{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T04:01:55Z","timestamp":1762056115931,"version":"build-2065373602"},"reference-count":18,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2022,6,24]],"date-time":"2022-06-24T00:00:00Z","timestamp":1656028800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this article, we present generalized conditions of three-step iterative schemes for solving nonlinear equations. The convergence order is shown using Taylor series, but the existence of high-order derivatives is assumed. However, only the first derivative appears on these schemes. Therefore, the hypotheses limit the utilization of the schemes to operators that are at least nine times differentiable, although the schemes may converge. To the best of our knowledge, no semi-local convergence has been given in the setting of a Banach space. Our goal is to extend the applicability of these schemes in both the local and semi-local convergence cases. Moreover, we use our idea of recurrent functions and conditions only on the derivative or divided differences of order one that appear in these schemes. This idea can be applied to extend other high convergence multipoint and multistep schemes. Numerical applications where the convergence criteria are tested complement this article.<\/jats:p>","DOI":"10.3390\/axioms11070307","type":"journal-article","created":{"date-parts":[[2022,6,25]],"date-time":"2022-06-25T10:39:13Z","timestamp":1656153553000},"page":"307","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Numerical Processes for Approximating Solutions of 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"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"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"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"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"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Christopher I.","family":"Argyros","sequence":"additional","affiliation":[{"name":"Department of Computing and Technology, Cameron University, Lawton, OK 73505, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,24]]},"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. 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New Trends in Fractional Programming, Nova Science Publisher."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/7\/307\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:39:31Z","timestamp":1760139571000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/7\/307"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,24]]},"references-count":18,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2022,7]]}},"alternative-id":["axioms11070307"],"URL":"https:\/\/doi.org\/10.3390\/axioms11070307","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2022,6,24]]}}}