Periodic Solutions of Li´enard Equations with Singular Forces of Repulsive Type*

In this paper we will use the coincidence degree to give an existence result of periodic solutions for the scalar Li´enard equations with singular forces of repulsive type. The main result of this paper clearly describes what balance conditions between the singular force at the singularity and at the infinity should be imposed if the equation has a periodic solution. As an application, we will improve the results of Ding, Ye, and Wang for the Brillouin electron beam focusing system.