{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T00:22:51Z","timestamp":1760228571776,"version":"build-2065373602"},"reference-count":45,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2022,5,19]],"date-time":"2022-05-19T00:00:00Z","timestamp":1652918400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Thailand Science Research and Innovation (TSRI)","award":["FRB65E0632M.1"],"award-info":[{"award-number":["FRB65E0632M.1"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Symmetries play a vital role in the study of physical phenomena in diverse areas such as dynamic systems, optimization, physics, scientific computing, engineering, mathematical biology, chemistry, and medicine, to mention a few. These phenomena specialize mostly in solving equilibria-like problems in abstract spaces. Motivated by these facts, this research provides two innovative modifying extragradient strategies for solving pseudomonotone equilibria problems in real Hilbert space with the Lipschitz-like bifunction constraint. Such strategies make use of multiple step-size concepts that are modified after each iteration and are reliant on prior iterations. The excellence of these strategies comes from the fact that they were developed with no prior knowledge of Lipschitz-type parameters or any line search strategy. Mild assumptions are required to prove strong convergence theorems for proposed strategies. Various numerical tests have been reported to demonstrate the numerical behavior of the techniques and then contrast them with others.<\/jats:p>","DOI":"10.3390\/sym14051045","type":"journal-article","created":{"date-parts":[[2022,5,20]],"date-time":"2022-05-20T00:18:11Z","timestamp":1653005891000},"page":"1045","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On Strengthened Extragradient Methods Non-Convex Combination with Adaptive Step Sizes Rule for Equilibrium Problems"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1454-2962","authenticated-orcid":false,"given":"Meshal","family":"Shutaywi","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science & Arts, King Abdulaziz University, P.O. Box 344, Rabigh 21911, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8773-4821","authenticated-orcid":false,"given":"Wiyada","family":"Kumam","sequence":"additional","affiliation":[{"name":"Applied Mathematics for Science and Engineering Research Unit (AMSERU), Program in Applied Statistics, Department of Mathematics and Computer Science, Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi (RMUTT), Pathum Thani 12110, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2659-8226","authenticated-orcid":false,"given":"Habib ur","family":"Rehman","sequence":"additional","affiliation":[{"name":"Department of Mathematics, King Mongkut\u2019s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8775-4790","authenticated-orcid":false,"given":"Kamonrat","family":"Sombut","sequence":"additional","affiliation":[{"name":"Applied Mathematics for Science and Engineering Research Unit (AMSERU), Department of Mathematics and Computer Science, Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi (RMUTT), Pathum Thani 12110, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,5,19]]},"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":"Shisha, O. (1972). A Minimax Inequality and Applications, Inequalities III, Academic Press."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ejor.2012.11.037","article-title":"Existence and solution methods for equilibria","volume":"227","author":"Bigi","year":"2013","journal-title":"Eur. J. Oper. Res."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1159","DOI":"10.1016\/0362-546X(92)90159-C","article-title":"Convergence of an adaptive penalty scheme for finding constrained equilibria","volume":"18","author":"Muu","year":"1992","journal-title":"Nonlinear Anal. Theory Methods Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"6121","DOI":"10.1016\/j.na.2011.05.091","article-title":"The Tikhonov regularization extended to equilibrium problems involving pseudomonotone bifunctions","volume":"74","author":"Hung","year":"2011","journal-title":"Nonlinear Anal. Theory Methods Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1023\/B:JOTA.0000005448.12716.24","article-title":"Application of the Proximal Point Method to Nonmonotone Equilibrium Problems","volume":"119","author":"Konnov","year":"2003","journal-title":"J. Optim. Theory Appl."},{"key":"ref_7","first-page":"91","article-title":"Proximal point algorithm extended to equilibrium problems","volume":"15","author":"Moudafi","year":"1999","journal-title":"J. Nat. Geom."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"336","DOI":"10.1016\/j.jmaa.2012.12.034","article-title":"A Tikhonov-type regularization for equilibrium problems in Hilbert spaces","volume":"401","author":"Oliveira","year":"2013","journal-title":"J. Math. Anal. Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1081","DOI":"10.1080\/00207160.2021.1949711","article-title":"An Inertial Extragradient Method for Iteratively Solving Equilibrium Problems in Real Hilbert Spaces","volume":"99","author":"Rehman","year":"2021","journal-title":"Int. J. Comput. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"38","DOI":"10.22436\/jmcs.022.01.04","article-title":"A modified extra-gradient method for a family of strongly pseudomonotone equilibrium problems in real Hilbert spaces","volume":"22","author":"Rehman","year":"2021","journal-title":"J. Math. Comput. Sci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1515\/dema-2021-0011","article-title":"Strong convergence inertial projection algorithm with self-adaptive step size rule for pseudomonotone variational inequalities in Hilbert spaces","volume":"54","author":"Wairojjana","year":"2021","journal-title":"Demonstr. Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1007\/s40314-020-1093-0","article-title":"The extragradient algorithm with inertial effects extended to equilibrium problems","volume":"39","author":"Rehman","year":"2020","journal-title":"Comput. Appl. Math."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Rehman, H.U., Kumam, P., Kumam, W., Shutaywi, M., and Jirakitpuwapat, W. (2020). The Inertial Sub-Gradient Extra-Gradient Method for a Class of Pseudo-Monotone Equilibrium Problems. Symmetry, 12.","DOI":"10.3390\/sym12030463"},{"key":"ref_14","first-page":"93","article-title":"Regularization method for nonmonotone equilibrium problems","volume":"10","author":"Konnov","year":"2009","journal-title":"J. Nonlinear Convex Anal."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"373","DOI":"10.1080\/10556780500094838","article-title":"Partial proximal point method for nonmonotone equilibrium problems","volume":"21","author":"Konnov","year":"2006","journal-title":"Optim. Methods Softw."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"111095","DOI":"10.1016\/j.chaos.2021.111095","article-title":"Asymptotic stability of fractional order (1,2] stochastic delay differential equations in Banach spaces","volume":"150","author":"Singh","year":"2021","journal-title":"Chaos Solitons Fractals"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Patel, R., Shukla, A., and Jadon, S.S. (2020). Existence and optimal control problem for semilinear fractional order (1,2] control system. Math. Methods Appl. Sci.","DOI":"10.1002\/mma.6662"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"306","DOI":"10.3182\/20140313-3-IN-3024.00107","article-title":"Controllability of Semilinear Stochastic System with Multiple Delays in Control","volume":"47","author":"Shukla","year":"2014","journal-title":"IFAC Proc. Vol."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"4246","DOI":"10.1007\/s00034-021-01680-2","article-title":"Existence and Optimal Control Results for Second-Order Semilinear System in Hilbert Spaces","volume":"40","author":"Shukla","year":"2021","journal-title":"Circuits Syst. Signal Process."},{"key":"ref_20","first-page":"215","article-title":"Convergence of relaxed inertial methods for equilibrium problems","volume":"3","author":"Duong","year":"2021","journal-title":"J. Appl. Numer. Optim."},{"key":"ref_21","first-page":"239","article-title":"The projection method with inertial extrapolation for solving split equilibrium problems in Hilbert spaces","volume":"3","author":"Ogbuisi","year":"2021","journal-title":"Appl. Set-Valued Anal. Optim."},{"key":"ref_22","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":"ref_23","doi-asserted-by":"crossref","unstructured":"Rehman, H.u., Kumam, P., Argyros, I.K., Shutaywi, M., and Shah, Z. (2020). Optimization Based Methods for Solving the Equilibrium Problems with Applications in Variational Inequality Problems and Solution of Nash Equilibrium Models. Mathematics, 8.","DOI":"10.3390\/math8050822"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Rehman, H.u., Kumam, P., Shutaywi, M., Alreshidi, N.A., and Kumam, W. (2020). Inertial Optimization Based Two-Step Methods for Solving Equilibrium Problems with Applications in Variational Inequality Problems and Growth Control Equilibrium Models. Energies, 13.","DOI":"10.3390\/en13123292"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2675","DOI":"10.1080\/02331934.2020.1797026","article-title":"A new Popov\u2019s subgradient extragradient method for two classes of equilibrium programming in a real Hilbert space","volume":"70","author":"Rehman","year":"2020","journal-title":"Optimization"},{"key":"ref_26","first-page":"2011","article-title":"Modified subgradient extragradient method for a family of pseudomonotone equilibrium problems in real a Hilbert space","volume":"21","author":"Rehman","year":"2020","journal-title":"J. Nonlinear Convex Anal."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1007\/BF02614504","article-title":"Equilibrium programming using proximal-like algorithms","volume":"78","author":"Antipin","year":"1996","journal-title":"Math. Program."},{"key":"ref_28","first-page":"747","article-title":"The extragradient method for finding saddle points and other problems","volume":"12","author":"Korpelevich","year":"1976","journal-title":"Matecon"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"749","DOI":"10.1080\/02331930601122876","article-title":"Extragradient algorithms extended to equilibrium problems","volume":"57","author":"Tran","year":"2008","journal-title":"Optimization"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1007\/s10092-019-0308-5","article-title":"Explicit iterative algorithms for solving equilibrium problems","volume":"56","author":"Hieu","year":"2019","journal-title":"Calcolo"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"2247","DOI":"10.1080\/02331934.2018.1523404","article-title":"Modified subgradient extragradient algorithms for solving monotone variational inequalities","volume":"67","author":"Yang","year":"2018","journal-title":"Optimization"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1677","DOI":"10.2298\/FIL1906677W","article-title":"New extragradient methods with non-convex combination for pseudomonotone equilibrium problems with applications in Hilbert spaces","volume":"33","author":"Wang","year":"2019","journal-title":"Filomat"},{"key":"ref_33","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":"2010","journal-title":"J. Optim. Theory Appl."},{"key":"ref_34","unstructured":"Tiel, J.V. (1984). Convex Analysis: An Introductory Text, Wiley. [1st ed.]."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1112\/blms\/17.3.293","article-title":"UNIFORM CONVEXITY, HYPERBOLIC GEOMETRY, AND NONEXPANSIVE MAPPINGS (Pure and Applied Mathematics: A Series of Monographs & Textbooks, 83) By K. Goebel and S. Reich: Pp. 192. SFr.96.-. (Marcel Dekker Inc, U.S.A., 1984)","volume":"17","author":"Bushell","year":"1985","journal-title":"Bull. Lond. Math. Soc."},{"key":"ref_36","unstructured":"Kreyszig, E. (1989). Introductory Functional Analysis with Applications, Wiley Classics Library, Wiley. [1st ed.]."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Bauschke, H.H., and Combettes, P.L. (2017). Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer International Publishing. [2nd ed.]. CMS Books in Mathematics.","DOI":"10.1007\/978-3-319-48311-5"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1017\/S0004972700020116","article-title":"Another control condition in an iterative method for nonexpansive mappings","volume":"65","author":"Xu","year":"2002","journal-title":"Bulletin of the Australian Mathematical Society"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"899","DOI":"10.1007\/s11228-008-0102-z","article-title":"Strong Convergence of Projected Subgradient Methods for Nonsmooth and Nonstrictly Convex Minimization","volume":"16","year":"2008","journal-title":"Set-Valued Anal."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1007\/BF02192244","article-title":"Generalized monotone bifunctions and equilibrium problems","volume":"90","author":"Bianchi","year":"1996","journal-title":"J. Optim. Theory Appl."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Mastroeni, G. (2003). On Auxiliary Principle for Equilibrium Problems. Nonconvex Optimization and Its Applications, Springer.","DOI":"10.1007\/978-1-4613-0239-1_15"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"282","DOI":"10.1186\/s13660-019-2233-1","article-title":"Weak convergence of explicit extragradient algorithms for solving equilibirum problems","volume":"2019","author":"Rehman","year":"2019","journal-title":"J. Inequalities Appl."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1016\/0022-247X(67)90085-6","article-title":"Construction of fixed points of nonlinear mappings in Hilbert space","volume":"20","author":"Browder","year":"1967","journal-title":"J. Math. Anal. Appl."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1080\/10556788.2020.1734805","article-title":"Modified Popov\u2019s explicit iterative algorithms for solving pseudomonotone equilibrium problems","volume":"36","author":"Rehman","year":"2020","journal-title":"Optim. Methods Softw."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1007\/s10589-016-9857-6","article-title":"Modified hybrid projection methods for finding common solutions to variational inequality problems","volume":"66","author":"Anh","year":"2017","journal-title":"Comput. Optim. Appl."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/5\/1045\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:14:56Z","timestamp":1760138096000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/5\/1045"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,19]]},"references-count":45,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2022,5]]}},"alternative-id":["sym14051045"],"URL":"https:\/\/doi.org\/10.3390\/sym14051045","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,5,19]]}}}