On the convergence of Newton-type methods under mild differentiability conditions

We introduce the new idea of recurrent functions to provide a new semilocal convergence analysis for Newton-type methods, under mild differentiability conditions. It turns out that our sufficient convergence conditions are weaker, and the error bounds are tighter than in earlier studies in some interesting cases (Chen, Ann Inst Stat Math 42:387–401, 1990; Chen, Numer Funct Anal Optim 10:37–48, 1989; Cianciaruso, Numer Funct Anal Optim 24:713–723, 2003; Cianciaruso, Nonlinear Funct Anal Appl 2009; Dennis 1971; Deuflhard 2004; Deuflhard, SIAM J Numer Anal 16:1–10, 1979; Gutiérrez, J Comput Appl Math 79:131–145, 1997; Hernández, J Optim Theory Appl 109:631–648, 2001; Hernández, J Comput Appl Math 115:245–254, 2000; Huang, J Comput Appl Math 47:211–217, 1993; Kantorovich 1982; Miel, Numer Math 33:391–396, 1979; Miel, Math Comput 34:185–202, 1980; Moret, Computing 33:65–73, 1984; Potra, Libertas Mathematica 5:71–84, 1985; Rheinboldt, SIAM J Numer Anal 5:42–63, 1968; Yamamoto, Numer Math 51: 545–557, 1987; Zabrejko, Numer Funct Anal Optim 9:671–684, 1987; Zinc̆ko 1963). Applications and numerical examples, involving a nonlinear integral equation of Chandrasekhar-type, and a differential equation are also provided in this study.

[1]  Louis B. Rall,et al.  Nonlinear Functional Analysis and Applications , 1971 .

[2]  G. Miel,et al.  Majorizing sequences and error bounds for iterative methods , 1980 .

[3]  G. Miel Unified error analysis for Newton-type methods , 1979 .

[4]  I. Argyros Convergence and Applications of Newton-type Iterations , 2008 .

[5]  G. Rybicki Radiative transfer , 2019, Climate Change and Terrestrial Ecosystem Modeling.

[6]  M. J. Rubio,et al.  Secant-like methods for solving nonlinear integral equations of the Hammerstein type , 2000 .

[7]  P. P. Zabrejko,et al.  The majorant method in the theory of newton-kantorovich approximations and the pták error estimates , 1987 .

[8]  P. Deuflhard Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms , 2011 .

[9]  tetsuro Yammoto,et al.  A convergence theorem for Newton-like methods in Banach spaces , 1987 .

[10]  Filomena Cianciaruso,et al.  Newton–Kantorovich Approximations When the Derivative Is Hölderian: Old and New Results , 2003 .

[11]  Some approximate methods of solving operator equations , 1972 .

[12]  J. Dennis Toward a Unified Convergence Theory for Newton-Like Methods , 1971 .

[13]  Igor Moret A note on Newton type iterative methods , 2005, Computing.

[14]  Huang Zhengda,et al.  A note on the Kantorovich theorem for Newton iteration , 1993 .

[15]  P. Deuflhard,et al.  Affine Invariant Convergence Theorems for Newton’s Method and Extensions to Related Methods , 1979 .

[16]  W. Rheinboldt A unified convergence theory for a class of iterative processes. , 1968 .

[17]  Xiaojun Chen,et al.  Convergence domains of certain iterative methods for solving nonlinear equations , 1989 .

[18]  M. A. Hernández The Newton Method for Operators with Hölder Continuous First Derivative , 2001 .

[19]  Jose M. Gutikez A new semilocal convergence theorem for Newton's method , 1997 .

[20]  Ioannis K. Argyros,et al.  A unifying local–semilocal convergence analysis and applications for two-point Newton-like methods in Banach space , 2004 .

[21]  Ioannis K. Argyros,et al.  On the Newton-Kantorovich hypothesis for solving equations , 2004 .

[22]  Peter Deuflhard,et al.  Newton Methods for Nonlinear Problems , 2004 .

[23]  Xiaojun Chen,et al.  On the convergence of Broyden-like methods for nonlinear equations with nondifferentiable terms , 1990 .

[24]  Ioannis K. Argyros,et al.  On a class of Newton-like methods for solving nonlinear equations , 2009 .