{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,2]],"date-time":"2026-04-02T22:05:55Z","timestamp":1775167555555,"version":"3.50.1"},"reference-count":36,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2025,9,28]],"date-time":"2025-09-28T00:00:00Z","timestamp":1759017600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["www.mdpi.com"],"crossmark-restriction":true},"short-container-title":["Axioms"],"abstract":"<jats:p>The local convergence analysis of the m+1-step Newton-Jarratt composite scheme with order 2m+1 has been shown previously. But the convergence order 2m+1 is obtained using Taylor series and assumptions on the existence of at least the fifth derivative of the mapping involved, which is not present in the method. These assumptions limit the applicability of the method. A priori error estimates or the radius of convergence or uniqueness of the solution results have not been given either. These drawbacks are addressed in this paper. In particular, the convergence is based only on the operators on the method, which are the operator and its first derivative. Moreover, the radius of convergence is established, a priori estimates and the isolation of the solution is discussed using generalized continuity assumptions on the derivative. Furthermore, the more challenging semi-local convergence analysis, not previously studied, is presented using majorizing sequences. The convergence for both analyses depends on the generalized continuity of the Jacobian of the mapping involved, which is used to control it and sharpen the error distances. Numerical examples validate the sufficient convergence conditions presented in the theory.<\/jats:p>","DOI":"10.3390\/axioms14100734","type":"journal-article","created":{"date-parts":[[2025,9,30]],"date-time":"2025-09-30T09:12:23Z","timestamp":1759223543000},"page":"734","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Extending the Applicability of Newton-Jarratt-like Methods with Accelerators of Order 2m + 1 for Solving Nonlinear Systems"],"prefix":"10.3390","volume":"14","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"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"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"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mykhailo","family":"Shakhov","sequence":"additional","affiliation":[{"name":"Department of Theory of Optimal Processes, Ivan Franko National University of Lviv, Universytetska Str. 1, 79000 Lviv, Ukraine"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,9,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Argyros, I.K., and Shakhno, S. (2019). Extended Local Convergence for the Combined Newton-Kurchatov Method Under the Generalized Lipschitz Conditions. Mathematics, 7.","DOI":"10.3390\/math7020207"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1007\/s100920050020","article-title":"An iterative method for the computation of the solutions of nonlinear equations","volume":"36","author":"Costabile","year":"1999","journal-title":"Calcolo"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1007\/s40324-015-0050-0","article-title":"An improved bisection Newton-like method for enclosing simple zeros of nonlinear equations","volume":"72","author":"Nisha","year":"2015","journal-title":"SeMA J."},{"key":"ref_4","unstructured":"Ortega, J.M., and Rheinboldt, W.C. (1970). Iterative Solution of Nonlinear Equations in Several Variables, Academic Press."},{"key":"ref_5","unstructured":"Traub, J.F. (1964). Iterative Methods for the Solution of Equations, Prentice-Hall."},{"key":"ref_6","unstructured":"Ostrowski, A.M. (1960). Solution of Equations and Systems of Equations, Academic Press."},{"key":"ref_7","first-page":"1678","article-title":"Super cubic iterative methods to solve systems of nonlinear equations","volume":"188","author":"Darvishi","year":"2007","journal-title":"Appl. Math. Comput."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1016\/j.cam.2003.12.041","article-title":"A modified Newton method with cubic convergence: The multivariate case","volume":"169","author":"Homeier","year":"2004","journal-title":"J. Comput. Appl. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/j.camwa.2008.10.067","article-title":"Some iterative methods for solving a system of nonlinear equations","volume":"57","author":"Noor","year":"2009","journal-title":"Comput. Math. Appl."},{"key":"ref_10","first-page":"300","article-title":"A new class of methods with higher order of convergence for solving systems of nonlinear equations","volume":"264","author":"Xiao","year":"2016","journal-title":"Appl. Math. Comput."},{"key":"ref_11","first-page":"70","article-title":"Stable high-order iterative methods for solving nonlinear models","volume":"303","author":"Behl","year":"2017","journal-title":"Appl. Math. Comput."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"893","DOI":"10.1007\/s10910-014-0464-4","article-title":"A new fourth-order family for solving nonlinear problems and its dynamics","volume":"53","author":"Cordero","year":"2015","journal-title":"J. Math. Chem."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2369","DOI":"10.1016\/j.aml.2012.07.005","article-title":"Increasing the convergence order of an iterative method for nonlinear systems","volume":"25","author":"Cordero","year":"2012","journal-title":"Appl. Math. Lett."},{"key":"ref_14","first-page":"1093","article-title":"An efficient three-step method to solve system of non linear equations","volume":"266","author":"Esmaeili","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_15","first-page":"98","article-title":"An improved Newton-Traub composition for solving systems of nonlinear equations","volume":"290","author":"Sharma","year":"2016","journal-title":"Appl. Math. Comput."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1007\/s40324-015-0055-8","article-title":"An efficient derivative-free family of seventh order methods for systems of nonlinear equations","volume":"73","author":"Sharma","year":"2016","journal-title":"SeMA J."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/S0893-9659(00)00100-2","article-title":"A Variant of Newton\u2019s Method with Accelerated Third-Order Convergence","volume":"13","author":"Weerakoon","year":"2000","journal-title":"Appl. Math. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1007\/s10092-015-0149-9","article-title":"Increasing the order of convergence for iterative methods to solve nonlinear systems","volume":"53","author":"Xiao","year":"2016","journal-title":"Calcolo"},{"key":"ref_19","first-page":"8","article-title":"Accelerating the convergence speed of iterative methods for solving nonlinear systems","volume":"333","author":"Xiao","year":"2018","journal-title":"Appl. Math. Comput."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1007\/s40314-021-01739-5","article-title":"A class of accurate Newton\u2013Jarratt-like methods with applications to nonlinear models","volume":"41","author":"Sharma","year":"2022","journal-title":"Comput. Appl. Math."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Argyros, I.K. (2022). The Theory and Application of Iteration Methods, Taylor and Francis. [2nd ed.].","DOI":"10.1201\/9781003128915"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1007\/s40819-020-0784-y","article-title":"Extended Two-Step-Kurchatov Method for Solving Banach Space Valued Nondifferentiable Equations","volume":"6","author":"Argyros","year":"2020","journal-title":"Int. J. Appl. Comput. Math."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Argyros, I.K., Shakhno, S., and Yarmola, H. (2019). Two-Step Solver for Nonlinear Equations. Symmetry, 11.","DOI":"10.3390\/sym11020128"},{"key":"ref_24","first-page":"80","article-title":"Some higher-order iteration functions for solving nonlinear models","volume":"334","author":"Alzahrani","year":"2018","journal-title":"Appl. Math. Comput."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1921","DOI":"10.1080\/00207160.2014.946412","article-title":"Some new efficient multipoint iterative methods for solving nonlinear systems of equations","volume":"92","author":"Cordero","year":"2015","journal-title":"Int. J. Comput. Math."},{"key":"ref_26","first-page":"322","article-title":"An integral equation formalism for solving the nonlinear Klein\u2013Gordon equation","volume":"243","author":"Jang","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Madhu, K., Elango, A., Landry, R.J., and Al-arydah, M. (2020). New Multi-Step Iterative Methods for Solving Systems of Nonlinear Equations and Their Application on GNSS Pseudorange Equations. Sensors, 20.","DOI":"10.3390\/s20215976"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1145\/355934.355936","article-title":"Testing Unconstrained Optimization Software","volume":"7","author":"Garbow","year":"1981","journal-title":"ACM Trans. Math. Softw."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1145\/355815.355820","article-title":"Numerical solution of nonlinear equations","volume":"5","author":"Cosnard","year":"1979","journal-title":"ACM Trans. Math. Softw."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Motsa, S.S., and Shateyi, S. (2012). New Analytic Solution to the Lane\u2013Emden Equation of Index 2. Math. Probl. Eng., 614796.","DOI":"10.1155\/2012\/614796"},{"key":"ref_31","unstructured":"Regmi, S. (2021). Optimized Iterative Methods with Applications in Diverse Disciplines, Nova Science Publisher."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1007\/s40324-016-0085-x","article-title":"Improved Newton-like methods for solving systems of nonlinear equations","volume":"74","author":"Sharma","year":"2016","journal-title":"SeMA J."},{"key":"ref_33","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_34","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_35","first-page":"972","article-title":"Higher order multi-step iterative method for computing the numerical solution of systems of nonlinear equations: Application to nonlinear PDEs and ODEs","volume":"269","author":"Ullah","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"103394","DOI":"10.1016\/j.mex.2025.103394","article-title":"A new iterative multi-step method for solving nonlinear equation","volume":"15","author":"Usman","year":"2025","journal-title":"MethodsX"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/10\/734\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,30]],"date-time":"2025-09-30T09:31:16Z","timestamp":1759224676000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/10\/734"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,28]]},"references-count":36,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2025,10]]}},"alternative-id":["axioms14100734"],"URL":"https:\/\/doi.org\/10.3390\/axioms14100734","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,9,28]]}}}