Periodic solutions of Lienard equations with asymmetric nonlinearities at resonance

The existence of 2pi-periodic solutions of the second-order differential equation x" + f(x)x' + ax(+) - bx(-) + g(x) = p(t), n E N, where a, b satisfy 1/roota + 1/rootb = 2/n and p(t) = p(t + 2pi), t is an element of R, is examined. Assume that limits lim(x-->+/-infinity). F(x) = F(+/-infinity) (F(x) = integral(0)(x)f(u)du) and lim(x-->+/-infinity) g(x) = g(+/-infinity) exist and are finite. It is proved that the equation has at least one 2pi-periodic solution provided that the zeros of the function Sigma(1) are simple and the zeros of the functions Sigma(1),Sigma(2) are different and the signs of Sigma(2) at the zeros of Sigma(1) in [0, 2pi/n) do not change or change more than two times, where Sigma(1) and Sigma(2) are defined as follows: Sigma(1)(theta) = n/pi[g(+infinity)/a - g(-infinity)/b] - 1/2pi integral(0)(2pi)p(t)phi(t + theta)dt, theta is an element of [0,2pi/n], Sigma(2)(theta) = n/pi[F(+infinity) - F(-infinity)] - 1/2piintegral(0)(2pi)p(t)phi'(t + theta)dt, theta is an element of [0,2pi/n]. Moreover, it is also proved that the given equation has at least one 2pi-periodic solution provided that the following conditions hold: -infinity < lim inf(x-->+/-infinity) F(x)/\x\(p-2)x less than or equal to lim sup Fx-->+/-infinity(x)/\x\(p-2)x < +infinity, 0 < lim inf (x-->+/-infinity) g(x)/\x\(q-2)x less than or equal to lim sup g(x)/\x\(q-2)x < +infinity, with 1 less than or equal to p < q < 2.