A McShane integral for multifunctions

Here we introduce a new kind of multivalued integral which does not need a priori the notion of measurability; this fact looks interesting for example in differential inclusions. The idea comes out from a discussionwith Prof. Jan Andres during a congress of Nonlinear Analysis. Our starting point is a paper by Jarnık and Kurzweil in which the authors proposed a new definition based on Kurzweil-Henstock ”selections” for Rn-valued multifunctions, defined in a bounded interval of R. Jarnık and Kurzweil applied it to differential inclusions and showed that under suitable conditions (namely compactness of values) this integral coincides with the Aumann’s one. Here we extend these results in two directions: we consider in fact multifunctions defined in the whole real line and moreover taking values in a Banach space not necessarily separable. In particular in section 3 we introduce the (*)-integral by using McShane integrable single valued functions and then we compare it with the Aumann integral. Finally, in section 4, making use of the Radstrom embedding theorem, the McShane multivalued integral is introduced and compared with the (*) and Aumann integrals. When the McShane multivalued integral exists, then the (*)-integral exists too and it coincides with it, and so all the properties of the single-valued McShane integral are inherited by the multivalued one.