Concerning the “terra incognita” between convergence regions of two Newton methods

The majorizing principle is used to show local and semilocal convergence of Newton methods to a locally unique solution of a nonlinear operator in a Banach space, when the Fréchet derivative of the operator involved satisfies a center-Hölder and a Hölder continuity condition. Then we investigate an unknown area (“terra incognita”) between the convergence regions of Newton's method, and the corresponding modified Newton's method. Our approach compares favorably with other corresponding ones in this direction.