{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:56:00Z","timestamp":1760147760486,"version":"build-2065373602"},"reference-count":14,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2023,3,1]],"date-time":"2023-03-01T00:00:00Z","timestamp":1677628800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>A local and semi-local convergence is developed of a class of iterative methods without derivatives for solving nonlinear Banach space valued operator equations under the classical Lipschitz conditions for first-order divided differences. Special cases of this method are well-known iterative algorithms, in particular, the Secant, Kurchatov, and Steffensen methods as well as the Newton method. For the semi-local convergence analysis, we use a technique of recurrent functions and majorizing scalar sequences. First, the convergence of the scalar sequence is proved and its limit is determined. It is then shown that the sequence obtained by the proposed method is bounded by this scalar sequence. In the local convergence analysis, a computable radius of convergence is determined. Finally, the results of the numerical experiments are given that confirm obtained theoretical estimates.<\/jats:p>","DOI":"10.3390\/computation11030049","type":"journal-article","created":{"date-parts":[[2023,3,1]],"date-time":"2023-03-01T03:28:06Z","timestamp":1677641286000},"page":"49","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Unified Convergence Criteria of Derivative-Free Iterative Methods for Solving Nonlinear Equations"],"prefix":"10.3390","volume":"11","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 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\/0000-0002-8986-2509","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":[[2023,3,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Argyros, I.K., and Magre\u00f1\u00e1n, \u00c1.A. (2017). Iterative Methods and Their Dynamics with Applications: A Contemporary Study, CRC Press.","DOI":"10.1201\/9781315153469"},{"key":"ref_2","unstructured":"Dennis, J.E., and Schnabel, R.B. (1983). Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1153","DOI":"10.1080\/00207160412331284123","article-title":"On the local convergence of secant-type methods","volume":"81","author":"Amat","year":"2004","journal-title":"Intern. J. Comput. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1016\/S0893-9659(01)00150-1","article-title":"The Secant method for nondifferentiable operators","volume":"15","author":"Hernandez","year":"2002","journal-title":"Appl. Math. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1016\/j.jmaa.2006.09.075","article-title":"A Kantorovich-type analysis for a fast iterative method for solving nonlinear equations","volume":"332","author":"Argyros","year":"2007","journal-title":"J. Math. Anal. Appl."},{"key":"ref_6","first-page":"524","article-title":"On a method of linear interpolation for the solution of functional equations","volume":"198","author":"Kurchatov","year":"1971","journal-title":"Dokl. Akad. Nauk SSSR"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"650","DOI":"10.1002\/pamm.200410306","article-title":"On a Kurchatov\u2019s method of linear interpolation for solving nonlinear equations","volume":"4","author":"Shakhno","year":"2004","journal-title":"Pamm Proc. Appl. Math. Mech."},{"key":"ref_8","unstructured":"Shakhno, S.M. (2004, January 24\u201328). Nonlinear majorants for investigation of methods of linear interpolation for the solution of nonlinear equations. 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Issues Anal."},{"key":"ref_12","first-page":"18","article-title":"A study of the local convergence of a derivative free method in Banach spaces","volume":"10","author":"Sharma","year":"2022","journal-title":"J Anal."},{"key":"ref_13","unstructured":"Traub, J.F. (1964). Iterative Methods for the Solution of Equations, Prentice Hall."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1016\/j.cam.2018.06.042","article-title":"Highly efficient family of iterative methods for solving nonlinear models","volume":"346","author":"Behl","year":"2019","journal-title":"J. Comput. Appl. 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