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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: 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 $\varepsilon$-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

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.

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