Parametric Borwein-Preiss variational principle and applications

Author:
Pando Gr. Georgiev

Journal:
Proc. Amer. Math. Soc. **133** (2005), 3211-3225

MSC (2000):
Primary 49J35, 49J52; Secondary 46N10

Published electronically:
June 20, 2005

MathSciNet review:
2161143

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Abstract | References | Similar Articles | Additional Information

Abstract: A parametric version of the Borwein-Preiss smooth variational principle is presented, which states that under suitable assumptions on a given convex function depending on a parameter, the minimum point of a smooth convex perturbation of it depends continuously on the parameter. Some applications are given: existence of a Nash equilibrium and a solution of a variational inequality for a system of partially convex functions, perturbed by arbitrarily small smooth convex perturbations when one of the functions has a non-compact domain; a parametric version of the Kuhn-Tucker theorem which contains a parametric smooth variational principle with constraints; existence of a continuous selection of a subdifferential mapping depending on a parameter.

The tool for proving this parametric smooth variational principle is a useful lemma about continuous -minimizers of quasi-convex functions depending on a parameter, which has independent interest since it allows direct proofs of Ky Fan's minimax inequality, minimax equalities for quasi-convex functions, Sion's minimax theorem, etc.

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Additional Information

**Pando Gr. Georgiev**

Affiliation:
Department of Mathematics and Informatics, Sofia University “St. Kl. Ohridski", 5 James Bourchier Blvd., 1126 Sofia, Bulgaria

Address at time of publication:
Department of Electrical and Computer Engineering and Computer Science, University of Cincinnati, ML 0030, Cincinnati, Ohio 45221-0030

Email:
pgeorgie@ececs.uc.edu

DOI:
https://doi.org/10.1090/S0002-9939-05-07853-6

Keywords:
Borwein-Preiss variational principle,
Ky Fan's inequality,
continuous selections,
minimax problems.

Received by editor(s):
May 31, 1999

Published electronically:
June 20, 2005

Communicated by:
Jonathan M. Borwein

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.