On a theorem of L.V. Kantorovich concerning Newtonʹs method

We study the problem of approximating a locally unique solution of an operator equation using Newtonʹs method. The well-known convergence theorem of L.V. Kantorovich involves a bound on the second Fréchet-derivative or the Lipschitz–Fréchet-differentiability of the operator involved on some neighborhood of the starting point. Here we provide local and semilocal convergence theorems for Newtonʹs method assuming the Fréchet-differentiability only at a point which is a weaker assumption. A numerical example is provided to show that our result can apply to solve a scalar equation where the above-mentioned ones may not.