{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,17]],"date-time":"2025-12-17T18:03:52Z","timestamp":1765994632622,"version":"3.37.3"},"reference-count":31,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2019,7,11]],"date-time":"2019-07-11T00:00:00Z","timestamp":1562803200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2019,7,11]],"date-time":"2019-07-11T00:00:00Z","timestamp":1562803200000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100011093","name":"University of Phayao","doi-asserted-by":"crossref","award":["UoE62001"],"award-info":[{"award-number":["UoE62001"]}],"id":[{"id":"10.13039\/501100011093","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2020,6]]},"DOI":"10.1007\/s11075-019-00768-w","type":"journal-article","created":{"date-parts":[[2019,7,11]],"date-time":"2019-07-11T14:21:45Z","timestamp":1562854905000},"page":"537-565","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["A new modified three-step iteration method for G-nonexpansive mappings in Banach spaces with a graph"],"prefix":"10.1007","volume":"84","author":[{"given":"Damrongsak","family":"Yambangwai","sequence":"first","affiliation":[]},{"given":"Sukanya","family":"Aunruean","sequence":"additional","affiliation":[]},{"given":"Tanakit","family":"Thianwan","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2019,7,11]]},"reference":[{"key":"768_CR1","doi-asserted-by":"publisher","first-page":"133","DOI":"10.4064\/fm-3-1-133-181","volume":"3","author":"S Banach","year":"1922","unstructured":"Banach, S.: Sur les oprations dans les ensembles abstraits et leur application aux quations intgrales. Fund. Math. 3, 133\u2013181 (1922)","journal-title":"Fund. Math."},{"issue":"4","key":"768_CR2","doi-asserted-by":"publisher","first-page":"1359","DOI":"10.1090\/S0002-9939-07-09110-1","volume":"136","author":"J Jachymski","year":"2008","unstructured":"Jachymski, J.: The contraction principle for mappings on a metric space with a graph. Proc. Amer. Math. Soc. 136(4), 1359\u20131373 (2008)","journal-title":"Proc. Amer. Math. Soc."},{"key":"768_CR3","doi-asserted-by":"publisher","first-page":"511","DOI":"10.2140\/pjm.1967.21.511","volume":"21","author":"RP Kelisky","year":"1967","unstructured":"Kelisky, R.P., Rivlin, T.J.: Iterates of Bernstein polynomials. Pac. J. Math. 21, 511\u2013520 (1967)","journal-title":"Pac. J. Math."},{"key":"768_CR4","doi-asserted-by":"publisher","first-page":"659","DOI":"10.1016\/j.topol.2011.10.013","volume":"159","author":"SMA Aleomraninejad","year":"2012","unstructured":"Aleomraninejad, S.M.A., Rezapour, S., Shahzad, N.: Some fixed point result on a metric space with a graph. Topol. Appl. 159, 659\u2013663 (2012)","journal-title":"Topol. Appl."},{"key":"768_CR5","doi-asserted-by":"publisher","unstructured":"Alfuraidan, M.R., Khamsi, M.A.: Fixed points of monotone nonexpansive mappings on a hyperbolic metric space with a graph. Fixed Point Theory Appl. https:\/\/doi.org\/10.1186\/s13663-015-0294-5 (2015)","DOI":"10.1186\/s13663-015-0294-5"},{"key":"768_CR6","doi-asserted-by":"publisher","unstructured":"Alfuraidan, M.R.: Fixed points of monotone nonexpansive mappings with a graph. Fixed Point Theory Appl. https:\/\/doi.org\/10.1186\/s13663-015-0299-0 (2015)","DOI":"10.1186\/s13663-015-0299-0"},{"key":"768_CR7","doi-asserted-by":"publisher","unstructured":"Tiammee, J., Kaewkhao, A., Suantai, S.: On Browder\u2019s convergence theorem and Halpern iteration process for G-nonexpansive mappings in Hilbert spaces endowed with graphs. Fixed Point Theory Appl. https:\/\/doi.org\/10.1186\/s13663-015-0436-9 (2015)","DOI":"10.1186\/s13663-015-0436-9"},{"key":"768_CR8","doi-asserted-by":"publisher","unstructured":"Tripak, O.: Common fixed points of G-nonexpansive mappings on Banach spaces with a graph. Fixed Point Theory Appl. https:\/\/doi.org\/10.1186\/s13663-016-0578-4 (2016)","DOI":"10.1186\/s13663-016-0578-4"},{"issue":"1","key":"768_CR9","doi-asserted-by":"publisher","first-page":"217","DOI":"10.1006\/jmaa.2000.7042","volume":"251","author":"MA Noor","year":"2000","unstructured":"Noor, M.A.: New approximation schemes for general variational inequalities. J. Math. Anal. Appl. 251(1), 217\u2013229 (2000)","journal-title":"J. Math. Anal. Appl."},{"key":"768_CR10","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611970838","volume-title":"Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanic","author":"R Glowinski","year":"1989","unstructured":"Glowinski, R., Tallec, P.L.: Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanic. SIAM, Philadelphia (1989)"},{"key":"768_CR11","doi-asserted-by":"publisher","first-page":"645","DOI":"10.1023\/A:1022646327085","volume":"97","author":"S Haubruge","year":"1998","unstructured":"Haubruge, S., Nguyen, V.H., Strodiot, J.J.: Convergence analysis and applications of the Glowinski Le Tallec splitting method for finding a zero of the sum of two maximal monotone operators. J. Optim. Theory Appl. 97, 645\u2013673 (1998)","journal-title":"J. Optim. Theory Appl."},{"key":"768_CR12","doi-asserted-by":"publisher","unstructured":"Sridarat, P., Suparaturatorn, R., Suantai, S., Cho, Y.J.: Covergence analysis of SP-iteration for G-nonexpansive mappings with directed graphs. Bull. Malays. Math. Sci Soc. https:\/\/doi.org\/10.1007\/s40840-018-0606-0 (2017)","DOI":"10.1007\/s40840-018-0606-0"},{"key":"768_CR13","unstructured":"Johnsonbaugh, R.: Discrete Mathematics. New Jersey (1997)"},{"key":"768_CR14","doi-asserted-by":"publisher","first-page":"591","DOI":"10.1090\/S0002-9904-1967-11761-0","volume":"73","author":"Z Opial","year":"1967","unstructured":"Opial, Z.: Weak convergence of successive approximations for nonexpansive mappings. Bull. Amer. Math. Soc. 73, 591\u2013597 (1967)","journal-title":"Bull. Amer. Math. Soc."},{"issue":"3","key":"768_CR15","doi-asserted-by":"crossref","first-page":"529","DOI":"10.1515\/GMJ.2006.529","volume":"13","author":"S Shahzad","year":"2006","unstructured":"Shahzad, S., Al-Dubiban, R.: Approximating common fixed points of nonexpansive mappings in Banach spaces. Georgian Math. J. 13(3), 529\u2013537 (2006)","journal-title":"Georgian Math. J."},{"key":"768_CR16","doi-asserted-by":"publisher","first-page":"301","DOI":"10.1006\/jmaa.1993.1309","volume":"178","author":"KK Tan","year":"1993","unstructured":"Tan, K.K., Xu, H.K.: Approximating fixed points of nonexpansive mapping by the Ishikawa iteration process. J. Math. Anal. Appl. 178, 301\u2013308 (1993)","journal-title":"J. Math. Anal. Appl."},{"issue":"1","key":"768_CR17","doi-asserted-by":"publisher","first-page":"153","DOI":"10.1017\/S0004972700028884","volume":"43","author":"J Schu","year":"1991","unstructured":"Schu, J.: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bull. Aust. Math. Soc. 43(1), 153\u2013159 (1991)","journal-title":"Bull. Aust. Math. Soc."},{"key":"768_CR18","doi-asserted-by":"publisher","first-page":"506","DOI":"10.1016\/j.jmaa.2005.03.002","volume":"331","author":"S Suantai","year":"2005","unstructured":"Suantai, S.: Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings. J. Math. Anal. Appl. 331, 506\u2013517 (2005)","journal-title":"J. Math. Anal. Appl."},{"key":"768_CR19","volume-title":"Fixed Point Theory for Lipschitzian-type Mappings with Applications","author":"RP Agarwal","year":"2009","unstructured":"Agarwal, R.P., O\u2019Regan, D., Sahu, D.R.: Fixed Point Theory for Lipschitzian-type Mappings with Applications. Springer, New York (2009)"},{"issue":"4","key":"768_CR20","doi-asserted-by":"publisher","first-page":"671","DOI":"10.1142\/S1793557111000551","volume":"4","author":"MG Sangago","year":"2011","unstructured":"Sangago, M.G.: Convergence of iterative schemes for nonexpansive mappings. Asian-European J. Math. 4(4), 671\u2013682 (2011)","journal-title":"Asian-European J. Math."},{"key":"768_CR21","doi-asserted-by":"crossref","unstructured":"Wajtaszczyk, P.: Banach Spaces for Analysts. Cambridge Univ. Press (1991)","DOI":"10.1017\/CBO9780511608735"},{"issue":"2","key":"768_CR22","doi-asserted-by":"publisher","first-page":"741","DOI":"10.1016\/0022-247X(76)90038-X","volume":"56","author":"BE Rhoades","year":"1976","unstructured":"Rhoades, B.E.: Comments on two fixed point iteration method. J. Math. Anal. Appl. 56(2), 741\u2013750 (1976)","journal-title":"J. Math. Anal. Appl."},{"key":"768_CR23","doi-asserted-by":"publisher","first-page":"3006","DOI":"10.1016\/j.cam.2010.12.022","volume":"235","author":"W Phuengrattana","year":"2011","unstructured":"Phuengrattana, W., Suantai, S.: On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval. J. Comput. Appl. Math. 235, 3006\u20133014 (2011)","journal-title":"J. Comput. Appl. Math."},{"key":"768_CR24","volume-title":"Numerical Analysis, 9th edn.","author":"RL Burden","year":"2010","unstructured":"Burden, R.L., Faires, J.D.: Numerical Analysis, 9th edn. Brooks\/Cole Cengage Learning, Boston (2010)"},{"key":"768_CR25","volume-title":"Iterative Approximation of Fixed Points","author":"V Berinde","year":"2002","unstructured":"Berinde, V.: Iterative Approximation of Fixed Points. Editura Efemeride, Baia Mare (2002)"},{"key":"768_CR26","doi-asserted-by":"publisher","unstructured":"Suparatulatorn, R., Cholamjiak, W., Suantai, S.: A modified S-iteration process for G-nonexpansive mappings in Banach spaces with graphs. Numer Algorithm. https:\/\/doi.org\/10.1007\/s11075-017-0324-y (2017)","DOI":"10.1007\/s11075-017-0324-y"},{"key":"768_CR27","doi-asserted-by":"publisher","first-page":"44","DOI":"10.1007\/s11784-019-0681-3","volume":"21","author":"T Thianwan","year":"2019","unstructured":"Thianwan, T., Yambangwai, D.: Convergence analysis for a new two-step iteration process for G-nonexpansive mappings with directed graphs. J. Fixed Point Theory Appl. 21, 44 (2019). https:\/\/doi.org\/10.1007\/s11784-019-0681-3","journal-title":"J. Fixed Point Theory Appl."},{"key":"768_CR28","doi-asserted-by":"crossref","unstructured":"Byrne, C.L.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Problems, 20 (2004)","DOI":"10.1088\/0266-5611\/20\/1\/006"},{"key":"768_CR29","doi-asserted-by":"publisher","first-page":"517","DOI":"10.1364\/JOSAA.7.000517","volume":"7","author":"CI Podilchuk","year":"1990","unstructured":"Podilchuk, C.I., Mammone, R.J.: Image recovery by convex projections using a least squares constraint. J. Optical Soc. Am. A 7, 517\u2013521 (1990)","journal-title":"J. Optical Soc. Am. A"},{"key":"768_CR30","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-5020-3","volume-title":"Nonlinear Functional Analysis and its Applications: Variational Methods and Applications","author":"E Zeidler","year":"1985","unstructured":"Zeidler, E.: Nonlinear Functional Analysis and its Applications: Variational Methods and Applications. Springer, New York (1985)"},{"key":"768_CR31","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-69512-4","volume-title":"Diffierential Inclusions: Set-valued Maps and Viability Theory","author":"JP Aubin","year":"1984","unstructured":"Aubin, J.P., Cellina, A.: Diffierential Inclusions: Set-valued Maps and Viability Theory. Springer, Berlin (1984)"}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-019-00768-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11075-019-00768-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-019-00768-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,9,23]],"date-time":"2022-09-23T17:44:53Z","timestamp":1663955093000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11075-019-00768-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,7,11]]},"references-count":31,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2020,6]]}},"alternative-id":["768"],"URL":"https:\/\/doi.org\/10.1007\/s11075-019-00768-w","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"type":"print","value":"1017-1398"},{"type":"electronic","value":"1572-9265"}],"subject":[],"published":{"date-parts":[[2019,7,11]]},"assertion":[{"value":"3 April 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 June 2019","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 July 2019","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}