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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, $f\left(x\right)=0$ 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 ${f}^{\text{'}}{\left({x}^{*}\right)}^{-1}{f}^{\text{'}}$ 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.

##### MSC:
 65J15 Equations with nonlinear operators (numerical methods) 47J25 Iterative procedures (nonlinear operator equations)