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摘要
This paper is concerned with the existence of response solutions for a class of quasi-periodically forced harmonic oscillators @@@ (x)double over dot + lambda(2)x = epsilon f (t, x, (x)over dot) @@@ where lambda is an element of O with O being a closed interval not containing zero, the forcing term f is of class C-l (l > 2n) and quasi-periodic in t with the frequency omega = (omega(1), center dot center dot center dot , omega(n)) is an element of R-n. In addition, the forced term f is reversible with respect to the involution G : (x, y) -> (-x, y). In order to study this equation, we investigate the existence of invariant tori for the following elliptic lower dimensional reversible system @@@ (theta)over dot = omega, (x)over dot = Omega y + f (theta, x, y, Omega), (y)over dot = -Omega x+ g(theta, x, y, Omega), @@@ where (theta, x, y) is an element of T-n x R-m x R-m, omega is an element of R-n is fixed satisfying the diophantine condition and Omega = (Omega(1), center dot center dot center dot , Omega(n)) is an element of R-m are real parameters. The perturbations f and g are of class C-l (l > 2n). Moreover, this system is reversible with respect to the involution M : (theta, x, y) -> (-theta, x, -y). By KAM iterative method, we prove the existence of invariant tori for the above reversible system. By applying this result, we show that there exists a positive Lebesgue measure set of lambda contained in O such that the harmonic oscillators has response solutions.
