{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:08:20Z","timestamp":1760144900640,"version":"build-2065373602"},"reference-count":49,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2024,5,28]],"date-time":"2024-05-28T00:00:00Z","timestamp":1716854400000},"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>We develop a new Kantorovich-like convergence analysis of Newton-type methods to solve nonsingular and singular nonlinear equations in Banach spaces. The outer or generalized inverses are exchanged by a finite sum of linear operators making the implementation of these methods easier than in earlier studies. The analysis uses relaxed generalized continuity of the derivatives of operators involved required to control the derivative and on real majorizing sequences. The same approach can also be implemented on other iterative methods with inverses. The examples complement the theory by verifying the convergence conditions and demonstrating the performance of the methods.<\/jats:p>","DOI":"10.3390\/axioms13060358","type":"journal-article","created":{"date-parts":[[2024,5,28]],"date-time":"2024-05-28T07:38:39Z","timestamp":1716881919000},"page":"358","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On the Kantorovich Theory for Nonsingular and Singular Equations"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9189-9298","authenticated-orcid":false,"given":"Ioannis K.","family":"Argyros","sequence":"first","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"}]},{"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 77205, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Michael I.","family":"Argyros","sequence":"additional","affiliation":[{"name":"Department of Computer Sciences, University of Oklahoma, Norman, OK 73501, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,5,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"545","DOI":"10.1007\/BF01400355","article-title":"A convergence theorem for Newton-like algorithms in Banach spaces","volume":"51","author":"Yamamoto","year":"1987","journal-title":"Numer. 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