{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,31]],"date-time":"2026-01-31T01:05:39Z","timestamp":1769821539859,"version":"3.49.0"},"reference-count":22,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2024,1,15]],"date-time":"2024-01-15T00:00:00Z","timestamp":1705276800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"European Union\u2019s Horizon 2020 Research and Innovation Programme","award":["823731 CONMECH"],"award-info":[{"award-number":["823731 CONMECH"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Here, we consider a stationary inclusion in a real Hilbert space X, governed by a set of constraints K, a nonlinear operator A, and an element f\u2208X. Under appropriate assumptions on the data, the inclusion has a unique solution, denoted by u. We state and prove a covergence criterion, i.e., we provide necessary and sufficient conditions on a sequence {un}\u2282X, which guarantee its convergence to the solution u. We then present several applications that provide the continuous dependence of the solution with respect to the data K, A and f on the one hand, and the convergence of an associate penalty problem on the other hand. We use these abstract results in the study of a frictional contact problem with elastic materials that, in a weak formulation, leads to a stationary inclusion for the deformation field. Finally, we apply the abstract penalty method in the analysis of two nonlinear elastic constitutive laws.<\/jats:p>","DOI":"10.3390\/axioms13010052","type":"journal-article","created":{"date-parts":[[2024,1,15]],"date-time":"2024-01-15T07:25:07Z","timestamp":1705303507000},"page":"52","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Convergence Criterion for a Class of Stationary Inclusions in Hilbert Spaces"],"prefix":"10.3390","volume":"13","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"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2813-0419","authenticated-orcid":false,"given":"Domingo A.","family":"Tarzia","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica, FCE, Universidad Austral, Paraguay 1950, Rosario S2000FZF, Argentina"},{"name":"CONICET, Rosario S2000EZP, Argentina"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,1,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Hlav\u00e1\u010dek, I., Haslinger, J., Nec\u01ces, J., and Lov\u00ed\u0161ek, J. (1988). Solution of Variational Inequalities in Mechanics, Springer.","DOI":"10.1007\/978-1-4612-1048-1"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Panagiotopoulos, P.D. (1985). Inequality Problems in Mechanics and Applications, Birkhauser.","DOI":"10.1007\/978-1-4612-5152-1"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Sofonea, M. (2023). Well-Posed Nonlinear Problems: A Study of Mathematical Models of Contact, Birkhauser.","DOI":"10.1007\/978-3-031-41416-9"},{"key":"ref_4","first-page":"309","article-title":"Convergence criteria, well-posedness concepts and applications","volume":"15","author":"Sofonea","year":"2023","journal-title":"Math. Its Appl. Ann. AOSR"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"761","DOI":"10.1137\/17M1120658","article-title":"An implicit sweeping process approach to quasistatic evolution variational inequalities","volume":"50","author":"Adly","year":"2018","journal-title":"SIAM J. Math. Anal."},{"key":"ref_6","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_7","first-page":"497","article-title":"Inertial extrapolation method for solving systems of monotone variational inclusion and fixed point problems using Bregman distance approach","volume":"28","author":"Abass","year":"2023","journal-title":"Nonlinear Funct. Anal. Appl."},{"key":"ref_8","first-page":"175","article-title":"Inertial proximal and contraction methods for solving monotone variational inclusion and fixed point problems","volume":"28","author":"Abuchu","year":"2023","journal-title":"Nonlinear Funct. Anal. Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"103335","DOI":"10.1016\/j.nonrwa.2021.103335","article-title":"Analysis and control of stationary inclusions in Contact Mechanics","volume":"61","author":"Sofonea","year":"2021","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Bauschke, H.H., and Combettes, P.L. (2011). Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer. CMS Books in Mathematics.","DOI":"10.1007\/978-1-4419-9467-7"},{"key":"ref_11","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_12","doi-asserted-by":"crossref","unstructured":"Hiriart-Urruty, J.-B., and Lemar\u00e9chal, C. (1993). Convex Analysis and Minimization Algorithms, I, II, Springer.","DOI":"10.1007\/978-3-662-02796-7"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Zeidler, E. (1986). Nonlinear Functional Analysis and Its Applications, Springer. Volume I: Fixed-Point Theorems.","DOI":"10.1007\/978-1-4612-4838-5"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Zeidler, E. (1990). Nonlinear Functional Analysis and Applications II A\/B, Springer.","DOI":"10.1007\/978-1-4612-0981-2"},{"key":"ref_15","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":"1968","journal-title":"Adv. Math."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Sofonea, M., and Matei, A. (2012). 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