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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)

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