the corresponding system is non-Hamiltonian, we transform the original system into a new reversible one, the Poincaré mapping of which satisfies the twist theorem for quasi-periodic reversible mappings of low smoothness, or is close to its linear part by normal form theorem. We then obtain results concerning the boundedness of solutions based on the recently work. The above two cases need some smooth and growth assumptions for f and p, which are precisely the innovations of this paper.