{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,1]],"date-time":"2026-03-01T05:22:29Z","timestamp":1772342549861,"version":"3.50.1"},"reference-count":42,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2025,4,15]],"date-time":"2025-04-15T00:00:00Z","timestamp":1744675200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,4,15]],"date-time":"2025-04-15T00:00:00Z","timestamp":1744675200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"name":"Natural Science Foundation of Guangxi Grant","award":["2021GXNSFFA196004 and 2024GXNSFBA010337"],"award-info":[{"award-number":["2021GXNSFFA196004 and 2024GXNSFBA010337"]}]},{"name":"NNSF of China Grant","award":["12371312"],"award-info":[{"award-number":["12371312"]}]},{"name":"Natural Science Foundation of Chongqing Grant","award":["CSTB2024NSCQ-JQX0033"],"award-info":[{"award-number":["CSTB2024NSCQ-JQX0033"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Glob Optim"],"published-print":{"date-parts":[[2025,8]]},"DOI":"10.1007\/s10898-025-01487-4","type":"journal-article","created":{"date-parts":[[2025,4,15]],"date-time":"2025-04-15T02:55:27Z","timestamp":1744685727000},"page":"1021-1043","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Existence and stability for a class of variational\u2013hemivariational inequalities involving multivalued Brezis pseudomonotone operators"],"prefix":"10.1007","volume":"92","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1818-842X","authenticated-orcid":false,"given":"Shengda","family":"Zeng","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xi","family":"Chen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jinxia","family":"Cen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0320-7026","authenticated-orcid":false,"given":"Patrick","family":"Winkert","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,4,15]]},"reference":[{"key":"1487_CR1","volume-title":"Infinite Dimensional Analysis. A Hitchhiker\u2019s Guide","author":"CD Aliprantis","year":"2006","unstructured":"Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis. A Hitchhiker\u2019s Guide. Springer-Verlag, Berlin, Heidelberg (2006)"},{"key":"1487_CR2","first-page":"111","volume":"33","author":"K Bartosz","year":"2015","unstructured":"Bartosz, K.: Numerical methods for evolution hemivariational inequalities. Adv. Var. Hemivar. Inequal. Adv. Mech. Math. 33, 111\u2013144 (2015)","journal-title":"Adv. Var. Hemivar. Inequal. Adv. Mech. Math."},{"issue":"2","key":"1487_CR3","doi-asserted-by":"publisher","first-page":"841","DOI":"10.1016\/j.jmaa.2014.09.078","volume":"423","author":"K Bartosz","year":"2015","unstructured":"Bartosz, K., Cheng, X., Kalita, P., Yu, Y., Zheng, C.: Rothe method for parabolic variational\u2013hemivariational inequalities. J. Math. Anal. Appl. 423(2), 841\u2013862 (2015)","journal-title":"J. Math. Anal. Appl."},{"issue":"1","key":"1487_CR4","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1007\/s11228-020-00543-3","volume":"29","author":"M Bianchi","year":"2021","unstructured":"Bianchi, M., Kassay, G., Pini, R.: Regularization of Brezis pseudomonotone variational inequalities. Set-Valued Var. Anal. 29(1), 175\u2013190 (2021)","journal-title":"Set-Valued Var. Anal."},{"issue":"2","key":"1487_CR5","doi-asserted-by":"publisher","first-page":"627","DOI":"10.1016\/j.jmaa.2011.03.015","volume":"381","author":"G Bonanno","year":"2011","unstructured":"Bonanno, G., Motreanu, D., Winkert, P.: Variational-hemivariational inequalities with small perturbations of nonhomogeneous Neumann boundary conditions. J. Math. Anal. Appl. 381(2), 627\u2013637 (2011)","journal-title":"J. Math. Anal. Appl."},{"issue":"2","key":"1487_CR6","doi-asserted-by":"publisher","first-page":"493","DOI":"10.12775\/TMNA.2014.029","volume":"43","author":"G Bonanno","year":"2014","unstructured":"Bonanno, G., Winkert, P.: Multiplicity results to a class of variational\u2013hemivariational inequalities. Topol. Methods Nonlinear Anal. 43(2), 493\u2013516 (2014)","journal-title":"Topol. Methods Nonlinear Anal."},{"issue":"6","key":"1487_CR7","first-page":"293","volume":"4","author":"H Brezis","year":"1972","unstructured":"Brezis, H., Nirenberg, L., Stampacchia, G.: A remark on Ky Fan\u2019s minimax principle. Boll. Un. Mat. Ital. 4(6), 293\u2013300 (1972)","journal-title":"Boll. Un. Mat. Ital."},{"issue":"1","key":"1487_CR8","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1007\/s00605-013-0502-5","volume":"172","author":"S Carl","year":"2013","unstructured":"Carl, S.: Existence and extremal solutions of parabolic variational\u2013hemivariational inequalities. Monatsh. Math. 172(1), 29\u201354 (2013)","journal-title":"Monatsh. Math."},{"key":"1487_CR9","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-009-2121-4","volume-title":"Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems","author":"I Cioranescu","year":"1990","unstructured":"Cioranescu, I.: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer Academic Publishers Group, Dordrecht (1990)"},{"issue":"3","key":"1487_CR10","doi-asserted-by":"publisher","first-page":"1314","DOI":"10.1137\/16M1080859","volume":"27","author":"C Clason","year":"2017","unstructured":"Clason, C., Valkonen, T.: Primal-dual extragradient methods for nonlinear nonsmooth PDE-constrained optimization. SIAM J. Optim. 27(3), 1314\u20131339 (2017)","journal-title":"SIAM J. Optim."},{"issue":"4","key":"1487_CR11","doi-asserted-by":"publisher","first-page":"743","DOI":"10.1007\/s10898-011-9706-1","volume":"52","author":"N Costea","year":"2012","unstructured":"Costea, N., R\u0103dulescu, V.D.: Inequality problems of quasi-hemivariational type involving set-valued operators and a nonlinear term. J. Glob. Optim. 52(4), 743\u2013756 (2012)","journal-title":"J. Glob. Optim."},{"key":"1487_CR12","doi-asserted-by":"publisher","first-page":"257","DOI":"10.1016\/j.nonrwa.2016.02.005","volume":"31","author":"C Fang","year":"2016","unstructured":"Fang, C., Han, W., Mig\u00f3rski, S., Sofonea, M.: A class of hemivariational inequalities for nonstationary Navier\u2013Stokes equations. Nonlinear Anal. Real World Appl. 31, 257\u2013276 (2016)","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"1487_CR13","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4899-1358-6","volume-title":"Variational Inequalities and Network Equilibrium Problems","author":"F Giannessi","year":"1995","unstructured":"Giannessi, F., Maugeri, A.: Variational Inequalities and Network Equilibrium Problems. Springer, New York, NY (1995)"},{"issue":"2","key":"1487_CR14","doi-asserted-by":"publisher","first-page":"1549","DOI":"10.1137\/19M1282490","volume":"52","author":"W Han","year":"2020","unstructured":"Han, W.: Singular perturbations of variational\u2013hemivariational inequalities. SIAM J. Math. Anal. 52(2), 1549\u20131566 (2020)","journal-title":"SIAM J. Math. Anal."},{"key":"1487_CR15","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-14490-0","volume-title":"Advances in Variational and Hemivariational Inequalities","author":"W Han","year":"2015","unstructured":"Han, W., Mig\u00f3rski, S., Sofonea, M.: Advances in Variational and Hemivariational Inequalities. Springer International Publishing, Switzerland (2015)"},{"key":"1487_CR16","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4615-6359-4","volume-title":"Handbook of Multivalued Analysis.","author":"S Hu","year":"1997","unstructured":"Hu, S., Papageorgiou, N.S.: Handbook of Multivalued Analysis., vol. I. Kluwer Academic Publishers, Dordrecht (1997)"},{"issue":"4","key":"1487_CR17","doi-asserted-by":"publisher","first-page":"495","DOI":"10.1007\/s10898-003-2686-z","volume":"32","author":"N-J Huang","year":"2005","unstructured":"Huang, N.-J., Fang, Y.-P.: On vector variational inequalities in reflexive Banach spaces. J. Glob. Optim. 32(4), 495\u2013505 (2005)","journal-title":"J. Glob. Optim."},{"issue":"3","key":"1487_CR18","doi-asserted-by":"publisher","first-page":"1007","DOI":"10.1007\/s10957-022-02126-8","volume":"195","author":"PD Khanh","year":"2022","unstructured":"Khanh, P.D., Nguyen, B.T.: Continuous Fr\u00e9chet differentiability of the Moreau envelope of convex functions on Banach spaces. J. Optim. Theory Appl. 195(3), 1007\u20131018 (2022)","journal-title":"J. Optim. Theory Appl."},{"key":"1487_CR19","doi-asserted-by":"crossref","unstructured":"Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, (2000)","DOI":"10.1137\/1.9780898719451"},{"key":"1487_CR20","doi-asserted-by":"publisher","first-page":"493","DOI":"10.1002\/cpa.3160200302","volume":"20","author":"J-L Lions","year":"1967","unstructured":"Lions, J.-L., Stampacchia, G.: Variational inequalities. Comm. Pure Appl. Math. 20, 493\u2013519 (1967)","journal-title":"Comm. Pure Appl. Math."},{"issue":"17","key":"1487_CR21","doi-asserted-by":"publisher","first-page":"9543","DOI":"10.1002\/mma.6622","volume":"43","author":"Y Liu","year":"2020","unstructured":"Liu, Y., Liu, Z., Motreanu, D.: Existence and approximated results of solutions for a class of nonlocal elliptic variational\u2013hemivariational inequalities. Math. Methods Appl. Sci. 43(17), 9543\u20139556 (2020)","journal-title":"Math. Methods Appl. Sci."},{"issue":"2","key":"1487_CR22","doi-asserted-by":"publisher","first-page":"2037","DOI":"10.1007\/s00245-020-09703-1","volume":"84","author":"Y Liu","year":"2021","unstructured":"Liu, Y., Liu, Z., Wen, C.-F., Yao, J.-C., Zeng, S.: Existence of solutions for a class of noncoercive variational\u2013hemivariational inequalities arising in contact problems. Appl. Math. Optim. 84(2), 2037\u20132059 (2021)","journal-title":"Appl. Math. Optim."},{"issue":"7","key":"1487_CR23","doi-asserted-by":"publisher","first-page":"1741","DOI":"10.1088\/0951-7715\/23\/7\/012","volume":"23","author":"Z Liu","year":"2010","unstructured":"Liu, Z., Motreanu, D.: A class of variational\u2013hemivariational inequalities of elliptic type. Nonlinearity 23(7), 1741\u20131752 (2010)","journal-title":"Nonlinearity"},{"issue":"1","key":"1487_CR24","doi-asserted-by":"publisher","first-page":"85","DOI":"10.1023\/B:JOGO.0000035017.75703.7c","volume":"29","author":"Z Liu","year":"2004","unstructured":"Liu, Z.: Some convergence results for evolution hemivariational inequalities. J. Global Optim. 29(1), 85\u201395 (2004)","journal-title":"J. Global Optim."},{"issue":"2","key":"1487_CR25","doi-asserted-by":"publisher","DOI":"10.1002\/zamm.201900112","volume":"100","author":"S Mig\u00f3rski","year":"2020","unstructured":"Mig\u00f3rski, S., Dudek, S.: A new class of variational\u2013hemivariational inequalities for steady Oseen flow with unilateral and frictional type boundary conditions. ZAMM Z. Angew. Math. Mech. 100(2), e201900112 (2020)","journal-title":"ZAMM Z. Angew. Math. Mech."},{"issue":"1\u20134","key":"1487_CR26","doi-asserted-by":"publisher","first-page":"285","DOI":"10.1023\/A:1026555014562","volume":"17","author":"S Mig\u00f3rski","year":"2000","unstructured":"Mig\u00f3rski, S., Ochal, A.: Optimal control of parabolic hemivariational inequalities. J. Glob. Optim. 17(1\u20134), 285\u2013300 (2000)","journal-title":"J. Glob. Optim."},{"issue":"6","key":"1487_CR27","doi-asserted-by":"publisher","first-page":"3384","DOI":"10.1016\/j.nonrwa.2011.06.002","volume":"12","author":"S Mig\u00f3rski","year":"2011","unstructured":"Mig\u00f3rski, S., Ochal, A., Sofonea, M.: History-dependent subdifferential inclusions and hemivariational inequalities in contact mechanics. Nonlinear Anal. Real World Appl. 12(6), 3384\u20133396 (2011)","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"1487_CR28","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4614-4232-5","volume-title":"Nonlinear Inclusions and Hemivariational Inequalities","author":"S Mig\u00f3rski","year":"2013","unstructured":"Mig\u00f3rski, S., Ochal, A., Sofonea, M.: Nonlinear Inclusions and Hemivariational Inequalities. Springer, New York (2013)"},{"key":"1487_CR29","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-011-2178-1","volume-title":"Network Economics: A Variational Inequality Approach","author":"A Nagurney","year":"1993","unstructured":"Nagurney, A.: Network Economics: A Variational Inequality Approach. Kluwer Academic Publishers Group, Dordrecht (1993)"},{"key":"1487_CR30","volume-title":"Mathematical Theory of Hemivariational Inequalities and Applications","author":"Z Naniewicz","year":"1995","unstructured":"Naniewicz, Z., Panagiotopoulos, P.D.: Mathematical Theory of Hemivariational Inequalities and Applications. Marcel Dekker Inc, New York (1995)"},{"key":"1487_CR31","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-5152-1","volume-title":"Inequality Problems in Mechanics and Applications","author":"PD Panagiotopoulos","year":"1985","unstructured":"Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Applications. Birkh\u00e4user Boston Inc, Boston, MA (1985)"},{"issue":"2","key":"1487_CR32","doi-asserted-by":"publisher","first-page":"293","DOI":"10.1007\/s10957-017-1173-5","volume":"175","author":"NS Papageorgiou","year":"2017","unstructured":"Papageorgiou, N.S., R\u0103dulescu, V.D., Repov\u0161, D.D.: Nonhomogeneous hemivariational inequalities with indefinite potential and Robin boundary condition. J. Optim. Theory Appl. 175(2), 293\u2013323 (2017)","journal-title":"J. Optim. Theory Appl."},{"issue":"1","key":"1487_CR33","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00245-016-9392-y","volume":"78","author":"NS Papageorgiou","year":"2018","unstructured":"Papageorgiou, N.S., R\u0103dulescu, V.D., Repov\u0161, D.D.: Nonlinear elliptic inclusions with unilateral constraint and dependence on the gradient. Appl. Math. Optim. 78(1), 1\u201323 (2018)","journal-title":"Appl. Math. Optim."},{"key":"1487_CR34","doi-asserted-by":"publisher","DOI":"10.1515\/9783111286952","volume-title":"Applied Nonlinear Functional Analysis. An Introduction","author":"NS Papageorgiou","year":"2024","unstructured":"Papageorgiou, N.S., Winkert, P.: Applied Nonlinear Functional Analysis. An Introduction, 2nd edn. De Gruyter, Berlin (2024)","edition":"2"},{"key":"1487_CR35","doi-asserted-by":"publisher","first-page":"124","DOI":"10.1016\/j.apnum.2016.03.002","volume":"105","author":"N Saito","year":"2016","unstructured":"Saito, N., Sugitani, Y., Zhou, G.: Unilateral problem for the Stokes equations: the well-posedness and finite element approximation. Appl. Numer. Math. 105, 124\u2013147 (2016)","journal-title":"Appl. Numer. Math."},{"key":"1487_CR36","doi-asserted-by":"publisher","DOI":"10.1007\/b99799","volume-title":"Models and Analysis of Quasistatic Contact","author":"M Shillor","year":"2004","unstructured":"Shillor, M., Sofonea, M., Telega, J.J.: Models and Analysis of Quasistatic Contact. Springer-Verlag, Berlin, Heidelberg (2004)"},{"issue":"2","key":"1487_CR37","doi-asserted-by":"publisher","first-page":"605","DOI":"10.1007\/s10898-012-9884-5","volume":"56","author":"G-J Tang","year":"2013","unstructured":"Tang, G.-J., Huang, N.-J.: Existence theorems of the variational\u2013hemivariational inequalities. J. Glob. Optim. 56(2), 605\u2013622 (2013)","journal-title":"J. Glob. Optim."},{"issue":"7","key":"1487_CR38","doi-asserted-by":"publisher","first-page":"1469","DOI":"10.1007\/s11425-020-1838-2","volume":"65","author":"Y-B Xiao","year":"2022","unstructured":"Xiao, Y.-B., Liu, M.-T., Chen, T., Huang, N.-J.: Stability analysis for evolutionary variational\u2013hemivariational inequalities with constraint sets. Sci. China Math. 65(7), 1469\u20131484 (2022)","journal-title":"Sci. China Math."},{"issue":"2","key":"1487_CR39","doi-asserted-by":"publisher","first-page":"789","DOI":"10.1007\/s00245-019-09563-4","volume":"83","author":"Y-B Xiao","year":"2021","unstructured":"Xiao, Y.-B., Sofonea, M.: Generalized penalty method for elliptic variational\u2013hemivariational inequalities. Appl. Math. Optim. 83(2), 789\u2013812 (2021)","journal-title":"Appl. Math. Optim."},{"key":"1487_CR40","doi-asserted-by":"publisher","first-page":"34","DOI":"10.1016\/j.cnsns.2017.07.016","volume":"56","author":"S Zeng","year":"2018","unstructured":"Zeng, S., Mig\u00f3rski, S.: A class of time-fractional hemivariational inequalities with application to frictional contact problem. Commun. Nonlinear Sci. Numer. Simul. 56, 34\u201348 (2018)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"issue":"2","key":"1487_CR41","doi-asserted-by":"publisher","first-page":"293","DOI":"10.1080\/00036811.2018.1491037","volume":"99","author":"B Zeng","year":"2020","unstructured":"Zeng, B., Mig\u00f3rski, S.: Variational\u2013hemivariational inverse problems for unilateral frictional contact. Appl. Anal. 99(2), 293\u2013312 (2020)","journal-title":"Appl. Anal."},{"issue":"2","key":"1487_CR42","doi-asserted-by":"publisher","first-page":"1246","DOI":"10.1137\/19M1282210","volume":"59","author":"S Zeng","year":"2021","unstructured":"Zeng, S., Mig\u00f3rski, S., Khan, A.A.: Nonlinear quasi-hemivariational inequalities: existence and optimal control. SIAM J. Control Optim. 59(2), 1246\u20131274 (2021)","journal-title":"SIAM J. Control Optim."}],"container-title":["Journal of Global Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10898-025-01487-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10898-025-01487-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10898-025-01487-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,6]],"date-time":"2025-09-06T12:22:43Z","timestamp":1757161363000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10898-025-01487-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,15]]},"references-count":42,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2025,8]]}},"alternative-id":["1487"],"URL":"https:\/\/doi.org\/10.1007\/s10898-025-01487-4","relation":{},"ISSN":["0925-5001","1573-2916"],"issn-type":[{"value":"0925-5001","type":"print"},{"value":"1573-2916","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,4,15]]},"assertion":[{"value":"17 August 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 March 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 April 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"There is no Conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"Not applicable.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical approval"}}]}}