ABSTRACT
We present in this chapter Newton{type methods. We establish convergence results using Lipschitz{type conditions.
We provide in this section new semilocal convergence results for Newton{like method using outer inverses but no Lipschitz conditions in a Banach space setting. The rst is the Kantorovich{type approach, whereas the second uses our new concept of recurrent functions. Comparisons are given between the two techniques. Our results are compared favorably with earlier ones using the information and requiring the same computational cost. We are concerned with the problem of approximating a locally unique solution x? of equation
QF (x) = 0; (6.1) where F is a Frechet{dierentiable operator dened on an open convex subset D of a Banach space X with values in Banach space Y and Q 2 L(Y;X ).
