{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,10]],"date-time":"2026-02-10T07:24:02Z","timestamp":1770708242472,"version":"3.49.0"},"reference-count":50,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2020,3,15]],"date-time":"2020-03-15T00:00:00Z","timestamp":1584230400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this article, we focus on improving the sub-gradient extra-gradient method to find a solution to the problems of pseudo-monotone equilibrium in a real Hilbert space. The weak convergence of our method is well-established based on the standard assumptions on a bifunction. We also present the application of our results that enable to solve numerically the pseudo-monotone and monotone variational inequality problems, in addition to the particular presumptions required by the operator. We have used various numerical examples to support our well-proved convergence results, and we can show that the proposed method involves a considerable influence over-running time and the total number of iterations.<\/jats:p>","DOI":"10.3390\/sym12030463","type":"journal-article","created":{"date-parts":[[2020,3,20]],"date-time":"2020-03-20T07:29:07Z","timestamp":1584689347000},"page":"463","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":33,"title":["The Inertial Sub-Gradient Extra-Gradient Method for a Class of Pseudo-Monotone Equilibrium Problems"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2659-8226","authenticated-orcid":false,"given":"Habib ur","family":"Rehman","sequence":"first","affiliation":[{"name":"KMUTTFixed Point Research Laboratory, KMUTT-Fixed Point Theory and Applications Research Group, SCL 802 Fixed Point Laboratory, Department of Mathematics, Faculty of Science, King Mongkut\u2019s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5463-4581","authenticated-orcid":false,"given":"Poom","family":"Kumam","sequence":"additional","affiliation":[{"name":"KMUTTFixed Point Research Laboratory, KMUTT-Fixed Point Theory and Applications Research Group, SCL 802 Fixed Point Laboratory, Department of Mathematics, Faculty of Science, King Mongkut\u2019s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand"},{"name":"Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Science Laboratory Building, King Mongkut\u2019s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand"},{"name":"Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8773-4821","authenticated-orcid":false,"given":"Wiyada","family":"Kumam","sequence":"additional","affiliation":[{"name":"Program in Applied Statistics, Department of Mathematics and Computer Science, Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi, Thanyaburi, Pathumthani 12110, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1454-2962","authenticated-orcid":false,"given":"Meshal","family":"Shutaywi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science and Arts, King Abdulaziz University, P.O. Box 344, Rabigh 21911, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5160-0036","authenticated-orcid":false,"given":"Wachirapong","family":"Jirakitpuwapat","sequence":"additional","affiliation":[{"name":"KMUTTFixed Point Research Laboratory, KMUTT-Fixed Point Theory and Applications Research Group, SCL 802 Fixed Point Laboratory, Department of Mathematics, Faculty of Science, King Mongkut\u2019s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,3,15]]},"reference":[{"key":"ref_1","first-page":"123","article-title":"From optimization and variational inequalities to equilibrium problems","volume":"63","author":"Blum","year":"1994","journal-title":"Math. Stud."},{"key":"ref_2","unstructured":"Facchinei, F., and Pang, J.S. (2007). 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