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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.


MSC:
65J15Equations with nonlinear operators (numerical methods)
47J25Iterative procedures (nonlinear operator equations)