{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,17]],"date-time":"2026-02-17T04:22:15Z","timestamp":1771302135605,"version":"3.50.1"},"reference-count":27,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2021,7,26]],"date-time":"2021-07-26T00:00:00Z","timestamp":1627257600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100014440","name":"Ministerio de Ciencia, Innovaci\u00f3n y Universidades","doi-asserted-by":"publisher","award":["PGC2018-095896-B-C22 (MCIU\/AEI\/FEDER, UE)"],"award-info":[{"award-number":["PGC2018-095896-B-C22 (MCIU\/AEI\/FEDER, UE)"]}],"id":[{"id":"10.13039\/100014440","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100016968","name":"Fondo Nacional de Innovaci\u00f3n y Desarrollo Cient\u00edfico\u2013Tecnol\u00f3gico","doi-asserted-by":"publisher","award":["FONDOCYT 027 - 2018"],"award-info":[{"award-number":["FONDOCYT 027 - 2018"]}],"id":[{"id":"10.13039\/100016968","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this work, we use the technique of recurrence relations to prove the semilocal convergence in Banach spaces of the multidimensional extension of Chun\u2019s iterative method. This is an iterative method of fourth order, that can be transferred to the multivariable case by using the divided difference operator. We obtain the domain of existence and uniqueness by taking a suitable starting point and imposing a Lipschitz condition to the first Fr\u00e9chet derivative in the whole domain. Moreover, we apply the theoretical results obtained to a nonlinear integral equation of Hammerstein type, showing the applicability of our results.<\/jats:p>","DOI":"10.3390\/axioms10030161","type":"journal-article","created":{"date-parts":[[2021,7,26]],"date-time":"2021-07-26T09:25:52Z","timestamp":1627291552000},"page":"161","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Semilocal Convergence of the Extension of Chun\u2019s Method"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7462-9173","authenticated-orcid":false,"given":"Alicia","family":"Cordero","sequence":"first","affiliation":[{"name":"Instituto de Matem\u00e1tica Multidisciplinar, Universitat Polit\u00e8cnica de Val\u00e8ncia, Cno. de Vera s\/n, 46022 Val\u00e8ncia, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4594-071X","authenticated-orcid":false,"given":"Javier G.","family":"Maim\u00f3","sequence":"additional","affiliation":[{"name":"Instituto Tecnol\u00f3gico de Santo Domingo (INTEC), Santo Domingo 10602, Dominican Republic"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2869-4334","authenticated-orcid":false,"given":"Eulalia","family":"Mart\u00ednez","sequence":"additional","affiliation":[{"name":"Instituto de Matem\u00e1tica Multidisciplinar, Universitat Polit\u00e8cnica de Val\u00e8ncia, Cno. de Vera s\/n, 46022 Val\u00e8ncia, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9893-0761","authenticated-orcid":false,"given":"Juan R.","family":"Torregrosa","sequence":"additional","affiliation":[{"name":"Instituto de Matem\u00e1tica Multidisciplinar, Universitat Polit\u00e8cnica de Val\u00e8ncia, Cno. de Vera s\/n, 46022 Val\u00e8ncia, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7755-5635","authenticated-orcid":false,"given":"Mar\u00eda P.","family":"Vassileva","sequence":"additional","affiliation":[{"name":"Instituto Tecnol\u00f3gico de Santo Domingo (INTEC), Santo Domingo 10602, Dominican Republic"}]}],"member":"1968","published-online":{"date-parts":[[2021,7,26]]},"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","doi-asserted-by":"crossref","unstructured":"Petkovi\u0107, M.S., Neta, B., Petkovi\u0107, L.D., and D\u017duni\u0107, J. (2012). Multipoint Methods for Solving Nonlinear Equations, Academic Press.","DOI":"10.1016\/B978-0-12-397013-8.00002-9"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Amat, S., and Busquier, S. (2016). Advances in Iterative Methods for Nonlinear Equations, Springer.","DOI":"10.1007\/978-3-319-39228-8"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1007\/s40324-015-0040-2","article-title":"A study of optimization for Steffensen-type methods with frozen divided differences","volume":"70","author":"Ezquerro","year":"2015","journal-title":"SeMA"},{"key":"ref_5","first-page":"309","article-title":"Semilocal convergence of a k-step iterative process and its application for solving a special kind of conservative problems","volume":"6","author":"Teruel","year":"2017","journal-title":"Numer. Algor."},{"key":"ref_6","first-page":"1","article-title":"On the local convergence of a Newton-Kurchatov-type method for non-differentiable operators","volume":"304","author":"Rubio","year":"2017","journal-title":"Appl. Math. Comput."},{"key":"ref_7","first-page":"23","article-title":"A new convergence theorem for Steffensen\u2019s method on Banach spaces and applications","volume":"1","author":"Argyros","year":"1997","journal-title":"Southwest Pure Appl. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1016\/j.cam.2007.05.008","article-title":"A Steffensen\u2019s type method in Banach spaces with applications on boundary-value problems","volume":"216","author":"Amat","year":"2008","journal-title":"Comput. Appl. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1084","DOI":"10.1016\/j.jmaa.2005.12.078","article-title":"A two-step Steffensen\u2019s method under modified convergence conditions","volume":"324","author":"Amat","year":"2006","journal-title":"Math. Anal. Appl."},{"key":"ref_10","first-page":"819","article-title":"On a Steffensen\u2019s type method and its behavior for semismooth equations","volume":"177","author":"Amat","year":"2006","journal-title":"Appl. Math. Comput."},{"key":"ref_11","first-page":"203","article-title":"On a modified Secant method","volume":"8","author":"Potra","year":"1979","journal-title":"Anal. Number. Theor. Approx."},{"key":"ref_12","unstructured":"Potra, F.A., and Ptak, V. (1984). Nondiscrete Induction and Iterative Processes, Research Notes in Mathematics 103; Pitman."},{"key":"ref_13","first-page":"315","article-title":"On the convergence of a damped Secant-like method with modified right hand side vector","volume":"252","author":"Argyros","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_14","first-page":"3","article-title":"On existence of divided differences in linear spaces","volume":"2","author":"Balazs","year":"1973","journal-title":"Rev. Anal. Numer. Theor. Approx."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1016\/j.cam.2012.06.005","article-title":"On the approximation of derivatives using divided difference operators preserving the local convergence order of iterative methods","volume":"237","author":"Noguera","year":"2013","journal-title":"J. Comput. Appl. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1590\/S1807-03022010000100002","article-title":"Some derivative free quadratic and cubic convergence iterative formulas for solving nonlinear equations","volume":"29","author":"Dehghan","year":"2010","journal-title":"Comput. Appl. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"430","DOI":"10.1007\/s10910-014-0432-z","article-title":"Solving nonlinear problems by Ostrowski-Chun type parametric families","volume":"53","author":"Cordero","year":"2015","journal-title":"J. Math. Chem."},{"key":"ref_18","first-page":"133","article-title":"A family of seventh-order schemes for solving nonlinear systems","volume":"57","author":"Abad","year":"2014","journal-title":"Bull. Math. Soc. Sci. Math. Roum."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1007\/BF02241866","article-title":"Recurrence relations for rational cubic models I: The Halley method","volume":"44","author":"Candela","year":"1990","journal-title":"Computing"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1007\/BF02238803","article-title":"Recurrence relations for rational cubic models II: The Chebyshev method","volume":"45","author":"Candela","year":"1990","journal-title":"Computing"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1093\/imanum\/22.2.187","article-title":"Generalized differentiability conditions for Newton\u2019s method","volume":"22","author":"Ezquerro","year":"2002","journal-title":"IMA Numer. Anal."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"354","DOI":"10.1016\/j.apnum.2006.05.001","article-title":"Halley\u2019s method for operators with unbounded second derivative","volume":"57","author":"Ezquerro","year":"2007","journal-title":"Appl. Numer. Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1016\/j.jco.2009.04.001","article-title":"An optimization of Chebyshev\u2019s method","volume":"25","author":"Ezquerro","year":"2009","journal-title":"Complexity"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1016\/j.cam.2014.06.008","article-title":"Semilocal convergence by using recurrence relations for a fifth-order method in Banach spaces","volume":"273","author":"Cordero","year":"2015","journal-title":"Comput. Appl. Math."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1080\/00036818908839899","article-title":"Integral equations arising in the kinetic theory of gases","volume":"34","author":"Hu","year":"1989","journal-title":"Appl. Anal."},{"key":"ref_26","first-page":"45","article-title":"Application of fixed point method for solving nonlinear Volterra-Hammerstein Integral Equation","volume":"74","author":"Maleknejad","year":"2012","journal-title":"Univ. Politeh. Buchar. Sci. Bull. Ser. A"},{"key":"ref_27","first-page":"686","article-title":"Variants of Newton\u2019s method using fifth-order quadrature formulas","volume":"190","author":"Cordero","year":"2007","journal-title":"Appl. Math. Comput."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/10\/3\/161\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:34:56Z","timestamp":1760164496000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/10\/3\/161"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,26]]},"references-count":27,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2021,9]]}},"alternative-id":["axioms10030161"],"URL":"https:\/\/doi.org\/10.3390\/axioms10030161","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,7,26]]}}}