Semilocal convergence for Super-Halley’s method under $$\omega $$ ω -differentiability condition

Abstract

The semilocal convergence of Super-Halley’s method for solving nonlinear equations in Banach spaces is established using recurrence relations under the assumption that the second Frechet derivative satisfies the \(\omega \)-continuity condition. One 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 new family of recurrence relations based on two parameters are derived. A convergence theorem with R-order convergence of the method is shown to be equal to two. 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 (Ezquerro and Hernandez, J. Math. Anal. Appl. 303:591–601, 2005).