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A random variational principle with application to weak Hadamard differentiability of convex integral functionals

Authors: Francesco S. De Blasi and Pando Gr. Georgiev
Journal: Proc. Amer. Math. Soc. 129 (2001), 2253-2260
MSC (2000): Primary 28B20; Secondary 46B20
Published electronically: March 15, 2001
MathSciNet review: 1823907
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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 $E$ is a superreflexive Banach space, then any convex continuous integral functional on $L^1(T, \mu; E)$ from a certain class (in particular the usual $L^1$ norm), is weak Hadamard differentiable on a subset whose complement is $\sigma$-very porous. The second application is a random version of the Caristi fixed point theorem for multifunctions.

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Additional Information

Francesco S. De Blasi
Affiliation: Department of Mathematics, University of Roma II ‘Tor Vergata’, Via della Ricerca Scientifica, 00133 Roma, Italy

Pando Gr. Georgiev
Affiliation: 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
Published electronically: March 15, 2001
Additional Notes: 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.
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2001 American Mathematical Society