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On the convergence of Newton-type methods under mild differentiability conditions. (English) Zbl 1190.65091

The authors introduce the idea of recurrent functions to provide a new semilocal convergence analysis for a Newton-type method (NTM) \[ x_{n + 1} = x_n - A(x_n )^{ - 1}P(x_n ),\quad(n \geq 0),\;(x_0 \in D), \]
\[ P(x) = F(x) + G(x), \quad (x \in D) \]
for the following equation
\[ F(x) + G(x) = 0. \]
Here \(F\), \(G\) are Fréchet-differentiable and continuous operators respectively, \(F:D \to Y,\,\,G:D \to Y,\) where \(D \subset R\) is the convex subset and \(X\) and \(Y\)are Banach spaces.
The NTM has been used by several authors to generate a sequence \(\{x_n \}\) approximating a locally unique solution \(x^\ast \) of the considered equation. \(A(x)\) belongs to the space of bounded linear operators from \(X\) to \(Y\) and is an approximation to the Fréchet derivative \({F}'(x)\) of the operator \(F(x)\). At each step one operator evaluation \(P(x_n)\) is required, and one inverse \(A(x_n )^{ - 1}\).
Using the idea of recurrent functions a combination of Lipschits and center-Lipschitz conditions, instead of only Lipschitz conditions the authors get weaker sufficient convergence conditions and larger convergence domain than in earlier studies of many scientists.
Applications and numerical examples are given too.

MSC:

65J15 Numerical solutions to equations with nonlinear operators
47J25 Iterative procedures involving nonlinear operators
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[1] Argyros, I.K.: The theory and application of abstract polynomial equations. Mathematics Series. St. Lucie/CRC/Lewis, Boca Raton (1998) · Zbl 0967.65070
[2] Argyros, I.K.: On the Newton–Kantorovich hypothesis for solving equations. J. Comput. Appl. Math. 169, 315–332 (2004) · Zbl 1055.65066 · doi:10.1016/j.cam.2004.01.029
[3] Argyros, I.K.: A unifying local–semilocal convergence analysis and applications for two-point Newton–like methods in Banach space. J. Math. Anal. Appl. 298, 374–397 (2004) · Zbl 1057.65029 · doi:10.1016/j.jmaa.2004.04.008
[4] Argyros, I.K.: Convergence and applications of Newton–type iterations. Springer, New York (2008) · Zbl 1153.65057
[5] Argyros, I.K.: On a class of Newton-like methods for solving nonlinear equations. J. Comput. Appl. Math. 228, 115–122 (2009) · Zbl 1168.65349 · doi:10.1016/j.cam.2008.08.042
[6] Chandrasekhar, S.: Radiative transfer. Dover, New York (1960) · Zbl 0037.43201
[7] Chen, X.: On the convergence of Broyden-like methods for nonlinear equations with nondifferentiable terms. Ann. Inst. Stat. Math. 42, 387–401 (1990) · Zbl 0718.65039 · doi:10.1007/BF00050844
[8] Chen, X., Yamamoto, T.: Convergence domains of certain iterative methods for solving nonlinear equations. Numer. Funct. Anal. Optim. 10, 37–48 (1989) · Zbl 0645.65028 · doi:10.1080/01630568908816289
[9] Cianciaruso, F., De Pascale, E.: Newton–Kantorovich aproximations when the derivative is Hölderian: old and new results. Numer. Funct. Anal. Optim. 24, 713–723 (2003) · Zbl 1037.65059 · doi:10.1081/NFA-120026367
[10] Cianciaruso, F.: A further journey in the ”terra incognita” of the Newton–Kantorovich method. Nonlinear Funct. Anal. Appl. (2009, in press) · Zbl 1242.65113
[11] Dennis, J.E.: Toward a unified convergence theory for Newton–like methods. In: Rall, L.B. (ed.) Nonlinear Functional Analysis and Applications, pp. 425–472. Academic, New York (1971) · Zbl 0276.65029
[12] Deuflhard, P.: Newton methods for nonlinear problems. In: Affine Invariance and Adaptive Algorithms. Springer Series in Computational Mathematics, vol. 35. Springer, Berlin (2004) · Zbl 1056.65051
[13] Deuflhard, P., Heindl, G.: Affine invariant convergence theorems for Newton’s method and extensions to related methods. SIAM J. Numer. Anal. 16, 1–10 (1979) · Zbl 0395.65028 · doi:10.1137/0716001
[14] Gutiérrez, J.M.: A new semilocal convergence theorem for Newton’s method. J. Comput. Appl. Math. 79, 131–145 (1997) · Zbl 0872.65045 · doi:10.1016/S0377-0427(97)81611-1
[15] Hernández, M.A.: The Newton method for operators with Hölder continuous first derivatives. J. Optim. Theory Appl. 109(3), 631–648 (2001) · Zbl 1012.65052 · doi:10.1023/A:1017571906739
[16] Hernández, M.A., Rubio, M.J., Ezquerro, J.A.: Secant-like methods for slving nonlinear integral equations of the Hammerstein type. J. Comput. Appl. Math. 115, 245–254 (2000) · Zbl 0944.65146 · doi:10.1016/S0377-0427(99)00116-8
[17] Huang, Z.: A note of Kantorovich theorem for Newton iteration. J. Comput. Appl. Math. 47, 211–217 (1993) · Zbl 0782.65071 · doi:10.1016/0377-0427(93)90004-U
[18] Kantorovich, L.V., Akilov, G.P.: Functional analysis. Pergamon, Oxford (1982) · Zbl 0484.46003
[19] Miel, G.J.: Unified error analysis for Newton–type methods. Numer. Math. 33, 391–396 (1979) · Zbl 0402.65038 · doi:10.1007/BF01399322
[20] Miel, G.J.: Majorizing sequences and error bounds for iterative methods. Math. Comput. 34, 185–202 (1980) · Zbl 0425.65033 · doi:10.1090/S0025-5718-1980-0551297-4
[21] Moret, I.: A note on Newton type iterative methods. Computing 33, 65–73 (1984) · Zbl 0532.65045 · doi:10.1007/BF02243076
[22] Potra, F.A.: Sharp error bounds for a class of Newton–like methods. Libertas Mathematica 5, 71–84 (1985) · Zbl 0581.47050
[23] Rheinboldt, W.C.: A unified convergence theory for a class of iterative processes. SIAM J. Numer. Anal. 5, 42–63 (1968) · Zbl 0155.46701 · doi:10.1137/0705003
[24] Yamamoto, T.: A convergence theorem for Newton–like methods in Banach spaces. Numer. Math. 51, 545–557 (1987) · Zbl 0633.65049 · doi:10.1007/BF01400355
[25] Zabrejko, P.P., Nguen, D.F.: The majorant method in the theory of Newton–Kantorovich approximations and the Pták error estimates. Numer. Funct. Anal. Optim. 9, 671–684 (1987) · Zbl 0627.65069 · doi:10.1080/01630568708816254
[26] Zincenko, A.I.: Some approximate methods of solving equations with non-differentiable operators. (Ukrainian), Dopovidi Akad, pp. 156–161. Nauk Ukraïn. RSR (1963)
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