A class of generalized mixed variational–hemivariational inequalities I: Existence and uniqueness results. (15th May 2020)

• Record Type: Journal Article
• Authors: Bai, Yunru; Migórski, Stanisław; Zeng, Shengda
• Is Part Of: Computers & mathematics with applications. Volume 79:issue 10(2020)
• Page Start: 2897
• Page End: 2911
• Publication Date: 2020-05-15
• Subjects: Mixed variational–hemivariational inequality -- Fan–Knaster–Kuratowski–Mazurkiewicz principle -- Ladyzhenskaya–Babuška–Brezzi (LBB) condition -- Upper semicontinuity -- Existence -- Uniqueness; Electronic data processing -- Periodicals; Mathematics -- Data processing -- Periodicals; 510.28541
• DOI: 10.1016/j.camwa.2019.12.025
• Languages: English
• ISSNs: 0898-1221

Abstract

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–Babuška–Brezzi (LBB) condition. All results are obtained in a general functional framework in reflexive Banach spaces.