Generalized Kantorovich-type theorem for the Fixed Slope Iterations

Abstract

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.