An improved error analysis for Newton-like methods under generalized conditions (Q1405185)

The author is concerned with the problem of approximating a locally unique solution \(x^*\) of the equation \(F(x)=0\), where \(F\) is a Fréchet-differentiable operator. Using Newton-like methods in a Banach space setting, under new and very general conditions he provides different sufficient convergence conditions than before. After his introducing of more precise majorizing sequences, he obtains finer error estimates and a better information on the location of the solution. His results reduce to earlier ones for special choices of majorizing functions. Finally, as an application, he shows that in the case of Newton's method the famous Newton-Kantorovich hypothesis can be weakened under the same information.