{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,13]],"date-time":"2026-04-13T15:42:31Z","timestamp":1776094951684,"version":"3.50.1"},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2025,12,8]],"date-time":"2025-12-08T00:00:00Z","timestamp":1765152000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,12,8]],"date-time":"2025-12-08T00:00:00Z","timestamp":1765152000000},"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":["J. Appl. Math. Comput."],"published-print":{"date-parts":[[2026,1]]},"DOI":"10.1007\/s12190-025-02661-3","type":"journal-article","created":{"date-parts":[[2025,12,8]],"date-time":"2025-12-08T10:59:14Z","timestamp":1765191554000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["An improved strongly convergent algorithm for quasimonotone variational inequalities and fixed point problems"],"prefix":"10.1007","volume":"72","author":[{"ORCID":"https:\/\/orcid.org\/0009-0001-4682-959X","authenticated-orcid":false,"given":"Xingfang","family":"Wang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7955-0341","authenticated-orcid":false,"given":"Xia","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,12,8]]},"reference":[{"key":"2661_CR1","volume-title":"Ekeland, I.: applied Nonlinear Analysis","author":"J.P. Aubin","year":"1984","unstructured":"Aubin, J.P., Ekeland, I.: Applied nonlinear analysis. Dover Publications, Inc, Mineola, NY, New York (1984)"},{"issue":"5","key":"2661_CR2","doi-asserted-by":"publisher","first-page":"120","DOI":"10.1016\/0041-5553(87)90058-9","volume":"27","author":"E.N. Khobotov","year":"1989","unstructured":"Khobotov, E.N.: Modification of the extra-gradient method for solving variational inequalities and certain optimization problems. USSR Comput. Math. Phys. 27(5), 120\u2013127 (1989)","journal-title":"USSR Comput. Math. Phys."},{"key":"2661_CR3","volume-title":"Pure Appl. Math.","author":"D. Kinderlehrer","year":"1980","unstructured":"Kinderlehrer, D., Stampacchia, G.: An introduction to variational inequalities and their applications. In: Pure Appl. Math., vol. 88. Academic Press, New York (1980)"},{"issue":"2","key":"2661_CR4","doi-asserted-by":"publisher","first-page":"535","DOI":"10.1007\/s11075-017-0327-8","volume":"77","author":"G. Cai","year":"2018","unstructured":"Cai, G., Shehu, Y., Iyiola, O.S.: Strong convergence results for variational inequalities and fixed point problems using modified viscosity implicit rules. Numer. Algoritms. 77(2), 535\u2013558 (2018)","journal-title":"Numer. Algoritms."},{"issue":"1","key":"2661_CR5","doi-asserted-by":"publisher","first-page":"27","DOI":"10.1007\/s11784-015-0258-8","volume":"18","author":"C. Fang","year":"2016","unstructured":"Fang, C., Wang, Y., Yang, S.: Two algorithms for solving single-valued variational inequalities and fixed point problems. J. Fixed. Point. Theory. Appl. 18(1), 27\u201343 (2016)","journal-title":"J. Fixed. Point. Theory. Appl."},{"issue":"3","key":"2661_CR6","doi-asserted-by":"publisher","first-page":"1499","DOI":"10.1137\/060675319","volume":"47","author":"P.E. 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(3), 1499\u20131515 (2008)","journal-title":"SIAM J. Control. Optim."},{"issue":"1","key":"2661_CR7","doi-asserted-by":"publisher","first-page":"273","DOI":"10.1007\/s10957-012-0055-0","volume":"155","author":"Y. Yao","year":"2012","unstructured":"Yao, Y., Postolache, M.: Iterative methods for pseudomonotone variational inequalities and fixed point problems. J. Optim. Theory. Appl. 155(1), 273\u2013287 (2012)","journal-title":"J Optim Theory Appl."},{"issue":"6","key":"2661_CR8","doi-asserted-by":"publisher","first-page":"1987","DOI":"10.1080\/02331934.2019.1711087","volume":"69","author":"X. Zhao","year":"2020","unstructured":"Zhao, X., Yao, Y.: Modified extragradient algorithms for solving monotone variational inequalities and fixed point problems. Optimization 69(6), 1987\u20132002 (2020)","journal-title":"Optimization"},{"issue":"4","key":"2661_CR9","first-page":"747","volume":"12","author":"G.M. Korpelevich","year":"1976","unstructured":"Korpelevich, G.M.: An extragradient method for finding saddle points and for other problems. \u00c8konom. I Mat. Metody. 12(4), 747\u2013756 (1976)","journal-title":"\u00c8konom. I Mat. Metody."},{"issue":"1","key":"2661_CR10","doi-asserted-by":"publisher","first-page":"111","DOI":"10.1023\/A:1013096613105","volume":"112","author":"B.S. 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(1), 111\u2013128 (2002)","journal-title":"J Optim Theory Appl."},{"issue":"5\u20136","key":"2661_CR11","first-page":"845","volume":"28","author":"L.D. Popov","year":"1980","unstructured":"Popov, L.D.: A modification of the Arrow-Hurwicz method for searching for saddle points. Numer. Math. 28(5\u20136), 845\u2013848 (1980)","journal-title":"Numer. Math."},{"issue":"2","key":"2661_CR12","doi-asserted-by":"publisher","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(2), 431\u2013446 (2000)","journal-title":"SIAM J. Control. Optim."},{"issue":"2","key":"2661_CR13","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.: The subgradient extragradient method for solving variational inequalities in Hilbert space. J. Optim. Theory. Appl. 148(2), 318\u2013335 (2011)","journal-title":"J Optim Theory Appl."},{"issue":"3","key":"2661_CR14","doi-asserted-by":"publisher","first-page":"927","DOI":"10.1007\/s11075-017-0467-x","volume":"79","author":"Q.L. Dong","year":"2018","unstructured":"Dong, Q.L., Jiang, D., Gibali: A.: A modified subgradient extragradient method for solving the variational inequality problem. Numer. Algoritms. 79(3), 927\u2013940 (2018)","journal-title":"Numer. Algoritms."},{"issue":"3","key":"2661_CR15","doi-asserted-by":"publisher","first-page":"687","DOI":"10.1007\/s10898-017-0506-0","volume":"70","author":"Q.L. Dong","year":"2018","unstructured":"Dong, Q.L., Cho, J.Y., Zhong, L.L., Rassias, T.M.: Inertial projection and contraction algorithms for variational inequalities. J. Global. Optim. 70(3), 687\u2013704 (2018)","journal-title":"J. Global Optim."},{"issue":"1","key":"2661_CR16","doi-asserted-by":"publisher","first-page":"10","DOI":"10.1007\/s10915-021-01670-1","volume":"90","author":"G.N. Ogwo","year":"2022","unstructured":"Ogwo, G.N., Izuchukwu, C., Shehu, Y., Mewomo, O.T.: Convergence of relaxed inertial subgradient extragradient methods for quasimonotone variational inequality problems. J. Sci. Comput. 90(1), 10 (2022)","journal-title":"J. Sci. Comput."},{"key":"2661_CR17","doi-asserted-by":"publisher","first-page":"114517","DOI":"10.1016\/j.cam.2022.114517","volume":"415","author":"Y. Shehu","year":"2022","unstructured":"Shehu, Y., Dong, Q.L., Liu, L.L.: Fast alternated inertial projection algorithms for pseudo-monotone variational inequalities. J. Comput. Appl. Math. 415, 114517 (2022)","journal-title":"J. Comput. Appl. Math."},{"issue":"1","key":"2661_CR18","doi-asserted-by":"publisher","first-page":"184","DOI":"10.1007\/s10473-023-0112-9","volume":"43","author":"D.V. Thong","year":"2023","unstructured":"Thong, D.V., Dung, V.T.: A relaxed inertial factor of the modified subgradient extragradient method for solving pseudo monotone variational inequalities in Hilbert spaces. Acta. Math. Sci. Ser. B. Engl. Ed. 43(1), 184\u2013204 (2023)","journal-title":"Acta Math. Sci. Ser. B Engl. Ed."},{"issue":"5","key":"2661_CR19","doi-asserted-by":"publisher","first-page":"937","DOI":"10.1007\/s10114-022-0243-2","volume":"38","author":"G. Cai","year":"2022","unstructured":"Cai, G., Dong, Q.L., Peng, Y.: Inertial viscosity iterative method for solving pseudo-monotone variational inequality problems and fixed point problems. Acta. Math. Sin. Engl. Ser. 38(5), 937\u2013952 (2022)","journal-title":"Acta Math. Sin. Engl. Ser."},{"issue":"2","key":"2661_CR20","doi-asserted-by":"publisher","first-page":"1387","DOI":"10.1007\/s12190-021-01576-z","volume":"68","author":"B. Tan","year":"2022","unstructured":"Tan, B., Zhou, Z., Li, S.: Viscosity-type inertial extragradient algorithms for solving variational inequality problems and fixed point problems. J. Appl. Math. Comput. 68(2), 1387\u20131411 (2022)","journal-title":"J. Appl. Math. Comput."},{"issue":"3","key":"2661_CR21","doi-asserted-by":"publisher","first-page":"761","DOI":"10.1007\/s11075-018-0626-8","volume":"82","author":"D.V. Thong","year":"2019","unstructured":"Thong, D.V., Hieu, D.V.: Some extragradient-viscosity algorithms for solving variational inequality problems and fixed point problems. Numer. Algoritms. 82(3), 761\u2013789 (2019)","journal-title":"Numer. Algoritms."},{"issue":"3","key":"2661_CR22","doi-asserted-by":"publisher","first-page":"523","DOI":"10.1007\/s10898-021-01095-y","volume":"82","author":"B. Tan","year":"2022","unstructured":"Tan, B., Qin, X., Yao, J.C.: Strong convergence of inertial projection and contraction methods for pseudomonotone variational inequalities with applications to optimal control problems. J. Global Optim. 82(3), 523\u2013557 (2022)","journal-title":"J. Global Optim."},{"issue":"1","key":"2661_CR23","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1080\/02331934.2017.1377199","volume":"67","author":"D.V. Thong","year":"2018","unstructured":"Thong, D.V., Hieu, D.V.: Modified subgradient extragradient algorithms for variational inequality problems and fixed point problems. Optimization 67(1), 83\u2013102 (2018)","journal-title":"Optimization"},{"issue":"4","key":"2661_CR24","doi-asserted-by":"publisher","first-page":"259","DOI":"10.1007\/s40314-024-02785-5","volume":"43","author":"W. Singh","year":"2024","unstructured":"Singh, W., Chandok, S.: Mann-type extragradient algorithm for solving variational inequality and fixed point problems. Comput. Appl. Math. 43(4), 259 (2024)","journal-title":"Comput. Appl. Math."},{"issue":"2","key":"2661_CR25","doi-asserted-by":"publisher","first-page":"551","DOI":"10.1007\/s10473-024-0210-3","volume":"44","author":"Y. Yao","year":"2024","unstructured":"Yao, Y., Adamu, A., Shehu, Y.: Strongly convergent inertial forward-backwardforward algorithm without on-line rule for variational inequalities. Acta. Math. Sci. Ser. B. Engl. Ed. 44(2), 551\u2013566 (2024)","journal-title":"Acta Math. Sci. Ser. B Engl. Ed."},{"key":"2661_CR26","doi-asserted-by":"publisher","DOI":"10.1080\/02331934.2024.2424446","author":"C. Izuchukwu","year":"2024","unstructured":"Izuchukwu, C., Shehu, Y., Yao, J.C.: New strong convergence analysis for variational inequalities and fixed point problems. Optimization (2024). https:\/\/doi.org\/10.1080\/02331934.2024.2424446","journal-title":"Optimization"},{"issue":"9","key":"2661_CR27","doi-asserted-by":"publisher","first-page":"1732","DOI":"10.1080\/00036811.2024.2432527","volume":"104","author":"C. Izuchukwu","year":"2024","unstructured":"Izuchukwu, C., Shehu, Y., Yao, J.C.: Convergence results for proximal point algorithm with inertial and correction terms. Appl. Anal. 104(9), 1732\u20131752 (2024)","journal-title":"Appl Anal."},{"key":"2661_CR28","doi-asserted-by":"publisher","DOI":"10.1080\/02331934.2024.2368070","author":"P. Li","year":"2024","unstructured":"Li, P., Xu, H.K., Ke, S.: Viscosity extragradient algorithms with double inertial extrapolation steps for quasimonotone variational inequalities and fixed point problems. Optimization (2024). https:\/\/doi.org\/10.1080\/02331934.2024.2368070","journal-title":"Optimization"},{"issue":"1","key":"2661_CR29","first-page":"145","volume":"7","author":"O.T. Mewomo","year":"2023","unstructured":"Mewomo, O.T., Alakoya, T.O., Yao, J.C., Akinyemi, L.: Strong convergent inertial Tseng\u2019s extragradient method for solving non-Lipschitz quasimonotone variational inequalities in banach spaces. J. Nonlinear Var. Anal. 7(1), 145\u2013172 (2023)","journal-title":"J. Nonlinear Var. Anal."},{"issue":"3","key":"2661_CR30","first-page":"429","volume":"6","author":"O.T. Mewomo","year":"2024","unstructured":"Mewomo, O.T., Nwokoye, R.N., Alakoyo, T.O., Ogwo, G.N.: On fixed point iterative methods for solving non-lipschitz quasi-monotone variational inequalities. Appl. Numer. Optim. 6(3), 429\u2013446 (2024)","journal-title":"Appl. Numer. Optim."},{"issue":"1","key":"2661_CR31","doi-asserted-by":"publisher","first-page":"48","DOI":"10.1007\/s40314-023-02555-9","volume":"43","author":"O.T. Mewomo","year":"2024","unstructured":"Mewomo, O.T., Ogwo, G.N., Alakoya, T.O.: Strong convergent algorithm for finding minimum-norm solutions of quasimonotone variational inequalities with fixed point constraint and application. Comput. Appl. Math. 43(1), 48 (2024)","journal-title":"Comput. Appl. Math."},{"key":"2661_CR32","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-48311-5","volume-title":"Convex Analysis and Monotone Operator Theory in Hilbert Spaces","author":"H.H. Bauschke","year":"2017","unstructured":"Bauschke, H.H., Combettes, P.L.: Convex analysis and monotone operator theory in Hilbert spaces. Springer, Cham (2017)"},{"key":"2661_CR33","volume-title":"Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mapping","author":"K. Goebel","year":"1984","unstructured":"Goebel, K., Reich, S.: Uniform convexity, hyperbolic geometry and nonexpansive mapping. Marcet Dekker, Inc, New York (1984)"},{"issue":"2","key":"2661_CR34","first-page":"742","volume":"75","author":"S. Saejung","year":"2012","unstructured":"Saejung, S., Yotkaew, P.: Approximation of zeros of inverse strongly monotone operators in banach spaces. Numer. Math. 75(2), 742\u2013750 (2012)","journal-title":"Numer. Math."},{"issue":"1","key":"2661_CR35","doi-asserted-by":"publisher","first-page":"391","DOI":"10.24193\/fpt-ro.2022.1.25","volume":"23","author":"B. Tan","year":"2022","unstructured":"Tan, B., Petru\u015fel, A., Qin, X., Yao, J.C.: Global and linear convergence of alternated inertial single projection algorithms for pseudo-monotone variational inequalities. Fixed Point Theory 23(1), 391\u2013425 (2022)","journal-title":"Fixed Point Theory"},{"key":"2661_CR36","doi-asserted-by":"publisher","first-page":"114739","DOI":"10.1016\/j.cam.2022.114739","volume":"418","author":"D.V. Thong","year":"2023","unstructured":"Thong, D.V., Liu, L.L., Dong, Q.L., Luong, V.L., Tuan, P.A.: Fast relaxed inertial Tseng\u2019s method-based algorithm for solving variational inequality and fixed point problems in Hilbert spaces. J. Comput. Appl. Math. 418, 114739 (2023)","journal-title":"J. Comput. Appl. Math."}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-025-02661-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12190-025-02661-3","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-025-02661-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T16:47:00Z","timestamp":1767890820000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12190-025-02661-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,12,8]]},"references-count":36,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,1]]}},"alternative-id":["2661"],"URL":"https:\/\/doi.org\/10.1007\/s12190-025-02661-3","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"value":"1598-5865","type":"print"},{"value":"1865-2085","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,12,8]]},"assertion":[{"value":"10 May 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 August 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 October 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 December 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"Not applicable.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethics approval and consent to participate"}},{"value":"The consent of all authors has been obtained for the publication of identifiable details in this journal and article.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Consent for publication"}},{"value":"The authors declare that they have no conflict of interest.","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"39"}}