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Newton’s method under weak Kantorovich conditions. (English) Zbl 0965.65081
For solving nonlinear equations $$x= F(x)$$ in a Banach space, the Newton-Kantorovich method is well-known. Unfortunately, the classical theorem provides convergence only if the Fréchet derivative of $$F$$ is Lipschitz continuous.
The authors prove convergence results and error estimates under the weaker assumption $\|F'(x)- F'(x_0)\|\leq L\|x- x_0\|,\quad L>0,$ for some given $$x_0$$. The results are then illustrated for a nonlinear integral equation.

MSC:
 65J15 Numerical solutions to equations with nonlinear operators 47J25 Iterative procedures involving nonlinear operators 45G10 Other nonlinear integral equations 65R20 Numerical methods for integral equations
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