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Convergence of Newton’s method and uniqueness of the solution of equations in Banach space. (English) Zbl 0942.65057
For the well-known Newton-Kantorovich method for solving nonlinear equations in Banach spaces, \(f(x)= 0\) in \(X\), the author gives exact estimates of convergence and uniqueness balls.
If \(f\) is continuously differentiable in some ball around an exact solution \(x^*\) and if \(f'(x^*)^{-1}f'\) satisfies a so-called radius Lipschitz condition with the \(L\) average, then the method is shown to be convergent for all starting points chosen in this ball. The optimal choice of the radius of this ball is also analyzed.
Under a so-called centre Lipschitz condition with the \(L\) average, the author obtains uniqueness and again demonstrates the optimal choice of the radius.
Reviewer: E.Emmrich (Berlin)

65J15 Numerical solutions to equations with nonlinear operators
47J25 Iterative procedures involving nonlinear operators
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