Existence of Positive Periodic Solutions for the Lienard Differential Equations with Weakly Repulsive Singularity

In this paper we discuss the existence of positive T-periodic solutions for the following second order differential equation x + f (x)x (over dot) + g(x) = c(t), where f, g : R+ = (0, +infinity) -> R are continuous functions, g has a repulsive singularity at the origin and c(t) is a continuous T-periodic function. The novelty of the present article is that for the first time we show that a weakly repulsive singularity enables the achievement of a new existence criterion of positive T-periodic solutions through a basic application of the coincidence degree theory. Recent results in the literature are generalized and significantly improved.