{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,17]],"date-time":"2026-04-17T22:17:16Z","timestamp":1776464236436,"version":"3.51.2"},"reference-count":51,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2023,3,1]],"date-time":"2023-03-01T00:00:00Z","timestamp":1677628800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,3,1]],"date-time":"2023-03-01T00:00:00Z","timestamp":1677628800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2023,3]]},"DOI":"10.1007\/s40314-023-02244-7","type":"journal-article","created":{"date-parts":[[2023,3,1]],"date-time":"2023-03-01T15:03:48Z","timestamp":1677683028000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["An algorithm for split equilibrium and fixed-point problems using inertial extragradient techniques"],"prefix":"10.1007","volume":"42","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8633-4121","authenticated-orcid":false,"given":"Jeremiah N.","family":"Ezeora","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Cyril D.","family":"Enyi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Francis O.","family":"Nwawuru","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Richard C.","family":"Ogbonna","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,3,1]]},"reference":[{"key":"2244_CR1","first-page":"126","volume":"18","author":"P Anantachai","year":"2020","unstructured":"Anantachai P, Kumam P, Parin C, Shehu Y (2020) Convergence of inertia modified Krasnoselski-Mann iteration with application to image recovery. Thai J Maths 18:126\u2013142","journal-title":"Thai J Maths"},{"key":"2244_CR2","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1080\/02331934.2011.607497","volume":"62","author":"PN Anh","year":"2013","unstructured":"Anh PN (2013) A hybrid extragradient method extended to fixed point problems and equilibrium problems. Optimization 62:271\u2013283","journal-title":"Optimization"},{"key":"2244_CR3","first-page":"512","volume":"2020","author":"Y Arfat","year":"2020","unstructured":"Arfat Y, Kumam P, Ngiamsunthorn PS, Khan MAA, Sarwar H, Fukhar-ud-Din H (2020) Approximation results for split equilibrium problems and fixed point problems of nonexpansive semigroup in Hilbert spaces. Adv Cont Discrete Models 2020:512","journal-title":"Adv Cont Discrete Models"},{"key":"2244_CR4","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1137\/080716542","volume":"2","author":"A Beck","year":"2009","unstructured":"Beck A, Teboulle M (2009) A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J Imaging Sci 2:183\u2013202","journal-title":"SIAM J Imaging Sci"},{"key":"2244_CR5","first-page":"123","volume":"63","author":"E Blum","year":"1994","unstructured":"Blum E, Oettli W (1994) From optimization and variational inequalities to equilibrium problems. Math Program 63:123\u2013145","journal-title":"Math Program"},{"key":"2244_CR6","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1088\/0266-5611\/18\/2\/310","volume":"18","author":"C Byrne","year":"2002","unstructured":"Byrne C (2002) Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl 18:441\u2013453","journal-title":"Inverse Probl"},{"key":"2244_CR7","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1088\/0266-5611\/20\/1\/006","volume":"20","author":"C Byrne","year":"2004","unstructured":"Byrne C (2004) A unified treatment for some iterative algorithms in signal processing and image reconstruction. Inverse Probl 20:103\u2013120","journal-title":"Inverse Probl"},{"key":"2244_CR8","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1007\/BF02142692","volume":"8","author":"Y Censor","year":"1994","unstructured":"Censor Y, Elfving T (1994) A multiprojection algorithm using Bregman projections in product space. Numer Alg 8:221\u2013239","journal-title":"Numer Alg"},{"key":"2244_CR9","first-page":"587","volume":"16","author":"Y Censor","year":"2009","unstructured":"Censor Y, Segal A (2009) The split common fixed point problem for directed operators. J Convex Anal 16:587\u2013600","journal-title":"J Convex Anal"},{"key":"2244_CR10","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1007\/s11075-011-9490-5","volume":"59","author":"Y Censor","year":"2012","unstructured":"Censor Y, Gibali A, Reich S (2012) Algorithms for the split variational inequality problem. Numer Alg 59:301\u2013323","journal-title":"Numer Alg"},{"key":"2244_CR11","doi-asserted-by":"crossref","unstructured":"Chidume CE (2009) Geometric properties of banach spaces and nonlinear iterations, vol 1965 of lecture notes in mathematics. Springer, London","DOI":"10.1007\/978-1-84882-190-3"},{"key":"2244_CR12","doi-asserted-by":"crossref","first-page":"1487","DOI":"10.2298\/FIL2005487C","volume":"34","author":"SY Cho","year":"2020","unstructured":"Cho SY (2020) A monotone Bregman projection algorithm for fixed point and equilibrium problems in a reflexive Banach space. Filomat 34:1487\u20131497","journal-title":"Filomat"},{"key":"2244_CR13","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1007\/BF00934554","volume":"32","author":"G Cohen","year":"1980","unstructured":"Cohen G (1980) Auxiliary problem principle and decomposition of optimization problems. J Optimiz Theory Appl 32:277\u2013305","journal-title":"J Optimiz Theory Appl"},{"key":"2244_CR14","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1007\/BF00938316","volume":"59","author":"G Cohen","year":"1988","unstructured":"Cohen G (1988) Auxiliary principle extended to variational inequalities. J Optimiz Theory Appl 59:325\u2013333","journal-title":"J Optimiz Theory Appl"},{"issue":"4","key":"2244_CR15","doi-asserted-by":"crossref","first-page":"901","DOI":"10.1080\/02331934.2019.1625899","volume":"69","author":"V Dadash","year":"2020","unstructured":"Dadash V, Iyiola OS, Shehu Y (2020) The subgradient extragradient method for pseudomonotone equilibrium problems. Optimization 69(4):901\u2013923","journal-title":"Optimization"},{"key":"2244_CR16","doi-asserted-by":"crossref","first-page":"3933","DOI":"10.1002\/mma.8024","volume":"45","author":"CD Enyi","year":"2022","unstructured":"Enyi CD, Shehu Y, Iyiola OS, Yao JC (2022) Convergence analysis of modified inertial forward-backward splitting scheme with applications. Math Meth Appl Sci 45:3933\u20133948","journal-title":"Math Meth Appl Sci"},{"issue":"2","key":"2244_CR17","doi-asserted-by":"publisher","first-page":"30","DOI":"10.52968\/28307336","volume":"7","author":"JN Ezeora","year":"2021","unstructured":"Ezeora JN, Bazuaye FE (2021) Iterative approximation of split feasibility problem in real Hilbert spaces. Int J Math Anal Optim Theory Appl 7(2):30\u201347. https:\/\/doi.org\/10.52968\/28307336","journal-title":"Int J Math Anal Optim Theory Appl"},{"key":"2244_CR18","doi-asserted-by":"crossref","first-page":"359","DOI":"10.22436\/jnsa.014.05.06","volume":"14","author":"JN Ezeora","year":"2021","unstructured":"Ezeora JN, Jackreece PC (2021) Iterative solution of split equilibrium and fixed point problems in real Hilbert spaces. J Nonlinear Sci Appl 14:359\u2013371","journal-title":"J Nonlinear Sci Appl"},{"issue":"3","key":"2244_CR19","first-page":"233","volume":"70","author":"JN Ezeora","year":"2018","unstructured":"Ezeora JN, Ogbonna RC (2018) Split feasibility problem for countable family of multi-valued nonliear mappings. Mate Vesnik 70(3):233\u2013242","journal-title":"Mate Vesnik"},{"key":"2244_CR20","volume-title":"Finite-dimensional variational inequalities and complementary problems","author":"F Facchinei","year":"2003","unstructured":"Facchinei F, Pang JS (2003) Finite-dimensional variational inequalities and complementary problems. Springer, New York"},{"issue":"2","key":"2244_CR21","first-page":"243","volume":"2","author":"A Gibali","year":"2017","unstructured":"Gibali A (2017) A new split inverse problem and application to least intensity feasible solutions. Pure Appl Funct Anal 2(2):243\u2013258","journal-title":"Pure Appl Funct Anal"},{"key":"2244_CR22","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1007\/s00025-020-01306-0","volume":"75","author":"A Gibali","year":"2020","unstructured":"Gibali A, Jolaoso LO, Mewomo OT, Taiwo A (2020) Fast and simple Bregman projection methods for solving variational inequalities and related problems in Banach spaces. Results Math 75:179","journal-title":"Results Math"},{"key":"2244_CR23","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511526152","volume-title":"Topics in metric fixed point theory. Cambridge studies in advanced mathematics","author":"K Goebel","year":"1990","unstructured":"Goebel K, Kirk WA (1990) Topics in metric fixed point theory. Cambridge studies in advanced mathematics, vol 28. Cambridge University Press, Cambridge"},{"key":"2244_CR24","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1186\/s13662-020-02800-z","volume":"2020","author":"SA Harisa","year":"2020","unstructured":"Harisa SA, Khan MAA, Mumtaz F, Farid N, Morsy A, Nisar KS, Ghaffar A (2020) Shrinking Cesaro means method for the split equilibrium and fixed point problems in Hilbert spaces. Adv Differ Equ 2020:345","journal-title":"Adv Differ Equ"},{"key":"2244_CR25","doi-asserted-by":"crossref","first-page":"162","DOI":"10.1186\/1029-242X-2012-162","volume":"2012","author":"Z He","year":"2012","unstructured":"He Z (2012) The split equilibrium problem and its convergence algorithms. J Inequal Appl 2012:162","journal-title":"J Inequal Appl"},{"key":"2244_CR26","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1023\/B:JOTA.0000026134.57920.e1","volume":"121","author":"TT Hue","year":"2004","unstructured":"Hue TT, Strodiot JJ, Nguyen VH (2004) Convergence of the approximate auxiliary problem method for solving generalized variational inequalities. J Optimiz Theory Appl 121:119\u2013145","journal-title":"J Optimiz Theory Appl"},{"key":"2244_CR27","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1080\/10556788.2021.1924715","volume":"2021","author":"OS Iyiola","year":"2021","unstructured":"Iyiola OS, Enyi CD, Shehu Y (2021) Reflected three-operator splitting method for monotone inclusion problem. Optim Methods Softw 2021:1\u201339. https:\/\/doi.org\/10.1080\/10556788.2021.1924715","journal-title":"Optim Methods Softw"},{"issue":"3","key":"2244_CR28","doi-asserted-by":"crossref","first-page":"583","DOI":"10.1080\/02331934.2020.1808648","volume":"71","author":"C Izuchukwu","year":"2022","unstructured":"Izuchukwu C, Ogwo GN, Mewomo OT (2022) An inertial method for solving generalized split feasibility problems over the solution set of monotone variational inclusions. Optimization 71(3):583\u2013611","journal-title":"Optimization"},{"key":"2244_CR29","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1016\/j.joems.2012.10.009","volume":"21","author":"KR Kazmi","year":"2013","unstructured":"Kazmi KR, Rizvi SH (2013) Iterative approximation of a common solution of a split equilibrium problem, a variational inequality problem and a fixed point problem. J Egypt Math Soc 21:44\u201351","journal-title":"J Egypt Math Soc"},{"key":"2244_CR30","doi-asserted-by":"crossref","first-page":"741","DOI":"10.1007\/s11075-017-0338-5","volume":"77","author":"DS Kim","year":"2018","unstructured":"Kim DS, Dinh BV (2018) Parallel extragradient algorithms for multiple set split equilibrium problems in Hilbert spaces. Numer Alg 77:741\u2013761","journal-title":"Numer Alg"},{"key":"2244_CR31","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1023\/B:JOTA.0000005448.12716.24","volume":"119","author":"IV Konnov","year":"2003","unstructured":"Konnov IV (2003) Application of the proximal point method to nonmonotone equilibrium problems. J Optimiz Theory Appl 119:317\u2013333","journal-title":"J Optimiz Theory Appl"},{"key":"2244_CR32","volume-title":"Equilibrium models and variational inequalities","author":"IV Konnov","year":"2007","unstructured":"Konnov IV (2007) Equilibrium models and variational inequalities. Elsevier, Amsterdam"},{"key":"2244_CR33","first-page":"747","volume":"12","author":"GM Korpelevich","year":"1976","unstructured":"Korpelevich GM (1976) Extragradient method for finding saddle points and other problems. Matecon 12:747\u2013756","journal-title":"Matecon"},{"key":"2244_CR34","doi-asserted-by":"crossref","first-page":"469","DOI":"10.1016\/j.jmaa.2005.12.066","volume":"325","author":"PE Mainge","year":"2007","unstructured":"Mainge PE (2007) Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces. J Math Anal Appl 325:469\u2013479","journal-title":"J Math Anal Appl"},{"key":"2244_CR35","doi-asserted-by":"crossref","first-page":"899","DOI":"10.1007\/s11228-008-0102-z","volume":"16","author":"PE Mainge","year":"2008","unstructured":"Mainge PE (2008) Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal 16:899\u2013912","journal-title":"Set-Valued Anal"},{"key":"2244_CR36","first-page":"154","volume":"4","author":"B Martinet","year":"1970","unstructured":"Martinet B (1970) Regularisation d inequations variationelles par approximations successives. Rev Fr Automat Inf 4:154\u2013159","journal-title":"Rev Fr Automat Inf"},{"key":"2244_CR37","first-page":"1244","volume":"3","author":"G Mastroeni","year":"2000","unstructured":"Mastroeni G (2000) On auxiliary principle for equilibrium problems. Publ Dipartim Math dell\u2019Univ Pisa 3:1244\u20131258","journal-title":"Publ Dipartim Math dell\u2019Univ Pisa"},{"key":"2244_CR38","first-page":"91","volume":"15","author":"A Moudafi","year":"1999","unstructured":"Moudafi A (1999) Proximal point algorithm extended to equilibrium problem. J Nat Geometry 15:91\u2013100","journal-title":"J Nat Geometry"},{"key":"2244_CR39","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1007\/s10957-011-9814-6","volume":"150","author":"A Moudafi","year":"2011","unstructured":"Moudafi A (2011) Split monotone variational inclusions. J Optim Theory Appl 150:275\u2013283","journal-title":"J Optim Theory Appl"},{"key":"2244_CR40","doi-asserted-by":"crossref","first-page":"1159","DOI":"10.1016\/0362-546X(92)90159-C","volume":"18","author":"D Muul","year":"1992","unstructured":"Muul D, Oettli W (1992) Convergence of an adaptive penalty scheme for finding constraint equilibria. Nonlinear Anal 18:1159\u20131166","journal-title":"Nonlinear Anal"},{"key":"2244_CR41","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1186\/s13660-019-2086-7","volume":"2019","author":"P Narin","year":"2019","unstructured":"Narin P, Mohsen R, Manatchanok K, Vahid D (2019) A new extragradient algorithm for split equilibrium problems and fixed point problems. J Ineq Appl 2019:137","journal-title":"J Ineq Appl"},{"key":"2244_CR42","first-page":"372","volume":"27","author":"Y Nesterov","year":"1983","unstructured":"Nesterov Y (1983) A method of solving a convex programming problem with convergence rate $$O(1\/k^22)$$. Soviet Math Doklady 27:372\u2013376","journal-title":"Soviet Math Doklady"},{"key":"2244_CR43","doi-asserted-by":"crossref","first-page":"475","DOI":"10.1023\/A:1023989403613","volume":"117","author":"MA Noor","year":"2003","unstructured":"Noor MA (2003) Extragradient methods for pseudomonotone variational inequalities. J Optimiz Theory Appl 117:475\u2013488","journal-title":"J Optimiz Theory Appl"},{"key":"2244_CR44","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1023\/B:JOTA.0000042526.24671.b2","volume":"122","author":"MA Noor","year":"2004","unstructured":"Noor MA (2004) Auxiliary principle technique for equilibrium problems. J Optimiz Theory Appl 122:371\u2013386","journal-title":"J Optimiz Theory Appl"},{"key":"2244_CR45","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0041-5553(64)90137-5","volume":"4","author":"BT Polyak","year":"1964","unstructured":"Polyak BT (1964) Some methods of speeding up the convergence of iteration methods. Comput Math Math Phys 4:1\u201317","journal-title":"Comput Math Math Phys"},{"issue":"6","key":"2244_CR46","doi-asserted-by":"crossref","first-page":"749","DOI":"10.1080\/02331930601122876","volume":"57","author":"TD Quoc","year":"2008","unstructured":"Quoc TD, Le DM, Nguyen VH (2008) Extragradient algorithms extended to equilibrium problems. Optimization 57(6):749\u2013776","journal-title":"Optimization"},{"key":"2244_CR47","doi-asserted-by":"crossref","first-page":"877","DOI":"10.1137\/0314056","volume":"14","author":"RT Rockafellar","year":"1976","unstructured":"Rockafellar RT (1976) Monotone operators and the proximal point algorithm. SIAM J Control Optim 14:877\u2013898","journal-title":"SIAM J Control Optim"},{"issue":"1","key":"2244_CR48","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/10556788.2019.1619729","volume":"36","author":"Y Shehu","year":"2021","unstructured":"Shehu Y, Vuong PT, Zemkoho A (2021) An inertial extrapolation method for convex simple bilevel optimization. Optim Methods Softw 36(1):1\u201319","journal-title":"Optim Methods Softw"},{"key":"2244_CR49","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1007\/s00186-020-00730-w","volume":"93","author":"Y Shehu","year":"2021","unstructured":"Shehu Y, Iyiola OS, Thong DV, Nguyen TCV (2021) An inertial subgradient extragradient algorithm extended to pseudomonotone equilibrium problems. Math Meth Oper Res 93:213\u2013242","journal-title":"Math Meth Oper Res"},{"issue":"5","key":"2244_CR50","doi-asserted-by":"crossref","first-page":"1791","DOI":"10.1088\/0266-5611\/21\/5\/017","volume":"21","author":"J Zhao","year":"2005","unstructured":"Zhao J, Yang Q (2005) Several solution methods for the split feasibility problem. Inverse Probl 21(5):1791\u20131799","journal-title":"Inverse Probl"},{"key":"2244_CR51","doi-asserted-by":"publisher","first-page":"6624321","DOI":"10.1155\/2021\/6624321","volume":"2021","author":"LJ Zhu","year":"2021","unstructured":"Zhu LJ, Kuo HC, Wen CF (2021) Strong convergence results of split equilibrium problems and fixed point problems. J Math 2021:6624321. https:\/\/doi.org\/10.1155\/2021\/6624321","journal-title":"J Math"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-023-02244-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-023-02244-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-023-02244-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,9]],"date-time":"2023-03-09T11:25:38Z","timestamp":1678361138000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-023-02244-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3]]},"references-count":51,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2023,3]]}},"alternative-id":["2244"],"URL":"https:\/\/doi.org\/10.1007\/s40314-023-02244-7","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,3]]},"assertion":[{"value":"18 June 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 November 2022","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 February 2023","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 March 2023","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"There is no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"Not applicable.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethics approval"}},{"value":"All the authors participated in the work.","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Consent to participate"}},{"value":"All the authors consented to publication of this work.","order":5,"name":"Ethics","group":{"name":"EthicsHeading","label":"Consent for publication"}}],"article-number":"103"}}