The Penalty Method for Variational Inequalities with Nonsmooth Unbounded Operators in Banach Space

TL;DR

This paper investigates the stability, convergence, and existence of solutions for the penalty method applied to variational inequalities with nonsmooth, unbounded operators in Banach spaces, extending previous results.

Contribution

It generalizes and improves upon Lions' earlier results, providing new stability theorems applicable even in Hilbert spaces for variational inequalities with complex operators.

Findings

Established existence of solutions under perturbations

Proved convergence and stability of the penalty method

Extended stability results to Banach and Hilbert spaces

Abstract

The existence of a solution, convergence and stability of the penalty method for variational inequalities with nonsmooth unbounded uniformly and properly monotone operators in Banach spase $B$ are investigated. All the objects of the inequality - the operator A, "the right-hand part" $f$ and the set of constrains $\Omega $ - are to be perturbed. The stability theorems are formulated in terms of geometric characteristics of the spaces $B$ and $B^*$. The results of this paper are continuity and generalization of the Lions' ones, published earlier in \cite{l}. They are new even in Hilbert spaces.