Ball convergence for combined three-step methods under generalized conditions in Banach space

Abstract

We give a local convergence analysis for an eighth-order convergent method in order to approximate a locally unique solution of nonlinear equation for Banach space valued operators. In contrast to the earlier studies using hypotheses up to the seventh Fréchet-derivative, we only use hypotheses on the first-order Fréchet-derivative and Lipschitz constants. Therefore, we not only expand the applicability of these methods but also provide the computable radius of convergence of these
methods. Finally, numerical examples show that our results apply to solve those nonlinear equations but earlier results cannot be used.