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

Title data

Baier, Robert ; Farkhi, Elza:
Regularity of set-valued maps and their selections through set differences. Part 1: Lipschitz continuity.
In: Serdica Mathematical Journal. Vol. 39 (2013) Issue 3-4 . - pp. 365-390.
ISSN 1310-6600

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Official URL: Volltext

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

Abstract in another language

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: Article in a journal
Refereed: Yes
Additional notes: 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)
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
Result of work at the UBT: Yes
DDC Subjects: 500 Science > 510 Mathematics
Date Deposited: 10 Feb 2015 07:45
Last Modified: 04 Sep 2015 05:59
URI: https://eref.uni-bayreuth.de/id/eprint/6652

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