{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T10:17:56Z","timestamp":1774347476264,"version":"3.50.1"},"reference-count":25,"publisher":"Springer Science and Business Media LLC","issue":"8","license":[{"start":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T00:00:00Z","timestamp":1774310400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T00:00:00Z","timestamp":1774310400000},"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":[[2026,9]]},"DOI":"10.1007\/s40314-026-03725-1","type":"journal-article","created":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T06:40:15Z","timestamp":1774334415000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Adaptive relaxation algorithms for split equality problems via projections onto intersections of half-spaces"],"prefix":"10.1007","volume":"45","author":[{"given":"Yi","family":"Zhang","sequence":"first","affiliation":[]},{"given":"Haichuan","family":"Ma","sequence":"additional","affiliation":[]},{"given":"Luoyi","family":"Shi","sequence":"additional","affiliation":[]},{"given":"Yasong","family":"Chen","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2026,3,24]]},"reference":[{"key":"3725_CR1","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4419-9467-7","volume-title":"Convex analysis and monotone operator theory in Hilbert spaces","author":"HH Bauschke","year":"2011","unstructured":"Bauschke HH, Combettes PL (2011) Convex analysis and monotone operator theory in Hilbert spaces. Springer, New York"},{"issue":"8","key":"3725_CR2","first-page":"1485","volume":"18","author":"CL Byrne","year":"2017","unstructured":"Byrne CL, Moudafi A (2017) Extensions of the CQ algorithm for the split feasibility and split equality problems. J Nonlinear Convex Anal 18(8):1485\u20131496","journal-title":"J Nonlinear Convex Anal"},{"issue":"2\u20134","key":"3725_CR3","doi-asserted-by":"publisher","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 a product space. Numer Algorithms 8(2\u20134):221\u2013239","journal-title":"Numer Algorithms"},{"key":"3725_CR4","first-page":"1","volume":"1","author":"S Chang","year":"2014","unstructured":"Chang S (2014) Agarwal RP (2014) Strong convergence theorems of general split equality problems for quasi-nonexpansive mappings. J Inequal Appl 1:1\u201314","journal-title":"J Inequal Appl"},{"issue":"9","key":"3725_CR5","doi-asserted-by":"publisher","first-page":"1887","DOI":"10.1080\/02331934.2014.895897","volume":"64","author":"QL Dong","year":"2015","unstructured":"Dong QL, He S, Zhao J (2015) Solving the split equality problem without prior knowledge of operator norms. Optimization 64(9):1887\u20131906","journal-title":"Optimization"},{"issue":"1","key":"3725_CR6","first-page":"53","volume":"22","author":"QL Dong","year":"2021","unstructured":"Dong QL, Peng Y, Yao Y (2021) Alternated inertial projection methods for the split equality problem. J Nonlinear Convex Anal 22(1):53\u201367","journal-title":"J Nonlinear Convex Anal"},{"issue":"1","key":"3725_CR7","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1287\/opre.31.1.101","volume":"31","author":"M Fukushima","year":"1983","unstructured":"Fukushima M (1983) An outer approximation algorithm for solving general convex programs. Oper Res 31(1):101\u2013113","journal-title":"Oper Res"},{"issue":"1","key":"3725_CR8","doi-asserted-by":"publisher","first-page":"58","DOI":"10.1007\/BF01589441","volume":"35","author":"M Fukushima","year":"1986","unstructured":"Fukushima M (1986) A relaxed projection method for variational inequalities. Math Program 35(1):58\u201370","journal-title":"Math Program"},{"key":"3725_CR9","first-page":"1","volume":"1","author":"S He","year":"2013","unstructured":"He S (2013) Yang C (2013) Solving the variational inequality problem defined on intersection of finite level sets. Abstr Appl Anal 1:1\u20138","journal-title":"Abstr Appl Anal"},{"issue":"1","key":"3725_CR10","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s40314-019-1014-2","volume":"39","author":"LO Jolaoso","year":"2020","unstructured":"Jolaoso LO, Taiwo A, Alakoya TO, Mewomo OT (2020) A unified algorithm for solving variational inequality and fixed point problems with application to the split equality problem. Comput Appl Math 39(1):1\u201328","journal-title":"Comput Appl Math"},{"issue":"8","key":"3725_CR11","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s40314-023-02486-5","volume":"42","author":"T Ling","year":"2023","unstructured":"Ling T, Tong X, Shi LY (2023) Two general splitting methods with alternated inertia for solving split equality problem in Hilbert spaces. Comput Appl Math 42(8):1\u201325","journal-title":"Comput Appl Math"},{"issue":"1","key":"3725_CR12","doi-asserted-by":"publisher","first-page":"117","DOI":"10.1016\/j.na.2012.11.013","volume":"79","author":"A Moudafi","year":"2013","unstructured":"Moudafi A (2013) A relaxed alternating CQ-algorithm for convex feasibility problems. Nonlinear Anal 79(1):117\u2013121","journal-title":"Nonlinear Anal"},{"issue":"4","key":"3725_CR13","first-page":"809","volume":"15","author":"A Moudafi","year":"2014","unstructured":"Moudafi A (2014) Alternating CQ-algorithms for convex feasibility and split fixed-point problems. J Nonlinear Convex Anal 15(4):809\u2013818","journal-title":"J Nonlinear Convex Anal"},{"issue":"15","key":"3725_CR14","doi-asserted-by":"publisher","first-page":"4423","DOI":"10.1080\/02331934.2021.1945053","volume":"71","author":"S Reich","year":"2022","unstructured":"Reich S, Tuyen TM (2022) A new approach to solving split equality problems in Hilbert spaces. Optimization 71(15):4423\u20134445","journal-title":"Optimization"},{"key":"3725_CR15","first-page":"1","volume":"1","author":"LY Shi","year":"2014","unstructured":"Shi LY, Chen R (2014) Wu Y (2014) Strong convergence of iterative algorithms for the split equality problem. J Inequal Appl 1:1\u201319","journal-title":"J Inequal Appl"},{"issue":"12","key":"3725_CR16","doi-asserted-by":"publisher","first-page":"2347","DOI":"10.1080\/02331934.2018.1545124","volume":"67","author":"LY Shi","year":"2018","unstructured":"Shi LY, Ansari QH, Yao JC, Wen CF (2018) Linear convergence of gradient projection algorithm for split equality problems. Optimization 67(12):2347\u20132358","journal-title":"Optimization"},{"key":"3725_CR17","first-page":"1","volume":"1","author":"T Tian","year":"2019","unstructured":"Tian T, Shi LY (2019) Chen R (2019) Linear convergence of the relaxed gradient projection algorithm for solving the split equality problems in Hilbert spaces. J Inequal Appl 1:1\u201312","journal-title":"J Inequal Appl"},{"issue":"1","key":"3725_CR18","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s40840-022-01443-5","volume":"46","author":"TM Tuyen","year":"2023","unstructured":"Tuyen TM (2023) Regularization methods for the split equality problems in Hilbert spaces. Bull Malays Math Sci Soc 46(1):1\u201320","journal-title":"Bull Malays Math Sci Soc"},{"issue":"11","key":"3725_CR19","doi-asserted-by":"publisher","first-page":"2321","DOI":"10.1080\/02331934.2014.967237","volume":"64","author":"PT Vuong","year":"2015","unstructured":"Vuong PT, Strodiot JJ, Nguyen VH (2015) A gradient projection method for solving split equality and split feasibility problems in Hilbert spaces. Optimization 64(11):2321\u20132341","journal-title":"Optimization"},{"issue":"1","key":"3725_CR20","first-page":"436","volume":"12","author":"YH Wang","year":"2022","unstructured":"Wang YH, Li XT, Jiang BN (2022) Two new inertial relaxed gradient CQ algorithms on the split equality problem. J Appl Anal Comput 12(1):436\u2013454","journal-title":"J Appl Anal Comput"},{"issue":"4","key":"3725_CR21","doi-asserted-by":"publisher","first-page":"1261","DOI":"10.1088\/0266-5611\/20\/4\/014","volume":"20","author":"Q Yang","year":"2004","unstructured":"Yang Q (2004) The relaxed CQ algorithm for solving the split feasibility problem. Inverse Probl 20(4):1261\u20131266","journal-title":"Inverse Probl"},{"issue":"5","key":"3725_CR22","doi-asserted-by":"publisher","first-page":"1417","DOI":"10.1080\/02331934.2022.2158036","volume":"73","author":"H Yu","year":"2024","unstructured":"Yu H, Wang F (2024) A new relaxed method for the split feasibility problem in Hilbert spaces. Optimization 73(5):1417\u20131432","journal-title":"Optimization"},{"issue":"2","key":"3725_CR23","doi-asserted-by":"publisher","first-page":"34","DOI":"10.1007\/s41980-024-00961-1","volume":"51","author":"W Zhan","year":"2025","unstructured":"Zhan W, Yu H (2025) A novel relaxed method for the split feasibility problem in Hilbert spaces. Bull Iran Math Soc 51(2):34","journal-title":"Bull Iran Math Soc"},{"issue":"12","key":"3725_CR24","doi-asserted-by":"publisher","first-page":"2619","DOI":"10.1080\/02331934.2014.883515","volume":"64","author":"J Zhao","year":"2015","unstructured":"Zhao J (2015) Solving split equality fixed-point problem of quasi-nonexpansive mappings without prior knowledge of operators norms. Optimization 64(12):2619\u20132630","journal-title":"Optimization"},{"issue":"12","key":"3725_CR25","doi-asserted-by":"publisher","first-page":"2925","DOI":"10.1080\/02331934.2022.2081082","volume":"72","author":"J Zhao","year":"2023","unstructured":"Zhao J, Li Y, Wang X (2023) An accelerate algorithm for the split equality common fixed-point problem of directed operators. Optimization 72(12):2925\u20132950","journal-title":"Optimization"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-026-03725-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-026-03725-1","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-026-03725-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T06:40:17Z","timestamp":1774334417000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-026-03725-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,3,24]]},"references-count":25,"journal-issue":{"issue":"8","published-print":{"date-parts":[[2026,9]]}},"alternative-id":["3725"],"URL":"https:\/\/doi.org\/10.1007\/s40314-026-03725-1","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,3,24]]},"assertion":[{"value":"26 June 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 February 2026","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 March 2026","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 March 2026","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"324"}}