A random variational principle with application to weak Hadamard differentiability of convex integral functionals
Francesco S. De Blasi and Pando Gr. Georgiev
Proc. Amer. Math. Soc. 129 (2001), 2253-2260
Primary 28B20; Secondary 46B20
March 15, 2001
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We present a random version of the Borwein-Preiss smooth variational principle, stating that under suitable conditions, the set of minimizers of a perturbed function depending on a random variable, admits a measurable selection. Two applications are given. The first one shows that if is a superreflexive Banach space, then any convex continuous integral functional on from a certain class (in particular the usual norm), is weak Hadamard differentiable on a subset whose complement is -very porous. The second application is a random version of the Caristi fixed point theorem for multifunctions.
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Francesco S. De Blasi
Department of Mathematics, University of Roma II ‘Tor Vergata’, Via della Ricerca Scientifica, 00133 Roma, Italy
Pando Gr. Georgiev
Department of Mathematics and Informatics, University of Sofia, 5 James Bourchier Blvd., 1126 Sofia, Bulgaria
Address at time of publication:
Laboratory for Advanced Brain Signal Processing, Brain Science Institute, The Institute of Physical and Chemical Research (RIKEN), 2-1, Hirosawa, Wako-shi, Saitama, 351-0198, Japan
Received by editor(s):
October 18, 1999
March 15, 2001
This work was partially supported by the project ‘Geometrical functional analysis in Banach spaces: variational principles and global approximation’ between Italy and Bulgaria, and partially by the National Foundation for Scientific Investigation in Bulgaria under contract number MM 703/1997
The second named author thanks University Roma II for their hospitality, where a part of this work was done during his stay as a Visiting Professor in July 1998. A part of this work was presented at the international conferences Analysis and Logic, August 1997, Mons, Belgium, and Functional Analysis and Approximation, Gargnano, Italy, October 1998.
Jonathan M. Borwein
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