We study the differential equationwith T -periodic right-hand side, which models e.g. a mechanical system with one degree of freedom subjected to dry friction and periodic external force. If, in particular, the damping term g is present and acts, up to a bounded difference, like a linear damping, we get existence of a T -periodic solution.In the more difficult case g = 0, we concentrate on the model equationand obtain sufficient conditions for the existence of a T -periodic solution by application of Brouwer's fixed point theorem. For this purpose we show that a certain associated autonomous differential equation admits a periodic orbit such that the surrounded set (minus some neighborhood of the equilibria) is forward invariant for the equation above. Under additional assumptions on µ we prove boundedness of all solutions. Finally, we provide a principle of linearized stability for periodic solutions without deadzones, where the "linearized" differential equation is an impulsive Hill equation.Mathematics Subject Classification (1991). 34A60, 34C25, 70K20, 70K30.