{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:15:44Z","timestamp":1760238944121,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2020,9,17]],"date-time":"2020-09-17T00:00:00Z","timestamp":1600300800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003086","name":"Basque Government","doi-asserted-by":"publisher","award":["IT1207-19"],"award-info":[{"award-number":["IT1207-19"]}],"id":[{"id":"10.13039\/501100003086","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The purpose of this research work is to prove some weak and strong convergence results for maps satisfying (E)-condition through three-step Thakur (J. Inequal. Appl.2014, 2014:328.) iterative process in Banach spaces. We also present a new example of maps satisfying (E)-condition, and prove that its three-step Thakur iterative process is more efficient than the other well-known three-step iterative processes. At the end of the paper, we apply our results for finding solutions of split feasibility problems. The presented research work updates some of the results of the current literature.<\/jats:p>","DOI":"10.3390\/axioms9030110","type":"journal-article","created":{"date-parts":[[2020,9,17]],"date-time":"2020-09-17T08:29:43Z","timestamp":1600331383000},"page":"110","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Some New Results on a Three-Step Iteration Process"],"prefix":"10.3390","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6991-4287","authenticated-orcid":false,"given":"Kifayat","family":"Ullah","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Science and Technology, Bannu 28100, Khyber Pakhtunkhwa, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8119-9546","authenticated-orcid":false,"given":"Junaid","family":"Ahmad","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Science and Technology, Bannu 28100, Khyber Pakhtunkhwa, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9320-9433","authenticated-orcid":false,"given":"Manuel","family":"de la Sen","sequence":"additional","affiliation":[{"name":"Institute of Research and Development of Processes, University of the Basque Country, Campus of Leioa (Bizkaia), P.O. Box 644-Bilbao, Barrio Sarriena, 48940 Leioa, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,9,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"133","DOI":"10.4064\/fm-3-1-133-181","article-title":"Sur les operations dans les ensembles abstraits et leur application aux equations integrales","volume":"3","author":"Banach","year":"1922","journal-title":"Fund. Math."},{"key":"ref_2","first-page":"794","article-title":"Un teorema generale sull\u2019 esistenza di elementi uniti in una transformazione funzionale","volume":"11","author":"Caccioppoli","year":"1930","journal-title":"Rend. Accad. Lincei"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1041","DOI":"10.1073\/pnas.54.4.1041","article-title":"Nonexpansive nonlinear operators in a Banach space","volume":"54","author":"Browder","year":"1965","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1002\/mana.19650300312","article-title":"Zum Prinzip der Kontraktiven Abbildung","volume":"30","author":"Gohde","year":"1965","journal-title":"Math. Nachr."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1004","DOI":"10.2307\/2313345","article-title":"A fixed point theorem for mappings which do not increase distance","volume":"72","author":"Kirk","year":"1965","journal-title":"Am. Math. Mon."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1088","DOI":"10.1016\/j.jmaa.2007.09.023","article-title":"Fixed point theorems and convergence theorems for some generalized non-expansive mapping","volume":"340","author":"Suzuki","year":"2008","journal-title":"J. Math. Anal. Appl."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1016\/j.jmaa.2010.08.069","article-title":"Fixed point theory for a class of generalized nonexpansive mappings","volume":"375","author":"Suzuki","year":"2011","journal-title":"J. Math. Anal. Appl."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Usurelu, G.I., Bejenaru, A., and Postolache, M. (2020). Operators with property (E) as concerns numerical anaysis and visualization. Numer. Func. Anal. Optim.","DOI":"10.1080\/01630563.2020.1775252"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"506","DOI":"10.1090\/S0002-9939-1953-0054846-3","article-title":"Mean value methods in iteration","volume":"4","author":"Mann","year":"1953","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1090\/S0002-9939-1974-0336469-5","article-title":"Fixed points by a new iteration method","volume":"44","author":"Ishikawa","year":"1974","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_11","first-page":"61","article-title":"Iterative construction of fixed points of nearly asymtotically non-expansive mappings","volume":"8","author":"Agarwal","year":"2007","journal-title":"J. Nonlinear Convex Anal."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1006\/jmaa.2000.7042","article-title":"New approximation schemes for general variational inequalities","volume":"251","author":"Noor","year":"2000","journal-title":"J. Math. Anal. Appl."},{"key":"ref_13","first-page":"223","article-title":"A new faster iteration process applied to constrained minimization and feasibility problems","volume":"66","author":"Abbas","year":"2014","journal-title":"Math. Vesnik"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"328","DOI":"10.1186\/1029-242X-2014-328","article-title":"New iteration scheme for numerical reckoning fixed points of nonexpansive mappings","volume":"2014","author":"Thakur","year":"2014","journal-title":"J. Inequal. Appl."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Maniu, G. (2020). On a three-step iteration process for Suzuki mappings with qualitative study. Numer. Funct. Anal. Optim.","DOI":"10.1080\/01630563.2020.1719415"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Agarwal, R.P., O\u2019Regan, D., and Sahu, D.R. (2009). Fixed Point Theory for Lipschitzian-Type Mappings with Applications Series, Springer. Topological Fixed Point Theory and Its Applications.","DOI":"10.1155\/2009\/439176"},{"key":"ref_17","unstructured":"Takahashi, W. (2000). Nonlinear Functional Analysis, Yokohoma Publishers."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"396","DOI":"10.1090\/S0002-9947-1936-1501880-4","article-title":"Uniformly convex spaces","volume":"40","author":"Clarkson","year":"1936","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"591","DOI":"10.1090\/S0002-9904-1967-11761-0","article-title":"Weak and strong convergence of the sequence of successive approximations for non-expansive mappings","volume":"73","author":"Opial","year":"1967","journal-title":"Bull. Am. Math. Soc."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1017\/S0004972700028884","article-title":"Weak and strong convergence to fixed points of asymptotically non-expansive mappings","volume":"43","author":"Schu","year":"1991","journal-title":"Bull. Austral. Math. Soc."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1090\/S0002-9939-1974-0346608-8","article-title":"Approximating fixed points of non-expansive mappings","volume":"44","author":"Senter","year":"1974","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_22","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. Algor."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"411","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 Prob."},{"key":"ref_24","first-page":"93","article-title":"A new three-step iterative algorithm for solving the split feasibility problem","volume":"81","author":"Feng","year":"2019","journal-title":"Univ. Politeh. Buch. Ser. A"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2021","DOI":"10.1088\/0266-5611\/22\/6\/007","article-title":"A variable Krasnoselskii-Mann algorithm and the multiple-set split feasibility problem","volume":"22","author":"Xu","year":"2006","journal-title":"Inverse Probl."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"105018","DOI":"10.1088\/0266-5611\/26\/10\/105018","article-title":"Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces","volume":"26","author":"Xu","year":"2010","journal-title":"Inverse Probl."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/9\/3\/110\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:10:53Z","timestamp":1760177453000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/9\/3\/110"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,9,17]]},"references-count":26,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2020,9]]}},"alternative-id":["axioms9030110"],"URL":"https:\/\/doi.org\/10.3390\/axioms9030110","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2020,9,17]]}}}