A Survey for a Generalized Li\'enard Equation Without the Massera Condition

Abstract

Massera's method [13] is powerful for the criterion of the unique existence of limit cycles of a classical Liénard equation as is seen in the Van der Pol equation. However, by the reason of which the condition of f (x) is strong (it is the condition that there does not exist the solutions satisfying the equation f ′ (x) = 0 except x = 0), we cannot apply it to many examples. Our purpose is to give several theorems such as can be applied to the generalized Liénard equation without Massera's condition. As an application, the unique existence of limit cycles of the Duff-Levinson system which is well-known in [3] or [15] is discussed as an improvement of [7].