Topological approach to differential inclusions

Abstract

The purpose of these lectures is to show how the topological degree theory for (non-convex) multivalued mappings can be usefully applied to differential inclusions. Namely, we shall apply it to get a topological characterization of the set of solutions and periodic solutions for some differential inclusions. We discuss these problems in the case when the considered differential inclusions are defined on Banach spaces or on proximate retracts. Recall that a proximate retract is a compact subset A of the Euclidean space ℝn such that there exists an open neighbourhood U of A in ℝn and a metric retraction r: U → A. It is well known that any compact convex subset A of ℝn or any compact C 2-manifold M ⊂ ℝn is a proximate retract. Moreover, a topological degree method for implicit differential equations and differential inclusions is presented.