Limit cycles of the generalized Li'enard differential equation via averaging theory

We apply the averaging theory of first and second order to a generalized Lienard differential equation. Our main result shows that for any $n,m geq 1$ there are differential equations $ddot{x}+f(x,dot{x})dot{x}+ g(x)=0$, with f and g polynomials of degree n and m respectively, having at most $[n/2]$ and $max{[(n-1)/2]+[m/2], [n+(-1)^{n+1}/2]}$ limit cycles, where $[cdot]$ denotes the integer part function.