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Vol. 49 No. 1–3 (2023)
Serdica Mathematical Journal

Articles

Variational principles for monotone variational inequalities: The single-valued case

https://doi.org/10.55630/serdica.2023.49.77-96

Published 20-11-2023

  • Pando Gr. Georgiev

Pando Gr. Georgiev
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences

Abstract

We consider a parameterized variational inequality \((A,Y)\) in a Banach space \(E\) defined on a closed, convex and bounded subset \(Y\) of \(E\) by a monotone operator \(A\) depending on a parameter. We prove that under suitable conditions, there exists an arbitrarily small monotone perturbation of \(A\) such that the perturbed variational inequality has a solution which is a continuous function of the parameter, and is near to a given approximate solution. In the nonparametric case this can be considered as a variational principle for variational inequalities, an analogue of the Borwein-Preiss smooth variational principle.

Some applications are given: an analogue of the Nash equilibrium problem, defined by a partially monotone operator, and a variant of the parametric Borwein-Preiss variational principle for Gâteaux differentiable convex functions under relaxed assumtions.

The tool for proving the main result is a useful lemma about existence of continuous \(\varepsilon\)-solutions of a variational inequality depending on a parameter. It has an independent interest and allows a direct proof of an analogue of Ky Fan's inequality for monotone operators, introduced here, which leads to a new proof of the Schauder fixed point theorem in Gâteaux smooth Banach spaces.

2020 Mathematics Subject Classification:

49J40, 47J20

Keywords

variational inequalities, monotone operators, variational principles, Nash equilibrium, Schauder fixed point theorem

Full text

Author Details

Pando Gr. Georgiev

Institute of Mathematics and Informatics
Bulgarian Academy of Sciences
Acad. G. Bonchev Str., Bl. 8
1113 Sofia, Bulgaria
pandogeorgiev2020@gmail.com

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