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

DOI:
https://doi.org/10.1090/S0002-9939-01-05990-1

Published electronically:
March 15, 2001

MathSciNet review:
1823907

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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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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

Email:
georgiev@bsp.brain.riken.go.jp, georgiev@bsp.brain.riken.go.jp

DOI:
https://doi.org/10.1090/S0002-9939-01-05990-1

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