On dependence of sets of functions on the mean value of their elements

Abstract

The paper considers, for a given closed bounded set M ⊂ R ^m and K = (0,1) ⁿ ⊂ R ⁿ , the set M = {h ϵ L₂ (K;R ^m ) | h(x) ϵ M a.e.x ϵ K} and its subsets

It is shown that, if a sequence {h_k } ⊂ coM converges to an element h_k ϵ M(h_k ) there is h‘_k ϵ M(h_o ) such that h'_k - hk → 0 as k → ∞ . If, in addition, the set M is finite or M is the convex hull of a finite set of elements, then the multivalued mapping h → M(h) is lower semicontinuous on coM.

First published online: 14 Oct 2010