Bernstein–Doetsch-type criteria for the continuity and Lipschitz continuity of convex set-valued mappings

The Bernstein–Doetsch criterion (for convex and midconvex functionals) has been repeatedly generalized to convex and midconvex set-valued mappings F : X → 2 Y ; continuity and local Lipschitz continuity were understood in the sense of the Hausdorff distance. However, all such results imposed restric...

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Bibliographic Details
Published inDoklady. Mathematics Vol. 94; no. 3; pp. 667 - 669
Main Author Marinov, A. V.
Format Journal Article
LanguageEnglish
Published Moscow Pleiades Publishing 01.11.2016
Springer Nature B.V
Subjects
Continuity
Criteria
Functionals
Mathematics
Mathematics and Statistics
Vector spaces
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Summary:The Bernstein–Doetsch criterion (for convex and midconvex functionals) has been repeatedly generalized to convex and midconvex set-valued mappings F : X → 2 Y ; continuity and local Lipschitz continuity were understood in the sense of the Hausdorff distance. However, all such results imposed restrictive additional boundedness-type conditions on the images F ( x ). In this paper, the Bernstein–Doetsch criterion is generalized to arbitrary convex and midconvex set-valued mappings acting on normed linear spaces X , Y .
ISSN:1064-5624
1531-8362
DOI:10.1134/S1064562416060193