A new semilocal convergence theorem for Newton's method in Banach space using hypotheses on the second Frechet-derivative

A new global Kantorovich-type convergence theorem for Newton's method in Banach space is provided for approximating a solution of a nonlinear equation. It is assumed that a solution exists and the second Frechet-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. (C) 2001 Elsevier Science B.V. All rights reserved.