Convergence of Newton-Kantorovich Approximations to an Approximate Zero

In this paper, we prove an a-posteriori and an a-priori convergence theorem for Newton-Kantorovich approximations starting from an initial point $x_0$. We apply these results to operators that are analytic at interior points of a closed ball centered at $x_0$ and of radius $R$. We obtain some theorems on approximate zeros and on approximate zeros of second kind for these operators which improve previous results.