Measure differential inclusions - between continuous and discrete

The paper is devoted to the study of the measure-driven differential inclusions d x ( t ) ∈ G ( t , x ( t ) ) d μ ( t ) , x ( 0 ) = x 0 for arbitrary finite Borel measure μ . This type of results allows one to treat in a similar manner differential and difference inclusions, as well as impulsive problems and therefore to study the evolution of hybrid systems with very complex (including Zeno) behavior. Our method is based on viewing the Borel measures as Lebesgue-Stieltjes measures. We thus obtain, under very general assumptions, the existence of regulated or bounded variation solutions of the considered problem and we indicate some advantages of our approach. MSC: 49N25, 34A60, 93C30, 49J53, 37N35, 34A37.