Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem

This paper is devoted to study the existence of periodic solutions of the second-order equation x″=f(t,x), where f is a Carathéodory function, by combining some new properties of Greenʹs function together with Krasnoselskii fixed point theorem on compression and expansion of cones. As applications, we get new existence results for equations with jumping nonlinearities as well as equations with a repulsive or attractive singularity. In this latter case, our results cover equations with weak singularities and are compared with some recent results by I. Rachunková, M. Tvrdý and I. Vrko .