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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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