We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.
MiS Preprint
46/2001
Few remarks on differential inclusions
Mikhail A. Sytchev
Abstract
In this paper we analyse the methodology of the theory of differential inclusions. First we emphasize that any sequence of piece-wise affine functions with successive elements obtained by perturbations of preceding ones in the sets of their affinity converges strongly. This gives a simple algorithm to construct sequences of approximate solutions which converge to exact ones (neither specific choice suggested by the method of convex integration nor Baire category methodology is required). Then we suggest a functional which is defined in the set of admissible functions and which measures maximal oscillations produced by sequences of admissible functions weakly convergent to a given one. This functional can be used to prove that the set of stable solutions is dense (residual) in the closure of the set of admissible functions both via the Baire category lemma or via specific choice of strictly convergent sequences.
