| Abstrakt: |
In the very general framework of a (possibly infinite dimensional) Banach space X X , we are concerned with the existence of bounded variation solutions for measure differential inclusions d x (t) ∈ G (t , x (t)) d g (t) , x (0) = x 0 , (1) d x (t) ∈ G (t , x (t)) d g (t) , x (0) = x 0 , (1) where d g d g 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 d g d g and the solution sets associated to some sequence of measures d g n d g n strongly convergent to d g d g is also investigated. The multifunction G : [ 0 , 1 ] × X → P (X) G : [ 0 , 1 ] × X → 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 Y (t) ⊂ F (t , Y (t)) , Y (0) = Y 0. (2) Y (t) ⊂ F (t , Y (t)) , Y (0) = Y 0. (2) It is shown that a recent parametrized version of Banach's Contraction Theorem given by V.V. Chistyakov follows from our result. [ABSTRACT FROM AUTHOR] |
