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Existence Results for a Class of Variational Quasi-Mixed Hemivariational-Like Inequalities

Existence Results for a Class of Variational Quasi-Mixed Hemivariational-Like Inequalities The paper aims to explore the existence results for a class of variational quasi-mixed hemivariational-like inequality problems with nonlinear terms in reflexive Banach spaces, which contain variational and hemivariational inequalities. We make use of stable (η,ψ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(\eta ,\psi )$$\end{document}-quasimonotonicity, KKM theorem, Clarke’s generalized directional derivative and Clarke’s generalized gradient to derive the existence theorems for the condition of the constrained set being bounded. Further, we obtain the solution’s existence results when the constrained set is unbounded by utilizing suitable coercive conditions. Moreover, we present some sufficient conditions to assure the boundedness of the solutions set. Besides, we also demonstrate a necessary and sufficient criteria for a restricted class of variational quasi-mixed hemivariational-like inequality problems. Several applications of the main results are illustrated. The new developments improve and generalize some well-known works. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the Malaysian Mathematical Sciences Society Springer Journals

Existence Results for a Class of Variational Quasi-Mixed Hemivariational-Like Inequalities

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Publisher
Springer Journals
Copyright
Copyright © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2022
ISSN
0126-6705
eISSN
2180-4206
DOI
10.1007/s40840-021-01214-8
Publisher site
See Article on Publisher Site

Abstract

The paper aims to explore the existence results for a class of variational quasi-mixed hemivariational-like inequality problems with nonlinear terms in reflexive Banach spaces, which contain variational and hemivariational inequalities. We make use of stable (η,ψ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(\eta ,\psi )$$\end{document}-quasimonotonicity, KKM theorem, Clarke’s generalized directional derivative and Clarke’s generalized gradient to derive the existence theorems for the condition of the constrained set being bounded. Further, we obtain the solution’s existence results when the constrained set is unbounded by utilizing suitable coercive conditions. Moreover, we present some sufficient conditions to assure the boundedness of the solutions set. Besides, we also demonstrate a necessary and sufficient criteria for a restricted class of variational quasi-mixed hemivariational-like inequality problems. Several applications of the main results are illustrated. The new developments improve and generalize some well-known works.

Journal

Bulletin of the Malaysian Mathematical Sciences SocietySpringer Journals

Published: Jan 17, 2022

Keywords: Variational quasi-mixed hemivariational-like inequality problem; KKM theorem; Existence results; Stable (ηψ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\eta \psi )$$\end{document}-quasimonotone; Coercivity conditions; 49J40; 47J20

References