A Newton–Kantorovich theorem for equations involving m-Fréchet differentiable operators and applications in radiative transfer

In this study we approximate a locally unique solution of a nonlinear operator equation in Banach space using Newtonʹs method. A complete error analysis showing the quadratic convergence of our method is also given. Our new theorem uses Lipschitz or Hölder continuity assumptions on m-Fréchet-differentiable operators where m⩾2 is a positive integer. A numerical example is given to show that our results provide a better information on the location of the solution as well as finer error bounds on the distances involved than earlier results. A second numerical example shows how to solve a nonlinear integral equation appearing in radiative transfer.