The number of limit cycles for generalized Abel equations with periodic coefficients of definite sign

Abstract

We study the number of limit cycles (isolated periodic solutions in the set of all periodic solutions) for the generalized Abel equation $x'=a(t)x^{n_a}+b(t)x^{n_b}+c(t)x^{n_c}+d(t)x$, where $n_a > n_b > n_c > 1$, $a(t),b(t),c(t), d(t)$ are $2\pi$-periodic continuous functions, and two of $a(t),b(t),c(t)$ have definite sign. &nbsp We obtain examples with at least seven limit cycles, and some sufficient conditions for the equation to have at most one or at most two positive limit cycles.