Set-valued problems under bounded variation assumptions involving the Hausdorff excess

In the very general framework of a (possibly infinite dimensional) Banach space $ X $, we are concerned with the existence of bounded variation solutions for measure differential inclusions

$ \begin{equation} \begin{split} &dx(t) \in G(t, x(t)) dg(t),\\ &x(0) = x_0, \end{split} \end{equation}\;\;\;\;\;\;(1) $

where $ dg $ is the Stieltjes measure generated by a nondecreasing left-continuous function.

This class of differential problems covers a wide variety of problems occuring when studying the behaviour of dynamical systems, such as: differential and difference inclusions, dynamic inclusions on time scales and impulsive differential problems. The connection between the solution set associated to a given measure $ dg $ and the solution sets associated to some sequence of measures $ dg_n $ strongly convergent to $ dg $ is also investigated.

The multifunction $ G : [0,1] \times X \to \mathcal{P}(X) $ with compact values is assumed to satisfy excess bounded variation conditions, which are less restrictive comparing to bounded variation with respect to the Hausdorff-Pompeiu metric, thus the presented theory generalizes already known existence and continuous dependence results. The generalization is two-fold, since this is the first study in the setting of infinite dimensional spaces.

Next, by using a set-valued selection principle under excess bounded variation hypotheses, we obtain solutions for a functional inclusion

$ \begin{equation} \begin{split} &Y(t)\subset F(t,Y(t)),\\ &Y(0) = Y_0. \end{split} \end{equation}\;\;\;\;(2) $

It is shown that a recent parametrized version of Banach's Contraction Theorem given by V.V. Chistyakov follows from our result.