LIMIT CYCLES FOR SOME ABEL EQUATIONS HAVING COEFFICIENTS WITHOUT FIXED SIGNS

Abstract

We prove that some 2π-periodic generalized Abel equations of the form x′ = A(t)xn+ B(t)xm+ C(t)x, with n ≠ 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)x3+ B(t)x2, where the functions A and B are trigonometric polynomials of degree one.