Periodic bouncing solutions for nonlinear impact oscillators

We prove the existence of a periodic solution to a nonlinear impact oscillator, whose restor- ing force has an asymptotically linear behavior. To this aim, after regularizing the problem, we use phase-plane analysis, and apply the Poincare ́–Bohl fixed point Theorem to the as- sociated Poincare ́ map, so to find a periodic solution of the regularized problem. Passing to the limit, we eventually find the “bouncing solution” we are looking for