<?xml version="1.0" encoding="ISO-8859-1"?><article xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">
<front>
<journal-meta>
<journal-id>0034-7426</journal-id>
<journal-title><![CDATA[Revista Colombiana de Matemáticas]]></journal-title>
<abbrev-journal-title><![CDATA[Rev.colomb.mat.]]></abbrev-journal-title>
<issn>0034-7426</issn>
<publisher>
<publisher-name><![CDATA[Universidad Nacional de Colombia y Sociedad Colombiana de Matemáticas]]></publisher-name>
</publisher>
</journal-meta>
<article-meta>
<article-id>S0034-74262021000200167</article-id>
<article-id pub-id-type="doi">10.15446/recolma.v55n2.102622</article-id>
<title-group>
<article-title xml:lang="en"><![CDATA[New linearization method for nonlinear problems in Hilbert space]]></article-title>
<article-title xml:lang="es"><![CDATA[Nuevo método de linealización para problemas no lineales en espacios de Hilbert]]></article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Bouazila]]></surname>
<given-names><![CDATA[Nada]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Guebbai]]></surname>
<given-names><![CDATA[Hamza]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Merchela]]></surname>
<given-names><![CDATA[Wassim]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
</contrib-group>
<aff id="Af1">
<institution><![CDATA[,Université 8 Mai 1945  ]]></institution>
<addr-line><![CDATA[Guelma ]]></addr-line>
<country>Algeria</country>
</aff>
<aff id="Af2">
<institution><![CDATA[,Derzhavin Tambov State University  ]]></institution>
<addr-line><![CDATA[Tambov ]]></addr-line>
<country>Russia</country>
</aff>
<pub-date pub-type="pub">
<day>00</day>
<month>12</month>
<year>2021</year>
</pub-date>
<pub-date pub-type="epub">
<day>00</day>
<month>12</month>
<year>2021</year>
</pub-date>
<volume>55</volume>
<numero>2</numero>
<fpage>167</fpage>
<lpage>175</lpage>
<copyright-statement/>
<copyright-year/>
<self-uri xlink:href="http://www.scielo.org.co/scielo.php?script=sci_arttext&amp;pid=S0034-74262021000200167&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.org.co/scielo.php?script=sci_abstract&amp;pid=S0034-74262021000200167&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.org.co/scielo.php?script=sci_pdf&amp;pid=S0034-74262021000200167&amp;lng=en&amp;nrm=iso"></self-uri><abstract abstract-type="short" xml:lang="en"><p><![CDATA[Abstract: In this paper, we build a Newton-like sequence to approach the zero of a nonlinear Fréchet differentiable function defined in Hilbert space. This new iterative sequence uses the concept of the adjoint operator, which makes it more manageable in practice compared to the one developed by Kantorovich which requires the calculation of the inverse operator in each iteration. Because the calculation of the adjoint operator is easier compared to the calculation of the inverse operator which requires in practice solving a system of linear equations, our new method makes the calculation of the term of our new sequence easier and more convenient for numerical approximations. We provide an a priori convergence theorem of this sequence, where we use hypotheses equivalent to those constructed by Kantorovich, and we show that our new iterative sequence converges towards the solution.]]></p></abstract>
<abstract abstract-type="short" xml:lang="es"><p><![CDATA[Resumen: En este artículo, construimos una sucesión similar a la de Newton para acercarnos al cero de una función diferenciable en el sentido Fréchet no lineal definida en un espacio de Hilbert. Esta nueva sucesión utiliza el concepto del adjunto del operador, que hace que el proceso iterativo sea más manejable en la práctica en comparación al desarrollado por Kantorovich que requiere el cálculo del operador inverso en cada iteración. Dado que el cálculo del operador adjunto es más fácil en comparación con el cálculo del operador inverso que en la práctica equivale a resolver un sistema de ecuaciones, nuestra nuevo método hace que el cálculo del término de nuestra nueva sucesión sea más fácil y conveniente para la aproximación numérica. Proporcionamos un teorema de convergencia a priori de esta sucesión, donde usamos unas hipótesis equivalentes a las construidas por Kantorovich, y mostramos que nuestra nueva sucesión iterativa converge hacia la solución.]]></p></abstract>
<kwd-group>
<kwd lng="en"><![CDATA[Nonlinear problems]]></kwd>
<kwd lng="en"><![CDATA[Newton-like method]]></kwd>
<kwd lng="en"><![CDATA[Fréchet differentiability]]></kwd>
<kwd lng="en"><![CDATA[Adjoint Operator]]></kwd>
<kwd lng="es"><![CDATA[Problemas no lineales]]></kwd>
<kwd lng="es"><![CDATA[método tipo Newton]]></kwd>
<kwd lng="es"><![CDATA[diferenciabilidad de Fréchet]]></kwd>
<kwd lng="es"><![CDATA[operador adjunto]]></kwd>
</kwd-group>
</article-meta>
</front><back>
<ref-list>
<ref id="B1">
<label>1</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Ahues]]></surname>
<given-names><![CDATA[M.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[A note on perturbed fixed slope iterations]]></article-title>
<source><![CDATA[Appl. Math. Letters]]></source>
<year>2005</year>
<volume>18</volume>
<numero>4</numero>
<issue>4</issue>
<page-range>375-80</page-range></nlm-citation>
</ref>
<ref id="B2">
<label>2</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Argyros]]></surname>
<given-names><![CDATA[I. K.]]></given-names>
</name>
<name>
<surname><![CDATA[Magre&#324;án]]></surname>
<given-names><![CDATA[Á. A.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Extended convergence results for the newton-kantorovich iteration]]></article-title>
<source><![CDATA[J. of Comp. and Appl. Math]]></source>
<year>2015</year>
<volume>286</volume>
<numero>3</numero>
<issue>3</issue>
<page-range>54-67</page-range></nlm-citation>
</ref>
<ref id="B3">
<label>3</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Cianciaruso]]></surname>
<given-names><![CDATA[F.]]></given-names>
</name>
<name>
<surname><![CDATA[De Pascale]]></surname>
<given-names><![CDATA[E.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Estimates of majorizing sequences in the newton kantorovich method: A further improvement]]></article-title>
<source><![CDATA[J. of Math. Anal. And Appl]]></source>
<year>2006</year>
<volume>322</volume>
<numero>1</numero>
<issue>1</issue>
<page-range>329-35</page-range></nlm-citation>
</ref>
<ref id="B4">
<label>4</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Eshkuvatov]]></surname>
<given-names><![CDATA[Z. K.]]></given-names>
</name>
<name>
<surname><![CDATA[Hameed]]></surname>
<given-names><![CDATA[H. H.]]></given-names>
</name>
<name>
<surname><![CDATA[Nik Long]]></surname>
<given-names><![CDATA[N. M. A.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[One dimensional nonlinear integral operator with newton kantorovich method]]></article-title>
<source><![CDATA[J. of King Saud University - Science]]></source>
<year>2016</year>
<volume>28</volume>
<numero>2</numero>
<issue>2</issue>
<page-range>172-7</page-range></nlm-citation>
</ref>
<ref id="B5">
<label>5</label><nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Ezquerro]]></surname>
<given-names><![CDATA[A. J.]]></given-names>
</name>
<name>
<surname><![CDATA[Hernández]]></surname>
<given-names><![CDATA[V.]]></given-names>
</name>
<name>
<surname><![CDATA[Angel]]></surname>
<given-names><![CDATA[M.]]></given-names>
</name>
</person-group>
<source><![CDATA[Newton's Method: an Updated Approach of Kantorovich's Theory]]></source>
<year>2017</year>
<publisher-name><![CDATA[Birkhäuser]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B6">
<label>6</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Grammont]]></surname>
<given-names><![CDATA[L.]]></given-names>
</name>
<name>
<surname><![CDATA[Ahues]]></surname>
<given-names><![CDATA[M.]]></given-names>
</name>
<name>
<surname><![CDATA[D'Almeida]]></surname>
<given-names><![CDATA[F. D.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[For nonlinear infinite dimensional equations, which to begin with: Linearization or discretization?]]></article-title>
<source><![CDATA[J. Integral Equations Applications]]></source>
<year>2014</year>
<volume>26</volume>
<numero>3</numero>
<issue>3</issue>
<page-range>413-36</page-range></nlm-citation>
</ref>
<ref id="B7">
<label>7</label><nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Kantorovich]]></surname>
<given-names><![CDATA[L. V.]]></given-names>
</name>
<name>
<surname><![CDATA[Akilov]]></surname>
<given-names><![CDATA[G. P.]]></given-names>
</name>
</person-group>
<source><![CDATA[Functional Analysis]]></source>
<year>1982</year>
<publisher-name><![CDATA[Pergamon Press]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B8">
<label>8</label><nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Kantorovich]]></surname>
<given-names><![CDATA[L. V.]]></given-names>
</name>
<name>
<surname><![CDATA[Krilov]]></surname>
<given-names><![CDATA[V. I.]]></given-names>
</name>
</person-group>
<source><![CDATA[Approximate Methods of Higher Analysis]]></source>
<year>1958</year>
<publisher-name><![CDATA[Interscience Publishers Inc]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B9">
<label>9</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Khellaf]]></surname>
<given-names><![CDATA[A.]]></given-names>
</name>
<name>
<surname><![CDATA[Merchela]]></surname>
<given-names><![CDATA[W.]]></given-names>
</name>
<name>
<surname><![CDATA[Benarab]]></surname>
<given-names><![CDATA[S.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[New numerical process solving nonlinear infinite-dimensional equations]]></article-title>
<source><![CDATA[Comp. Appl. Math]]></source>
<year>2020</year>
<volume>39</volume>
<numero>93</numero>
<issue>93</issue>
</nlm-citation>
</ref>
</ref-list>
</back>
</article>
