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Convergence of Newton’s method and uniqueness of the solution of equations in Banach spaces. II. (English) Zbl 1027.65078
The authors continue their work on the Newton-Kantorowitsch method for solving nonlinear equations in a Banach space. Convergence and uniqueness results are studied under the assumption that the operator’s derivative fulfills a so-called radius or center Lipschitz condition with the weak $L$ average. Part I of this work has been published by X. Wang [IMA J. Numer. Anal. 20, No. 1, 123-134 (2000; Zbl 0942.65057)]. The reader should also consult X. Wang, C. Li, and M.-J. Lai [BIT 42, 206-213 (2002; Zbl 0998.65057)].

##### MSC:
 65J15 Equations with nonlinear operators (numerical methods) 47J25 Iterative procedures (nonlinear operator equations)
##### References:
 [1] Wang, X. H.: Convergence of Newton’s method and uniqueness of the solution of equations in Banach space. IMA Journal of Numerical Analysis, 19(2), 123–134 (2000) · Zbl 0942.65057 · doi:10.1093/imanum/20.1.123 [2] Smale, S.: Newton’s method estimates from data at one point. In The Merging of Disciplines: New Directions in Pure, Applied and Computational Mathematics, Ewing, R., Gross, K., Martin, C. eds, New York: Spring-Verlag, 185–196 (1986) [3] Dedieu, J. P.: Estimations for the separation number of a polynomial system. Journal of Symbolic Computation, to appear [4] Traub, J. F., Wozniakowski, H.: Convergence and complexity of Newton iteration. J. Assoc. For Comp. Math., 29(2), 250–258 (1979) [5] Wang, X. H.: The convergence on Newton’s method. KeXue TongBao (A Special Issue of Mathematics, Physics and Chemistry), 25(1), 36–37 (1980)