{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:45:13Z","timestamp":1760240713586,"version":"build-2065373602"},"reference-count":13,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2019,8,3]],"date-time":"2019-08-03T00:00:00Z","timestamp":1564790400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004054","name":"King Abdulaziz University","doi-asserted-by":"publisher","award":["247-253-D1440)"],"award-info":[{"award-number":["247-253-D1440)"]}],"id":[{"id":"10.13039\/501100004054","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Problems from numerous disciplines such as applied sciences, scientific computing, applied mathematics, engineering to mention some can be converted to solving an equation. That is why, we suggest higher-order iterative method to solve equations with Banach space valued operators. Researchers used the suppositions involving seventh-order derivative by Chen, S.P. and Qian, Y.H. But, here, we only use suppositions on the first-order derivative and Lipschitz constrains. In addition, we do not only enlarge the applicability region of them but also suggest computable radii. Finally, we consider a good mixture of numerical examples in order to demonstrate the applicability of our results in cases not covered before.<\/jats:p>","DOI":"10.3390\/sym11081002","type":"journal-article","created":{"date-parts":[[2019,8,5]],"date-time":"2019-08-05T03:25:22Z","timestamp":1564975522000},"page":"1002","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Ball Convergence for Combined Three-Step Methods Under Generalized Conditions in Banach Space"],"prefix":"10.3390","volume":"11","author":[{"given":"R. A.","family":"Alharbey","sequence":"first","affiliation":[{"name":"Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ioannis K.","family":"Argyros","sequence":"additional","affiliation":[{"name":"Department of Mathematics Sciences, Cameron University, Lawton, OK 73505, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ramandeep","family":"Behl","sequence":"additional","affiliation":[{"name":"Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,8,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"22","DOI":"10.12691\/ajams-5-1-5","article-title":"A family of combined iterative methods for solving nonlinear equations","volume":"5","author":"Chen","year":"2017","journal-title":"Am. J. Appl. Math. Stat."},{"key":"ref_2","unstructured":"Argyros, I.K. (2008). Convergence and Application of Newton-Type Iterations, Springer."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Rheinboldt, W.C. (1978). An Adaptive Continuation Process for Solving Systems of Nonlinear Equations, Polish Academy of Science, Banach Center Publications.","DOI":"10.4064\/-3-1-129-142"},{"key":"ref_4","unstructured":"Traub, J.F. (1964). Iterative Methods for the Solution of Equations, Prentice-Hall."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Argyros, I.K., and Hilout, S. (2013). Computational Methods in Nonlinear Analysis, World Scientific Publishing Company.","DOI":"10.1142\/8475"},{"key":"ref_6","first-page":"117","article-title":"A third-order modification of Newton method for systems of nonlinear equations","volume":"191","author":"Kou","year":"2007","journal-title":"Appl. Math. Comput."},{"key":"ref_7","unstructured":"Petkovic, M.S., Neta, B., Petkovic, L., and D\u017euni\u010d, J. (2013). Multipoint Methods for Solving Nonlinear Equations, Elsevier."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1007\/s11075-012-9585-7","article-title":"An efficient fourth-order weighted-Newton method for system of nonlinear equations","volume":"62","author":"Sharma","year":"2013","journal-title":"Numer. Algor."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1007\/s10543-009-0226-z","article-title":"New iterations of R-order four with reduced computational cost","volume":"49","author":"Ezquerro","year":"2009","journal-title":"BIT Numer. Math."},{"key":"ref_10","first-page":"164","article-title":"A modified Chebyshev\u2019s iterative method with at least sixth order of convergence","volume":"206","author":"Amat","year":"2008","journal-title":"Appl. Math. Comput."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1007\/s40324-015-0047-8","article-title":"Ball convergence theorems and the convergence planes of an iterative methods for nonlinear equations","volume":"71","author":"Argyros","year":"2015","journal-title":"SeMA"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1016\/j.cam.2012.11.024","article-title":"Increasing the order of convergence of iterative schemes for solving nonlinear system","volume":"252","author":"Cordero","year":"2012","journal-title":"J. Comput. Appl. Math."},{"key":"ref_13","unstructured":"Potra, F.A., and Pt\u00e1k, V. (1984). Nondiscrete Introduction and Iterative Process. Research Notes in Mathematics, Pitman Advanced Publishing Program."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/8\/1002\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:13:02Z","timestamp":1760188382000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/8\/1002"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,8,3]]},"references-count":13,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2019,8]]}},"alternative-id":["sym11081002"],"URL":"https:\/\/doi.org\/10.3390\/sym11081002","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2019,8,3]]}}}