{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,16]],"date-time":"2026-04-16T23:36:28Z","timestamp":1776382588717,"version":"3.51.2"},"reference-count":44,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2018,11,5]],"date-time":"2018-11-05T00:00:00Z","timestamp":1541376000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2019,11]]},"DOI":"10.1007\/s11075-018-0626-8","type":"journal-article","created":{"date-parts":[[2018,11,5]],"date-time":"2018-11-05T00:46:15Z","timestamp":1541378775000},"page":"761-789","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":45,"title":["Some extragradient-viscosity algorithms for solving variational inequality problems and fixed point problems"],"prefix":"10.1007","volume":"82","author":[{"given":"Duong Viet","family":"Thong","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dang","family":"Van Hieu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2018,11,5]]},"reference":[{"key":"626_CR1","first-page":"1164","volume":"12","author":"AS Antipin","year":"1976","unstructured":"Antipin, A.S.: On a method for convex programs using a symmetrical modification of the Lagrange function. Ekonomika i Mat. Metody 12, 1164\u20131173 (1976)","journal-title":"Ekonomika i Mat. Metody"},{"key":"626_CR2","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1007\/s11075-011-9490-5","volume":"56","author":"Y Censor","year":"2012","unstructured":"Censor, Y., Gibali, A., Reich, S.: Algorithms for the split variational inequality problem. Numer. Algorithms. 56, 301\u2013323 (2012)","journal-title":"Numer. Algorithms."},{"key":"626_CR3","doi-asserted-by":"crossref","first-page":"318","DOI":"10.1007\/s10957-010-9757-3","volume":"148","author":"Y Censor","year":"2011","unstructured":"Censor, Y., Gibali, A, Reich, S.: The subgradient extragradient method for solving variational inequalities in Hilbert space. J. Optim. Theory Appl. 148, 318\u2013335 (2011)","journal-title":"J. Optim. Theory Appl."},{"key":"626_CR4","doi-asserted-by":"crossref","first-page":"827","DOI":"10.1080\/10556788.2010.551536","volume":"26","author":"Y Censor","year":"2011","unstructured":"Censor, Y., Gibali, A., Reich, S.: Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space. Optim. Meth. Softw. 26, 827\u2013845 (2011)","journal-title":"Optim. Meth. Softw."},{"key":"626_CR5","doi-asserted-by":"crossref","first-page":"1119","DOI":"10.1080\/02331934.2010.539689","volume":"61","author":"Y Censor","year":"2011","unstructured":"Censor, Y., Gibali, A., Reich, S.: Extensions of Korpelevich\u2019s extragradient method for the variational inequality problem in Euclidean space. Optimization 61, 1119\u20131132 (2011)","journal-title":"Optimization"},{"key":"626_CR6","doi-asserted-by":"crossref","first-page":"635","DOI":"10.1007\/s10898-009-9454-7","volume":"46","author":"LC Ceng","year":"2010","unstructured":"Ceng, L.C., Hadjisavvas, N., Wong, N.C.: Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems. J. Glob. Optim. 46, 635\u2013646 (2010)","journal-title":"J. Glob. Optim."},{"key":"626_CR7","doi-asserted-by":"crossref","first-page":"861","DOI":"10.1016\/j.cam.2010.01.050","volume":"234","author":"CE Chidume","year":"2010","unstructured":"Chidume, C.E., Maruster, S.: Iterative methods for the computation of fixed points of demicontractive mappings. J. Comput. Appl. Math. 234, 861\u2013882 (2010)","journal-title":"J. Comput. Appl. Math."},{"key":"626_CR8","doi-asserted-by":"crossref","first-page":"757","DOI":"10.1007\/s10559-015-9768-z","volume":"51","author":"SV Denisov","year":"2015","unstructured":"Denisov, S.V., Semenov, V.V., Chabak, L.M.: Convergence of the modified extragradient method for variational inequalities with non-Lipschitz operators. Cybern. Syst. Anal. 51, 757\u2013765 (2015)","journal-title":"Cybern. Syst. Anal."},{"key":"626_CR9","volume-title":"Uniform convexity, hyperbolic geometry, and nonexpansive mappings","author":"K Goebel","year":"1984","unstructured":"Goebel, K., Reich, S.: Uniform convexity, hyperbolic geometry, and nonexpansive mappings. Marcel Dekker, New York (1984)"},{"key":"626_CR10","first-page":"265","volume":"26","author":"PT Harker","year":"1990","unstructured":"Harker, P.T., Pang, J.S.: A damped-Newton method for the linear complementarity problem. Lect. Appl. Math. 26, 265\u2013284 (1990)","journal-title":"Lect. Appl. Math."},{"key":"626_CR11","doi-asserted-by":"crossref","first-page":"498","DOI":"10.1016\/0022-247X(77)90076-2","volume":"59","author":"TL Hicks","year":"1997","unstructured":"Hicks, T.L., Kubicek, J.D.: On the Mann iteration process in a Hilbert space. J. Math. Anal. Appl. 59, 498\u2013504 (1997)","journal-title":"J. Math. Anal. Appl."},{"key":"626_CR12","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1007\/s10589-016-9857-6","volume":"66","author":"DV Hieu","year":"2017","unstructured":"Hieu, D.V., Anh, P.K., Muu, L.D.: Modified hybrid projection methods for finding common solutions to variational inequality problems. Comput. Optim. Appl. 66, 75\u201396 (2017)","journal-title":"Comput. Optim. Appl."},{"key":"626_CR13","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1007\/s10898-017-0564-3","volume":"70","author":"DV Hieu","year":"2018","unstructured":"Hieu, D.V., Thong, D.V.: New extragradient-like algorithms for strongly pseudomonotone variational inequalities. J. Glob. Optim. 70, 385\u2013399 (2018)","journal-title":"J. Glob. Optim."},{"key":"626_CR14","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1023\/A:1013096613105","volume":"112","author":"BS He","year":"2002","unstructured":"He, B.S., Liao, L.Z.: Improvements of some projection methods for monotone nonlinear variational inequalities. J. Optim. Theory Appl. 112, 111\u2013128 (2002)","journal-title":"J. Optim. Theory Appl."},{"key":"626_CR15","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1016\/j.na.2003.07.023","volume":"61","author":"H Iiduka","year":"2005","unstructured":"Iiduka, H., Takahashi, W.: Strong convergence theorems for nonexpansive mappings and inverse strongly monotone mappings. Nonlinear Anal. TMA. 61, 341\u2013350 (2005)","journal-title":"Nonlinear Anal. TMA."},{"key":"626_CR16","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1007\/s10898-013-0042-5","volume":"58","author":"PD Khanh","year":"2014","unstructured":"Khanh, P.D., Vuong, P.T.: Modified projection method for strongly pseudomonotone variational inequalities. J. Global Optim. 58, 341\u2013350 (2014)","journal-title":"J. Global Optim."},{"key":"626_CR17","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1007\/s40306-015-0150-z","volume":"41","author":"PD Khanh","year":"2016","unstructured":"Khanh, P.D.: A modifed extragradient method for infnite-dimensional variational inequalities. Acta Math Vietnam. 41, 251\u2013263 (2016)","journal-title":"Acta Math Vietnam."},{"key":"626_CR18","doi-asserted-by":"crossref","first-page":"399","DOI":"10.1007\/s10957-013-0494-2","volume":"163","author":"R Kraikaew","year":"2014","unstructured":"Kraikaew, R., Saejung, S.: Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces. J. Optim. Theory Appl. 163, 399\u2013412 (2014)","journal-title":"J. Optim. Theory Appl."},{"key":"626_CR19","first-page":"747","volume":"12","author":"GM Korpelevich","year":"1976","unstructured":"Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekonomikai Matematicheskie Metody. 12, 747\u2013756 (1976)","journal-title":"Ekonomikai Matematicheskie Metody."},{"key":"626_CR20","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1006\/jmaa.1995.1289","volume":"194","author":"LS Liu","year":"1995","unstructured":"Liu, L.S.: Ishikawa and Mann iteration process with errors for nonlinear strongly accretive mappings in Banach space. J. Math. Anal. Appl. 194, 114\u2013125 (1995)","journal-title":"J. Math. Anal. Appl."},{"key":"626_CR21","doi-asserted-by":"crossref","first-page":"631","DOI":"10.1007\/s10559-011-9343-1","volume":"47","author":"SI Lyashko","year":"2011","unstructured":"Lyashko, S.I., Semenov, V.V., Voitova, T.A.: Low-cost modification of Korpelevich\u2019s methods for monotone equilibrium problems. Cybern. Syst. Anal. 47, 631\u2013640 (2011)","journal-title":"Cybern. Syst. Anal."},{"key":"626_CR22","doi-asserted-by":"crossref","first-page":"1499","DOI":"10.1137\/060675319","volume":"47","author":"PE Maing\u00e9","year":"2008","unstructured":"Maing\u00e9, P.E.: A hybrid extragradient-viscosity method for monotone operators and fixed point problems. SIAM J. Control Optim. 47, 1499\u20131515 (2008)","journal-title":"SIAM J. Control Optim."},{"key":"626_CR23","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1007\/s10898-014-0150-x","volume":"61","author":"YV Malitsky","year":"2015","unstructured":"Malitsky, Y.V., Semenov, V.V.: A hybrid method without extrapolation step for solving variational inequality problems. J. Glob. Optim. 61, 193\u2013202 (2015)","journal-title":"J. Glob. Optim."},{"key":"626_CR24","doi-asserted-by":"crossref","first-page":"502","DOI":"10.1137\/14097238X","volume":"25","author":"YV Malitsky","year":"2015","unstructured":"Malitsky, Y.V.: Projected reflected gradient methods for monotone variational inequalities. SIAM J. Optim. 25, 502\u2013520 (2015)","journal-title":"SIAM J. Optim."},{"key":"626_CR25","doi-asserted-by":"crossref","first-page":"506","DOI":"10.1090\/S0002-9939-1953-0054846-3","volume":"4","author":"WR Mann","year":"1953","unstructured":"Mann, W.R.: Mean value methods in iteration. Proc. Amer. Math. Soc. 4, 506\u2013510 (1953)","journal-title":"Proc. Amer. Math. Soc."},{"key":"626_CR26","first-page":"1065","volume":"16","author":"C Mongkolkeha","year":"2013","unstructured":"Mongkolkeha, C., Cho, Y.J., Kumam, P.: Convergence theorems for k-dimeicontactive mappings in Hilbert spaces. Math. Inequal. Appl. 16, 1065\u20131082 (2013)","journal-title":"Math. Inequal. Appl."},{"key":"626_CR27","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1006\/jmaa.1999.6615","volume":"241","author":"A Moudafi","year":"2000","unstructured":"Moudafi, A.: Viscosity approximating methods for fixed point problems. J. Math. Anal. Appl. 241, 46\u201355 (2000)","journal-title":"J. Math. Anal. Appl."},{"key":"626_CR28","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1007\/s10957-005-7564-z","volume":"128","author":"N Nadezhkina","year":"2006","unstructured":"Nadezhkina, N., Takahashi, W.: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 128, 191\u2013201 (2006)","journal-title":"J. Optim. Theory Appl."},{"key":"626_CR29","doi-asserted-by":"crossref","first-page":"1230","DOI":"10.1137\/050624315","volume":"16","author":"N Nadezhkina","year":"2006","unstructured":"Nadezhkina, N., Takahashi, W.: Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings. SIAM J. Optim. 16, 1230\u20131241 (2006)","journal-title":"SIAM J. Optim."},{"key":"626_CR30","first-page":"335","volume-title":"Constructive Techniques for Accretive and Monotone Operators. Applied Nonlinear Analysis","author":"S Reich","year":"1979","unstructured":"Reich, S.: Constructive Techniques for Accretive and Monotone Operators. Applied Nonlinear Analysis, pp 335\u2013345. Academic Press, New York (1979)"},{"key":"626_CR31","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1007\/s11075-016-0253-1","volume":"76","author":"Y Shehu","year":"2017","unstructured":"Shehu, Y., Iyiola, O.S.: Strong convergence result for monotone variational inequalities. Numer. Algorithms. 76, 259\u2013282 (2017)","journal-title":"Numer. Algorithms."},{"key":"626_CR32","doi-asserted-by":"crossref","first-page":"765","DOI":"10.1137\/S0363012997317475","volume":"37","author":"MV Solodov","year":"1999","unstructured":"Solodov, M.V., Svaiter, B.F.: A new projection method for variational inequality problems. SIAM J. Control Optim. 37, 765\u2013776 (1999)","journal-title":"SIAM J. Control Optim."},{"key":"626_CR33","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1023\/A:1025407607560","volume":"118","author":"W Takahashi","year":"2003","unstructured":"Takahashi, W., Toyoda, M.: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 118, 417\u2013428 (2003)","journal-title":"J. Optim. Theory Appl."},{"key":"626_CR34","doi-asserted-by":"crossref","first-page":"1481","DOI":"10.1007\/s11784-016-0323-y","volume":"19","author":"DV Thong","year":"2017","unstructured":"Thong, D.V.: Viscosity approximation methods for solving fixed point problems and split common fixed point problems. J. Fixed Point Theory Appl. 19, 1481\u20131499 (2017)","journal-title":"J. Fixed Point Theory Appl."},{"key":"626_CR35","doi-asserted-by":"crossref","first-page":"1045","DOI":"10.1007\/s11075-017-0412-z","volume":"78","author":"DV Thong","year":"2108","unstructured":"Thong, D.V., Hieu, D.V.: Weak and strong convergence theorems for variational inequality problems. Numer. Algorithms. 78, 1045\u20131060 (2108)","journal-title":"Numer. Algorithms."},{"key":"626_CR36","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1080\/02331934.2017.1377199","volume":"67","author":"DV Thong","year":"2018","unstructured":"Thong, D.V., Hieu, D.V.: Modified subgradient extragradient algorithms for variational inequality problems and fixed point problems. Optimization 67, 83\u2013102 (2018)","journal-title":"Optimization"},{"issue":"2","key":"626_CR37","doi-asserted-by":"publisher","first-page":"597","DOI":"10.1007\/s11075-017-0452-4","volume":"79","author":"Duong Viet Thong","year":"2017","unstructured":"Thong, D.V., Hieu, D.V.: Modified subgradient extragradient method for variational inequality problems. Numer. Algorithms. \nhttps:\/\/doi.org\/10.1007\/s11075-017-0452-4\n\n (2017)","journal-title":"Numerical Algorithms"},{"key":"626_CR38","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1016\/j.cam.2018.03.019","volume":"341","author":"DV Thong","year":"2018","unstructured":"Thong, D.V., Hieu, D.V.: Inertial extragradient algorithms for strongly pseudomonotone variational inequalities. J. Comput. Appl. Math. 341, 80\u201398 (2018)","journal-title":"J. Comput. Appl. Math."},{"key":"626_CR39","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1137\/S0363012998338806","volume":"38","author":"P Tseng","year":"2000","unstructured":"Tseng, P.: A modified forward-backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 38, 431\u2013446 (2000)","journal-title":"SIAM J. Control Optim."},{"key":"626_CR40","doi-asserted-by":"crossref","first-page":"1125","DOI":"10.11650\/twjm\/1500406682","volume":"16","author":"FH Wang","year":"2012","unstructured":"Wang, F.H., Xu, H.K.: Weak and strong convergence theorems for variational inequality and fixed point problems with Tseng\u2019s extragradient method. Taiwanese J. Math. 16, 1125\u20131136 (2012)","journal-title":"Taiwanese J. Math."},{"key":"626_CR41","doi-asserted-by":"crossref","first-page":"240","DOI":"10.1112\/S0024610702003332","volume":"66","author":"HK Xu","year":"2002","unstructured":"Xu, H.K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240\u2013256 (2002)","journal-title":"J. Lond. Math. Soc."},{"issue":"3","key":"626_CR42","doi-asserted-by":"publisher","first-page":"741","DOI":"10.1007\/s11075-018-0504-4","volume":"80","author":"Jun Yang","year":"2018","unstructured":"Yang, J., Liu, H.: Strong convergence result for solving monotone variational inequalities in Hilbert space, Numer Algorithms. \nhttps:\/\/doi.org\/10.1007\/s11075-018-0504-4\n\n (2018)","journal-title":"Numerical Algorithms"},{"key":"626_CR43","doi-asserted-by":"crossref","first-page":"559","DOI":"10.1080\/02331934.2012.674947","volume":"63","author":"Y Yao","year":"2014","unstructured":"Yao, Y., Marino, G., Muglia, L.: A modified Korpelevich\u2019s method convergent to the minimum-norm solution of a variational inequality. Optimization 63, 559\u2013569 (2014)","journal-title":"Optimization"},{"key":"626_CR44","doi-asserted-by":"crossref","first-page":"1293","DOI":"10.11650\/twjm\/1500557303","volume":"10","author":"LC Zeng","year":"2006","unstructured":"Zeng, L.C., Yao, J.C.: Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems. Taiwan J. Math. 10, 1293\u20131303 (2006)","journal-title":"Taiwan J. Math."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-018-0626-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11075-018-0626-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-018-0626-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,3,3]],"date-time":"2020-03-03T01:08:44Z","timestamp":1583197724000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11075-018-0626-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,11,5]]},"references-count":44,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2019,11]]}},"alternative-id":["626"],"URL":"https:\/\/doi.org\/10.1007\/s11075-018-0626-8","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,11,5]]},"assertion":[{"value":"24 April 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 October 2018","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 November 2018","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}