{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,28]],"date-time":"2026-01-28T12:13:14Z","timestamp":1769602394841,"version":"3.49.0"},"reference-count":28,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2021,1,11]],"date-time":"2021-01-11T00:00:00Z","timestamp":1610323200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,1,11]],"date-time":"2021-01-11T00:00:00Z","timestamp":1610323200000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2021,2]]},"DOI":"10.1007\/s40314-020-01393-3","type":"journal-article","created":{"date-parts":[[2021,1,12]],"date-time":"2021-01-12T02:17:06Z","timestamp":1610417826000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":24,"title":["Self-adaptive inertial extragradient algorithms for solving variational inequality problems"],"prefix":"10.1007","volume":"40","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1509-1809","authenticated-orcid":false,"given":"Bing","family":"Tan","sequence":"first","affiliation":[]},{"given":"Jingjing","family":"Fan","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7911-2758","authenticated-orcid":false,"given":"Songxiao","family":"Li","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,1,11]]},"reference":[{"key":"1393_CR1","doi-asserted-by":"publisher","first-page":"189","DOI":"10.1007\/s10898-019-00834-6","volume":"76","author":"NT An","year":"2020","unstructured":"An NT, Nam NM, Qin X (2020) Solving $$ k $$-center problems involving sets based on optimization techniques. J Glob Optim 76:189\u2013209","journal-title":"J Glob Optim"},{"key":"1393_CR2","doi-asserted-by":"publisher","first-page":"340","DOI":"10.1080\/00036811.2018.1495329","volume":"99","author":"QH Ansari","year":"2020","unstructured":"Ansari QH, Islam M, Yao JC (2020) Nonsmooth variational inequalities on Hadamard manifolds. Appl Anal 99:340\u2013358","journal-title":"Appl Anal"},{"key":"1393_CR3","doi-asserted-by":"publisher","first-page":"172","DOI":"10.1080\/10556788.2018.1437159","volume":"34","author":"A Beck","year":"2019","unstructured":"Beck A, Guttmann-Beck N (2019) FOM\u2014a MATLAB toolbox of first-order methods for solving convex optimization problems. Optim Methods Softw 34:172\u2013193","journal-title":"Optim Methods Softw"},{"key":"1393_CR4","doi-asserted-by":"publisher","first-page":"318","DOI":"10.1007\/s10957-010-9757-3","volume":"148","author":"Y Censor","year":"2011","unstructured":"Censor Y, Gibali A, Reich S (2011) The subgradient extragradient method for solving variational inequalities in Hilbert space. J Optim Theory Appl 148:318\u2013335","journal-title":"J Optim Theory Appl"},{"key":"1393_CR5","doi-asserted-by":"publisher","first-page":"1607","DOI":"10.1016\/S0252-9602(12)60127-1","volume":"32","author":"SY Cho","year":"2012","unstructured":"Cho SY, Kang SM (2012) Approximation of common solutions of variational inequalities via strict pseudocontractions. Acta Math Sci 32:1607\u20131618","journal-title":"Acta Math Sci"},{"key":"1393_CR6","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1186\/1029-242X-2013-199","volume":"2013","author":"SY Cho","year":"2013","unstructured":"Cho SY, Li W, Kang SM (2013) Convergence analysis of an iterative algorithm for monotone operators. J Inequal Appl 2013:199","journal-title":"J Inequal Appl"},{"key":"1393_CR7","doi-asserted-by":"publisher","first-page":"2199","DOI":"10.1080\/02331934.2019.1625355","volume":"69","author":"J Fan","year":"2020","unstructured":"Fan J, Liu L, Qin X (2020) A subgradient extragradient algorithm with inertial effects for solving strongly pseudomonotone variational inequalities. Optimization 69:2199\u20132215","journal-title":"Optimization"},{"key":"1393_CR8","first-page":"747","volume":"12","author":"GM Korpelevich","year":"1976","unstructured":"Korpelevich GM (1976) The extragradient method for finding saddle points and other problems. Ekonomikai Matematicheskie Metody 12:747\u2013756","journal-title":"Ekonomikai Matematicheskie Metody"},{"key":"1393_CR9","doi-asserted-by":"publisher","first-page":"399","DOI":"10.1007\/s10957-013-0494-2","volume":"163","author":"R Kraikaew","year":"2014","unstructured":"Kraikaew R, Saejung S (2014) Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces. J Optim Theory Appl 163:399\u2013412","journal-title":"J Optim Theory Appl"},{"key":"1393_CR10","doi-asserted-by":"publisher","first-page":"114","DOI":"10.1006\/jmaa.1995.1289","volume":"194","author":"LS Liu","year":"1995","unstructured":"Liu LS (1995) Ishikawa and Mann iteration process with errors for nonlinear strongly accretive mappings in Banach space. J Math Anal Appl 194:114\u2013125","journal-title":"J Math Anal Appl"},{"key":"1393_CR11","first-page":"471","volume":"20","author":"L Liu","year":"2019","unstructured":"Liu L (2019) A hybrid steepest descent method for solving split feasibility problems involving nonexpansive mappings. J Nonlinear Convex Anal 20:471\u2013488","journal-title":"J Nonlinear Convex Anal"},{"key":"1393_CR12","doi-asserted-by":"publisher","DOI":"10.1080\/02331934.2019.1613404","author":"L Liu","year":"2019","unstructured":"Liu L, Qin X, Agarwal RP (2019) Iterative methods for fixed points and zero points of nonlinear mappings with applications. Optimization. https:\/\/doi.org\/10.1080\/02331934.2019.1613404","journal-title":"Optimization"},{"key":"1393_CR13","doi-asserted-by":"publisher","first-page":"1499","DOI":"10.1137\/060675319","volume":"47","author":"PE Maing\u00e9","year":"2008","unstructured":"Maing\u00e9 PE (2008) A hybrid extragradient-viscosity method for monotone operators and fixed point problems. SIAM J Control Optim 47:1499\u20131515","journal-title":"SIAM J Control Optim"},{"key":"1393_CR14","doi-asserted-by":"publisher","first-page":"821","DOI":"10.1007\/s10589-019-00124-7","volume":"74","author":"X Qin","year":"2019","unstructured":"Qin X, An NT (2019) Smoothing algorithms for computing the projection onto a Minkowski sum of convex sets. Comput Optim Appl 74:821\u2013850","journal-title":"Comput Optim Appl"},{"key":"1393_CR15","first-page":"2325","volume":"21","author":"X Qin","year":"2020","unstructured":"Qin X, Wang L, Yao JC (2020) Inertial splitting method for maximal monotone mappings. J Nonlinear Convex Anal 21:2325\u20132333","journal-title":"J Nonlinear Convex Anal"},{"key":"1393_CR16","doi-asserted-by":"publisher","DOI":"10.1080\/02331934.2019.1702040","author":"DR Sahu","year":"2020","unstructured":"Sahu DR, Yao JC, Verma M, Shukla KK (2020) Convergence rate analysis of proximal gradient methods with applications to composite minimization problems. Optimization. https:\/\/doi.org\/10.1080\/02331934.2019.1702040","journal-title":"Optimization"},{"key":"1393_CR17","doi-asserted-by":"publisher","first-page":"259","DOI":"10.1007\/s11075-016-0253-1","volume":"76","author":"Y Shehu","year":"2017","unstructured":"Shehu Y, Iyiola OS (2017) Strong convergence result for monotone variational inequalities. Numer Algorithms 76:259\u2013282","journal-title":"Numer Algorithms"},{"key":"1393_CR18","doi-asserted-by":"publisher","first-page":"161","DOI":"10.1007\/s40314-019-0955-9","volume":"38","author":"Y Shehu","year":"2019","unstructured":"Shehu Y, Iyiola OS, Li XH, Dong Q-L (2019) Convergence analysis of projection method for variational inequalities. Comput Appl Math 38:161","journal-title":"Comput Appl Math"},{"key":"1393_CR19","first-page":"337","volume":"4","author":"B Tan","year":"2020","unstructured":"Tan B, Li S (2020) Strong convergence of inertial Mann algorithms for solving hierarchical fixed point problems. J Nonlinear Var Anal 4:337\u2013355","journal-title":"J Nonlinear Var Anal"},{"key":"1393_CR20","first-page":"171","volume":"2","author":"B Tan","year":"2020","unstructured":"Tan B, Xu S (2020) Strong convergence of two inertial projection algorithms in Hilbert spaces. J Appl Numer Optim 2:171\u2013186","journal-title":"J Appl Numer Optim"},{"key":"1393_CR21","first-page":"871","volume":"21","author":"B Tan","year":"2020","unstructured":"Tan B, Xu S, Li S (2020) Inertial shrinking projection algorithms for solving hierarchical variational inequality problems. J Nonlinear Convex Anal 21:871\u2013884","journal-title":"J Nonlinear Convex Anal"},{"key":"1393_CR22","doi-asserted-by":"publisher","first-page":"136","DOI":"10.1007\/s40314-019-0899-0","volume":"38","author":"DV Thong","year":"2019","unstructured":"Thong DV, Hieu DV (2019) Strong convergence of extragradient methods with a new step size for solving variational inequality problems. Comput Appl Math 38:136","journal-title":"Comput Appl Math"},{"key":"1393_CR23","doi-asserted-by":"publisher","first-page":"431","DOI":"10.1137\/S0363012998338806","volume":"38","author":"P Tseng","year":"2000","unstructured":"Tseng P (2000) A modified forward\u2013backward splitting method for maximal monotone mappings. SIAM J Control Optim 38:431\u2013446","journal-title":"SIAM J Control Optim"},{"key":"1393_CR24","first-page":"225","volume":"3","author":"F Wang","year":"2019","unstructured":"Wang F, Pham H (2019) On a new algorithm for solving variational inequality and fixed point problems. J Nonlinear Var Anal 3:225\u2013233","journal-title":"J Nonlinear Var Anal"},{"key":"1393_CR25","first-page":"333","volume":"3","author":"X Wang","year":"2019","unstructured":"Wang X, Ou X, Zhang T, Chen JW (2019) An alternate minimization method beyond positive definite proximal regularization: convergence and complexity. J Nonlinear Var Anal 3:333\u2013355","journal-title":"J Nonlinear Var Anal"},{"key":"1393_CR26","doi-asserted-by":"publisher","first-page":"240","DOI":"10.1112\/S0024610702003332","volume":"66","author":"HK Xu","year":"2002","unstructured":"Xu HK (2002) Iterative algorithms for nonlinear operators. J Lond Math Soc 66:240\u2013256","journal-title":"J Lond Math Soc"},{"key":"1393_CR27","doi-asserted-by":"publisher","first-page":"741","DOI":"10.1007\/s11075-018-0504-4","volume":"80","author":"J Yang","year":"2019","unstructured":"Yang J, Liu H (2019) Strong convergence result for solving monotone variational inequalities in Hilbert space. Numer Algorithms 80:741\u2013752","journal-title":"Numer Algorithms"},{"key":"1393_CR28","doi-asserted-by":"publisher","first-page":"220","DOI":"10.1007\/s40314-020-01237-0","volume":"39","author":"Z Zhou","year":"2020","unstructured":"Zhou Z, Tan B, Li S (2020) A new accelerated self-adaptive stepsize algorithm with excellent stability for split common fixed point problems. Comput Appl Math 39:220","journal-title":"Comput Appl Math"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-020-01393-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s40314-020-01393-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-020-01393-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,2,11]],"date-time":"2021-02-11T20:22:57Z","timestamp":1613074977000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s40314-020-01393-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,11]]},"references-count":28,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2021,2]]}},"alternative-id":["1393"],"URL":"https:\/\/doi.org\/10.1007\/s40314-020-01393-3","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1,11]]},"assertion":[{"value":"6 June 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 November 2020","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 December 2020","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 January 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"19"}}