Semilocal convergence for the Super-Halley’s method

Abstract

The semilocal convergence of Super-Halley’s method for solving nonlinear equations in Banach spaces is established under the assumption that the second Frechet derivative satisfies the ω-continuity condition. This condition is milder than the well-known Lipschitz and Holder continuity conditions. The importance of our work lies in the fact that numerical examples can be given to show that our approach is successful even in cases where the Lipschitz and the Holder continuity conditions fail. The difficult computation of second Frechet derivative is also avoided by replacing it with the divided difference containing only the first Frechet derivatives. A number of recurrence relations based on two parameters are derived. A convergence theorem is established to estimate a priori error bounds along with the domains of existence and uniqueness of the solutions. The R-order convergence of the method is shown to be at least three. Two numerical examples are worked out to demonstrate the efficacy of our method. It is observed that in both examples the existence and uniqueness regions of solution are improved when compared with those obtained in [7].