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

Abstract

<p style='text-indent:20px;'>In the very general framework of a (possibly infinite dimensional) Banach space <inline-formula><tex-math id="M1">\begin{document}$ X $\end{document}</tex-math></inline-formula>, we are concerned with the existence of bounded variation solutions for measure differential inclusions</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE100"> \begin{document}$ \begin{equation} \begin{split} &amp;dx(t) \in G(t, x(t)) dg(t),\\ &amp;x(0) = x_0, \end{split} \end{equation}\;\;\;\;\;\;(1) $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M2">\begin{document}$ dg $\end{document}</tex-math></inline-formula> is the Stieltjes measure generated by a nondecreasing left-continuous function.</p><p style='text-indent:20px;'>This class of differential problems covers a wide variety of problems occuring when studying the behaviour of dynamical systems, such as: differential and difference inclusions, dynamic inclusions on time scales and impulsive differential problems. The connection between the solution set associated to a given measure <inline-formula><tex-math id="M3">\begin{document}$ dg $\end{document}</tex-math></inline-formula> and the solution sets associated to some sequence of measures <inline-formula><tex-math id="M4">\begin{document}$ dg_n $\end{document}</tex-math></inline-formula> strongly convergent to <inline-formula><tex-math id="M5">\begin{document}$ dg $\end{document}</tex-math></inline-formula> is also investigated.</p><p style='text-indent:20px;'>The multifunction <inline-formula><tex-math id="M6">\begin{document}$ G : [0,1] \times X \to \mathcal{P}(X) $\end{document}</tex-math></inline-formula> with compact values is assumed to satisfy excess bounded variation conditions, which are less restrictive comparing to bounded variation with respect to the Hausdorff-Pompeiu metric, thus the presented theory generalizes already known existence and continuous dependence results. The generalization is two-fold, since this is the first study in the setting of infinite dimensional spaces.</p><p style='text-indent:20px;'>Next, by using a set-valued selection principle under excess bounded variation hypotheses, we obtain solutions for a functional inclusion</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE102"> \begin{document}$ \begin{equation} \begin{split} &amp;Y(t)\subset F(t,Y(t)),\\ &amp;Y(0) = Y_0. \end{split} \end{equation}\;\;\;\;(2) $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>It is shown that a recent parametrized version of Banach's Contraction Theorem given by V.V. Chistyakov follows from our result.</p>