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

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.