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

Abstract

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 $f^{\prime }$ be Hölder continuous. The case of $f^{\prime }$ being Hölder continuous in a generalized sense is analyzed as well.