{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T10:44:10Z","timestamp":1767869050107,"version":"3.49.0"},"reference-count":21,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2026,1,6]],"date-time":"2026-01-06T00:00:00Z","timestamp":1767657600000},"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 parametric family of two-step methods, with its special cases, has been introduced in various papers. However, in most cases, the local convergence analysis relies on the existence of derivatives of orders that the method does not require. Moreover, the more challenging semi-local convergence analysis was not introduced for this class of methods. These drawbacks are considered in this paper. We determine the radius of convergence and the uniqueness of the solution based on generalized continuity conditions. We also present the semi-local convergence analysis for this family of methods, which has not been studied before, using majorizing sequences. Numerical experiments and basins of attraction are included to validate the theoretical conditions and demonstrate the stability of the methods.<\/jats:p>","DOI":"10.3390\/axioms15010041","type":"journal-article","created":{"date-parts":[[2026,1,6]],"date-time":"2026-01-06T15:09:36Z","timestamp":1767712176000},"page":"41","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Extended Parametric Family of Two-Step Methods with Applications for Solving Nonlinear Equations or Systems"],"prefix":"10.3390","volume":"15","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"}]},{"ORCID":"https:\/\/orcid.org\/0009-0008-2386-8298","authenticated-orcid":false,"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":[[2026,1,6]]},"reference":[{"key":"ref_1","unstructured":"Ortega, J.M., and Rheinboldt, W.C. (1970). Iterative Solution of Nonlinear Equations in Several Variables, Academic Press."},{"key":"ref_2","unstructured":"Traub, J.F. (1982). Iterative Methods for the Solution of Equations, Chelsea Publishing Company."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Argyros, I.K., and Shakhno, S. (2019). Extending the applicability of two-step solvers for solving equations. Mathematics, 7.","DOI":"10.3390\/math7010062"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"32","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_5","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_6","doi-asserted-by":"crossref","first-page":"1706841","DOI":"10.1155\/2020\/1706841","article-title":"A new high-order and efficient family of iterative techniques for nonlinear models","volume":"2020","author":"Behl","year":"2020","journal-title":"Complexity"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"125849","DOI":"10.1016\/j.amc.2020.125849","article-title":"Higher order Jarratt-like iterations for solving systems of nonlinear equations","volume":"395","author":"Zhanlav","year":"2021","journal-title":"Appl. Math. Comput."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1007\/s40314-022-02159-9","article-title":"Simple yet highly efficient numerical techniques for systems of nonlinear equations","volume":"42","author":"Singh","year":"2023","journal-title":"Comput. Appl. Math."},{"key":"ref_9","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_10","doi-asserted-by":"crossref","first-page":"1802","DOI":"10.1016\/j.mcm.2010.11.063","article-title":"Approximation of artificial satellites\u2019 preliminary orbits: The efficiency challenge","volume":"54","author":"Arroyo","year":"2011","journal-title":"Math. Comput. Model."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Cordero, A., Rojas-Hiciano, R.V., Torregrosa, J.R., and Vassileva, M.P. (2023). Fractal complexity of a new biparametric family of fourth optimal order based on the Ermakov-Kalitkin scheme. Fractal Fract., 7.","DOI":"10.3390\/fractalfract7060459"},{"key":"ref_12","first-page":"69","article-title":"Solving system of non-linear equations using family of Jarratt methods","volume":"12","author":"Khirallah","year":"2013","journal-title":"Int. J. Differ. Equ. Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1565","DOI":"10.1515\/ijnsns-2021-0202","article-title":"Higher order Traub-Steffensen type methods and their convergence analysis in Banach spaces","volume":"24","author":"Kumar","year":"2023","journal-title":"Int. J. Nonlinear Sci. Numer. Simul."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1017","DOI":"10.1080\/00207170500197605","article-title":"Numerical approach to the non-linear diofantic equations with applications to the controllability of infinite dimensional dynamical systems","volume":"78","author":"Respondek","year":"2005","journal-title":"Int. J. Control"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1879","DOI":"10.1007\/s11075-023-01631-9","article-title":"A highly efficient class of optimal fourth-order methods for solving nonlinear systems","volume":"95","author":"Cordero","year":"2024","journal-title":"Numer. Algorithms"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"101907","DOI":"10.1016\/j.jco.2024.101907","article-title":"A two-point Newton-like method of optimal fourth order convergence for systems of nonlinear equations","volume":"86","author":"Singh","year":"2025","journal-title":"J. Complex."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2377","DOI":"10.1016\/j.amc.2011.08.011","article-title":"Ostrowski type methods for solving systems of nonlinear equations","volume":"218","author":"Grau","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"709","DOI":"10.1007\/s40010-019-00617-4","article-title":"On a Three-Step Efficient Fourth-Order Method for Computing the Numerical Solution of System of Nonlinear Equations and Its Applications","volume":"90","author":"Singh","year":"2020","journal-title":"Proc. Natl. Acad. Sci. India Sect. A Phys. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1007\/s11075-012-9585-7","article-title":"An efficient fourth order weighted-Newton method for systems of nonlinear equations","volume":"62","author":"Sharma","year":"2013","journal-title":"Numer. Algorithms"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1007\/s11075-022-01412-w","article-title":"A simple yet efficient two-step fifth-order weighted-Newton method for nonlinear models","volume":"93","author":"Singh","year":"2023","journal-title":"Numer. Algorithms"},{"key":"ref_21","unstructured":"Ostrowski, A.M. (1960). Solution of Equation and Systems of Equations, Academic Press."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/15\/1\/41\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T05:20:52Z","timestamp":1767849652000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/15\/1\/41"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1,6]]},"references-count":21,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,1]]}},"alternative-id":["axioms15010041"],"URL":"https:\/\/doi.org\/10.3390\/axioms15010041","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1,6]]}}}