Weaker convergence for Newton's method under Hölder differentiability

Abstract

We use more precise majorizing sequences than in earlier studies such as [J. Appell, E. De Pascale, J.V. Lysenko, and P.P. Zabrejko, New results on Newton–Kantorovich approximations with applications to nonlinear integral equations, Numer. Funct. Anal. Optim. 18 (1997), pp. 1–17; I.K. Argyros, Concerning the ‘terra incognita’ between convergence regions of two Newton methods, Nonlinear Anal. 62 (2005), pp. 179–194; F. Cianciaruso, A further journey in the ‘terra incognita’ of the Newton–Kantorovich method, Nonlinear Funct. Anal. Appl. 15 (2010), pp. 173–183; F. Cianciaruso and E. De Pascale, Newton–Kantorovich approximations when the derivative is Hölderian: Old and new results, Numer. Funct. Anal. Optim. 24 (2003), pp. 713–723; F. Cianciaruso, E. De Pascale, and P.P. Zabrejko, Some remarks on the Newton–Kantorovich approximations, Atti Sem. Mat. Fis. Univ. Modena 48 (2000), pp. 207–215; E. De Pascale and P.P. Zabrejko, Convergence of the Newton–Kantorovich method under Vertgeim conditions: A new improvement, Z. Anal. Anwendvugen 17 (1998), pp. 271–280; J.A. Ezquerro and M.A. Hernández, On the R-order of convergence of Newton's method under mild differentiability conditions, J. Comput. Appl. Math. 197 (2006), pp. 53–61; J.V. Lysenko, Conditions for the convergence of the Newton–Kantorovich method for nonlinear equations with Hölder linearizations (in Russian), Dokl. Akad. Nauk BSSR 38 (1994), pp. 20–24; P.D. Proinov, New general convergence theory for iterative processes and its applications to Newton–Kantorovich type theorems, J. Complexity 26 (2010), pp. 3–42; J. Rokne, Newton's method under mild differentiability conditions with error analysis, Numer. Math. 18 (1971/72), pp. 401–412; B.A. Vertgeim, On conditions for the applicability of Newton's method, (in Russian), Dokl. Akad. N., SSSR 110 (1956), pp. 719–722; B.A. Vertgeim, On some methods for the approximate solution of nonlinear functional equations in Banach spaces, Uspekhi Mat. Nauk 12 (1957), pp. 166–169 (in Russian); English transl.: Amer. Math. Soc. Transl. 16 (1960), pp. 378–382; P.P. Zabrejko and D.F. Nguen, The majorant method in the theory of Newton–Kantorovich approximations and the Pták error estimates, Numer. Funct. Anal. Optim. 9 (1987), pp. 671–684; A.I. Zinc˘enko, Some approximate methods of solving equations with non-differentiable operators (Ukrainian), Dopovidi Akad. Nauk Ukraïn. RSR (1963), pp. 156–161] to provide a semilocal convergence analysis for Newton's method under Hölder differentiability conditions. Our sufficient convergence conditions are also weaker even in the Lipschitz differentiability case. Moreover, the results are obtained under the same or less computational cost. Numerical examples are provided where earlier conditions do not hold but for which the new conditions are satisfied.

Keywords:

• Newton's method
• Banach space
• Hölder continuity
• Lipschitz continuity
• semilocal convergence
• Kantorovich hypothesis

2000 AMS Subject Classification:

• 65H10
• 65G99
• 65J15
• 47H17
• 49M15