Limit Cycles of the Generalized Polynomial Liénard Differential Equations

We apply the averaging theory of first, second and third order to the class of generalized polynomial Liénard differential equations. Our main result shows that for any n, m greater than or equal to 1 there aredifferential equations of the form $\ddot{x}+f(x) \dot{x}+g(x)=0$, with f and g polynomials of degree n and m respectively, having at least [(n+m-1)/2] limit cycles, where [.] denotes the integer part function.