The number of limit cycles in planar systems and generalized Abel equations with monotonous hyperbolicity

We extend some previous results on the maximum number of isolated periodic solutions of generalized Abel equation and rigid systems. The key hypothesis is a monotonicity assumption on any stability operator (for instance, the divergence) along the solutions of a suitable transversal system. In such a case, at most two isolated periodic solutions exist. Under a simple additional assumption, we also prove a uniqueness result for limit cycles of rigid systems. Our results are easily applicable to special classes of equations, since the hypotheses hold when a suitable convexity property is satisfied.