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New general convergence theory for iterative processes and its applications to Newton-Kantorovich type theorems. (English) Zbl 1185.65095
The author proposes a lot of new general convergence theorems for the Picard iteration, applied to a mapping \(T\) in a complete metric space. To elaborate this new theory, he uses the concepts of quasi-homogeneous functions, gauge functions of high order, a function of initial conditions of the mapping \(T\), a convergence function of the mapping \(T\) and the initial points of a mapping. The function of the initial conditions of a mapping represents a generalization of the concept of contraction.
Four new convergence theorems for the Picard iteration are proved (Theorems 5.4, 5.5, 5.6, 5.7); each of these theorems gives the radius of the convergence ball, error estimates (a priori and a posteriori) and the existence of a fixed point for the mapping \(T\). These results are then applied to obtain fixed point theorems for the iterated contraction mapping (with respect to a function of initial conditions). Also, these results are applied to study the convergence of the Newton-Kantorovich method for operator equations in Banach spaces. Three Newton-Kantorovich type theorems which generalize, extend, or complete some results from the literature are proved.
In the last section, the theory is applied to Newton’s iteration for the zeros of an analytic function and also, many published results are extended (especially the results of S. Smale, Newton’s method estimates from data at one point. The merging of disciplines: new directions in pure, applied, and computational mathematics, Proc. Symp. Honor G. S. Young, Laramie/Wyo. 1985, 185–196 (1986; Zbl 0613.65058)).

MSC:
65J15 Numerical solutions to equations with nonlinear operators (do not use 65Hxx)
47J05 Equations involving nonlinear operators (general)
47H10 Fixed-point theorems
47J25 Iterative procedures involving nonlinear operators
65H05 Numerical computation of solutions to single equations
65E05 General theory of numerical methods in complex analysis (potential theory, etc.)
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
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