On forced nonlinear oscillations

The existence of a 1-periodic solution of the generalized Liénard equation x″ + g(x)x′ + f(t, x) = e(t), where g(x) is continuous, e(t) is continuous, periodic of period 1 and with mean value 0 and f is continuous, periodic of period 1 in t, is proved under one of the following conditions: (i) there exists M ? 0 such that $\text{f(t, x)x ? 0}\text{for}\text{¦x¦? M}$ and

$\text{lim sup}\text{|x|?+∞}\text{|f(t,x)|}\text{| x |}\text{<}\text{4π}{}^2\text{2π + 1}$

(ii) there exists M ? 0 such that $\text{f(t, x)x ? 0}\text{for}\text{¦x¦? M}$. Earlier results of A. C. Lazer, J. Mawhin and R. Reissig are obtained as particular cases.