Response solutions for quasi-periodically forced harmonic oscillators

We prove that the non-linear quasi-periodically forced harmonic oscillator with two frequencies ( 1 , α ) (1,\alpha ) has at least one response solution if the forcing is small. No arithmetic condition on the frequency is assumed and the smallness of the non-linear forcing does not depend on α \alpha . The result strengthens the existing results in the literature where the frequency is assumed to be Diophantine. The proof is based on a modified KAM theory for the lower dimensional tori.