Weaker conditions for the convergence of Newton's method

The Newton–Kantorovich hypothesis (15) has been used for a long time as a sufficient condition for convergence of Newton’s method to a locally unique solution of a nonlinear equation in a Banach space setting. Recently in [3], [4] we showed that this hypothesis can always be replaced by a condition weaker in general (see (18), (19) or (20)) whose verification requires the same computational cost. Moreover, finer error bounds and at least as precise information on the location of the solution can be obtained this way. Here we show that we can further weaken conditions (18)–(20) and still improve on the error bounds given in [3], [4] (see Remark 1(c)).