Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal (Nonlinear Anal. Real World Appl.) (20230101), 69, Paper No 103709, 25~pp. ISSN: 1468-1218 (print).eISSN: 1878-5719.
Subject
47 Operator theory -- 47H Nonlinear operators and their properties 47H05 Monotone operators and generalizations 47H10 Fixed-point theorems
49 Calculus of variations and optimal control; optimization -- 49J Existence theories 49J40 Variational methods including variational inequalities
In the literature, there have been relatively few research results focused on quasi-variational inequality problems, as compared to classical variational inequality problems. Among the former, quasi-variational-hemivariational inequalities have received particularly little attention. These types of problems are significantly more challenging than their classical counterparts, due to their non-constant constraint set. \par In this paper, the authors study the solvability and regularization of a class of multivalued quasi-variational-hemivariational inequalities (MQHI) in reflexive Banach spaces. The novelty of this paper is threefold. Firstly, the obtained solvability result for MQHI generalizes the study of single-valued variational-hemivariational inequalities in [S. Migórski and S. Dudek, Comput. Math. Appl. {\bf 100} (2021), 51--61; MR4310460]. The approach used here reduces the requirement of continuity of the multivalued operator, as compared with the method used in [S. Zeng, S. Migórski and A.~A. Khan, SIAM J. Control Optim. {\bf 59} (2021), no.~2, 1246--1274; MR4236852] for the solvability of a relevant problem. Secondly, the authors construct a regularized problem for perturbations of MQHI, using the regularization technique. They prove a weak convergence result for the obtained regularized solution sequence under mild conditions. Finally, as an application, the theoretical results are applied to a static frictional contact problem involving a class of elastic materials and some mechanical interpretations are provided. \par It is noteworthy that the results presented in this paper are interesting, and they demonstrate the potential for further research in this area.