  In this report, we address differential systems with Lipschitz non
linearities; this study is motivated by the subject of vibrations of structures
with unilateral springs or non linear stress-strain law close to the linear
case. We consider existence and solution with fixed point methods; this method
is constructive and provides a numerical algorithm which is under study. We
describe the method for a static case example and we address periodic solutions
of differential systems arising in the vibration of structures.
