{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:38:58Z","timestamp":1760233138175,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2022,12,21]],"date-time":"2022-12-21T00:00:00Z","timestamp":1671580800000},"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>A plethora of quantum physics problems are related to symmetry principles. Moreover, by using symmetry theory and mathematical modeling, these problems reduce to solving iteratively finite differences and systems of nonlinear equations. In particular, Newton-type methods are introduced to generate sequences approximating simple solutions of nonlinear equations in the setting of Banach spaces. Specializations of these methods include the modified Newton method, Newton\u2019s method, and other single-step methods. The convergence of these methods is established with similar conditions. However, the convergence region is not large in general. That is why a unified semilocal convergence analysis is developed that can be used to handle these methods under even weaker conditions that are not previously considered. The approach leads to the extension of the applicability of these methods in cases not covered before but without new conditions. The idea is to replace the Lipschitz parameters or other parameters used by smaller ones to force convergence in cases not possible before. It turns out that the error analysis is also extended. Moreover, the new idea does not depend on the method. That is why it can also be applied to other methods to also extend their applicability. Numerical applications illustrate and test the convergence conditions.<\/jats:p>","DOI":"10.3390\/sym15010015","type":"journal-article","created":{"date-parts":[[2022,12,21]],"date-time":"2022-12-21T05:42:53Z","timestamp":1671601373000},"page":"15","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Newton-Type Methods for Solving Equations in Banach Spaces: A Unified Approach"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9189-9298","authenticated-orcid":false,"given":"Ioannis K.","family":"Argyros","sequence":"first","affiliation":[{"name":"Department of Computing and Mathematical Sciences, Cameron University, Lawton, OK 73505, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3845-6260","authenticated-orcid":false,"given":"Stepan","family":"Shakhno","sequence":"additional","affiliation":[{"name":"Department of Theory of Optimal Processes, Ivan Franko National University of Lviv, Universytetska Str. 1, 79000 Lviv, Ukraine"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0035-1022","authenticated-orcid":false,"given":"Samundra","family":"Regmi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Houston, Houston, TX 77204, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8986-2509","authenticated-orcid":false,"given":"Halyna","family":"Yarmola","sequence":"additional","affiliation":[{"name":"Department of Computational Mathematics, Ivan Franko National University of Lviv, Universytetska Str. 1, 79000 Lviv, Ukraine"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Argyros, I.K. (2022). The Theory and Applications of Iterative Methods, CRC-Taylor and Francis Publ. Group. [2nd ed.].","DOI":"10.1201\/9781003128915-7"},{"key":"ref_2","unstructured":"Ortega, J.M., and Rheinboldt, W.C. (1970). Iterative Solution of Nonlinear Equations in Several Variables, Academic Press."},{"key":"ref_3","unstructured":"Dennis, J.E., and Schnabel, R.B. (1983). Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall."},{"key":"ref_4","unstructured":"Traub, J.F. (1964). Iterative Methods for the Solution of Equations, Prentice Hall."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Ezquerro, J.A., and Hern\u00e1ndez-Ver\u00f3n, M.A. (2017). Newton\u2019s Method: An Updated Approach of Kantorovich\u2019s Theory. Frontiers in Mathematics, Birkh\u00e4user\/Springer.","DOI":"10.1007\/978-3-319-55976-6"},{"key":"ref_6","unstructured":"Verma, R. (2019). New Trends in Fractional Programming, Nova Science Publisher."},{"key":"ref_7","unstructured":"Potra, F.A., and Pt\u00e1k, V. (1984). Nondiscrete induction and iterative processes. Research Notes in Mathematics, Pitman (Advanced Publishing Program)."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0377-0427(00)00417-9","article-title":"Historical developments in convergence analysis for Newton\u2019s and Newton-like methods","volume":"124","author":"Yamamoto","year":"2000","journal-title":"J. Comput. Appl. Math."},{"key":"ref_9","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_10","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-Kantorovich type theorems","volume":"26","author":"Proinov","year":"2010","journal-title":"J. Complex."},{"key":"ref_11","unstructured":"Kantorovich, L.V., and Akilov, G.P. (1982). Functional Analysis, Pergamon Press."},{"key":"ref_12","first-page":"372","article-title":"On an improved convergence analysis of Newton\u2019s scheme","volume":"225","author":"Argyros","year":"2013","journal-title":"Appl. Math. Comput."},{"key":"ref_13","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 scheme","volume":"28","author":"Argyros","year":"2012","journal-title":"J. Complex."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1704","DOI":"10.1080\/00207160.2019.1652739","article-title":"High order iterations for systems of nonlinear equations","volume":"97","author":"Zhanlav","year":"2020","journal-title":"Int. J. Comput. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"451","DOI":"10.1007\/s10092-015-0157-9","article-title":"Simple yet efficient Newton-like method for systems of nonlinear equations","volume":"53","author":"Sharma","year":"2016","journal-title":"Calcolo"},{"key":"ref_16","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_17","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_18","doi-asserted-by":"crossref","first-page":"378","DOI":"10.1016\/j.cam.2013.11.018","article-title":"Convergence of the two-step combined method and uniqueness of the solution of nonlinear operator equations","volume":"261","author":"Shakhno","year":"2014","journal-title":"J. Comput. Appl. Math."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"222","DOI":"10.1016\/j.cam.2009.02.010","article-title":"On an iterative algorithm with superquadratic convergence for solving nonlinear operator equations","volume":"231","author":"Shakhno","year":"2009","journal-title":"J. Comput. Appl. Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"710","DOI":"10.1016\/j.cam.2017.04.021","article-title":"An Ostrowski-type method with memory using a novel self-accelerating parameters","volume":"330","author":"Wang","year":"2018","journal-title":"J. Comput. Appl. Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"240","DOI":"10.1016\/j.jmaa.2018.08.007","article-title":"On a two-step optimal Steffensen-type method: Relaxed local and semi-local convergence analysis and dynamical stability","volume":"468","author":"Moccari","year":"2018","journal-title":"J. Math. Anal. Appl."},{"key":"ref_22","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":"Comput. Appl. Math."},{"key":"ref_23","first-page":"62","article-title":"On Newton\u2019s method with accelerated convergence","volume":"6","author":"Bartish","year":"1987","journal-title":"Vest. Kiev Univ. Model. Complex Syst."},{"key":"ref_24","first-page":"77","article-title":"Newton-like methods for the Computation of Fixed Points","volume":"10","author":"Werner","year":"1984","journal-title":"Comput. Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/1\/15\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:45:16Z","timestamp":1760147116000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/1\/15"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,21]]},"references-count":24,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1]]}},"alternative-id":["sym15010015"],"URL":"https:\/\/doi.org\/10.3390\/sym15010015","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,12,21]]}}}