{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T09:51:56Z","timestamp":1775037116389,"version":"3.50.1"},"reference-count":21,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2025,11,8]],"date-time":"2025-11-08T00:00:00Z","timestamp":1762560000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,11,8]],"date-time":"2025-11-08T00:00:00Z","timestamp":1762560000000},"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 Optim Theory Appl"],"published-print":{"date-parts":[[2026,1]]},"DOI":"10.1007\/s10957-025-02872-5","type":"journal-article","created":{"date-parts":[[2025,11,8]],"date-time":"2025-11-08T06:53:39Z","timestamp":1762584819000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["CQ-algorithms for the related problems of split equality problem in Hilbert spaces"],"prefix":"10.1007","volume":"208","author":[{"given":"Yu","family":"Cao","sequence":"first","affiliation":[]},{"given":"Yishuo","family":"Peng","sequence":"additional","affiliation":[]},{"given":"Yasong","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Luoyi","family":"Shi","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,11,8]]},"reference":[{"issue":"3","key":"2872_CR1","first-page":"485","volume":"15","author":"H Attouch","year":"2008","unstructured":"Attouch, H., Bolte, J., Redont, P., Soubeyran, A.: Alternating proximal algorithms for weakly coupled convex minimization problems. applications to dynamical games and PDE\u2019s. J. Convex Anal. 15(3), 485\u2013506 (2008)","journal-title":"J. Convex Anal."},{"key":"2872_CR2","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, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, Berlin (2011)"},{"issue":"1","key":"2872_CR3","first-page":"135","volume":"6","author":"Y Cao","year":"2024","unstructured":"Cao, Y., Peng, Y.S., Shi, L.S., Chen, Y.S.: Modified Relaxed CQ-Algorithms for a Split Equality Problem in Hilbert Spaces. J. Appl. Numer. Optim. 6(1), 135\u2013151 (2024)","journal-title":"J. Appl. Numer. Optim."},{"issue":"10","key":"2872_CR4","doi-asserted-by":"publisher","first-page":"2353","DOI":"10.1088\/0031-9155\/51\/10\/001","volume":"51","author":"Y Censor","year":"2006","unstructured":"Censor, Y., Bortfeld, T., Martin, B., Trofimov, A.: A unified approach for inversion problems in intensity-modulated radiation therapy. Phys. Med. Biol. 51(10), 2353\u20132365 (2006)","journal-title":"Phys. Med. Biol."},{"issue":"9","key":"2872_CR5","doi-asserted-by":"publisher","first-page":"1887","DOI":"10.1080\/02331934.2014.895897","volume":"64","author":"QL Dong","year":"2015","unstructured":"Dong, Q.L., He, S.N., Zhao, J.: Solving the split equality problem without prior knowledge of operator norms. Optimization 64(9), 1887\u20131906 (2015)","journal-title":"Optimization"},{"issue":"3","key":"2872_CR6","first-page":"591","volume":"23","author":"QL Dong","year":"2022","unstructured":"Dong, Q.L., Liu, L.L., Yao, Y.H.: Self-adaptive projection and contraction methods with alternated inertial terms for solving the split feasibility problem. J. Nonlinear Convex Anal. 23(3), 591\u2013605 (2022)","journal-title":"J. Nonlinear Convex Anal."},{"issue":"4","key":"2872_CR7","doi-asserted-by":"publisher","first-page":"3067","DOI":"10.1007\/s12190-023-01875-7","volume":"69","author":"T Ling","year":"2023","unstructured":"Ling, T., Tong, X.L., Shi, L.Y.: Modified relaxed CQ methods for the split feasibility problems in Hilbert spaces with applications. J. Appl. Math. Comput. 69(4), 3067\u20133094 (2023)","journal-title":"J. Appl. Math. Comput."},{"issue":"8","key":"2872_CR8","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/28\/8\/085004","volume":"28","author":"G Lopez","year":"2012","unstructured":"Lopez, G., Martin-Marquez, V., Wang, F.H., Xu, H.K.: Solving the split feasibility problem without prior knowledge of matrix norms. Inverse Probl. 28(8), 085004 (2012)","journal-title":"Inverse Probl."},{"issue":"4","key":"2872_CR9","first-page":"809","volume":"15","author":"A Moudafi","year":"2014","unstructured":"Moudafi, A.: Alternating CQ-algorithms for convex feasibility and split fixed-point problems. J. Nonlinear Convex Anal. 15(4), 809\u2013818 (2014)","journal-title":"J. Nonlinear Convex Anal."},{"key":"2872_CR10","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.: A relaxed alternating CQ-algorithm for convex feasibility problems. Nonlinear Anal. -Theory Methods Appl. 79, 117\u2013121 (2013)","journal-title":"Nonlinear Anal. -Theory Methods Appl."},{"issue":"2","key":"2872_CR11","doi-asserted-by":"publisher","first-page":"87","DOI":"10.4169\/amer.math.monthly.119.02.087","volume":"119","author":"BS Mordukhovich","year":"2012","unstructured":"Mordukhovich, B.S., Nam, N.M., Salinas, J.: Solving a generalized Heron problem by means of convex analysis. Am. Math. Monthly. 119(2), 87\u201399 (2012)","journal-title":"Am. Math. Monthly."},{"issue":"3","key":"2872_CR12","doi-asserted-by":"publisher","DOI":"10.1007\/s40314-024-02618-5","volume":"43","author":"YS Peng","year":"2024","unstructured":"Peng, Y.S., Cao, Y., Shi, L.Y., Chen, Y.S.: Convergence analysis for solving the split equality equilibrium problem in Hilbert spaces. Comput. Appl. Math. 43(3), 111 (2024)","journal-title":"Comput. Appl. Math."},{"issue":"2","key":"2872_CR13","volume":"115","author":"Y Shehu","year":"2021","unstructured":"Shehu, Y., Dong, Q.L., Liu, L.L.: Global and linear convergence of alternated inertial methods for split feasibility problems. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A-Math. 115(2), 53 (2021)","journal-title":"Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A-Math."},{"issue":"1","key":"2872_CR14","doi-asserted-by":"publisher","DOI":"10.1186\/1029-242X-2014-478","volume":"2014","author":"LY Shi","year":"2015","unstructured":"Shi, L.Y., Chen, R.D., Wu, Y.J.: Strong convergence of iterative algorithms for the split equality problem. J. Inequal. Appl. 2014(1), 478 (2015)","journal-title":"J. Inequal. Appl."},{"issue":"1","key":"2872_CR15","doi-asserted-by":"publisher","first-page":"78","DOI":"10.1007\/s10957-022-02113-z","volume":"196","author":"S Reich","year":"2023","unstructured":"Reich, S., Tuyen, T.M.: The Generalized Fermat-Torricelli problem in Hilbert spaces. J. Optim. Theory Appl. 196(1), 78\u201397 (2023)","journal-title":"J. Optim. Theory Appl."},{"issue":"2","key":"2872_CR16","doi-asserted-by":"publisher","first-page":"773","DOI":"10.1007\/s11075-023-01589-8","volume":"95","author":"B Tan","year":"2024","unstructured":"Tan, B., Qin, X.L., Wang, X.F.: Alternated inertial algorithms for split feasibility problems. Numerical Algorithms. 95(2), 773\u2013812 (2024)","journal-title":"Numerical Algorithms."},{"issue":"2","key":"2872_CR17","doi-asserted-by":"publisher","first-page":"1441","DOI":"10.1007\/s12190-024-02008-4","volume":"70","author":"XL Tong","year":"2024","unstructured":"Tong, X.L., Ling, T., Shi, L.Y.: Self-adaptive relaxed CQ algorithms for solving split feasibility problem with multiple output sets. J. Appl. Math. Comupt. 70(2), 1441\u20131469 (2024)","journal-title":"J. Appl. Math. Comupt."},{"issue":"10","key":"2872_CR18","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/26\/10\/105018","volume":"26","author":"HK Xu","year":"2010","unstructured":"Xu, H.K.: Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Problems. 26(10), 105018 (2010)","journal-title":"Inverse Problems."},{"issue":"3","key":"2872_CR19","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.W.: Strong convergence result for solving monotone variational inequalities in Hilbert space. Numerical Algorithms. 80(3), 741\u2013752 (2019)","journal-title":"Numerical Algorithms."},{"issue":"4","key":"2872_CR20","doi-asserted-by":"publisher","first-page":"1261","DOI":"10.1088\/0266-5611\/20\/4\/014","volume":"20","author":"QZ Yang","year":"2004","unstructured":"Yang, Q.Z.: The relaxed CQ algorithm solving the split feasibility problem. Inverse Prob. 20(4), 1261\u20131266 (2004)","journal-title":"Inverse Prob."},{"issue":"5","key":"2872_CR21","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.H.: A new relaxed method for the split feasibility problem in Hilbert spaces. Optimization 73(5), 1417\u20131432 (2024)","journal-title":"Optimization"}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-025-02872-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10957-025-02872-5","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-025-02872-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T07:23:38Z","timestamp":1775028218000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10957-025-02872-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,8]]},"references-count":21,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,1]]}},"alternative-id":["2872"],"URL":"https:\/\/doi.org\/10.1007\/s10957-025-02872-5","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"value":"0022-3239","type":"print"},{"value":"1573-2878","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,11,8]]},"assertion":[{"value":"29 September 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 October 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 November 2025","order":3,"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":"Ethical Approval"}},{"value":"The authors declare no competing interests.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"50"}}