New and Generalized Convergence Conditions for the Newton-kantorovich Method

We present new semilocal convergence theorems for Newton methods in a Banach space. Using earlier general conditions we find more precise error estimates on the distances involved using the majorant principle. Moreover we provide a better information on the location of the solution. In the special case of Newton’s method under Lipschitz conditions we show that the famous Newton–Kantorovich hypothesis having gone unchallenged for a long time can be weakened under the same hypotheses/computational cost.