Existence of periodic solutions of ordinary differential equations

We prove the existence of a periodic solution, y∈C1(R,Rℓ), of a first-order differential equation , where f is periodic with respect to t and admits a star-shaped compact set that is invariant under the Euler iterates of the equation with sufficiently small time-step. As in Peanoʼs Theorem for the Cauchy problem, the only required regularity condition on f is continuity. We present two nontrivial examples that illustrate the usefulness of this theorem in applications related to forced oscillations.