Examining the algorithm of Newton’s method, $$\displaystyle x_{n+1}=x_n-[F'(x_n)]^{-1}F(x_n),\quad n\geq 0, \quad \mbox{with } x_0\mbox{ given}, $$ we see that it involves only the operator F and its first Frechet derivative F′, suggests that trying to impose conditions only on the operators F and F′ to guarantee the convergence of Newton’s method. Thus, the continuity condition (K3) on F″(x) in the Newton-Kantorovich Theorem 1.1 can be easily replaced by a Lipschitz continuity condition on F′, i.e.