{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:52:00Z","timestamp":1760233920192,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2021,3,12]],"date-time":"2021-03-12T00:00:00Z","timestamp":1615507200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004396","name":"Thailand Research Fund","doi-asserted-by":"publisher","award":["MRG6280079"],"award-info":[{"award-number":["MRG6280079"]}],"id":[{"id":"10.13039\/501100004396","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We propose a modified extragradient method for solving the variational inequality problem in a Hilbert space. The method is a combination of the well-known subgradient extragradient with the Mann\u2019s mean value method in which the updated iterate is picked in the convex hull of all previous iterates. We show weak convergence of the mean value iterate to a solution of the variational inequality problem, provided that a condition on the corresponding averaging matrix is fulfilled. Some numerical experiments are given to show the effectiveness of the obtained theoretical result.<\/jats:p>","DOI":"10.3390\/sym13030462","type":"journal-article","created":{"date-parts":[[2021,3,15]],"date-time":"2021-03-15T02:51:48Z","timestamp":1615776708000},"page":"462","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Mean Extragradient Method for Solving Variational Inequalities"],"prefix":"10.3390","volume":"13","author":[{"given":"Apichit","family":"Buakird","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1640-8919","authenticated-orcid":false,"given":"Nimit","family":"Nimana","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6539-7107","authenticated-orcid":false,"given":"Narin","family":"Petrot","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand"},{"name":"Center of Excellence in Nonlinear Analysis and Optimization, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,3,12]]},"reference":[{"key":"ref_1","unstructured":"Kinderlehrer, D., and Stampacchia, G. 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