Periodic solutions of differential inclusions on compact subsets of \({\mathbb{R}}^ n\)
Let M be a compact subset of \({\mathbb{R}}^ n\) such that there exists an open set U (M\(\subset U)\) and a continuous function r: \(U\to M\) such that \(dist(x,M)=| x-r(x)|\) for every \(x\in U\). It is shown that the Bouligand contingent cone valued map \(T_ M: M\to {\mathbb{R}}^ n\) is lower semi-continuous. Suppose that the Euler characteristic of M is not equal to zero and a Carathéodory type convex valued map F: \([0,\infty)\times M\to {\mathbb{R}}^ n\) satisfies \(F(t,x)\cap T_ M(x)\neq \emptyset\) for all (t,x)\(\in [0,\infty)\times M\). If the map F is periodic with respect to the first variable then the differential inclusion \(x'(t)\in F(t,x(t))\) possesses a periodic solution. The problem of existence of periodic solutions is reduced to the problem of existence of fixed points of the set valued translation operator along trajectories of the differential inclusion.
scientific article; zbMATH DE number 4156961
scientific article