An improvement of Massera’s theorem for the existence and uniqueness of a periodic solution for the Liénard equation

In this paper we prove the existence and uniqueness of a periodic solution for the Liénard equation x¨ + f (x) x˙ + x = 0. The classical Massera’s monotonicity assumptions, which are required in the whole line, are relaxed to the interval (alfa, delta), where alfa and delta can be easily determined. In the final part of the paper a simple perturbation criterion of uniqueness is presented.