{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:35:50Z","timestamp":1760146550684,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2024,11,18]],"date-time":"2024-11-18T00:00:00Z","timestamp":1731888000000},"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>The convergence order of an iterative method used to solve equations is usually determined by using Taylor series expansions, which in turn require high-order derivatives, which are not necessarily present in the method. Therefore, such convergence analysis cannot guarantee the theoretical convergence of the method to a solution if these derivatives do not exist. However, the method can converge. This indicates that the most sufficient convergence conditions required by the Taylor approach can be replaced by weaker ones. Other drawbacks exist, such as information on the isolation of simple solutions or the number of iterations that must be performed to achieve the desired error tolerance. This paper positively addresses all these issues by considering a technique that uses only the operators on the method and \u03a9-generalized continuity to control the derivative. Moreover, both local and semi-local convergence analyses are presented for Banach space-valued operators. The technique can be used to extend the applicability of other methods along the same lines. A large number of concrete examples are shown in which the convergence conditions are fulfilled.<\/jats:p>","DOI":"10.3390\/axioms13110802","type":"journal-article","created":{"date-parts":[[2024,11,19]],"date-time":"2024-11-19T11:58:07Z","timestamp":1732017487000},"page":"802","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Extension of an Eighth-Order Iterative Technique to Address Non-Linear Problems"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2791-6230","authenticated-orcid":false,"given":"Higinio","family":"Ramos","sequence":"first","affiliation":[{"name":"Escuela Polit\u00e9cnica Superior de Zamora, Universidad de Salamanca, Avda. de Requejo 33, 49029 Zamora, Spain"},{"name":"Scientific Computing Group, Universidad de Salamanca, Plaza de la Merced, 37008 Salamanca, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ioannis K.","family":"Argyros","sequence":"additional","affiliation":[{"name":"Department of Computing and Mathematical Sciences, Cameron University, Lawton, OK 73505, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1505-8945","authenticated-orcid":false,"given":"Ramandeep","family":"Behl","sequence":"additional","affiliation":[{"name":"Mathematical Modelling and Applied Computation Research Group (MMAC), Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia"},{"name":"Department of Mathematics, Saveetha School of Engineering, SIMATS, Chennai 602105, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hashim","family":"Alshehri","sequence":"additional","affiliation":[{"name":"Mathematical Modelling and Applied Computation Research Group (MMAC), Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,11,18]]},"reference":[{"key":"ref_1","unstructured":"Argyros, G.I., Regmi, S., Argyros, I.K., and George, S. (2024). Contemporary Algorithms, Nova Publisher. [4th ed.]."},{"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","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_4","unstructured":"Argyros, I.K. (2022). Theory and Applications of Iterative Methods, 2nd Edition Engineering Series, CRC Press-Taylor and Francis Group."},{"key":"ref_5","unstructured":"Ostrowski, A.M. (1966). Solutions of Equations and Systems of Equations, Academic Press."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"591","DOI":"10.1016\/j.camwa.2013.12.004","article-title":"An efficient fifth order method for solving systems of nonlinear equations","volume":"67","author":"Sharma","year":"2014","journal-title":"Comput. Math. Appl."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1347","DOI":"10.1080\/09720502.2021.1884393","article-title":"Family of optimal two-step fourth order iterative method and its extension for solving nonlinear equations","volume":"24","author":"Ogbereyivwe","year":"2021","journal-title":"J. Interdiscip. Math."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Akram, S., Khalid, M., Junjua MU, D., Altaf, S., and Kumar, S. (2023). Extension of King\u2019s iterative scheme by means of memory for nonlinear equations. Symmetry, 15.","DOI":"10.3390\/sym15051116"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Panday, S., Mittal, S.K., Stoenoiu, C.E., and J\u00e4ntschi, L. (2024). A New Adaptive Eleventh-Order Memory Algorithm for Solving. Nonlinear Equations. Math., 12.","DOI":"10.3390\/math12121809"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"12549","DOI":"10.1002\/mma.9197","article-title":"A modified Chebyshev-Halley-type iterative family with memory for solving nonlinear equations and its stability analysis","volume":"46","author":"Sharma","year":"2023","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Wang, X., and Tao, Y. (2020). A new Newton method with memory for solving nonlinear equations. Mathematics, 8.","DOI":"10.3390\/math8010108"},{"key":"ref_12","first-page":"1007","article-title":"A two-step method adaptive with memory with eighth-order for solving nonlinear equations and its dynamic","volume":"10","author":"Torkashvand","year":"2022","journal-title":"Comput. Methods Differ. Equat."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Thangkhenpau, G., Panday, S., Mittal, S.K., and J\u00e4ntschi, L. (2023). Novel parametric families of with and without memory iterative methods for multiple roots of nonlinear equations. Mathematics, 11.","DOI":"10.3390\/math11092036"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1111","DOI":"10.3390\/a8041111","article-title":"An optimal biparametric multipoint family and its self- acceleration with memory for solving nonlinear equations","volume":"8","author":"Zheng","year":"2015","journal-title":"Algorithms"},{"key":"ref_15","first-page":"3754","article-title":"Sixteenth-order method for nonlinear equations","volume":"215","author":"Li","year":"2010","journal-title":"Appl. Math. Comput."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1007\/s11075-009-9345-5","article-title":"A family of modified Ostrowski\u2019s methods with accelerated eighth order convergence","volume":"54","author":"Sharma","year":"2010","journal-title":"Numer. Algorithms"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2278","DOI":"10.1016\/j.cam.2009.10.012","article-title":"A family of three-point methods of optimal order for solving nonlinear equations","volume":"233","author":"Thukral","year":"2010","journal-title":"J. Comput. Appl. Math."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2969","DOI":"10.1016\/j.cam.2010.04.009","article-title":"New modifications of Potra-Ptak\u2019s method with optimal fourth and eighth order of convergence","volume":"234","author":"Cordero","year":"2010","journal-title":"J. Comput. Appl. Math."},{"key":"ref_19","first-page":"3449","article-title":"Eighth-order methods with high efficiency index for solving nonlinear equations","volume":"215","author":"Liu","year":"2010","journal-title":"Appl. Math. Comput."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1611","DOI":"10.1016\/j.cam.2010.03.002","article-title":"New eighth-order methods for solving nonlinear equations","volume":"234","author":"Liu","year":"2010","journal-title":"J. Comput. Appl. Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"244","DOI":"10.1016\/j.cam.2007.10.054","article-title":"A family of multi-point iterative methods for nonlinear equations","volume":"222","author":"Nedzhibov","year":"2008","journal-title":"J. Comput. Appl. Math."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1007\/s11075-009-9359-z","article-title":"A modified Newton-Jarrat\u2019s composition","volume":"55","author":"Cordero","year":"2010","journal-title":"Numer. Algorithms"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"146","DOI":"10.1016\/j.cam.2006.10.072","article-title":"Some modification of Newton\u2019s method with fifth-order convergence","volume":"209","author":"Kou","year":"2007","journal-title":"J. Comput. Appl. Math."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Wang, X. (2021). Fixed-point iterative method with eight-order constructed by undermined paramater technique for solving nonlinear systems. Symmetry, 13.","DOI":"10.3390\/sym13050863"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"472","DOI":"10.1016\/j.aml.2009.12.006","article-title":"On some computational orders of convergence","volume":"23","author":"Noguera","year":"2010","journal-title":"Appl. Math. Lett."},{"key":"ref_26","first-page":"125849","article-title":"Higher order Jarratt-like iterations for solving systems of nonlinear equations","volume":"395","author":"Zhanlav","year":"2021","journal-title":"Appl. Math. Comput."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/11\/802\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:34:46Z","timestamp":1760114086000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/11\/802"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,11,18]]},"references-count":26,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2024,11]]}},"alternative-id":["axioms13110802"],"URL":"https:\/\/doi.org\/10.3390\/axioms13110802","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,11,18]]}}}