{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T03:00:23Z","timestamp":1760151623035,"version":"build-2065373602"},"reference-count":27,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2022,4,6]],"date-time":"2022-04-06T00:00:00Z","timestamp":1649203200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this work, we suggest a differential variational inequality in reflexive Banach spaces and construct a sequence with a set of constraints and a penalty parameter. We use the penalty method to prove a unique solution to the problem and make suitable assumptions to prove the convergence of the sequence. The proof is based on arguments for compactness, symmetry, pseudomonotonicity, Mosco convergence, inverse strong monotonicity and Lipschitz continuity. Finally, we discuss the boundary value problem for the differential variational inequality problem as an application.<\/jats:p>","DOI":"10.3390\/sym14040760","type":"journal-article","created":{"date-parts":[[2022,4,7]],"date-time":"2022-04-07T22:23:23Z","timestamp":1649370203000},"page":"760","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["The Convergence Results of Differential Variational Inequality Problems"],"prefix":"10.3390","volume":"14","author":[{"given":"Shih-Sen","family":"Chang","sequence":"first","affiliation":[{"name":"Center for General Education, China Medical University, Taichung 40402, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"family":"Salahuddin","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Jazan University, Jazan 45142, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8137-2401","authenticated-orcid":false,"given":"Lin","family":"Wang","sequence":"additional","affiliation":[{"name":"College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming 650221, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhaoli","family":"Ma","sequence":"additional","affiliation":[{"name":"College of Public Foundation, Yunnan Open University, Kunming 650221, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,4,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1007\/s10107-006-0052-x","article-title":"Differential variational inequalities","volume":"113","author":"Pang","year":"2008","journal-title":"Math. 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Equ."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1017","DOI":"10.1080\/02331934.2019.1571057","article-title":"Existence results of semilinear differential variational inequalities without compactness","volume":"68","author":"Lu","year":"2019","journal-title":"Optimization"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1080\/00036811.2020.1745780","article-title":"Generalized penalty method for semilinear differential variational inequalities","volume":"101","author":"Li","year":"2022","journal-title":"Appl. Anal."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1503","DOI":"10.1002\/zamm.201700348","article-title":"Penalty and regularization method for variational hemivariational inequalities with application to frictional contact","volume":"98","author":"Zeng","year":"2018","journal-title":"Z. Angew. Math. Mech."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2561","DOI":"10.1016\/j.camwa.2017.12.018","article-title":"A penalty method for history-dependent variational-hemivariational inequalities","volume":"75","author":"Sofonea","year":"2018","journal-title":"Comput. Math. Appl."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Glowinski, R. (1984). Numerical Methods for Nonlinear Variational Problems, Springer.","DOI":"10.1007\/978-3-662-12613-4"},{"key":"ref_16","first-page":"1574","article-title":"Penalty method for a class of differential variational inequalities","volume":"19","author":"Liu","year":"2019","journal-title":"Appl. Anal."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"743","DOI":"10.1080\/02331934.2019.1647539","article-title":"A class of history-dependent differential variational inequalities with application to contact problems","volume":"69","author":"Liu","year":"2020","journal-title":"Optimization"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Chang, S.S., Wang, L., Wang, G., and Zhao, Y.H. (2021). Existence and convergence results for generalized mixed quasi-variationa Hemivariational inequality problem. Symmetry, 13.","DOI":"10.3390\/sym13101882"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Chang, S.S., Ahmadini, A.A.H., Liu, M., and Tang, J.F. (2022). The optimal control problems for generalized elliptic Quasivariational inequalities. Symmetry, 14.","DOI":"10.3390\/sym14020199"},{"key":"ref_20","first-page":"82","article-title":"On penalty method for non-stationary general set valued equilibrium problems","volume":"23","author":"Salahuddin","year":"2016","journal-title":"Commun. Appl. Nonlinear Anal."},{"key":"ref_21","first-page":"469","article-title":"General nonconvex split variational inequality problems","volume":"25","author":"Kim","year":"2017","journal-title":"Korean J. Math."},{"key":"ref_22","first-page":"25","article-title":"Existence solution for the generalized relaxed pseudomonotone variational inequalities","volume":"25","author":"Kim","year":"2020","journal-title":"Nonlinear Funct. Anal. Appl."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"510","DOI":"10.1016\/0001-8708(69)90009-7","article-title":"Convergence of convex sets and of solutions of variational inequalities","volume":"3","author":"Mosco","year":"1969","journal-title":"Adv. Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"6787","DOI":"10.1016\/j.jde.2016.01.012","article-title":"Evolutionary problems driven by variational inequalities","volume":"260","author":"Liu","year":"2016","journal-title":"J. Differ. Equ."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Pazy, A. (1983). Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer.","DOI":"10.1007\/978-1-4612-5561-1"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Denkowski, Z., Mig\u00f3rski, S., and Papageorgiou, N.S. (2003). An Introduction to Nonlinear Analysis: Theory, Kluwer Academic\/Plenum Publishers.","DOI":"10.1007\/978-1-4419-9158-4"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Mig\u00f3rski, S., Ochal, A., and Sofonea, M. (2013). Nonlinear inclusions and hemivariational inequalities. Models and Analysis of Contact Problems, Springer.","DOI":"10.1007\/978-1-4614-4232-5"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/4\/760\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:49:26Z","timestamp":1760136566000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/4\/760"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,6]]},"references-count":27,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2022,4]]}},"alternative-id":["sym14040760"],"URL":"https:\/\/doi.org\/10.3390\/sym14040760","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,4,6]]}}}