{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:57:13Z","timestamp":1760144233980,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2024,3,28]],"date-time":"2024-03-28T00:00:00Z","timestamp":1711584000000},"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>Weak sharp type solutions are analyzed for a variational integral inequality defined by a convex functional of the multiple integral type. A connection with the sufficiency property associated with the minimum principle is formulated, as well. Also, an illustrative numerical application is provided.<\/jats:p>","DOI":"10.3390\/axioms13040225","type":"journal-article","created":{"date-parts":[[2024,3,28]],"date-time":"2024-03-28T12:22:46Z","timestamp":1711628566000},"page":"225","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Weak Sharp Type Solutions for Some Variational Integral Inequalities"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8209-3869","authenticated-orcid":false,"given":"Savin","family":"Trean\u0163\u0103","sequence":"first","affiliation":[{"name":"Faculty of Applied Sciences, National University of Science and Technology Politehnica Bucharest, 060042 Bucharest, Romania"},{"name":"Academy of Romanian Scientists, 54 Splaiul Independentei, 050094 Bucharest, Romania"},{"name":"Fundamental Sciences Applied in Engineering-Research Center, National University of Science and Technology Politehnica Bucharest, 060042 Bucharest, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0170-5286","authenticated-orcid":false,"given":"Tareq","family":"Saeed","sequence":"additional","affiliation":[{"name":"Financial Mathematics and Actuarial Science (FMAS)-Research Group, Department of Mathematics, Faculty of Sciences, King Abdulaziz University, Jeddah 21589, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,3,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1340","DOI":"10.1137\/0331063","article-title":"Weak sharp minima in mathematical programming","volume":"31","author":"Burke","year":"1993","journal-title":"SIAM J. Control Optim."},{"key":"ref_2","unstructured":"Patriksson, M. (1993). A Unified Framework of Descent Algorithms for Nonlinear Programs and Variational Inequalities. [Ph.D. Thesis, Department of Mathematics, Link\u00f6ping Institute of Technology]."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1137\/S1052623496309867","article-title":"Weak sharp solutions of variational inequalities","volume":"9","author":"Marcotte","year":"1998","journal-title":"SIAM J. Optim."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1016\/j.jmaa.2010.08.062","article-title":"Weak sharp solutions for variational inequalities in Banach spaces","volume":"374","author":"Hu","year":"2011","journal-title":"J. Math. Anal. Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"563","DOI":"10.1007\/s11590-015-0882-7","article-title":"Characterization of weakly sharp solutions of a variational inequality by its primal gap function","volume":"10","author":"Liu","year":"2016","journal-title":"Optim. Lett."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1287","DOI":"10.1007\/s11590-015-0925-0","article-title":"Weak sharp efficiency in multiobjective optimization","volume":"10","author":"Zhu","year":"2016","journal-title":"Optim. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"805","DOI":"10.1007\/s11590-015-0906-3","article-title":"Minimum and maximum principle sufficiency properties for nonsmooth variational inequalities","volume":"10","author":"Alshahrani","year":"2016","journal-title":"Optim. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Clarke, F.H. (2013). Functional Analysis, Calculus of Variations and Optimal Control, Springer.","DOI":"10.1007\/978-1-4471-4820-3"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"547","DOI":"10.1016\/j.camwa.2017.09.033","article-title":"Higher-order Hamilton dynamics and Hamilton-Jacobi divergence PDE","volume":"75","year":"2018","journal-title":"Comput. Math. Appl."},{"key":"ref_10","first-page":"827","article-title":"On a class of differential quasi-variational-hemivariational inequalities in infinite-dimensional Banach spaces","volume":"11","year":"2021","journal-title":"Evol. Equations Control. Theory"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"6191","DOI":"10.1080\/00036811.2021.1919646","article-title":"On a class of controlled differential variational inequalities","volume":"101","year":"2022","journal-title":"Appl. Anal."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1007\/s11117-020-00765-7","article-title":"Weak sharp solutions associated with a multidimensional variational-type inequality","volume":"25","author":"Singh","year":"2021","journal-title":"Positivity"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Trean\u0163\u0103, S. (2021). On a class of differential variational inequalities in infinite-dimensional spaces. Mathematics, 9.","DOI":"10.3390\/math9030266"},{"key":"ref_14","first-page":"805","article-title":"Some results on (\u03c1, b, d)-variational inequalities","volume":"14","year":"2020","journal-title":"J. Math. Inequal."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1186\/s13660-020-02323-x","article-title":"On weak sharp solutions in (\u03c1, b, d)-variational inequalities","volume":"2020","year":"2020","journal-title":"J. Inequal. Appl."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1007\/s10479-017-2700-3","article-title":"Characterization of weakly sharp solutions of a variational-type inequality with convex functional","volume":"269","author":"Jayswal","year":"2018","journal-title":"Ann. Oper. Res."},{"key":"ref_17","unstructured":"Kassay, G., and R\u0103dulescu, V. (2018). Equilibrium Problems and Applications, Elsevier."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"647","DOI":"10.1007\/s12190-017-1126-z","article-title":"Efficiency conditions in vector control problems governed by multiple integrals","volume":"57","author":"Mititelu","year":"2018","journal-title":"J. Appl. Math. Comput."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"505","DOI":"10.7153\/jmi-04-45","article-title":"On Bergstrom inequality for commuting gramian normal operators","volume":"4","author":"Ciurdariu","year":"2010","journal-title":"J. Math. Ineq."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Ciurdariu, L., and Grecu, E. (2023). Several Quantum Hermite\u2013Hadamard-Type Integral Inequalities for Convex Functions. Fractal Fract., 7.","DOI":"10.3390\/fractalfract7060463"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1186\/1029-242X-2014-119","article-title":"A generalized form of Gr\u00fcss type inequality and other integral inequalities","volume":"2014","author":"Minculete","year":"2014","journal-title":"J. Ineq. Appl."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Khan, M.B., Zaini, H.G., Trean\u0163\u0103, S., Santos-Garcia, G., Macias-Diaz, J.E., and Soliman, M.S. (2022). Fractional Calculus for Convex Functions in Interval-Valued Settings and Inequalities. Symmetry, 14.","DOI":"10.3390\/sym14020341"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Saeed, T., Afzal, W., Abbas, M., Trean\u0163\u0103, S., and De la Sen, M. (2022). Some New Generalizations of Integral Inequalities for Harmonical cr-(h1, h2)-Godunova-Levin Functions and Applications. Mathematics, 10.","DOI":"10.3390\/math10234540"},{"key":"ref_24","first-page":"100324","article-title":"New Classes of Interval-Valued Variational Problems and Inequalities","volume":"13","author":"Saeed","year":"2023","journal-title":"Res. Control Optim."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"701","DOI":"10.1007\/s10957-013-0460-z","article-title":"On finite convergence of iterative methods for variational inequalities in Hilbert spaces","volume":"161","author":"Matsushita","year":"2014","journal-title":"J. Optim. Theory Appl."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF01581071","article-title":"Minimum principle sufficiency","volume":"57","author":"Ferris","year":"1992","journal-title":"Math. Program."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Hiriart-Urruty, J.-B., and Lemar\u00e9chal, C. (2001). Fundamentals of Convex Analysis, Springer.","DOI":"10.1007\/978-3-642-56468-0"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"480","DOI":"10.1016\/j.jde.2021.07.013","article-title":"On well-posed isoperimetric-type constrained variational control problems","volume":"298","year":"2021","journal-title":"J. Differ. Equ."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/4\/225\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:20:11Z","timestamp":1760106011000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/4\/225"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,3,28]]},"references-count":28,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2024,4]]}},"alternative-id":["axioms13040225"],"URL":"https:\/\/doi.org\/10.3390\/axioms13040225","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,3,28]]}}}