Regularity of set-valued maps and their selections through set differences. Part 2: One-sided Lipschitz properties

R. Baier, E. Farkhi: Regularity of set-valued maps and their selections through set differences. Part 2: One-sided Lipschitz properties
Serdica Mathematical Journal 39 (3-4), 391 - 422, 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/2013/2013-391-422.pdf
MR Nummer: MR3205353
Zentralblattnummer: 06456240
Keywords: one-sided Lipschitzian set-valued maps; selections; generalized Steiner selection; metric selection; set differences; Demyanov difference; metric difference
Mathematics Subject Classification Code: 47H06 (54C65 47H04 54C60 26E25)
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Abstract:

We introduce one-sided Lipschitz (OSL) conditions of set-valued maps with respect to given set differences. The existence of selections of such maps that pass through any point of their graphs and inherit uniformly their OSL constants is studied. We show that the OSL property of a convex-valued set-valued map with respect to the Demyanov difference with a given constant is characterized by the same property of the generalized Steiner selections. We prove that an univariate OSL map with compact images in IR1 has OSL selections with the same OSL constant. For such a multifunction which is OSL with respect to the metric difference, one-sided Lipschitz metric selections exist through every point of its graph with the same OSL constant.

Contents:

1. Introduction
2. Set differences and their properties
3. OSL multimaps through set differences
3.1 One-sided Lipschitz conditions
3.2 Properties and relations between the different notions
4. One-sided Lipschitz selections
4.1 One-sided Lipschitz conditions
4.2 One-sided Lipschitz metric selections
4.3 One-sided Lipschitz selections of OSL maps
5. Examples
5.1 Examples illustrating strict inclusions in Theorem 3.17
5.2 Examples for Lipschitz selections
  Conclusions

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