On the number of periodic solutions for some polynomial differential equations with periodic coefficients

We have considered the differential equations (generalized Abel equations), which right-hand sides are polynomials of degree n>2 with continuous coefficients that periodically depend on the argument. Many authors estimated the number of periodic solutions for such equations. In this article, it is assumed that the coefficient for the highest degree of polynomial is either non-negative or nonpositive, and some of the coefficients are zero. In the case of odd n it is proved that the equation has at most three periodic solutions and their multiplicity is analyzed. In the case of even n and zero free term of the right-hand side, it has been established that the equation has at most four periodic solutions and the multiplicity of these solutions is also analyzed.