Expanding the applicability of a fifth-order convergent method in a Banach space under weak conditions

We present a local convergence analysis for a fifth-order convergent method in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. In contrast to the earlier studies using hypotheses up to the fifth Fréchet derivative, we only use hypotheses on the first-order Fréchet derivative and Lipschitz constants. In this way, we not only expand the applicability of the methods but also evaluate the theoretical radius of convergence of these methods. Finally, a variety of concrete numerical examples demonstrate that our results even apply to solve those nonlinear equations where earlier studies cannot apply.