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Regularity of set-valued maps and their selections through set differences. Part 1: Lipschitz continuity

URN to cite this document: urn:nbn:de:bvb:703-epub-1851-2

Title data

Baier, Robert ; Farkhi, Elza:
Regularity of set-valued maps and their selections through set differences. Part 1: Lipschitz continuity.
Bayreuth , 2013 . - 24 S.

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Project information

Project title:
Project's official titleProject's id
Hermann Minkowski Center for GeometryNo information
Marie-Curie Initial Training Network "Sensitivity Analysis for Deterministic Controller Design" (SADCO)264735-SADCO

Project financing: Andere
Hermann Minkowski Center for Geometry at Tel-Aviv University, Israel; European Union "FP7-People-ITN" programme

Abstract

We introduce Lipschitz continuity of set-valued maps with respect to a given set difference. The existence of Lipschitz selections that pass through any point of the graph of the map and inherit its Lipschitz constant is studied. We show that the Lipschitz property of the set-valued map with respect to the Demyanov difference with a given constant is characterized by the same property of its generalized Steiner selections. For a univariate multifunction with only compact values in $R^n$, we characterize its Lipschitz continuity in the Hausdorff metric (with respect to the metric difference) by the same property of its metric selections with the same constant.

Further data

Item Type: Preprint, postprint
Additional notes (visible to public): Special issue dedicated to the 65th anniversary of Professor Asen L. Dontchev and to the 60th anniversary of Professor Vladimir M. Veliov.
Contents:
1. Introduction
2. Set Differences and Their Properties
3. Regularity Notions for Multimaps through Set Differences
3.1 Lipschitz Continuity
3.2 Properties and Hierarchy of Lipschitz Maps
4. Lipschitz Generalized Steiner Selections
5. Lipschitz Metric Selections
6. Examples
6.1 Examples for Different Lipschitz Notions
6.2 Examples for Lipschitz Selections
Conclusions
Keywords: Lipschitz continuous set-valued maps; selections; generalized Steiner selection; metric selection; set differences; Demyanov metric; Demyanov difference; metric difference
Subject classification: Mathematics Subject Classification Code: 54C65 (54C60 26E25)
DDC Subjects: 500 Science > 510 Mathematics
Institutions of the University: Faculties
Faculties > Faculty of Mathematics, Physics und Computer Science
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics V (Applied Mathematics)
Profile Fields
Profile Fields > Advanced Fields
Profile Fields > Advanced Fields > Nonlinear Dynamics
Language: English
Originates at UBT: Yes
URN: urn:nbn:de:bvb:703-epub-1851-2
Date Deposited: 06 Feb 2015 07:46
Last Modified: 28 Mar 2019 10:36
URI: https://epub.uni-bayreuth.de/id/eprint/1851

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