A convergence analysis and applications for two-point Newton-like methods in Banach space under relaxed conditions

Summary.

We provide a local as well as a semilocal convergence analysis for two-point Newton- like methods in a Banach space setting under very relaxed conditions. Our equation contains a Fréchet differentiable operator F and another operator G whose differentiability is not assumed. Using more precise majorizing sequences than before we provide sufficient convergence conditions for Newton-like methods to a locally unique solution of equation F(x)+G(x) = 0. In the semilocal case we show under weaker conditions that our estimates on the distances involved are finer and the information on the location of the solution at least as precise as in earlier results. In the local case a larger radius of convergence is obtained. Several numerical examples are provided to show that our results compare favorably with earlier ones. As a special case we show that the famous Newton–Kantorovich hypothesis is weakened under the same hypotheses as the ones contained in the Newton–Kantorovich theorem.