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

R. Baier, E. Farkhi: Regularity of set-valued maps and their selections through set differences. Part 1: Lipschitz continuity
Serdica Mathematical Journal 39 (3-4), 365 - 390, 2013, Special issue dedicated to the 65th anniversary of Professor Asen L. Dontchev and to the 60th anniversary of Professor Vladimir M. Veliov.

Smart-Link: http://www.math.bas.bg/serdica/2013/2013-365-390.pdf
MR Nummer: MR3205352
Zentralblattnummer: 06456239
Keywords: nonconvex subdifferentials; directional derivatives; difference of convex (delta-convex, DC) functions; differences of sets
Mathematics Subject Classification Code: 54C65 (54C60 26E25)
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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 IRn, 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.

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

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