On the Semi-local Convergence Analysis of Higher Order Iterative Method in Two Folds

We study the semi-local convergence analysis of an existing well defined method of order eight in Banach spaces to get the solution of the nonlinear equations. In the existing references, semi-local convergence of this iterative method is established by assuming the bound on the norm of the third-order Fréchet derivative which satisfies either Lipschitz or Hölder or ω-continuity condition. The purpose of this study is in two-folds. In the first fold, we prove the semi-local convergence analysis of the method by inferring the bound on the norm of the second-order Fréchet derivative on using recurrence relation technique. Another one contains the presume of the bound on the norm of the third-order Fréchet derivative at an initial approximation instead of considering it on the given domain of the nonlinear operator for showing the convergence, existence and uniqueness of the solution along with its apriori error bound expression is given. Two illustrations are also included in the support of the theoretical discussion. © 2019, Springer Nature India Private Limited.