On Newton's method under mild differentiability conditions and applications

摘要

In this study we are concerned with the problem of approximating a locally unique solution of a nonlinear operator equation in a Banach space. We examine the convergence of the Newton–Kantorovich method by assuming that the Fréchet-derivative of the nonlinear operator involved is only Hölder continuous. Many results exist already in the literature to cover the stronger case when the operator is Lipschitz continuous. Moreover we use this study to counter some claims made by Galperin and Waksman on related work of ours. Furthermore we use our results to solve a two point boundary value problem appearing in kinetic theory of gasses, elasticity and other applied areas.