{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:55:37Z","timestamp":1760151337848,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2022,3,16]],"date-time":"2022-03-16T00:00:00Z","timestamp":1647388800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>We consider a variational\u2013hemivariational inequality in a real Hilbert space, which depends on two parameters. We prove that the inequality is governed by a maximal monotone operator, then we deduce various existence, uniqueness and equivalence results. The proofs are based on the theory of maximal monotone operators, fixed point arguments and the properties of the subdifferential, both in the sense of Clarke and in the sense of convex analysis. These results lay the background in the study of various classes of inequalities. We use them to prove existence, uniqueness and continuous dependence results for the solution of elliptic and history-dependent variational\u2013hemivariational inequalities. We also present some iterative methods in solving these inequalities, together with various convergence results.<\/jats:p>","DOI":"10.3390\/axioms11030136","type":"journal-article","created":{"date-parts":[[2022,3,16]],"date-time":"2022-03-16T22:09:58Z","timestamp":1647468598000},"page":"136","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Monotonicity Arguments for Variational\u2013Hemivariational Inequalities in Hilbert Spaces"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6110-1433","authenticated-orcid":false,"given":"Mircea","family":"Sofonea","sequence":"first","affiliation":[{"name":"Laboratoire de Math\u00e9matiques et Physique, University of Perpignan Via Domitia, 52 Avenue Paul Alduy, 66860 Perpignan, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,3,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1007\/BF01170410","article-title":"Nonconvex energy functions, hemivariational inequalities and substationarity principles","volume":"48","author":"Panagiotopoulos","year":"1983","journal-title":"Acta Mech."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1017\/S0962492919000023","article-title":"Numerical analysis of hemivariational inequalities in contact mechanics","volume":"28","author":"Han","year":"2019","journal-title":"Acta Numer."},{"key":"ref_3","unstructured":"Naniewicz, Z., and Panagiotopoulos, P.D. (1995). Mathematical Theory of Hemivariational Inequalities and Applications, Marcel Dekker."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Panagiotopoulos, P.D. (1993). Hemivariational Inequalities: Applications in Mechanics and Engineering, Springer.","DOI":"10.1007\/978-3-642-51677-1"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Sofonea, M., and Mig\u00f3rski, S. (2018). Variational\u2013Hemivariational Inequalities with Applications, Chapman & Hall\/CRC Press.","DOI":"10.1201\/9781315153261"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2388","DOI":"10.1007\/s10915-019-01090-2","article-title":"Virtual element method for an elliptic hemivariational inequality with applications to contact mechanics","volume":"81","author":"Feng","year":"2019","journal-title":"J. Sci. Comput."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"113330","DOI":"10.1016\/j.cam.2020.113330","article-title":"The virtual element method for general elliptic hemivariational inequalities","volume":"389","author":"Feng","year":"2021","journal-title":"J. Comput. Appl. Math."},{"key":"ref_8","unstructured":"Haslinger, J., Miettinen, M., and Panagiotopoulos, P.D. (2013). Finite Element Method for Hemivariational Inequalities: Theory, Methods and Applications, Springer Science & Business Media."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1023\/A:1008282922372","article-title":"Comparing nonsmooth nonconvex bundle methods in solving hemivariational inequalities","volume":"14","author":"Miettinen","year":"1999","journal-title":"J. Glob. Optim."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"364","DOI":"10.1016\/j.cam.2018.08.046","article-title":"Numerical analysis of history-dependent variational\u2013hemivariational inequalities with applications in contact mechanics","volume":"351","author":"Xu","year":"2019","journal-title":"J. Comput. Appl. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1007\/s00033-018-0980-3","article-title":"On convergence of solutions to variational-hemivariational inequalities","volume":"69","author":"Zeng","year":"2018","journal-title":"Z. Angew. Math. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Saramito, P. (2016). Complex Fluids. Modeling and Algorithms, Springer. Math\u00e9matiques & Applications 79.","DOI":"10.1007\/978-3-319-44362-1"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"103288","DOI":"10.1016\/j.ijnonlinmec.2019.103288","article-title":"Magnetorheological Fluids: Qualitative comparison between a mixture model in the extended Irreversible Thermodynamics framework and an Herschel\u2013Bulkley experimental elastoviscoplastic model","volume":"118","author":"Versaci","year":"2020","journal-title":"Int. J. Non-Linear Mech."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1007\/s10659-016-9600-7","article-title":"A class of variational\u2013hemivariational in- equalities in reflexive Banach spaces","volume":"127","author":"Ochal","year":"2017","journal-title":"J. Elast."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"103114","DOI":"10.1016\/j.nonrwa.2020.103114","article-title":"Minimization principles for elliptic hemivariational inequalities","volume":"54","author":"Han","year":"2020","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Clarke, F.H. (1990). Optimization and Nonsmooth Analysis, SIAM.","DOI":"10.1137\/1.9781611971309"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Denkowski, Z., Mig\u00f3rski, S., and Papageorgiou, N.S. (2003). An Introduction to Nonlinear Analysis: Applications, Kluwer Academic\/Plenum Publishers.","DOI":"10.1007\/978-1-4419-9156-0"},{"key":"ref_18","unstructured":"Kurdila, A.J., and Zabarankin, M. (2005). Convex Functional Analysis, Birkh\u00e4user."},{"key":"ref_19","unstructured":"Zeidler, E. (2013). Nonlinear Functional Analysis and Its Applications: II\/B: Nonlinear Monotone Operators, Springer Science & Business Media."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Berinde, V. (2007). Iterative Approximation of Fixed Points, Springer. Lecture Notes in Mathematics 1912.","DOI":"10.1109\/SYNASC.2007.49"},{"key":"ref_21","first-page":"91","article-title":"Regularization of non-coercive quasi variational inequalities","volume":"29","author":"Giannessi","year":"2020","journal-title":"Control Cybernet."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Bauschke, H.H., and Combettes, P.L. (2011). Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer.","DOI":"10.1007\/978-1-4419-9467-7"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/3\/136\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:37:31Z","timestamp":1760135851000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/3\/136"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,3,16]]},"references-count":22,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2022,3]]}},"alternative-id":["axioms11030136"],"URL":"https:\/\/doi.org\/10.3390\/axioms11030136","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2022,3,16]]}}}