  The extended modification of the Newton method is considered when the inverse
of the derivative (of the operator F(x) in the equation F(x)=0) is replaced by
an invertible bounded x-independent operator B. The continuity assumption is
relaxed to the requirement that F(x) is continuously Frechet-differentiable.
The Kantorovich majorization technique is adapted to formulate and prove the
corresponding generalization of the Kantorovich theorem originally stated for
the standard modified Newton method (MNM) when the derivative is Lipschitz
continuous. In the MNM case, the generalized theorem is shown to extend the
existing one due to Argyros. For a generic B and a Holder continuous
derivative, the proposed theorem leads to a weaker condition of the semilocal
convergence, larger uniqueness domain and finer error bounds compared to the
previous results of Ahues and Argyros.
