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On Newton’s method and nondiscrete mathematical induction. (English) Zbl 0642.65043
The author considers the equation (1) $$F(x)=0$$ where F is a nonlinear operator from a Banach space E into itself. The method of nondiscrete mathematical induction is used to find sharp error bounds for Newton’s method. It is assumed that the operator F has Hölder continuous derivatives. When the Fréchet-derivative of the operator F satisfies a Lipschitz condition the obtained results reduced to the ones obtained by Ptak and Potra in 1972.