On the uniqueness of the limit cycle for the Liénard equation with f(x) not sign-definite

The problem of uniqueness of limit cycles for the Liénard equation ẍ+f(x)ẋ+g(x)=0 is investigated. The classical assumption of sign-definiteness of f(x) is relaxed. The effectiveness of our result as a perturbation technique is illustrated by some constructive examples of small amplitude limit cycles, coming from bifurcation theory.