Abstract and Applied Analysis 
Volume 2008 (2008), Article ID 829701, 35 pages
doi:10.1155/2008/829701
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

Recommended by Yong Zhou

Abstract

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.