Approximation from the exterior of multifunctions with connected values
Let \(({\mathcal M}, d)\) be a metric space and \(\mathcal K\) be the family of all compact connected subsets of \(\mathbb{R}^n\). Equip \(\mathcal K\) with the Hausdorff metric \(h\). Then one can define the set \(\text{Lip}({\mathcal M}, {\mathcal K})\) of Lipschitz (multi-) functions \(f: ({\mathcal M}, d)\to ({\mathcal K}, h)\). Theorem. For every upper semicontinuous multi-function \(F: ({\mathcal M}, d)\to ({\mathcal K}, h)\) there exists a sequence \(\{F_n\}_{n\in \mathbb{N}}\subset \text{Lip}({\mathcal M}, {\mathcal K})\) such that \[ F(t)\leq F_{n+ 1}(t)\leq F_n(t)\tag{a} \] for all \(t\in {\mathcal M}\) and \(n\in \mathbb{N}\); \[ \lim_{n\to+ \infty} h(F_n(t), F(t))= 0\tag{b} \] for every \(t\in {\mathcal M}\). A function \(G: {\mathcal M}\to {\mathcal K}\) is said to be upper semicontinuous if for every \(t\in {\mathcal M}\) and \(\varepsilon> 0\) there is an open set \(U\ni t\) such that \(F(t')\) for every \(t'\in U\) lies in the \(\varepsilon\)- neighbourhood of \(F(t)\).
scientific article; zbMATH DE number 720903