Periodic motions in impact oscillators with perfectly elastic bounces

We consider oscillators x" + lambdax = p(t) with an obstacle at zero, i.e. the motion is restricted to the half-axis x greater than or equal to 0, and the moving particle bounces when it hits the position x = 0, driven by a 2pi-periodic forcing term. We present existence results for periodic solutions. In particular, for the resonant cases lambda = k(2)/4, where k is an integer, the existence depends on the number of zeros of the function Phi*(k,p)(theta) = integral(0)(2pi) p(t)\cos (k)/(2) (t + theta)\dt. Actually, we obtain the result for a more general equation, with p(t) replaced by a nonlinear term g(t, x). A non-existence result is also proposed, as well as a multiplicity result, which concerns the case where p(t) is negative for all t. In this last case, we obtain solutions which have an arbitrary large number of impacts, on the wall x = 0, in a period. The basic idea of our proofs consists in considering the impact oscillator as a limit case of an asymmetric oscillator.