{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:14:28Z","timestamp":1760058868881,"version":"build-2065373602"},"reference-count":37,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2025,5,7]],"date-time":"2025-05-07T00:00:00Z","timestamp":1746576000000},"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>In this work, we study a class of variational inequality problems defined over the intersection of sub-level sets of a countable family of convex functions. We propose a new iterative method for approximating the solution within the framework of Hilbert spaces. The method incorporates several strategies, including inertial effects, a self-adaptive step size, and a relaxation technique, to enhance convergence properties. Notably, it requires computing only a single projection onto a half space. Using some mild conditions, we prove that the sequence generated by our proposed method is strongly convergent to a minimum-norm solution to the problem. Finally, we present some numerical results that validate the applicability of our proposed method.<\/jats:p>","DOI":"10.3390\/axioms14050354","type":"journal-article","created":{"date-parts":[[2025,5,7]],"date-time":"2025-05-07T11:46:19Z","timestamp":1746618379000},"page":"354","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Totally Relaxed, Self-Adaptive Tseng Extragradient Method for Monotone Variational Inequalities"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1521-8355","authenticated-orcid":false,"given":"Olufemi Johnson","family":"Ogunsola","sequence":"first","affiliation":[{"name":"Department of Mathematics, Federal University of Agriculture, Alabata PMB 2240, Nigeria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4451-2126","authenticated-orcid":false,"given":"Olawale Kazeem","family":"Oyewole","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Tshwane University of Technology, Arcadia 0007, South Africa"},{"name":"Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Saveetha University, Chennai 602105, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6366-2728","authenticated-orcid":false,"given":"Seithuti Philemon","family":"Moshokoa","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Tshwane University of Technology, Arcadia 0007, South Africa"}]},{"given":"Hammed Anuoluwapo","family":"Abass","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, Pretoria 0204, South Africa"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,7]]},"reference":[{"key":"ref_1","first-page":"138","article-title":"Sul problem elastostatico di signorini con ambigue condizioni al contorno","volume":"34","author":"Fichera","year":"1963","journal-title":"Atti Accad. Naz. Lincei Cl Sci. Fis. Mat. Nat."},{"key":"ref_2","unstructured":"Stampacchia, G. (1968). Variational Inequalities. Theory and Applications of Monotone Operators, Proceedings of the NATO Advanced Study Institute, Venice, Italy, 17\u201330 June 1968, Edizioni Odersi."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1119","DOI":"10.1080\/02331934.2010.539689","article-title":"Extensions of Korpelevich\u2019s extragradient method for the variational inequality problem in Euclidean space","volume":"61","author":"Censor","year":"2012","journal-title":"Optimization"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1007\/BF01589441","article-title":"A relaxed projection method for variational inequalities","volume":"35","author":"Fukushima","year":"1986","journal-title":"Math. Program."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Gu, Z., Mani, G., Gnanaprakasam, A.J., and Li, Y. (2021). Solving a System of Nonlinear Integral Equations via Common Fixed Point Theorems on Bicomplex Partial Metric Space. Mathematics, 9.","DOI":"10.3390\/math9141584"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"942315","DOI":"10.1155\/2013\/942315","article-title":"Solving the variational inequality problem defined on intersection of finite level sets","volume":"2013","author":"He","year":"2013","journal-title":"Abstr. Appl. Anal."},{"key":"ref_7","first-page":"747","article-title":"An extragradient method for finding saddle points and for other problems","volume":"12","author":"Korpelevich","year":"1976","journal-title":"Ekon. Mat. Metody"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Nallaselli, G., Baazeem, A.S., Gnanaprakasam, A.J., Mani, G., Javed, K., Ameer, E., and Mlaiki, N. (2023). Fixed Point Theorems via Orthogonal Convex Contraction in Orthogonal b-Metric Spaces and Applications. Axioms, 12.","DOI":"10.3390\/axioms12020143"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1007\/s11784-021-00886-w","article-title":"Best proximity point of generalized F-proximal non-self contractions","volume":"23","author":"Beg","year":"2021","journal-title":"J. Fixed Point Theory Appl."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Gnanaprakasam, A.J., Nallaselli, G., Haq, A.U., Mani, G., Baloch, I.A., and Nonlaopon, K. (2022). Common Fixed-Points Technique for the Existence of a Solution to Fractional Integro-Differential Equations via Orthogonal Branciari Metric Spaces. Symmetry, 14.","DOI":"10.3390\/sym14091859"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Ramaswamy, R., Mani, G., Gnanaprakasam, A.J., Abdelnaby, O.A.A., Stojiljkovi\u0107, V., Radojevic, S., and Radenovi\u0107, S. (2022). Fixed Points on Covariant and Contravariant Maps with an Application. Mathematics, 10.","DOI":"10.3390\/math10224385"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11565-020-00354-2","article-title":"An iterative algorithm for solving variational inequality generalized mixed equilibrium, convex minimization and zeros problems for a class of nonexpansive-type mappings","volume":"67","author":"Alakoya","year":"2021","journal-title":"Ann. Univ. Ferrara Sez. VII Sci. Mat."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"687","DOI":"10.1007\/s10898-017-0506-0","article-title":"Inertial projection and contraction algorithms for variational inequalities","volume":"70","author":"Dong","year":"2017","journal-title":"J. Glob. Optim."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2773","DOI":"10.1007\/s13398-019-00658-9","article-title":"Relaxed projection and contraction methods for solving Lipschitz-continuous monotone variational inequalities","volume":"113","author":"He","year":"2019","journal-title":"Rev. De La Real Acad. De Cienc. Exactas Fis. Y Nat. Ser. A Mat."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"373","DOI":"10.3934\/naco.2021011","article-title":"A modified extragradient algorithm for a certain class of split pseudo-monotone variational inequality problem","volume":"12","author":"Ogwo","year":"2022","journal-title":"Numer. Algebra Control Optim."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"93","DOI":"10.24193\/fpt-ro.2020.1.07","article-title":"A modified inertial subgradient extragradient method for solving pseudomonotone variational inequalities and common fixed point problems","volume":"21","author":"Ceng","year":"2020","journal-title":"Fixed Point Theory"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1007\/s10898-010-9613-x","article-title":"Korpelevich\u2019s method for variational inequality problems in Banach spaces","volume":"50","author":"Iusem","year":"2011","journal-title":"J. Glob. Optim."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1419","DOI":"10.1007\/s11075-021-01081-1","article-title":"Inertial methods for finding minimum-norm solutions of the split variational inequality problem beyond monotonicity","volume":"88","author":"Ogwo","year":"2021","journal-title":"Numer. Algorithms"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1137\/S0363012998338806","article-title":"A Modified forward-backward splitting method for maximal monotone mappings","volume":"38","author":"Tseng","year":"2000","journal-title":"SIAM J. Control Optim."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"741","DOI":"10.1007\/s11075-018-0504-4","article-title":"Strong convergence result for solving monotone variational inequalities in Hilbert space","volume":"80","author":"Yang","year":"2019","journal-title":"Numer. Algorithms"},{"key":"ref_21","unstructured":"Censor, Y., Gibali, A., and Reich, S. (2010). The Split Variational Inequality Problem, The Technion-Israel Institute of Technology."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1007\/s002459900037","article-title":"A class of projection and contraction methods for monotone variational inequalities","volume":"35","author":"He","year":"1997","journal-title":"Appl. Math. Optim."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1023\/B:COAP.0000013058.17185.90","article-title":"Comparison of two kinds of prediction-correction methods for monotone variational inequalities","volume":"27","author":"He","year":"2004","journal-title":"Comput. Optim. Appl."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1814","DOI":"10.1137\/S0363012994268655","article-title":"Modified projection-type methods for monotone variational inequalities","volume":"34","author":"Solodov","year":"1996","journal-title":"SIAM J. Control Optim."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1007\/BF02192286","article-title":"A class of iterative methods for solving nonlinear projection equations","volume":"91","author":"Sun","year":"1996","journal-title":"J. Optim. Theory Appl."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0041-5553(64)90137-5","article-title":"Some methods of speeding up the convergence of iteration methods","volume":"4","author":"Polyak","year":"1964","journal-title":"USSR Comput. Math. Math. Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1237","DOI":"10.1080\/02331934.2019.1686632","article-title":"On the convergence of inertial two-subgradient extragradient method for solving variational inequality problems","volume":"69","author":"Cao","year":"2020","journal-title":"Optimization"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"773","DOI":"10.1137\/S1052623403427859","article-title":"Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in hilbert space","volume":"14","author":"Alvarez","year":"2004","journal-title":"SIAM J. Optim."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"547","DOI":"10.1007\/s00245-019-09584-z","article-title":"Convergence of a relaxed inertial forward\u2013backward algorithm for structured monotone inclusions","volume":"80","author":"Attouch","year":"2019","journal-title":"Appl. Math. Optim."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1080\/10556788.2017.1396601","article-title":"A generic online acceleration scheme for optimization algorithms via relaxation and inertia","volume":"34","author":"Iutzeler","year":"2019","journal-title":"Optim. Methods Softw."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1487","DOI":"10.1080\/02331934.2018.1476515","article-title":"Totally relaxed, self-adaptive algorithm for solving variational inequalities over the intersection of sub-level sets","volume":"67","author":"He","year":"2018","journal-title":"Optimization"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Bauschke, H.H., and Combettes, P.L. (2017). Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer. [2nd ed.].","DOI":"10.1007\/978-3-319-48311-5"},{"key":"ref_33","first-page":"2323","article-title":"The supporting hyperplane and an alternative to solutions of variational inequalities","volume":"16","author":"Nguyen","year":"2015","journal-title":"J. Nonlinear Convex Anal."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"742","DOI":"10.1016\/j.na.2011.09.005","article-title":"Approximation of zeros of inverse strongly monotone operators in Banach spaces","volume":"75","author":"Saejung","year":"2012","journal-title":"Nonlinear Anal. Theory Methods Appl."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1006\/jmaa.1993.1309","article-title":"Approximating Fixed Points of Nonexpansive Mappings by the Ishikawa Iteration Process","volume":"178","author":"Tan","year":"1993","journal-title":"J. Math. Anal. Appl."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1007\/s13160-018-00341-3","article-title":"Two strong convergence subgradient extragradient methods for solving variational inequalities in Hilbert spaces","volume":"36","author":"Thong","year":"2019","journal-title":"Jpn. J. Ind. Appl. Math."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1186\/s13660-023-03043-8","article-title":"Outer approximated projection and contraction method for solving variational inequalities","volume":"2023","author":"Uzor","year":"2023","journal-title":"J. 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