Properties of set-valued Young integrals and Young differential inclusions generated by sets of Hölder functions

Abstract

The present studies concern properties of set-valued Young integrals generated by families of \(\beta \)-Hölder functions and differential inclusions governed by such a type of integrals. These integrals differ from classical set-valued integrals of set-valued functions constructed in an Aumann’s sense. Integrals and inclusions considered in the manuscript contain as a particular case set-valued integrals and inclusions driven by a fractional Brownian motion. Our study is focused on topological properties of solutions to Young differential inclusions. In particular, we show that the set of all solutions is compact in the space of continuous functions. We also study its dependence on initial conditions as well as properties of reachable sets of solutions. The results obtained in the paper are finally applied to some optimality problems driven by Young differential inclusions. The properties of optimal solutions and their reachable sets are discussed.