Existence of non-trivial limit cycles in Abel equations with symmetries

We study the periodic solutions of the generalized Abel equation x' = a(1)A(1)(t)x(n1) + a(2)A(2)(t)x(n2) + a(3)A(3)(t)x(n3), where n(1), n(2), n(3) > 1 are distinct integers, a(1), a(2), a(3) is an element of R, and A(1), A(2), A(3) are 2 pi-periodic analytic functions such that A(1)(t) sin t, A(2)(t) cos t, A(3)(t) sin t cos t are pi-periodic positive even functions. When (n(3) - n(1))(n(3) - n(2)) < 0 we prove that the equation has no non-trivial (different from zero) limit cycle for any value of the parameters a(1), a(2), a(3). When (n(3) - n(1))(n(3) - n(2)) > 0 we obtain under additional conditions the existence of non-trivial limit cycles. In particular, we obtain limit cycles not detected by Abelian integrals. (C) 2013 Elsevier Ltd. All rights reserved.