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 equationequation(1)u″+f(u)u′+g(t,u)=e(t),u″+f(u)u′+g(t,u)=e(t), where the nonlinearity 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≡0f≡0 the existence of subharmonic solutions is proved utilizing a variational method and when f≠0f≠0 we prove the existence of a periodic solution using topological degree theory.