LIMIT CYCLES FOR SOME ABEL EQUATIONS HAVING COEFFICIENTS WITHOUT FIXED SIGNS

We prove that some 2 pi-periodic generalized Abel equations of the form x' = A(t)x(n) + B(t)x(m) + C(t)x, with n not equal m and n, m >= 2 have at most three limit cycles. The novelty of our result is that, in contrast with other results of the literature, our hypotheses allow the functions A, B, and C to change sign. Finally we study in more detail the Abel equation x' = A(t)x(3) + B(t)x(2), where the functions A and B are trigonometric polynomials of degree one.