On the weakening of the convergence of Newton's method using recurrent functions

We use Newton's method to approximate a locally unique solution of an equation in a Banach space setting. We introduce recurrent functions to provide a weaker semilocal convergence analysis for Newton's method than before [J. Appell, E. De Pascale, J.V. Lysenko, P.P. Zabrejko, New results on Newton-Kantorovich approximations with applications to nonlinear integral equations, Numer. Funct. Anal. Optim. 18 (1997) 1-17; I.K. Argyros, The theory and application of abstract polynomial equations, in: Mathematics Series, St. Lucie/CRC/Lewis Publ., Boca Raton, Florida, USA, 1998; I.K. Argyros, Concerning the "terra incognita" between convergence regions of two Newton methods, Nonlinear Anal. 62 (2005) 179-194; I.K. Argyros, Convergence and Applications of Newton-Type Iterations, Springer-Verlag Publ., New York, 2008: S. Chandrasekhar, Radiative Transfer, Dover Publ., New York. 1960; F. Cianciaruso, E. De Pascale, Newton-Kantorcivich approximations when the derivative is Holderian: Old and new results, Numer. Funct. Anal. Optim. 24 (2003) 713-723; N.T. Demidovich, P.P. Zabrejko, Ju.V. Lysenko, Some remarks on the Newton-Kantorovich method for nonlinear equations with Holder continuous linearizations, Izv. Akad. Nauk Belorus 3 (1993) 22-26. (in Russian): E. De Pascale, P.P. Zabrejko, Convergence of the Newton-Kantorovich method under Vertgeim conditions: A new improvement, Z. Anal, Anwendvugen 17 (1998) 271-280; LV. Kantorovich, G.P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982; J.V. Lysenko, Conditions for the convergence of the Newton-Kantorovich method for nonlinear equations with Hblder linearizations, Dokl. Akad. Nauk BSSR 38 (1994) 20-24. (in Russian); B.A. Vertgeim, On conditions for the applicability of Newton's method, (Russian), Dokl. Akad. Nauk., SSSR 110 (1956) 719-722: B.A. Vertgeim, On some methods for the approximate solution of nonlinear functional equations in Banach spaces, Uspekhi Mat. Nauk 12 (1957) 166-169. (in Russian): English trans].:: Amer. Math. Soc. Transl. 1 (1960) 378-382] provided that the Frechet-derivative of the operator involved is p-Holder continuous (p is an element of (0, 1]). Numerical examples involving integral and differential equations are also provided in this study. Crown Copyright (C) 2009 Published by Elsevier Inc. All rights reserved.