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

Abstract

We prove the existence of periodic solutions for the equation (1) 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 ≡ 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.