EULER APPROXIMATION OF NONCONVEX DISCONTINUOUS DIFFERENTIAL INCLUSIONS
Tzanko Donchev · 2002
In the paper we study two types of time-discretization of one sided Lipschitz differential inclusions which right-hand side is neither upper nor lower semicontinuous. In the first one the original right-hand side is used. In the second one we use its closed graph convex regularization. It is remarkable that the both schemes give O(h 1/2) approximation of the solution set of the regularized differential inclusion. In the last section we apply these results to investigate some qualitative properties of differential inclusions in Hilbert spaces. The paper is a natural extension of [6] (see also [8]). Let H be a Hilbert space and let I = [0, 1]. Consider the following differential inclusion: ˙x(t) ∈ F (t, x(t)), x(0) = x0. (1) Here x0 ∈ H and F is a multifunction from I × H into H with nonempty closed and bounded values. The corresponding to (1) discretized inclusion is: ˙y(t) ∈ F (t, y(ti)); y(ti) = lim y(t); y(0) = x0. (2) t↑ti The mesh points on I are 0 = t0 < t1 < · · · < tN = 1. The main advantage of (2) is that we require only that F (·, x) admits a (strongly) measurable selection. No assumptions for F (t, ·) have to be made.