A new semilocal convergence theorem for Newtonʹs method in banach space using hypotheses on the second Fréchet-derivative

A new global Kantorovich-type convergence theorem for Newtonʹs method in Banach space is provided for approximating a solution of a nonlinear equation. It is assumed that a solution exists and the second Fréchet-derivative of the operator involved satisfies a Lipschitz condition. Our convergence condition differs from earlier ones, and therefore it has theoretical and practical value. Finally, a simple numerical example is provided to show that our results apply, where earlier ones fail.