On the number of limit cycles of a class of polynomial systems of Liénard type

In this paper we study a planar system of the form ẋ=pk(y), ẏ=−gm(x)−εfn(x)y, where pk(y) is a polynomial of degree k in y, and fn(x),gm(x) are polynomials in x with degree of n and m, respectively. Let H(m,n,k) denote the maximal number of limit cycles having odd multiplicities of this system. We give some lower bounds of H(m,n,k).