Strong convergence of selections implied by weak
Tadeusz Rzeżuchowski · Bulletin of the Australian Mathematical Society · 1989
In some situations weak convergence inL1, implies strong convergence. LetP, L: T→C∘(ℝd) be measurable multifunctions (C∘(ℝd) being the set of closed, convex subsets of ℝd) the valuesL(t)affine sets andW(t)=P(t)∩L(t)extremal faces ofP(t). Letpkbe integrable selections ofP, the projection ofpk,(t)onL(t)andpk(t)onW(t). We prove that if converges weakly to zero thenpk− kconverges to zero in measure. We give also some extensions of this theorem. As applications to differential inclusions we investigate convergence of derivatives of convergent sequences of solutions and we describe solutions which are in some sense isolated. Finally we discuss what can be said about control functionsuwhen the corresponding trajectories of ẋ =f(t, x, u)are convergent to some trajectory.