{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T00:52:21Z","timestamp":1760230341082,"version":"build-2065373602"},"reference-count":21,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2022,7,20]],"date-time":"2022-07-20T00:00:00Z","timestamp":1658275200000},"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>Symmetries are vital in the study of physical phenomena such as quantum physics and the micro-world, among others. Then, these phenomena reduce to solving nonlinear equations in abstract spaces. These equations in turn are mostly solved iteratively. That is why the objective of this paper was to obtain a uniform way to study three-step iterative methods to solve equations defined on Banach spaces. The convergence is established by using information appearing in these methods. This is in contrast to earlier works which relied on derivatives of the higher order to establish the convergence. The numerical example completes this paper.<\/jats:p>","DOI":"10.3390\/sym14071484","type":"journal-article","created":{"date-parts":[[2022,7,21]],"date-time":"2022-07-21T03:34:40Z","timestamp":1658374480000},"page":"1484","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Extended Convergence of Three Step Iterative Methods for Solving Equations in Banach Space with Applications"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0035-1022","authenticated-orcid":false,"given":"Samundra","family":"Regmi","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Houston, Houston, TX 77204, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9189-9298","authenticated-orcid":false,"given":"Ioannis K.","family":"Argyros","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3530-5539","authenticated-orcid":false,"given":"Santhosh","family":"George","sequence":"additional","affiliation":[{"name":"Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka, Mangaluru 575 025, India"}]},{"given":"Christopher I.","family":"Argyros","sequence":"additional","affiliation":[{"name":"Department of Computing and Technology, Cameron University, Lawton, OK 73505, USA"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"364","DOI":"10.1016\/j.jco.2011.12.003","article-title":"Weaker conditions for the convergence of Newton\u2019s method","volume":"28","author":"Argyros","year":"2012","journal-title":"J. Complex."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Argyros, I.K. (2021). Unified Convergence Criteria for Iterative Banach Space Valued Methods with Applications. Mathematics, 9.","DOI":"10.3390\/math9161942"},{"key":"ref_3","unstructured":"Argyros, I.K., and Magr\u00e9\u00f1an, A.A. (2018). A Contemporary Study of Iterative Methods, Elsevier (Academic Press)."},{"key":"ref_4","unstructured":"Argyros, I.K. (2022). The Theory and Applications of Iteration Methods, CRC Press, Taylor and Francis Group. [2nd ed.]."},{"key":"ref_5","unstructured":"Ezquerro, J.A., and Hernandez, M.A. (2018). Newton\u2019s Method: An Updated Approach of Kantorovich\u2019s Theory, Springer."},{"key":"ref_6","unstructured":"Kantorovich, L.V., and Akilov, G.P. (1982). Functional Analysis, Pergamon Press."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1007\/BF01385696","article-title":"Convergence of Newton-like methods for singular operator equations using outer inverses","volume":"66","author":"Nashed","year":"1993","journal-title":"Numer. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1016\/j.jco.2009.05.001","article-title":"New general convergence theory for iterative processes and its applications to Newton\u2013Kantorovich type theorems","volume":"26","author":"Proinov","year":"2010","journal-title":"J. Complex."},{"key":"ref_9","first-page":"253","article-title":"On an iterative Method of order 1.839\u22ef for solving nonlinear least squares problems","volume":"161","author":"Shakhno","year":"2005","journal-title":"Appl. Math. Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"111095","DOI":"10.1016\/j.chaos.2021.111095","article-title":"Asymptotic stability of fractional order (1, 2] stochastic delay differential equations in Banach spaces","volume":"150","author":"Singh","year":"2021","journal-title":"Chaos Solitons Fractals"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1016\/j.cam.2003.12.041","article-title":"A modified Newton method with cubic convergence: The multivariate case","volume":"169","author":"Homeier","year":"2004","journal-title":"J. Comput. Appl. Math."},{"key":"ref_12","first-page":"686","article-title":"Variants of Newton\u2019s method using fifth-order quadrature formulas","volume":"190","author":"Cordero","year":"2007","journal-title":"Appl. Math. Comput."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/j.camwa.2008.10.067","article-title":"Some iterative methods for solving a system of nonlinear equations","volume":"57","author":"Noor","year":"2009","journal-title":"Comput. Math. Appl."},{"key":"ref_14","unstructured":"Verma, R. (2019). New Trends in Fractional Programming, Nova Science Publisher."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1007\/s40314-014-0193-0","article-title":"Efficient derivative-free numerical methods for solving systems of nonlinear equations","volume":"35","author":"Sharma","year":"2016","journal-title":"Comp. Appl. Math."},{"key":"ref_16","first-page":"251","article-title":"Achieving higher order of convergence for solving systems of nonlinear equations","volume":"311","author":"Xiao","year":"2017","journal-title":"Appl. Math. Comput."},{"key":"ref_17","first-page":"2377","article-title":"Ostrowski type methods for solving system of nonlinear equations","volume":"218","author":"Grau","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"536","DOI":"10.1016\/j.cnsns.2009.04.013","article-title":"Some eight order root finding three-step methods","volume":"15","author":"Kou","year":"2010","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_19","unstructured":"Traub, J.F. (1964). Iterative Methods for the Solution of Equations, Prentice Hall."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1367","DOI":"10.1080\/00036811.2017.1422727","article-title":"Controllability for a class of second-order evolution differential inclusions without compactness","volume":"98","author":"Vijayakumar","year":"2019","journal-title":"Appl. Anal."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Vijayakumar, V., Nisar, K.S., Chalishajar, D., Shukla, A., Malik, M., Alsaadi, A., and Aldosary, S.F. (2022). A Note on Approximate Controllability of Fractional Semilinear Integrodifferential Control Systems via Resolvent Operators. Fractal Fract., 6.","DOI":"10.3390\/fractalfract6020073"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/7\/1484\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:54:26Z","timestamp":1760140466000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/7\/1484"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,20]]},"references-count":21,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2022,7]]}},"alternative-id":["sym14071484"],"URL":"https:\/\/doi.org\/10.3390\/sym14071484","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,7,20]]}}}