Two-step Newton methods

We present sufficient convergence conditions for two-step Newton methods in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. The advantages of our approach under the same computational cost over other studies such as [9–25] for the semilocal convergence case are: weaker sufficient convergence conditions, more precise error bounds on the distances involved an at least as precise information on the location of the solution. In the local convergence case more precise error estimates are presented. Numerical examples involving Hammerstein nonlinear integral equations where the older convergence conditions are not satisfied but the new conditions are satisfied are also presented in this study for the semilocal convergence case. Moreover a larger convergence ball is obtained in the local convergence case. AMS Subject Classification. 65J15, 47H17 65H10, 65G99, 49M15.