{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:22:21Z","timestamp":1760059341105,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,6,6]],"date-time":"2025-06-06T00:00:00Z","timestamp":1749168000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001665","name":"French National Research Agency","doi-asserted-by":"publisher","award":["ANR-24-CE46-7619 MaNStArT"],"award-info":[{"award-number":["ANR-24-CE46-7619 MaNStArT"]}],"id":[{"id":"10.13039\/501100001665","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>We consider an evolutionary inclusion associated with a time-dependent convex in an abstract Hilbert space. We recall a unique solvability result obtained based on arguments of nonlinear equations with maximal monotone operators combined with a penalty method. Then, we state and prove two well-posedness results. Next, we provide three examples of such inclusions that arise in mechanics. The first one concerns an elastic\u2013perfectly plastic constitutive law, while the last two examples are mathematical models that describe the equilibrium of an elastic body and an elastic\u2013perfectly plastic body, respectively, in frictional contact with an obstacle. The contact is bilateral and the friction is modeled with the Tresca friction law. We use our abstract results in the study of these examples to provide the convergence of the solution with respect to the data.<\/jats:p>","DOI":"10.3390\/axioms14060448","type":"journal-article","created":{"date-parts":[[2025,6,6]],"date-time":"2025-06-06T09:02:03Z","timestamp":1749200523000},"page":"448","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Well-Posed Evolutionary Inclusion in Mechanics"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0009-0001-2019-0134","authenticated-orcid":false,"given":"Rawane","family":"Mansour","sequence":"first","affiliation":[{"name":"Laboratoire de Mod\u00e9lisation Pluridisciplinaire et Simulations, University of Perpignan Via Domitia, 52 Avenue Paul Alduy, 66860 Perpignan, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6110-1433","authenticated-orcid":false,"given":"Mircea","family":"Sofonea","sequence":"additional","affiliation":[{"name":"Laboratoire de Mod\u00e9lisation Pluridisciplinaire et Simulations, University of Perpignan Via Domitia, 52 Avenue Paul Alduy, 66860 Perpignan, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Capatina, A. (2014). Variational Inequalities and Frictional Contact Problems. Advances in Mechanics and Mathematics, Springer.","DOI":"10.1007\/978-3-319-10163-7"},{"key":"ref_2","unstructured":"Eck, C., Jaru\u0161ek, J., and Krbe\u010d, M. (2005). Unilateral Contact Problems: Variational Methods and Existence Theorems. Pure and Applied Mathematics, Chapman\/CRC Press."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Panagiotopoulos, P.D. (1985). Inequality Problems in Mechanics and Applications, Birkh\u00e4user.","DOI":"10.1007\/978-1-4612-5152-1"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Han, W., and Sofonea, M. (2002). Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity. Studies in Advanced Mathematics, American Mathematical Society, Providence, RI\u2013International Press.","DOI":"10.1090\/amsip\/030"},{"key":"ref_5","unstructured":"Ciarlet, P.G., and Lions, J.-L. (1996). Numerical methods for unilateral problems in solid mechanics. Handbook of Numerical Analysis, Volume IV, North-Holland."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1007\/s00245-023-09991-3","article-title":"A new system of differential quasi-hemivariational inequalities in contact mechanics","volume":"88","author":"Cai","year":"2023","journal-title":"Appl. Math. Optimiz."},{"key":"ref_7","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_8","doi-asserted-by":"crossref","first-page":"103871","DOI":"10.1016\/j.nonrwa.2023.103871","article-title":"A nonsmooth optimization approach for time-dependent hemivariational inequalities","volume":"73","author":"Jureczka","year":"2023","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"106563","DOI":"10.1016\/j.cnsns.2022.106563","article-title":"A survey of numerical methods for hemivariational inequalities with applications to Contact Mechanics","volume":"114","author":"Ochal","year":"2022","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"574","DOI":"10.1007\/s10957-020-01659-0","article-title":"Optimal control of history-dependent evolution inclusions with applications to frictional contact","volume":"185","year":"2020","journal-title":"J. Optim. Theory Appl. (JOTA)"},{"key":"ref_11","first-page":"431","article-title":"Existence theorems in plasticity","volume":"55","author":"Johnson","year":"1976","journal-title":"J. Math. Pures Appl."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Germain, P., and Nayroles, B. (1976). Application of convex analysis to the treatment of elasto-plastic systems. Applications of Methods of Functional Analysis to Problems in Mechanics, Springer. Lecture Notes in Mathematics.","DOI":"10.1007\/BFb0088742"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1090\/qam\/614549","article-title":"Evolution problems for a class of dissipative materials","volume":"38","author":"Suquet","year":"1981","journal-title":"Q. Appl. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1016\/S0362-546X(98)00100-X","article-title":"A quasistatic contact problem for an elastic perfectly plastic body with Tresca\u2019s friction","volume":"35","author":"Amassad","year":"1999","journal-title":"Nonlinear Anal. TMA"},{"key":"ref_15","first-page":"215","article-title":"A nonlinear evolution inclusion in perfect plasticity with friction","volume":"LXX","author":"Amassad","year":"2001","journal-title":"Acta Math. Univ.-Comen."},{"key":"ref_16","first-page":"631","article-title":"On the stability of functional optimization problems","volume":"6","author":"Tykhonov","year":"1966","journal-title":"USSR Comput. Math. Math. Phys."},{"key":"ref_17","first-page":"764","article-title":"Convergence of minimizing sequences in conditional extremum problem","volume":"7","author":"Levitin","year":"1966","journal-title":"Soviet Math. Dokl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1080\/01630568108816100","article-title":"A characterization of Tychonov well-posedness for minimum problems with applications to variational inequalities","volume":"3","author":"Lucchetti","year":"1981","journal-title":"Numer. Funct. Anal. Optim."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1080\/01630568308816145","article-title":"Some properties of \u201cwell-posedness\u201d variational inequalities governed by linear operators","volume":"5","author":"Lucchetti","year":"1983","journal-title":"Numer. Funct. Anal. Optim."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Dontchev, A.L., and Zolezzi, T. (1993). Well-Posed Optimization Problems, Springer. Lecture Notes Mathematics.","DOI":"10.1007\/BFb0084195"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Lucchetti, R. (2006). Convexity and Well-posed Problems. CMS Books in Mathematics, Springer.","DOI":"10.1007\/0-387-31082-7"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Drozdov, A.D. (1996). Finite Elasticity and Viscoelasticity\u2013A Course in the Nonlinear Mechanics of Solids, World Scientific.","DOI":"10.1142\/2905"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Duvaut, G., and Lions, J.-L. (1976). Inequalities in Mechanics and Physics, Springer.","DOI":"10.1007\/978-3-642-66165-5"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Sofonea, M., and Matei, A. (2012). Mathematical Models in Contact Mechanics, Cambridge University Press.","DOI":"10.1017\/CBO9781139104166"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/6\/448\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:47:45Z","timestamp":1760032065000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/6\/448"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,6]]},"references-count":24,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2025,6]]}},"alternative-id":["axioms14060448"],"URL":"https:\/\/doi.org\/10.3390\/axioms14060448","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,6,6]]}}}