On Newton's method for solving equations containing Fréchet-differentiable operators of order at least two
The author provides a semilocal convergence analysis for Newton's method approximating the locally unique solution \(x^{*} \) of \(F(x)=0\), where \(F\) is a Fréchet-differentiable operator defined on a convex subset \(\mathcal{D}\subset\mathcal{X}\) , \(\mathcal{X}\) is a Banach space, with values in the Banach space \(\mathcal{Y}\). Using a combination and of Lipschitz and center-Lipschitz conditions the author proves the following theorem: Let \(F\) be twice Fréchet-differentiable satisfying some Lipschitz type conditions. Then if \[ \overline{U}(x_{0},\frac{1}{L_{0}}) =\{x\in \mathcal{X}:\parallel x-x_{0}\parallel\leq \frac{1}{L_{0}}\}\subseteq\mathcal{D} \] the sequence generated by Newton's method is well defined and remains in \(\overline{U}(x_{0},t_{0}-t^{*})\) for all \(n>0\) and converges to a unique solution \(x^{*}\) in \(\overline{U}(x_{0},t_{0}-t^{*}).\) Moreover the following estimates hold for all \(n\) \[ \parallel x_{n}-x_{n-1}\parallel \leq t_{n}-t_{n-1} \] where the sequence \(t_{n}\)is defined recursively. Numerical examples are also provided to show that the results apply to solve nonlinear equations in cases in which the earlier couldn't.
nonlinear integral equation of Chandrasekhar-type
scientific article; zbMATH DE number 5632603