On the Newton–Kantorovich hypothesis for solving equations

The famous Newton–Kantorovich hypothesis has been used for a long time as a sufficient condition for the convergence of Newtonʹs method to a solution of an equation in connection with the Lipschitz continuity of the Fréchet-derivative of the operator involved. Here using Lipschitz and center-Lipschitz conditions we show that the Newton–Kantorovich hypothesis can be weakened. The error bounds obtained under our semilocal convergence result are more precise than the corresponding ones given by the dominating Newton–Kantorovich theorem.