Periodic Solutions of Discontinuous Second Order Differential Systems

We provide sufficient conditions por the existence of periodic solutions of some classes of autonomous and non–autonomous second order differential equations with discontinuous right–hand sides. In the plane the discontinuities considered are given by the straight lines either x = 0, or xy = 0. Two applications of these results are made, one to control systems with variable structure, and the other to small external periodic excitation of a discontinuous nonlinear oscillator.