Extended local convergence for Newton-type solver under weak conditions

Abstract

We present the local convergence of a Newton-type solver for equations involving Banach space valued operators. The eighth order of convergence was shown earlier in the special case of the $k-$dimensional Euclidean space, using hypotheses up to the eighth derivative although these derivatives do not appear in the method. We show convergence  using only the first derivative. This way we extend the applicability of the methods. Numerical examples are used to show the convergence conditions. Finally, the basins of attraction of the method, on some test problems are presented.