Existence of periodic solutions of pendulum-like ordinary and functional differential equations

The equation x ''(t) = a(t, x(t)) + b(t, x) + d(t, x)e(x' (t)) is considered, where a :R-2 -> R, b,d : R x C(R,R) -> R, e : R -> R are continuous, and a, b, d are T-periodic with respect to t. Using the Leray-Schauder degree theory we prove that a sign condition, in which a dominates b, is sufficient for the existence of a T-periodic solution. The main theorem is applied to the equation of the forced damped pendulum.