Period two implies all periods for a class of ODEs: A multivalued map approach

We present an elementary proof that, for a multivalued map $\varphi:\mathbb{R}\multimap \mathbb{R}$ with nonempty connected values and monotone margins, the existence of a periodic orbit of any order $k>1$ implies the existence of periodic orbits of all orders. This generalizes a very recent result of this type in terms of scalar ordinary differential equations without uniqueness, due to F. Obersnel and P. Omari, obtained by means of lower and upper solutions techniques.