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On upper semicontinuous multivalued mappings. (English. Russian original) Zbl 0656.54014

Sov. Math., Dokl. 35, No. 3, 587-590 (1987); translation from Dokl. Akad. Nauk SSSR 294, No. 4, 792-795 (1987).
Conditions are given that guarantees the existence of continuous selections of an upper semicontinuous multifunction \(F\cap W\to conv(R^ n)\), \(W\subset R^ n\). They are of the form \(a_ 1(x)+G(x)\subset F(x)\subset a_ 2(x)+2G(x)\), where G lower semicontinuous, with uniformly strictly convex values, \(G(x)=-G(x)\), and \(a_ k\) some mappings. Another theorem says that for any \(\epsilon\), with \(0\leq \epsilon \leq 1\), there is an upper semicontinuous multifunction \(F:[0,1]\to conv(R^ n)\) which is discontinuous on a set of measure \(\epsilon\) as well as every its selection.
Reviewer: T.Rzezuchowski

MSC:

54C65 Selections in general topology
54C08 Weak and generalized continuity
54C60 Set-valued maps in general topology
34A60 Ordinary differential inclusions
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