{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,31]],"date-time":"2025-10-31T14:27:54Z","timestamp":1761920874103,"version":"build-2065373602"},"reference-count":36,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2021,4,18]],"date-time":"2021-04-18T00:00:00Z","timestamp":1618704000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We study the local convergence of a family of fifth and sixth convergence order derivative free methods for solving Banach space valued nonlinear models. Earlier results used hypotheses up to the seventh derivative to show convergence. However, we only use the first divided difference of order one as well as the first derivative in our analysis. We also provide computable radius of convergence, error estimates, and uniqueness of the solution results not given in earlier studies. Hence, we expand the applicability of these methods. The dynamical analysis of the discussed family is also presented. Numerical experiments complete this article.<\/jats:p>","DOI":"10.3390\/sym13040715","type":"journal-article","created":{"date-parts":[[2021,4,19]],"date-time":"2021-04-19T21:59:49Z","timestamp":1618869589000},"page":"715","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["A Family of Fifth and Sixth Convergence Order Methods for Nonlinear Models"],"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\/0000-0001-8456-6391","authenticated-orcid":false,"given":"Debasis","family":"Sharma","sequence":"additional","affiliation":[{"name":"Department of Mathematics, IIIT Bhubaneswar, Odisha 751003, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Christopher I.","family":"Argyros","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of Oklahoma, Norman, OK 73071, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2489-4080","authenticated-orcid":false,"given":"Sanjaya Kumar","family":"Parhi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Fakir Mohan University, Odisha 756020, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shanta Kumari","family":"Sunanda","sequence":"additional","affiliation":[{"name":"Department of Mathematics, IIIT Bhubaneswar, Odisha 751003, India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,4,18]]},"reference":[{"key":"ref_1","unstructured":"Argyros, I.K. 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