On Dependence of Sets of Functions on the Mean Value of their Elements
Journal Title: Mathematical Modelling and Analysis - Year 2009, Vol 14, Issue 1
Abstract
The paper considers, for a given closed bounded set $Msubset {mathbb R}^m$ and $K=(0,1)^nsubset {mathbb R}^n$, the set ${mathcal M}= { hin L_2(K;{mathbb R}^m)mid h(x) in M,,a.e.,xin K}$ and its subsets [ {mathcal M}(hat h)=Big{ hin {mathcal M}mid int _K h(x)dx =hat h Big}. ] It is shown that, if a sequence ${ hat h_k} subset coM $ converges to an element $h_0$. $h_k in mathcal{M}(hat h_k)$ there is $h_k^{‘} in mathcal{M}(hat h_0)$ such that $h_k^{‘}-h_k rightarrow 0$ as $k rightarrow infty$. If, in addition, the set $M$ is finite or $M$ is the convex hull of a finite set of elements, then the multivalued mapping $hat h rightarrow {mathcal M}(hat h)$ is lower semicontinuous on $coM$.
Authors and Affiliations
U. Raitums
Keywords
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- EP ID EP83039
- DOI 10.3846/1392-6292.2009.14.91-98
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