Limit Cycles for Generalized Abel Equations

This paper deals with the problem of finding upper bounds on the number of periodic solutions of a class of one-dimensional non-autonomous differential equations: those with the right-hand sides being polynomials of degree n and whose coefficients are real smooth 1-periodic functions. The case n = 3 gives the so-called Abel equations which have been thoroughly studied and are quite understood. We consider two natural generalizations of Abel equations. Our results extend previous works of Lins Neto and Panov and try to step forward in the understanding of the case n > 3. They can be applied, as well, to control the number of limit cycles of some planar ordinary differential equations.