Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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

Abstract:

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.


References [Enhancements On Off] (What's this?)

  • [A-E] J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley and Sons, 1984. MR 87a:58002
  • [B-F] J. Borwein, S. Fitzpatrick, A weak Hadamard smooth renorming of $L^1$, Canad. Math. Bull., 36, (1993), 407-413. MR 94k:46035
  • [B-P] J.M. Borwein and D. Preiss, A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions, Trans. Amer. Math. Soc., 303, (1987), 517-527. MR 88k:49013
  • [D-G-Z1] R. Deville, G. Godefroy and V. Zizler, A smooth variational principle with applications to Hamilton-Jacobi equations in infinite dimensions, J. Funct. Anal., 111, (1993), 197-212. MR 94b:49010
  • [D-G-Z2] R. Deville, G. Godefroy and V. Zizler, Smoothness and Renormings in Banach Spaces, Pitman Monographs No. 64, London: Longman, 1993. MR 94d:46012
  • [D-U] J. Diestel, J.J. Uhl, Jr., Vector measures, Math. Surveys, No. 15, 1997, AMS, Providence, Rhode Island. MR 56:12216
  • [E1] I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47, (1974), 324-353. MR 49:11344
  • [E2] I. Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc., 1, (1979), 443-474. MR 80h:49007
  • [H] C.J. Himmelberg, Measurable relations, Fund. Math., 87, (1975), 53-72. MR 51:3384
  • [K-F] A. Kolmogorov, C. Fomin, Elements of function theory, Moscow, Nauka (1987).
  • [K-RN] K. Kuratowski, C. Ryll-Nardzewski, A general theorem on selectors, Bull. Acad. Polon. Sci., Ser. Sci. Math. Astron. Phys., 13, (1965), 397-403. MR 32:6421
  • [Ph] R. R. Phelps, Convex functions, Monotone Operators and Differentiability, Lecture Notes in Mathematics, Vol. 1364, 1989. MR 90g:46063
  • [P-P-N] D. Preiss, R. R. Phelps and I. Namioka, Smooth Banach spaces, weak Asplund spaces and monotone or USCO mappings, Israel J. Math., 72, (1990), 257-279. MR 92h:46021

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 28B20, 46B20

Retrieve articles in all journals with MSC (2000): 28B20, 46B20


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

American Mathematical Society