Recurrence relations for semilocal convergence of a Newton-like method in Banach spaces

The aim of this paper is to establish the semilocal convergence of a multipoint third order Newton-like method for solving F(x) = 0 in Banach spaces by using recurrence relations. The convergence of this method is studied under the assumption that the second Frechet derivative of F satisfies Holder continuity condition. This continuity condition is milder than the usual Lipschitz continuity condition. A new family of recurrence relations are defined based on the two new constants which depend on the operator F. These recurrence relations give a priori error bounds for the method. Two numerical examples are worked out to demonstrate the applicability of the method in cases where the Lipschitz continuity condition over second derivative of F fails but Holder continuity condition holds. (C) 2008 Elsevier Inc. All rights reserved.