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

We introduce new semilocal convergence theorems for Newton-like methods in a Banach space setting. Using new and very general conditions we provide different sufficient convergence conditions than before. This way we introduce more precise majorizing sequences, which in turn lead to finer error estimates and a better information on the location of the solution. Moreover for special choices of majorizing functions our results reduce to earlier ones. In the local case we obtain a larger convergence radius (ball). Finally, as an application, we show that in the case of Newtonʹs method the famous Newton–Kantorovich hypothesis can be weakened under the same information.