On the use of regularization to correct monotone variational inequalities given approximately (Q1311172)

This article deals with Browder-Tikhonov regularization in application to the variational inequality \[ \langle Fx- f, x-z\rangle\leq 0 \qquad (z\in Q) \tag \(*\) \] where \(F\) is a monotone operator acting from \(D(P) \subseteq \mathbb{X}\) into \(\mathbb{X}^*\), \(\mathbb{X}\) is a strictly convex Banach space with strictly convex conjugate space \(\mathbb{X}^*\) in the case when the solution set of \((*)\) is empty. The main result is a theorem about conditions under which the Browder-Tikhonov regularization defines the solution to the variational inequality \[ \langle Fx- f+v_ *, x-z\rangle \leq 0 \qquad (z\in Q) \tag \(**\) \] with the least \(v_ *\in \mathbb{X}^*\); further a theorem about conditions under which the finite-dimensional approximations of the Browder-Tikhonov regularization define the solution to \((**)\). As an example the elliptic variational inequality of second order of the Dirichlet problem is considered in detail.