{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T21:35:03Z","timestamp":1776893703963,"version":"3.51.2"},"reference-count":37,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2026,7,1]],"date-time":"2026-07-01T00:00:00Z","timestamp":1782864000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,7,1]],"date-time":"2026-07-01T00:00:00Z","timestamp":1782864000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2026,7,1]],"date-time":"2026-07-01T00:00:00Z","timestamp":1782864000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-017"},{"start":{"date-parts":[[2026,7,1]],"date-time":"2026-07-01T00:00:00Z","timestamp":1782864000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"},{"start":{"date-parts":[[2026,7,1]],"date-time":"2026-07-01T00:00:00Z","timestamp":1782864000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-012"},{"start":{"date-parts":[[2026,7,1]],"date-time":"2026-07-01T00:00:00Z","timestamp":1782864000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2026,7,1]],"date-time":"2026-07-01T00:00:00Z","timestamp":1782864000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-004"}],"funder":[{"DOI":"10.13039\/501100004706","name":"Zhejiang Normal University","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100004706","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Communications in Nonlinear Science and Numerical Simulation"],"published-print":{"date-parts":[[2026,7]]},"DOI":"10.1016\/j.cnsns.2026.109785","type":"journal-article","created":{"date-parts":[[2026,1,28]],"date-time":"2026-01-28T16:48:44Z","timestamp":1769618924000},"page":"109785","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":0,"special_numbering":"C","title":["Inertial subgradient\u2013extragradient schemes for variational inequalities in 2-uniformly convex spaces"],"prefix":"10.1016","volume":"158","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0526-4833","authenticated-orcid":false,"given":"Grace Nnennaya","family":"Ogwo","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2659-8226","authenticated-orcid":false,"given":"Habib ur","family":"Rehman","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0855-4097","authenticated-orcid":false,"given":"Jen-Chih","family":"Yao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"78","reference":[{"key":"10.1016\/j.cnsns.2026.109785_bib0001","doi-asserted-by":"crossref","first-page":"491","DOI":"10.1007\/s10589-020-00217-8","article-title":"Weak convergence of iterative methods for solving quasimonotone variational inequalities","volume":"77","author":"Liu","year":"2020","journal-title":"Comput Optim Appl"},{"key":"10.1016\/j.cnsns.2026.109785_bib0002","first-page":"627","article-title":"Convergence analysis of an inertial tseng\u2019s extragradient algorithm for solving pseudomonotone variational inequalities and applications","volume":"5","author":"Liu","year":"2021","journal-title":"J Nonlinear Var Anal"},{"key":"10.1016\/j.cnsns.2026.109785_bib0003","series-title":"On inertial type self-adaptive iterative algorithms for solving pseudomonotone equilibrium problems and fixed point problems","author":"Ogbuisi","year":"2021"},{"key":"10.1016\/j.cnsns.2026.109785_bib0004","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 Alg"},{"key":"10.1016\/j.cnsns.2026.109785_bib0005","first-page":"221","article-title":"Adaptive modified inertial projection and contraction methods for pseudomonotone variational inequalities","volume":"4","author":"Tan","year":"2022","journal-title":"J Appl Numer Optim"},{"key":"10.1016\/j.cnsns.2026.109785_bib0006","first-page":"747","article-title":"The extragradient method for finding saddle points and other problems","volume":"12","author":"Korpelevich","year":"1976","journal-title":"Ekonomika I Matematicheskie Metody"},{"key":"10.1016\/j.cnsns.2026.109785_bib0007","first-page":"1164","article-title":"On a method for convex programs using asymmetrical modification of the lagrange function","volume":"12","author":"Antipin","year":"1976","journal-title":"Ekonomika I Matematicheskie Metody"},{"key":"10.1016\/j.cnsns.2026.109785_bib0008","series-title":"Extensions of korpelevich\u2019s extragradient method for the variational inequality problem in euclidean space","first-page":"1119","author":"Censor","year":"2011"},{"key":"10.1016\/j.cnsns.2026.109785_bib0009","doi-asserted-by":"crossref","first-page":"318","DOI":"10.1007\/s10957-010-9757-3","article-title":"The subgradient extragradient method for solving variational inequalities in hilbert space","volume":"148","author":"Censor","year":"2011","journal-title":"J Optim Theory Appl"},{"key":"10.1016\/j.cnsns.2026.109785_bib0010","doi-asserted-by":"crossref","first-page":"827","DOI":"10.1080\/10556788.2010.551536","article-title":"Strong convergence of subgradient extragradient methods for the variational inequality problem in hilbert space","volume":"26","author":"Censor","year":"2011","journal-title":"Optim Meth Softw"},{"key":"10.1016\/j.cnsns.2026.109785_bib0011","doi-asserted-by":"crossref","first-page":"399","DOI":"10.1007\/s10957-013-0494-2","article-title":"Strong convergence of the halpern subgradient extragradient method for solving variational inequalities in hilbert spaces","volume":"163","author":"Kraikaew","year":"2014","journal-title":"J Optim Theory Appl"},{"key":"10.1016\/j.cnsns.2026.109785_bib0012","doi-asserted-by":"crossref","first-page":"2519","DOI":"10.1080\/02331934.2020.1860979","article-title":"The extragradient method for quasimonotone variational inequalities","volume":"71","author":"Salahuddin","year":"2022","journal-title":"Optimization"},{"issue":"2","key":"10.1016\/j.cnsns.2026.109785_bib0013","doi-asserted-by":"crossref","first-page":"248","DOI":"10.1287\/moor.26.2.248.10558","article-title":"A weak-to-strong convergence principle for fejermonotone methods in Hilbert spaces","volume":"26","author":"Bauschke","year":"2001","journal-title":"Math Oper Res"},{"issue":"1","key":"10.1016\/j.cnsns.2026.109785_bib0014","doi-asserted-by":"crossref","DOI":"10.1186\/s13663-018-0641-4","article-title":"Convergence theorems of subgradient extragradient algorithm for solving variational inequalities and a convex feasibility problem","volume":"16","author":"Chidume","year":"2018","journal-title":"Fixed Point Theory Appl"},{"key":"10.1016\/j.cnsns.2026.109785_bib0015","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1016\/j.cam.2014.12.048","article-title":"Convex optimization over fixed point sets of quasi-nonexpansive and nonexpansive mappings in utility-based bandwidth allocation problems with operational constraints","volume":"282","author":"Iiduka","year":"2015","journal-title":"J Comput Appl Math"},{"key":"10.1016\/j.cnsns.2026.109785_bib0016","series-title":"Utility-based multi-service bandwidth allocation in the 4g heterogeneous wireless networks","first-page":"1","author":"Luo","year":"2009"},{"key":"10.1016\/j.cnsns.2026.109785_bib0017","doi-asserted-by":"crossref","first-page":"395","DOI":"10.22436\/jnsa.010.02.06","article-title":"Strong convergence of the halpern subgradient extragradient method for solving variational inequalities in banach spaces","volume":"10","author":"Liu","year":"2017","journal-title":"J Nonlinear Sci Appl"},{"issue":"2","key":"10.1016\/j.cnsns.2026.109785_bib0018","first-page":"17","article-title":"A self-adaptive subgradient extragradient algorithm for variational inequality problems and fixed point problems in banach spaces","volume":"17","author":"Ogbuisi","year":"2020","journal-title":"Aust J Math Anal Appl"},{"issue":"5","key":"10.1016\/j.cnsns.2026.109785_bib0019","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 and Math Phys"},{"key":"10.1016\/j.cnsns.2026.109785_bib0020","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1007\/s10440-019-00297-7","article-title":"A novel inertial projection and contraction method for solving pseudomonotone variational inequality problems","volume":"169","author":"Cholamjiak","year":"2020","journal-title":"Acta Appl Math"},{"issue":"12","key":"10.1016\/j.cnsns.2026.109785_bib0021","doi-asserted-by":"crossref","first-page":"3485","DOI":"10.1080\/02331934.2021.1901289","article-title":"An inertial projection and contraction method with a line search technique for variational inequality and fixed point problems","volume":"71","author":"Jolaoso","year":"2021","journal-title":"Optimization"},{"issue":"5","key":"10.1016\/j.cnsns.2026.109785_bib0022","doi-asserted-by":"crossref","first-page":"981","DOI":"10.1080\/02331934.2019.1566825","article-title":"Accelerated hybrid and shrinking projection methods for variational inequality problems","volume":"68","author":"Thong","year":"2019","journal-title":"Optimization"},{"key":"10.1016\/j.cnsns.2026.109785_bib0023","doi-asserted-by":"crossref","first-page":"757","DOI":"10.1007\/s11590-020-01599-8","article-title":"An inertial popov\u2019s method for solving pseudomonotone variational inequalities","volume":"15","author":"Thong","year":"2021","journal-title":"Optim Lett"},{"key":"10.1016\/j.cnsns.2026.109785_bib0024","series-title":"Geometry of banach spaces. duality mappings and nonlinear problems","author":"Cioranescu","year":"1990"},{"key":"10.1016\/j.cnsns.2026.109785_bib0025","series-title":"Nonlinear functional analysis","author":"Takahashi","year":"2000"},{"key":"10.1016\/j.cnsns.2026.109785_bib0026","series-title":"Fixed point theory for Lipschitzian-type mappings with applications","author":"Agarwal","year":"2009"},{"key":"10.1016\/j.cnsns.2026.109785_bib0027","series-title":"Metric and generalized projection operators in Banach spaces: Properties and applications","volume":"vol. 178","author":"Alber","year":"1996"},{"key":"10.1016\/j.cnsns.2026.109785_bib0028","first-page":"251","article-title":"Strong convergence for gradient projection method and relatively nonexpansive mappings in banach spaces","volume":"271","author":"Nakajo","year":"2015","journal-title":"Appl Math Comput"},{"key":"10.1016\/j.cnsns.2026.109785_bib0029","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1016\/j.jat.2005.02.007","article-title":"A strong convergence theorem for relatively nonexpansive mappings in a banach space","volume":"134","author":"Matsushita","year":"2005","journal-title":"J Approx Theory"},{"key":"10.1016\/j.cnsns.2026.109785_bib0030","doi-asserted-by":"crossref","first-page":"668","DOI":"10.1016\/j.jmaa.2007.07.019","article-title":"Weak convergence of a projection algorithm for variational inequalities in a banach space","volume":"339","author":"Iiduka","year":"2008","journal-title":"J Math Anal Appl"},{"key":"10.1016\/j.cnsns.2026.109785_bib0031","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1007\/s10589-014-9659-7","article-title":"A double projection method for solving variational inequalities without monotonicity","volume":"60","author":"Ye","year":"2015","journal-title":"Comput Optim Appl"},{"key":"10.1016\/j.cnsns.2026.109785_bib0032","first-page":"6577","article-title":"Strong convergence theorems by halpern-mann iterations for relatively nonexpansive mappings in banach spaces","volume":"217","author":"Nilsrakoo","year":"2011","journal-title":"Appl Math Comput"},{"key":"10.1016\/j.cnsns.2026.109785_bib0033","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"},{"issue":"2","key":"10.1016\/j.cnsns.2026.109785_bib0034","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1006\/jmaa.1993.1309","article-title":"Approximating fixed points of nonexpensive mappings by the ishikawa iteration process","volume":"178","author":"Tan","year":"1993","journal-title":"J Math Anal Appl"},{"key":"10.1016\/j.cnsns.2026.109785_bib0035","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/j.ejor.2020.04.035","article-title":"The forward-backward-forward method from continuous and discrete perspective for pseudomonotone variational inequalities in Hilbert spaces","volume":"287","author":"Bot","year":"2020","journal-title":"Eur J Oper Res"},{"issue":"42","key":"10.1016\/j.cnsns.2026.109785_bib0036","article-title":"An inertial forward-backward splitting method for solving inclusion problems in Hilbert spaces","volume":"20","author":"Cholamjiak","year":"2018","journal-title":"J Fixed Point Theory Appl"},{"issue":"8","key":"10.1016\/j.cnsns.2026.109785_bib0037","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1007\/s40314-024-02929-7","article-title":"Three novel inertial subgradient extragradient methods for quasi-monotone variational inequalities in banach spaces","volume":"43","author":"Wang","year":"2024","journal-title":"Comput Appl Math"}],"container-title":["Communications in Nonlinear Science and Numerical Simulation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S1007570426001462?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S1007570426001462?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T20:35:26Z","timestamp":1776890126000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S1007570426001462"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,7]]},"references-count":37,"alternative-id":["S1007570426001462"],"URL":"https:\/\/doi.org\/10.1016\/j.cnsns.2026.109785","relation":{},"ISSN":["1007-5704"],"issn-type":[{"value":"1007-5704","type":"print"}],"subject":[],"published":{"date-parts":[[2026,7]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"Inertial subgradient\u2013extragradient schemes for variational inequalities in 2-uniformly convex spaces","name":"articletitle","label":"Article Title"},{"value":"Communications in Nonlinear Science and Numerical Simulation","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.cnsns.2026.109785","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright","label":"Copyright"}],"article-number":"109785"}}