On the existence and uniqueness of limit cycles in Liénard differential equations allowing discontinuities

In this paper we study the non-existence and the uniqueness of limit cycles for the Lienard differential equation of the form x'' − f(x)x' + g(x) = 0 where the functions f and g satisfy xf(x) > 0 and xg(x) > 0 for x ≠ 0 but can be discontinuous at x = 0.In particular, our results allow us to prove the non-existence of limit cycles under suitable assumptions, and also prove the existence and uniqueness of a limit cycle in a class of discontinuous Lienard systems which are relevant in engineering applications.

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