A unified approach for constructing fast two-step Newton-like methods

We introduce some new very general ways of constructing fast two-step Newton-like methods 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 the iterations involved using Newton-Kantorovich-type hypotheses and the majorant method. Our results depend on the existence of a Lipschitz function defined on a closed ball centered at a certain point and of a fixed radius and with values into the positive real axis. Special choices of this function lead to favorable comparisons with results already in the literature. Some applications to the solution of nonlinear integral equations appearing in radiative transfer as well as to the solution of integral equations of Uryson-type are also provided.