ON THE R-ORDER CONVERGENCE OF A THIRD ORDER METHOD IN BANACH SPACES UNDER MILD DIFFERENTIABILITY CONDITIONS

Abstract

The aim of this paper is to discuss the convergence of a third order method for solving nonlinear equations F(x)=0 in Banach spaces by using recurrence relations. The convergence of the method is established under the assumption that the second Fréchet derivative of F satisfies a condition that is milder than Lipschitz/Hölder continuity condition. A family of recurrence relations based on two parameters depending on F is also derived. An existence-uniqueness theorem is also given that establish convergence of the method and a priori error bounds. A numerical example is worked out to show that the method is successful even in cases where Lipschitz/Hölder continuity condition fails.