Regularity of setvalued maps and their selections through set differences. Part 2: Onesided Lipschitz properties
R. Baier, E. Farkhi:
Regularity of setvalued maps and their selections through set differences. Part 2: Onesided Lipschitz properties
Serdica Mathematical Journal
39
(34),
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.
SmartLink:
http://www.math.bas.bg/serdica/2013/2013391422.pdf
MR Nummer:
MR3205353
Zentralblattnummer:
06456240
Keywords: onesided Lipschitzian setvalued 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 onesided Lipschitz (OSL) conditions of setvalued 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 convexvalued setvalued 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 IR^{1} has OSL selections with the same OSL constant. For such a multifunction which is OSL with respect to the metric difference, onesided 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 
Onesided Lipschitz conditions 
3.2 
Properties and relations between the different notions 

4. 
Onesided Lipschitz selections
4.1 
Onesided Lipschitz conditions 
4.2 
Onesided Lipschitz metric selections 
4.3 
Onesided Lipschitz selections of OSL maps 

5. 
Examples
5.1 
Examples illustrating strict inclusions in Theorem 3.17 
5.2 
Examples for Lipschitz selections 


Conclusions 