On the comparison of a Kantorovich-type and Moore theorems

It was shown in Zhue and Wolfe (Nonlinear Anal. 15(2):229–232, 1990) that the hypotheses of the affine invariant Moore theorem for solving nonlinear equations using Newton’s method are always valid when those of the Kantorovich theorem due to Deuflhard and Heindl (SIAM J. Numer. Anal. 16:1–10, 1980) hold but not necessarily vice versa. Here we show that this result is not true in general for a weaker version of the Kantorovich theorem shown recently by us in Argyros (Advances in the Efficiency of Computational Methods and Applications, World Scientific, Singapore, 2000; Int. J. Comput. Math. 80:5, 2002) and Argyros and Szidarovszky (The Theory and Applications of Iteration Methods, CRC Press, Boca Raton, 1993).