Results on Newton methods. Part 1: A unified approach for constructing perturbed Newton-like methods in Banach space and their applications

Abstract

Many applied problems can be formulated to fit the model of a nonlinear equation. Undoubtedly, Newton methods are the most popular methods for solving such equations. Here we report the results of our recent investigations in a series of two papers. We unified Newton methods and provided convergence results to solve nonlinear equations. We used Newton–Kantorovich type hypotheses and the majorant method. Our results depend on the existence of a Lipschitz function with which we replace the Lipschitz constant usually appearing in the hypotheses of these investigations. We find our methods useful for many reasons. From the computational point of view we have allowed our iterations to be corrected at every step and still converge to a solution of the equation at hand. In most earlier results the iterates are assumed to be computed exactly to achieve convergence. Moreover we showed that under weaker hypotheses our iterations converge faster than others already in the literature. Finally we used our results to solve Uryson and Fredholm-type integral equations appearing in radiative transfer.