Existence of at Most 1, 2, or 3 Zeros of a Melnikov Function and Limit Cycles

We investigate the existence of at most one, two, or three limit cycles bifurcated from a periodic annulus of a Hamiltonian system under a class of perturbations and obtain some sufficient conditions which ensure that the corresponding Melnikov function has at most one, two, or three zeros in an open interval. We also give applications to some systems which appear in codimension two bifurcations and to some Lienard systems.