Existence of periodic solutions for second-order differential equations with singularities and the strong force condition

We prove the existence of periodic solutions for the equation u″ + f(u)u′ + g(t, u) = e(t), where the no nlinearity g has a repulsive singularity at the origin. In previous papers dealing with this kind of problem it is usually assumed a nonintegrability condition on g near the origin. We provide a weaker condition that substitutes the nonintegrability of g. If f ≡ 0 the existence of subharmonic solutions is proved utilizing a variational method and when f ≠ 0 we prove the existence of a periodic solution using topological degree theory.