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

Let f: B(xo, R) C X 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 NewtonKantorovich approximations toward a solution of the equation f(x) = 0, under the hypothesis that 1' be Hlder continuous. The case of f' being Holder continuous in a generalized sense is analyzed as well.