{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T17:24:14Z","timestamp":1777569854090,"version":"3.51.4"},"reference-count":48,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2020,8,31]],"date-time":"2020-08-31T00:00:00Z","timestamp":1598832000000},"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>A plethora of applications from mathematical programming, such as minimax, and mathematical programming, penalization, fixed point to mention a few can be framed as equilibrium problems. Most of the techniques for solving such problems involve iterative methods that is why, in this paper, we introduced a new extragradient-like method to solve equilibrium problems in real Hilbert spaces with a Lipschitz-type condition on a bifunction. The advantage of a method is a variable stepsize formula that is updated on each iteration based on the previous iterations. The method also operates without the previous information of the Lipschitz-type constants. The weak convergence of the method is established by taking mild conditions on a bifunction. For application, fixed-point theorems that involve strict pseudocontraction and results for pseudomonotone variational inequalities are studied. We have reported various numerical results to show the numerical behaviour of the proposed method and correlate it with existing ones.<\/jats:p>","DOI":"10.3390\/axioms9030101","type":"journal-article","created":{"date-parts":[[2020,8,31]],"date-time":"2020-08-31T08:11:19Z","timestamp":1598861479000},"page":"101","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["A General Inertial Projection-Type Algorithm for Solving Equilibrium Problem in Hilbert Spaces with Applications in Fixed-Point Problems"],"prefix":"10.3390","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8750-3732","authenticated-orcid":false,"given":"Nopparat","family":"Wairojjana","sequence":"first","affiliation":[{"name":"Applied Mathematics Program, Faculty of Science and Technology, Valaya Alongkorn Rajabhat University under the Royal Patronage (VRU), 1 Moo 20 Phaholyothin Road, Klong Neung, Klong Luang, Pathumthani 13180, 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":"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\/0000-0001-9320-9433","authenticated-orcid":false,"given":"Manuel","family":"De la Sen","sequence":"additional","affiliation":[{"name":"Institute of Research and Development of Processes IIDP, University of the Basque Country, 48940 Leioa, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0224-4661","authenticated-orcid":false,"given":"Nuttapol","family":"Pakkaranang","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":[[2020,8,31]]},"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","unstructured":"Facchinei, F., and Pang, J.S. (2007). Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer Science & Business Media."},{"key":"ref_4","unstructured":"Konnov, I. (2007). Equilibrium Models and Variational Inequalities, Elsevier."},{"key":"ref_5","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_6","doi-asserted-by":"crossref","first-page":"749","DOI":"10.1080\/02331930601122876","article-title":"Extragradient algorithms extended to equilibrium problems","volume":"57","author":"Quoc","year":"2008","journal-title":"Optimization"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1007\/s10898-011-9693-2","article-title":"Dual extragradient algorithms extended to equilibrium problems","volume":"52","author":"Quoc","year":"2011","journal-title":"J. Glob. Optim."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Lyashko, S.I., and Semenov, V.V. (2016). A New Two-Step Proximal Algorithm of Solving the Problem of Equilibrium Programming. Optimization and Its Applications in Control and Data Sciences, Springer International Publishing.","DOI":"10.1007\/978-3-319-42056-1_10"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"506","DOI":"10.1016\/j.jmaa.2006.08.036","article-title":"Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces","volume":"331","author":"Takahashi","year":"2007","journal-title":"J. Math. Anal. Appl."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"ur Rehman, H., Kumam, P., Cho, Y.J., and Yordsorn, P. (2019). Weak convergence of explicit extragradient algorithms for solving equilibirum problems. J. Inequalities Appl., 2019.","DOI":"10.1186\/s13660-019-2233-1"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1007\/s10898-015-0330-3","article-title":"On ergodic algorithms for equilibrium problems","volume":"64","author":"Anh","year":"2015","journal-title":"J. Glob. Optim."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Hieu, D.V., Quy, P.K., and Vy, L.V. (2019). Explicit iterative algorithms for solving equilibrium problems. Calcolo, 56.","DOI":"10.1007\/s10092-019-0308-5"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"811","DOI":"10.1080\/00036811.2017.1292350","article-title":"New extragradient method for a class of equilibrium problems in Hilbert spaces","volume":"97","author":"Hieu","year":"2017","journal-title":"Appl. Anal."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"ur Rehman, H., Kumam, P., Je Cho, Y., Suleiman, Y.I., and Kumam, W. (2020). Modified Popov\u2019s explicit iterative algorithms for solving pseudomonotone equilibrium problems. Optim. Methods Softw., 1\u201332.","DOI":"10.1080\/10556788.2020.1734805"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"ur Rehman, H., Kumam, P., Abubakar, A.B., and Cho, Y.J. (2020). The extragradient algorithm with inertial effects extended to equilibrium problems. Comput. Appl. Math., 39.","DOI":"10.1007\/s40314-020-1093-0"},{"key":"ref_16","first-page":"91","article-title":"An inexact subgradient algorithm for equilibrium problems","volume":"30","author":"Santos","year":"2011","journal-title":"Comput. Appl. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"823","DOI":"10.1007\/s13398-016-0328-9","article-title":"Halpern subgradient extragradient method extended to equilibrium problems","volume":"111","author":"Hieu","year":"2016","journal-title":"Revista de la Real Academia de Ciencias Exactas, F\u00edsicas y Naturales Serie A Matem\u00e1ticas"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"ur Rehman, H., 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_19","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1080\/02331934.2012.745528","article-title":"The subgradient extragradient method extended to equilibrium problems","volume":"64","author":"Anh","year":"2012","journal-title":"Optimization"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1007\/s10957-009-9529-0","article-title":"Regularization Algorithms for Solving Monotone Ky Fan Inequalities with Application to a Nash-Cournot Equilibrium Model","volume":"142","author":"Muu","year":"2009","journal-title":"J. Optim. Theory Appl."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"ur Rehman, H., Kumam, P., Argyros, I.K., Deebani, W., and Kumam, W. (2020). Inertial Extra-Gradient Method for Solving a Family of Strongly Pseudomonotone Equilibrium Problems in Real Hilbert Spaces with Application in Variational Inequality Problem. Symmetry, 12.","DOI":"10.3390\/sym12040503"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"ur Rehman, H., Kumam, P., Argyros, I.K., Alreshidi, N.A., Kumam, W., and Jirakitpuwapat, W. (2020). A Self-Adaptive Extra-Gradient Methods for a Family of Pseudomonotone Equilibrium Programming with Application in Different Classes of Variational Inequality Problems. Symmetry, 12.","DOI":"10.3390\/sym12040523"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"ur Rehman, H., 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":"Yordsorn, P., Kumam, P., ur Rehman, H., and Ibrahim, A.H. (2020). A Weak Convergence Self-Adaptive Method for Solving Pseudomonotone Equilibrium Problems in a Real Hilbert Space. Mathematics, 8.","DOI":"10.3390\/math8071165"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"313","DOI":"10.37193\/CJM.2020.02.15","article-title":"Modified two-step extragradient method for solving the pseudomonotone equilibrium programming in a real Hilbert space","volume":"36","author":"Yordsorn","year":"2020","journal-title":"Carpathian J. Math."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/1687-1847-2011-1","article-title":"On the Existence of Equilibrium Points, Boundedness, Oscillating Behavior and Positivity of a SVEIRS Epidemic Model under Constant and Impulsive Vaccination","volume":"2011","author":"Agarwal","year":"2011","journal-title":"Adv. Differ. Equ."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"La Sen, M.D., and Agarwal, R.P. (2011). Some fixed point-type results for a class of extended cyclic self-mappings with a more general contractive condition. Fixed Point Theory Appl., 2011.","DOI":"10.1186\/1687-1812-2011-59"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Wairojjana, N., ur Rehman, H., Argyros, I.K., and Pakkaranang, N. (2020). An Accelerated Extragradient Method for Solving Pseudomonotone Equilibrium Problems with Applications. Axioms, 9.","DOI":"10.3390\/axioms9030099"},{"key":"ref_29","first-page":"1","article-title":"On Best Proximity Point Theorems and Fixed Point Theorems for -Cyclic Hybrid Self-Mappings in Banach Spaces","volume":"2013","year":"2013","journal-title":"Abstr. Appl. Anal."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"ur Rehman, H., 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_31","doi-asserted-by":"crossref","unstructured":"Rehman, H.U., Kumam, P., Dong, Q.L., Peng, Y., and Deebani, W. (2020). A new Popov\u2019s subgradient extragradient method for two classes of equilibrium programming in a real Hilbert space. Optimization, 1\u201336.","DOI":"10.1080\/02331934.2020.1797026"},{"key":"ref_32","first-page":"13","article-title":"Parallel extragradient algorithms for a family of pseudomonotone equilibrium problems and fixed point problems of nonself-nonexpansive mappings in Hilbert space","volume":"2020","author":"Wang","year":"2020","journal-title":"J. Nonlinear Funct. Anal."},{"key":"ref_33","first-page":"189","article-title":"Convergence theorems of common solutions for fixed point, variational inequality and equilibrium problems, J","volume":"3","author":"Shahzad","year":"2019","journal-title":"Nonlinear Var. Anal."},{"key":"ref_34","first-page":"335","article-title":"The subgradient extragradient method for solving mixed equilibrium problems and fixed point problems in Hilbert spaces","volume":"1","author":"Farid","year":"2019","journal-title":"J. Appl. Numer. Optim."},{"key":"ref_35","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_36","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_37","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_38","doi-asserted-by":"crossref","first-page":"639","DOI":"10.1007\/s40306-019-00338-1","article-title":"Inertial Extragradient Algorithms for Solving Equilibrium Problems","volume":"44","author":"Vinh","year":"2019","journal-title":"Acta Math. Vietnam."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"2003","DOI":"10.1080\/02331934.2018.1505886","article-title":"Modified extragradient algorithms for solving equilibrium problems","volume":"67","author":"Hieu","year":"2018","journal-title":"Optimization"},{"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","unstructured":"Kreyszig, E. (1989). Introductory Functional Analysis with Applications, Wiley Classics Library, Wiley. [1st ed.]."},{"key":"ref_43","unstructured":"Tiel, J.V. (1984). Convex Analysis: An Introductory Text, Wiley. [1st ed.]."},{"key":"ref_44","unstructured":"Ioffe, A.D., and Tihomirov, V.M. (1979). Theory of Extremal Problems. Studies in Mathematics and Its Applications 6, North-Holland, Elsevier."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"591","DOI":"10.1090\/S0002-9904-1967-11761-0","article-title":"Weak convergence of the sequence of successive approximations for nonexpansive mappings","volume":"73","author":"Opial","year":"1967","journal-title":"Bull. Amer. Math. Soc."},{"key":"ref_46","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_47","doi-asserted-by":"crossref","unstructured":"Bauschke, H.H., and Combettes, P.L. (2011). Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer.","DOI":"10.1007\/978-1-4419-9467-7"},{"key":"ref_48","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."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/9\/3\/101\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:05:06Z","timestamp":1760177106000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/9\/3\/101"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,31]]},"references-count":48,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2020,9]]}},"alternative-id":["axioms9030101"],"URL":"https:\/\/doi.org\/10.3390\/axioms9030101","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,8,31]]}}}