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A new semilocal convergence theorem for Newton’s method. (English) Zbl 0872.65045

The well known Newton method for solving a nonlinear equation $F\left(x\right)=0$ in a Banach space is considered and a new semilocal convergence theorem is proved under different assumptions from those of the Kantorovich theorem. Here it is assumed that the second Fréchet derivative ${F}^{\text{'}\text{'}}$ exists and is continuous and bounded and that the condition

$\left|{F}^{\text{'}}{\left({x}_{0}\right)}^{-1}\left[{F}^{\text{'}\text{'}}\left(x\right)-{F}^{\text{'}\text{'}}\left({x}_{0}\right)\right]\right|\le k|x-{x}_{0}|$

is satisfied in a certain neighbourhood of ${x}_{0}$. The proof of convergence is similar to that of Huang, a suitable cubic polynomial is checked in the proof. The author shows uniqueness of the solution and estimates the errors. Two examples are added to show situations where the Kantorovich assumptions fail but those of the discussed theorem are fulfilled or vice versa.

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