Abstract and Applied Analysis
Volume 2008 (2008), Article ID 829701, 35 pages
Research Article

Generalized Solutions of Functional Differential Inclusions

Anna Machina,1,2 Aleksander Bulgakov,3 and Anna Grigorenko4

1Center for Integrative Genetics (CIGENE), Norwegian University of Life Sciences, Aas 1432, Norway
2Department of Mathematical Sciences and Technology, Norwegian University of Life Sciences, Aas 1432, Norway
3Department of Algebra and Geometry, Tambov State University, Tambov 392000, Russia
4Department of Higher Mathematics, Faculty of Electronics and Computer Sciences, Moscow State Forest University, Moscow 141005, Russia

Received 12 March 2007; Revised 4 July 2007; Accepted 12 September 2007

Academic Editor: Yong Zhou

Copyright © 2008 Anna Machina et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We consider the initial value problem for a functional differential inclusion with a Volterra multivalued mapping that is not necessarily decomposable in L1n[a,b]. The concept of the decomposable hull of a set is introduced. Using this concept, we define a generalized solution of such a problem and study its properties. We have proven that standard results on local existence and continuation of a generalized solution remain true. The question on the estimation of a generalized solution with respect to a given absolutely continuous function is studied. The density principle is proven for the generalized solutions. Asymptotic properties of the set of generalized approximate solutions are studied.