A Class of Generalized Mixed Variational-Hemivariational Inequalities I: Existence and Uniqueness Results
Abstract
We investigate a generalized Lagrange multiplier system in a Banach space, called a mixed variational-hemivariational inequality (MVHVI, for short), which contains a hemivariational inequality and a variational inequality. First, we employ the Minty technique and a monotonicity argument to establish an equivalence theorem, which provides three different equivalent formulations of the inequality problem. Without compactness for one of operators in the problem, a general existence theorem for (MVHVI) is proved by using the Fan-Knaster-Kuratowski-Mazurkiewicz principle combined with methods of nonsmooth analysis. Furthermore, we demonstrate several crucial properties of the solution set to (MVHVI) which include boundedness, convexity, weak closedness, and continuity. Finally, a uniqueness result with respect to the first component of the solution for the inequality problem is proved by using the Ladyzhenskaya-Babuska-Brezzi (LBB) condition. All results are obtained in a general functional framework in reflexive Banach spaces.
Keywords
Cite
@article{arxiv.1903.06566,
title = {A Class of Generalized Mixed Variational-Hemivariational Inequalities I: Existence and Uniqueness Results},
author = {Yunru Bai and Stanislaw Migorski and Shengda Zeng},
journal= {arXiv preprint arXiv:1903.06566},
year = {2019}
}