The midpoint method in Banach spaces and the Ptâk error estimates

摘要

We introduce the midpoint two-step method to approximate a locally unique solution of a nonlinear operator equation in a Banach space setting. We provide existence-uniqueness theorems as well as an error analysis for this iteration using Newton-Kantorovich-type assumptions and the majorant method. We assume that the Fréchet-derivative of the operator involved satisfies Ptâk-like estimates. Our results are also extended to include nonlinear operator equations with a nondifferentiable term. For special choices of the functions involved we show that our method is of order three, and compares favorably with other well-known Newton methods of order three. Our results can apply to solve nonlinear integral equations appearing in radiative transfer [3].