Convergence of the Newton-Kantorovich Method under Vertgeim Conditions: a New Improvement

Let f: B(x_0, R) \subset X \to Y be an operator from a closed ball of a Banach space X to a Banach space Y . We give new conditions to ensure the convergence of Newton-Kantorovich approximations toward a solution of the equation f(x) = 0 , under the hypothesis that f' be Hölder continuous. The case of f' being Hölder continuous in a generalized sense is analyzed as well.