{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,21]],"date-time":"2026-01-21T07:22:26Z","timestamp":1768980146962,"version":"3.49.0"},"reference-count":57,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2019,5,10]],"date-time":"2019-05-10T00:00:00Z","timestamp":1557446400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We introduce an iterative algorithm which converges strongly to a common element of fixed point sets of nonexpansive mappings and sets of zeros of maximal monotone mappings. Our iterative method is quite general and includes a large number of iterative methods considered in recent literature as special cases. In particular, we apply our algorithm to solve a general system of variational inequalities, convex feasibility problem, zero point problem of inverse strongly monotone and maximal monotone mappings, split common null point problem, split feasibility problem, split monotone variational inclusion problem and split variational inequality problem. Under relaxed conditions on the parameters, we derive some algorithms and strong convergence results to solve these problems. Our results improve and generalize several known results in the recent literature.<\/jats:p>","DOI":"10.3390\/sym11050655","type":"journal-article","created":{"date-parts":[[2019,5,13]],"date-time":"2019-05-13T03:57:07Z","timestamp":1557719827000},"page":"655","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":19,"title":["Iteration Process for Fixed Point Problems and Zeros of Maximal Monotone Operators"],"prefix":"10.3390","volume":"11","author":[{"given":"Ashish","family":"Nandal","sequence":"first","affiliation":[{"name":"Department of Mathematics, Pt NRS Government College, Rohtak 124001, India"}]},{"given":"Renu","family":"Chugh","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Maharshi Dayanand University, Rohtak 124001, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0738-787X","authenticated-orcid":false,"given":"Mihai","family":"Postolache","sequence":"additional","affiliation":[{"name":"Department of General Education, China Medical University, Taichung 40402, Taiwan"},{"name":"Iacob Institute of Mathematical Statistics and Applied Mathematics, Romanian Academy, Gh. Mihoc-C., 050711 Bucharest, Romania"},{"name":"Department of Mathematics and Informatics, University \u201cPolitehnica\u201d of Bucharest, 060042 Bucharest, Romania"}]}],"member":"1968","published-online":{"date-parts":[[2019,5,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Chugh, R., Malik, P., and Kumar, V. (2015). On a new faster implicit fixed point iterative scheme in convex metric spaces. J. Funct. Spaces, 2015.","DOI":"10.1155\/2015\/905834"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"3611","DOI":"10.2298\/FIL1712611K","article-title":"Random iterative algorithms and almost sure stability in Banach spaces","volume":"31","author":"Khan","year":"2017","journal-title":"Filomat"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2303","DOI":"10.2298\/FIL1708303H","article-title":"Jungck-type implicit iterative algorithms with numerical examples","volume":"31","author":"Kumar","year":"2017","journal-title":"Filomat"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1186\/1687-1812-2014-183","article-title":"Algorithms with strong convergence for the split common solution of the feasibility problem and fixed point problem","volume":"2014","author":"Yao","year":"2014","journal-title":"Fixed Point Theory Appl."},{"key":"ref_5","first-page":"875","article-title":"Mann-type iteration method for solving the split common fixed point problem","volume":"18","author":"Yao","year":"2017","journal-title":"J. Nonlinear Convex Anal."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1309","DOI":"10.1080\/02331934.2017.1390747","article-title":"Self-adaptive algorithms for the split problem of the demicontractive operators","volume":"67","author":"Yao","year":"2018","journal-title":"Optimization"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Dadashi, V., and Postolache, M. (2019). Forward-backward splitting algorithm for fixed point problems and zeros of the sum of monotone operators. Arab. J. Math., 1\u201311.","DOI":"10.1007\/s40065-018-0236-2"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"61","DOI":"10.3390\/math7010061","article-title":"An iterative algorithm for solving generalized variational inequalities and fixed points problems","volume":"7","author":"Yao","year":"2019","journal-title":"Mathematics"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Yao, Y., Postolache, M., and Zhu, Z. (2019). Gradient methods with selection technique for the multiple-sets split feasibility problem. Optimization.","DOI":"10.1080\/02331934.2019.1602772"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Yao, Y., Noor, M.A., Liou, Y.C., and Kang, S.M. (2012). Iterative algorithms for generalized variational inequalities. Abstr. Appl. Anal., 1\u201310.","DOI":"10.1155\/2012\/768272"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"887","DOI":"10.1137\/0314056","article-title":"Monotone operators and proximal point algorithm","volume":"14","author":"Rockafellar","year":"1976","journal-title":"SIAM J. Control Optim."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1137\/0329022","article-title":"On the convergence of the proximal point algorithm for convex minimization","volume":"29","year":"1991","journal-title":"SIAM J. Control Optim."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"226","DOI":"10.1006\/jath.2000.3493","article-title":"Approximating solutions of maximal monotone operators in Hilbert spaces","volume":"106","author":"Kamimura","year":"2000","journal-title":"J. Approx. Theory"},{"key":"ref_14","first-page":"69","article-title":"Strong convergence theorems for nonexpansive nonself-mappings and inverse-strongly-monotone mappings","volume":"11","author":"Iiduka","year":"2004","journal-title":"J. Convex Anal."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1080\/02331939608844217","article-title":"Combining the proximal algorithm and Tikhonov regularization","volume":"37","author":"Lehdili","year":"1996","journal-title":"Optimization"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1007\/s11075-011-9490-5","article-title":"Algorithms for the split variational inequality problem","volume":"59","author":"Censor","year":"2012","journal-title":"Numer. Algorithms"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"4596","DOI":"10.1016\/j.na.2012.01.021","article-title":"A von Neumann alternating method for finding common solutions to variational inequalities","volume":"75","author":"Censor","year":"2012","journal-title":"Nonlinear Anal."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1007\/s11228-011-0192-x","article-title":"Common solutions to variational inequalities","volume":"20","author":"Censor","year":"2012","journal-title":"Set-Valued Var. Anal."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"3614","DOI":"10.1109\/TSP.2006.879312","article-title":"Energy based sensor network source localization via projection onto convex sets (POCS)","volume":"54","author":"Blatt","year":"2006","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"607","DOI":"10.1088\/0266-5611\/4\/3\/006","article-title":"On the use of Cimmino\u2019s simultaneous projections method for computing a solution of the inverse problem in radiation therapy treatment planning","volume":"4","author":"Censor","year":"1988","journal-title":"Inverse Probl."},{"key":"ref_21","unstructured":"Herman, G.T. (2009). Fundamentals of Computerized Tomography: Image Reconstruction from Projections, Springer. [2nd ed.]."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1016\/S1076-5670(08)70157-5","article-title":"The convex feasibility problem in image recovery","volume":"95","author":"Combettes","year":"1996","journal-title":"Adv. Imaging Electron Phys."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1007\/BF02142692","article-title":"A multiprojection algorithm using bregman projections in a product space","volume":"8","author":"Censor","year":"1994","journal-title":"Numer. Algorithms"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Bauschke, H.H., and Combettes, P.L. (2011). Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer.","DOI":"10.1007\/978-1-4419-9467-7"},{"key":"ref_25","first-page":"471","article-title":"On a strongly nonexpansive sequence in Hilbert spaces","volume":"8","author":"Aoyama","year":"2007","journal-title":"J. Nonlinear Convex Anal."},{"key":"ref_26","first-page":"459","article-title":"Nonexpansive projections and resolvents of accretive operators in Banach spaces","volume":"3","author":"Bruck","year":"1977","journal-title":"Houston J. Math."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"240","DOI":"10.1112\/S0024610702003332","article-title":"Iterative algorithms for nonlinear operators","volume":"66","author":"Xu","year":"2002","journal-title":"J. Lond. Math. Soc."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1088\/0266-5611\/20\/1\/006","article-title":"A unified treatment of some iterative algorithms in signal processing and image reconstruction","volume":"20","author":"Byrne","year":"2004","journal-title":"Inverse Probl."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Huang, Y.Y., and Hong, C.C. (2013). A unified iterative treatment for solutions of problems of split feasibility and equilibrium in Hilbert spaces. Abstr. Appl. Anal., 613928.","DOI":"10.1155\/2013\/613928"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Goebel, K., and Kirk, W.A. (1990). Topics in Metric Fixed Point Theory, Cambridge University Press.","DOI":"10.1017\/CBO9780511526152"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Barbu, V. (1976). Nonlinear Semigroups and Differential Equations in Banach space, Noordhoff.","DOI":"10.1007\/978-94-010-1537-0"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"899","DOI":"10.1007\/s11228-008-0102-z","article-title":"Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization","volume":"16","author":"Mainge","year":"2008","journal-title":"Set-Valued Anal."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1016\/j.sysconle.2018.10.001","article-title":"Resilient consensus of switched multi-agent systems","volume":"122","author":"Shang","year":"2018","journal-title":"Syst. Control Lett."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1007\/s00186-007-0207-4","article-title":"Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities","volume":"67","author":"Ceng","year":"2008","journal-title":"Math. Methods Oper. Res."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1007\/BF01027691","article-title":"On the convergence of von Neumann\u2019s alternating projection algorithm","volume":"1","author":"Bauschke","year":"1993","journal-title":"Set-Valued Anal."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"498","DOI":"10.1137\/130919052","article-title":"Analysis of the convergence rate for the cyclic projection algorithm applied to basic semi-algebraic convex sets","volume":"24","author":"Borwein","year":"2014","journal-title":"SIAM J. Optim."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0041-5553(67)90113-9","article-title":"The method of projections for finding the common point of convex sets","volume":"7","author":"Gubin","year":"1967","journal-title":"USSR Comput. Math. Math. Phys."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1090\/S0002-9947-1956-0084194-4","article-title":"On the numerical solution of the heat conduction problem in 2 and 3 space variables","volume":"82","author":"Douglas","year":"1956","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1007\/BF01581204","article-title":"On the Douglas\u2013Rachford splitting method and the proximal point algorithm for maximal monotone operators","volume":"55","author":"Eckstein","year":"1992","journal-title":"Math. Program."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"964","DOI":"10.1137\/0716071","article-title":"Splitting algorithms for the sum of two nonlinear operators","volume":"16","author":"Lions","year":"1979","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Arag\u00f3n Artacho, F.J., Censor, Y., and Gibali, A. (2018). The cyclic Douglas-Rachford algorithm with-sets-Douglas- Rachford operators. Optim. Method Softw.","DOI":"10.1080\/10556788.2018.1504049"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jmaa.2014.06.075","article-title":"Linear and strong convergence of algorithms involving averaged nonexpansive operators","volume":"421","author":"Bauschke","year":"2015","journal-title":"J. Math. Anal. Appl."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10957-013-0381-x","article-title":"A cyclic Douglas\u2013Rachford iteration scheme","volume":"160","author":"Borwein","year":"2014","journal-title":"J. Optim. Theory Appl."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10957-013-0381-x","article-title":"Recent results on Douglas-Rachford methods for combinatorial optimization problems","volume":"163","author":"Borwein","year":"2014","journal-title":"J. Optim. Theory Appl."},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Cegielski, A. (2012). Iterative Methods for Fixed Point Problems in Hilbert Spaces, Springer. Lecture Notes in Mathematics.","DOI":"10.1007\/978-3-642-30901-4"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"1689","DOI":"10.1080\/02331934.2017.1359591","article-title":"Weak convergence theorem for zero points of inverse strongly monotone mapping and fixed points of nonexpansive mapping in Hilbert space","volume":"66","author":"Tian","year":"2017","journal-title":"Optimization"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1016\/j.na.2004.07.054","article-title":"The asymptotic behavior of the composition of two resolvents","volume":"60","author":"Bauschke","year":"2005","journal-title":"Nonlinear Anal."},{"key":"ref_48","first-page":"759","article-title":"The split common null point problem","volume":"13","author":"Byrne","year":"2012","journal-title":"J. Nonlinear Convex Anal."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"1481","DOI":"10.1007\/s11784-016-0323-y","article-title":"Viscosity approximation methods for solving fixed point problems and split common fixed point problems","volume":"19","author":"Thong","year":"2017","journal-title":"J. Fixed Point Theory Appl."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1088\/0266-5611\/18\/2\/310","article-title":"Iterative oblique projection onto convex sets and the split feasibility problem","volume":"18","author":"Byrne","year":"2002","journal-title":"Inverse Probl."},{"key":"ref_51","first-page":"497","article-title":"Characterization of the sub differentials of convex functions, Pac","volume":"17","author":"Rockafellar","year":"1966","journal-title":"J. Math."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"2021","DOI":"10.1088\/0266-5611\/22\/6\/007","article-title":"A variable Krasnosel\u2019skii-Mann algorithm and the multiple-set split feasibility problem","volume":"22","author":"Xu","year":"2006","journal-title":"Inverse Probl."},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"Deepho, J., and Kumam, P. (2015). A viscosity approximation method for the split feasibility problem. Trans. Engng. Tech., 69\u201377.","DOI":"10.1007\/978-94-017-9588-3_6"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1007\/s10957-011-9814-6","article-title":"Split monotone variational inclusions","volume":"150","author":"Moudafi","year":"2011","journal-title":"J. Optim. Theory Appl."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"503","DOI":"10.1007\/s13398-015-0245-3","article-title":"An iterative method for solving split monotone variational inclusion and fixed point problems","volume":"110","author":"Shehu","year":"2016","journal-title":"RACSAM."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1007\/s11228-014-0285-4","article-title":"Iterative methods for generalized split feasibility problems in Hilbert spaces","volume":"23","author":"Takahashi","year":"2015","journal-title":"Set-Valued Var. Anal."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"513","DOI":"10.1016\/j.jmaa.2014.01.068","article-title":"On split common fixed point problems","volume":"415","author":"Kraikaew","year":"2014","journal-title":"J. Math. Anal. Appl."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/5\/655\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:50:56Z","timestamp":1760187056000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/5\/655"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,5,10]]},"references-count":57,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2019,5]]}},"alternative-id":["sym11050655"],"URL":"https:\/\/doi.org\/10.3390\/sym11050655","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,5,10]]}}}