{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,7]],"date-time":"2026-02-07T19:17:50Z","timestamp":1770491870380,"version":"3.49.0"},"reference-count":43,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2020,6,20]],"date-time":"2020-06-20T00:00:00Z","timestamp":1592611200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,6,20]],"date-time":"2020-06-20T00:00:00Z","timestamp":1592611200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/100011512","name":"National Research Foundation","doi-asserted-by":"publisher","award":["120784"],"award-info":[{"award-number":["120784"]}],"id":[{"id":"10.13039\/100011512","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2020,9]]},"DOI":"10.1007\/s40314-020-01221-8","type":"journal-article","created":{"date-parts":[[2020,6,20]],"date-time":"2020-06-20T10:03:02Z","timestamp":1592647382000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["A modified contraction method for solving certain class of split monotone variational inclusion problems with application"],"prefix":"10.1007","volume":"39","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8262-8605","authenticated-orcid":false,"given":"C.","family":"Izuchukwu","sequence":"first","affiliation":[]},{"given":"J. N.","family":"Ezeora","sequence":"additional","affiliation":[]},{"given":"J.","family":"Martinez-Moreno","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,6,20]]},"reference":[{"key":"1221_CR1","doi-asserted-by":"publisher","first-page":"1825","DOI":"10.1016\/j.camwa.2004.10.044","volume":"29","author":"R Ahmad","year":"2005","unstructured":"Ahmad R, Ansari QH, Irfan SS (2005) Generalized variational inclusions and generalized resolvent equations in Banach spaces. Comput Math Appl 29:1825\u20131835","journal-title":"Comput Math Appl"},{"key":"1221_CR2","doi-asserted-by":"publisher","first-page":"137","DOI":"10.1007\/BF03007664","volume":"26","author":"JB Baillon","year":"1997","unstructured":"Baillon JB, Haddad G (1997) Quelques proprietes des operateurs angle-bornes et cycliquement monotones. Isr J Math 26:137\u2013150","journal-title":"Isr J Math"},{"issue":"1","key":"1221_CR3","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1137\/080716542","volume":"2","author":"A Beck","year":"2009","unstructured":"Beck A, Teboulle M (2009) A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J Imaging Sci 2(1):183\u2013202","journal-title":"SIAM J Imaging Sci"},{"key":"1221_CR4","volume-title":"Operateur maximaux monotones, in mathematics studies","author":"H Br\u00e9ziz","year":"1973","unstructured":"Br\u00e9ziz H (1973) Operateur maximaux monotones, in mathematics studies, vol 5. North-Holland, Amsterdam"},{"key":"1221_CR5","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1088\/0266-5611\/20\/1\/006","volume":"20","author":"C Byrne","year":"2004","unstructured":"Byrne C (2004) A unified treatment for some iterative algorithms in signal processing and image reconstruction. Inverse Probl 20:103\u2013120","journal-title":"Inverse Probl"},{"key":"1221_CR6","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 product space. Numer Algorithm 8:221\u2013239","journal-title":"Numer Algorithm"},{"key":"1221_CR7","doi-asserted-by":"publisher","first-page":"2071","DOI":"10.1088\/0266-5611\/21\/6\/017","volume":"21","author":"Y Censor","year":"2005","unstructured":"Censor Y, Elfving T, Kopf N, Bortfield T (2005) The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Probl 21:2071\u20132084","journal-title":"Inverse Probl"},{"key":"1221_CR8","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 (2006) A unified approach for inversion problems in intensity modulated radiation therapy. Phys Med Biol 51:2353\u20132365","journal-title":"Phys Med Biol"},{"key":"1221_CR9","doi-asserted-by":"publisher","first-page":"301","DOI":"10.1007\/s11075-011-9490-5","volume":"59","author":"Y Censor","year":"2012","unstructured":"Censor Y, Gibali A, Reich S (2012) Algorithms for the split variational inequality problem. Numer Algorithm 59:301\u2013323","journal-title":"Numer Algorithm"},{"issue":"5","key":"1221_CR10","doi-asserted-by":"publisher","first-page":"5750","DOI":"10.1007\/s40314-018-0661-z","volume":"37","author":"W Cholamjiak","year":"2017","unstructured":"Cholamjiak W, Pholasa N, Suantai S (2017) A modified inertial shrinking projection method for solving inclusion problems and quasi-nonexpansive multivalued mappings. Comput Appl Math 37(5):5750\u20135774","journal-title":"Comput Appl Math"},{"key":"1221_CR11","doi-asserted-by":"publisher","DOI":"10.1007\/s11784-018-0526-5","author":"W Cholamjiak","year":"2018","unstructured":"Cholamjiak W, Cholamjiak P, Suantai S (2018) An inertial forward\u2013backward splitting method for solving inclusion problems in Hilbert spaces. J Fixed Point Theory Appl. https:\/\/doi.org\/10.1007\/s11784-018-0526-5","journal-title":"J Fixed Point Theory Appl"},{"key":"1221_CR12","doi-asserted-by":"publisher","first-page":"3097","DOI":"10.1007\/s11784-017-0472-7","volume":"19","author":"QL Dong","year":"2017","unstructured":"Dong QL, Jiang D, Cholamjiak P, Shehu Y (2017) A strong convergence result involving an inertial forward\u2013backward algorithm for monotone inclusions. J Fixed Point Theory Appl 19:3097\u20133118","journal-title":"J Fixed Point Theory Appl"},{"issue":"8","key":"1221_CR13","doi-asserted-by":"publisher","first-page":"2921","DOI":"10.2298\/FIL1808921E","volume":"32","author":"JN Ezeora","year":"2019","unstructured":"Ezeora JN, Izuchukwu C (2019) Iterative approximation of solution of split variational inclusion problems. Filomat 32(8):2921\u20132932","journal-title":"Filomat"},{"key":"1221_CR14","first-page":"41","volume":"6","author":"A Gibali","year":"2015","unstructured":"Gibali A (2015) A new non-Lipschitzian projection method for solving variational inequalities in Euclidean spaces. J Nonlinear Anal Optim 6:41\u201351","journal-title":"J Nonlinear Anal Optim"},{"issue":"2","key":"1221_CR15","first-page":"243","volume":"2","author":"A Gibali","year":"2017","unstructured":"Gibali A (2017) A new split inverse problem and application to least intensity feasible solutions. Pure Appl Funct Anal 2(2):243\u2013258","journal-title":"Pure Appl Funct Anal"},{"key":"1221_CR16","doi-asserted-by":"publisher","DOI":"10.1007\/s13324-019-00330-w","author":"A Gibali","year":"2019","unstructured":"Gibali A, Thong DV, Tuan PA (2019) Two simple projection-type methods for solving variational inequalities. Anal Math Phys. https:\/\/doi.org\/10.1007\/s13324-019-00330-w","journal-title":"Anal Math Phys"},{"key":"1221_CR17","doi-asserted-by":"publisher","first-page":"2802","DOI":"10.1137\/09076773X","volume":"31","author":"JM Hendrickx","year":"2010","unstructured":"Hendrickx JM, Olshevsky A (2010) Matrix $$P$$-norms are NP-hard to approximate if $$P\\ne 1, 2,\\infty $$. SIAM J Matrix Anal Appl 31:2802\u20132812","journal-title":"SIAM J Matrix Anal Appl"},{"key":"1221_CR18","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1007\/s10589-016-9857-6","volume":"66","author":"DV Hieu","year":"2017","unstructured":"Hieu DV, Anh PK, Muu LD (2017) Modified hybrid projection methods for finding common solutions to variational inequality problems. Comput Optim Appl 66:75\u201396","journal-title":"Comput Optim Appl"},{"issue":"4","key":"1221_CR19","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s11784-018-0632-4","volume":"20","author":"C Izuchukwu","year":"2018","unstructured":"Izuchukwu C, Okeke CC, Isiogugu FO (2018) A viscosity iterative technique for split variational inclusion and fixed point problems between a Hilbert space and a Banach space. J Fixed Point Theory Appl 20(4):1\u201325","journal-title":"J Fixed Point Theory Appl"},{"key":"1221_CR20","doi-asserted-by":"publisher","unstructured":"Izuchukwu C, Isiogugu FO, Okeke CC (2019) A new viscosity-type iteration for a finite family of split variational inclusion and fixed point problems between Hilbert and Banach spaces. J Inequal Appl. https:\/\/doi.org\/10.1186\/s13660-019-2201-9","DOI":"10.1186\/s13660-019-2201-9"},{"issue":"2","key":"1221_CR21","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1080\/10556788.2017.1359265","volume":"34","author":"P Jailoka","year":"2019","unstructured":"Jailoka P, Suantai S (2019) Split common fixed point and null point problems for demicontractive operators in Hilbert spaces. Optim Methods Softw 34(2):1\u201316","journal-title":"Optim Methods Softw"},{"key":"1221_CR22","doi-asserted-by":"publisher","DOI":"10.1007\/s13398-018-0504-1","author":"SA Khan","year":"2018","unstructured":"Khan SA, Suantai S, Cholamjiak W (2018) Shrinking projection methods involving inertial forward\u2013backward splitting methods for inclusion problems. RACSAM. https:\/\/doi.org\/10.1007\/s13398-018-0504-1","journal-title":"RACSAM"},{"key":"1221_CR23","doi-asserted-by":"publisher","first-page":"154","DOI":"10.1007\/978-3-642-59073-3_11","volume":"452","author":"B Lemaire","year":"1997","unstructured":"Lemaire B (1997) Which fixed point does the iteration method select? Recent Adv Optim 452:154\u2013157","journal-title":"Recent Adv Optim"},{"key":"1221_CR24","doi-asserted-by":"publisher","first-page":"275","DOI":"10.1007\/s10957-011-9814-6","volume":"150","author":"A Moudafi","year":"2011","unstructured":"Moudafi A (2011) Split monotone variational inclusions. J Optim Theory Appl 150:275\u2013283","journal-title":"J Optim Theory Appl"},{"key":"1221_CR25","doi-asserted-by":"publisher","first-page":"1168","DOI":"10.1080\/10556788.2018.1472259","volume":"34","author":"CC Okeke","year":"2019","unstructured":"Okeke CC, Izuchukwu C (2019) A strong convergence theorem for monotone inclusion and minimization problems in complete CAT(0) spaces. Optim Methods Softw 34:1168\u20131183","journal-title":"Optim Methods Softw"},{"key":"1221_CR26","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1090\/S0002-9947-1970-0282272-5","volume":"149","author":"RT Rockafellar","year":"1970","unstructured":"Rockafellar RT (1970) On the maximality of sums of nonlinear monotone operators. Trans Am Math Soc 149:75\u2013288","journal-title":"Trans Am Math Soc"},{"key":"1221_CR27","doi-asserted-by":"publisher","first-page":"742","DOI":"10.1016\/j.na.2011.09.005","volume":"75","author":"S Saejung","year":"2012","unstructured":"Saejung S, Yotkaew P (2012) Approximation of zeros of inverse strongly monotone operators in Banach spaces. Nonlinear Anal 75:742\u2013750","journal-title":"Nonlinear Anal"},{"issue":"1","key":"1221_CR28","doi-asserted-by":"publisher","first-page":"206","DOI":"10.1007\/s11587-016-0262-3","volume":"65","author":"P Senakka","year":"2016","unstructured":"Senakka P, Cholamjiak P (2016) Approximation method for solving fixed point problem of Bregman strongly nonexpansive mappings in reflexive Banach spaces. Ricerche di Matematica 65(1):206\u2013220","journal-title":"Ricerche di Matematica"},{"key":"1221_CR29","doi-asserted-by":"publisher","unstructured":"Shehu Y, Cholamjiak P (2019) Iterative method with inertial for variational inequalities in Hilbert spaces. Calcolo. https:\/\/doi.org\/10.1007\/s10092-018-0300-5","DOI":"10.1007\/s10092-018-0300-5"},{"key":"1221_CR30","doi-asserted-by":"publisher","unstructured":"Shehu Y, Iyiola OS, Ogbuisi FU (2019a) Numer Algorithms. Iterative method with inertial terms for nonexpansive mappings: applications to compressed sensing. https:\/\/doi.org\/10.1007\/s11075-019-00727-5","DOI":"10.1007\/s11075-019-00727-5"},{"key":"1221_CR31","doi-asserted-by":"publisher","unstructured":"Shehu Y, Li XH, Dong QL (2019b) An efficient projection-type method for monotone variational inequalities in Hilbert spaces. Numer Algorithms. https:\/\/doi.org\/10.1007\/s11075-019-00758-y","DOI":"10.1007\/s11075-019-00758-y"},{"issue":"4","key":"1221_CR32","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1080\/10556788.2018.1472257","volume":"34","author":"S Suantai","year":"2019","unstructured":"Suantai S, Shehu Y, Cholamjiak P (2019) Nonlinear iterative methods for solving the split common null point problem in Banach spaces. Optim Methods Softw 34(4):1\u201322","journal-title":"Optim Methods Softw"},{"issue":"2","key":"1221_CR33","doi-asserted-by":"publisher","first-page":"205","DOI":"10.1007\/s11228-014-0285-4","volume":"23","author":"W Takahashi","year":"2015","unstructured":"Takahashi W, Xu HK, Yao JC (2015) Iterative methods for generalized split feasibility problems in Hilbert spaces. Set Valued Var Anal 23(2):205\u2013221","journal-title":"Set Valued Var Anal"},{"key":"1221_CR34","doi-asserted-by":"publisher","DOI":"10.1007\/s40314-019-0855-z","author":"DV Thong","year":"2019","unstructured":"Thong DV, Cholamjiak P (2019) Strong convergence of a forward-backward splitting method with a new step size for solving monotone inclusions. Comput Appl Math. https:\/\/doi.org\/10.1007\/s40314-019-0855-z","journal-title":"Comput Appl Math"},{"key":"1221_CR35","doi-asserted-by":"publisher","DOI":"10.1007\/s11075-017-0412-z","author":"DV Thong","year":"2017","unstructured":"Thong DV, Hieu DV (2017) Weak and strong convergence theorems for variational inequality problems. Numer Algorithm. https:\/\/doi.org\/10.1007\/s11075-017-0412-z","journal-title":"Numer Algorithm"},{"key":"1221_CR36","doi-asserted-by":"publisher","DOI":"10.1007\/s110784-018-0610-x","author":"DV Thong","year":"2018","unstructured":"Thong DV, Hieu DV (2018) New extragradient methods for solving variational inequality problems and fixed point problems. J Fixed Point Theory Appl. https:\/\/doi.org\/10.1007\/s110784-018-0610-x","journal-title":"J Fixed Point Theory Appl"},{"key":"1221_CR37","doi-asserted-by":"publisher","DOI":"10.1007\/s11075-019-00780-0","author":"DV Thong","year":"2019","unstructured":"Thong DV, Shehu Y, Iyiola OS (2019) Weak and strong convergence theorems for solving pseudo-monotone variational inequalities with non-Lipschitz mappings. Numer Algorithms. https:\/\/doi.org\/10.1007\/s11075-019-00780-0","journal-title":"Numer Algorithms"},{"key":"1221_CR38","doi-asserted-by":"publisher","DOI":"10.1186\/s13660-017-1397-9","author":"M Tian","year":"2017","unstructured":"Tian M, Jiang B-N (2017) Weak convergence theorem for a class of split variational inequality problems and applications in Hilbert space. J Inequal Appl. https:\/\/doi.org\/10.1186\/s13660-017-1397-9","journal-title":"J Inequal Appl"},{"key":"1221_CR39","doi-asserted-by":"publisher","first-page":"902","DOI":"10.1080\/01630563.2018.1564763","volume":"40","author":"M Tian","year":"2019","unstructured":"Tian M, Jiang B-N (2019) Viscosity approximation methods for a Class of generalized split feasibility problems with variational inequalities in Hilbert space. Numer Funct Anal Optim 40:902\u2013923","journal-title":"Numer Funct Anal Optim"},{"key":"1221_CR40","doi-asserted-by":"publisher","DOI":"10.1007\/s10957-019-01523-w","author":"TM Tuyen","year":"2019","unstructured":"Tuyen TM, Thuy NT, Trang NM (2019) A strong convergence theorem for a parallel iterative method for solving the split common null point problem in Hilbert Spaces. J Optim Theory Appl. https:\/\/doi.org\/10.1007\/s10957-019-01523-w","journal-title":"J Optim Theory Appl"},{"key":"1221_CR41","doi-asserted-by":"publisher","first-page":"279","DOI":"10.1016\/j.jmaa.2004.04.059","volume":"298","author":"HK Xu","year":"2004","unstructured":"Xu HK (2004) Viscosity approximation methods for nonexpansive mappings. J Math Anal Appl 298:279\u2013291","journal-title":"J Math Anal Appl"},{"key":"1221_CR42","doi-asserted-by":"publisher","first-page":"10518","DOI":"10.1088\/0266-5611\/26\/10\/105018","volume":"26","author":"HK Xu","year":"2010","unstructured":"Xu HK (2010) Iterative methods for split feasibility problem in infinite-dimensional Hilbert space. Inverse Probl. 26:10518","journal-title":"Inverse Probl."},{"issue":"2","key":"1221_CR43","doi-asserted-by":"publisher","first-page":"456","DOI":"10.1016\/j.na.2007.05.032","volume":"69","author":"H Zhou","year":"2008","unstructured":"Zhou H (2008) Convergence theorems of fixed points for $$\\kappa $$-strict pseudo-contractions in Hilbert spaces. Nonlinear Anal 69(2):456\u2013462","journal-title":"Nonlinear Anal"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-020-01221-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-020-01221-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-020-01221-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,6,19]],"date-time":"2021-06-19T23:19:05Z","timestamp":1624144745000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-020-01221-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,6,20]]},"references-count":43,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2020,9]]}},"alternative-id":["1221"],"URL":"https:\/\/doi.org\/10.1007\/s40314-020-01221-8","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,6,20]]},"assertion":[{"value":"22 March 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"31 May 2020","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 June 2020","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 June 2020","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Compliance with ethical standards"}},{"value":"The authors declare that they have no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"188"}}