Math. Nachr. zyxwvuts 172 (1995) 5-14 zyxwvut On the Two-step Newton Method for the Solution of Nonlinear Operator Equations By J. APPELL of Wiirzburg, E. DE PASCALE of Arcavacata di Rende, and N. A. EVKHUTA and P. P. ZABREJKO of Minsk (Received November 16, 1993) (Revised Version May 3, 1994) Abstract. Building on the method of Kantorovich majorants, we give convergence results and error estimates for the two-step Newton method for the approximate solution of a nonlinear operator equation. 1. The two-step Newton method It is well-known that the classical Newton method for the approximate solution of the equation zyxwvuts (1) zyxwvutsr f(x) = zyxwvu 0, where zyxwvutsr f is some nonlinear operator between two Banach spaces X and Y, converges much faster than the modified method (see e.g. [l - 171). On the other hand, applying the classical method requires inverting the derivative off on each step at a new point, while for applying the modified method it suffices to invert the derivative off only at the starting point. Thus the question arises whether or not one may find some kind of “intermediate method” between the classical and the modified method which combines both the high convergence speed of the classical method and the easy calculability of the modified method. One prominent example of such an intermediate method is the two-step Newton method which builds on the approximations (2) X”+1 = yn -f’(%)-lf(Y”) zyxwvu (n = 0,1,2, ...I (3) with zyxwvuts Yn = x, - f’(xn)-’f(&) (n = 0, 1, 2, ...) (see e.g. [6, 8,9]). There is a recent paper [3] where the author proposes a new variant of the two-step method which is based on the method of Kantorovich majorants given in [15]. Let us recall the main result of this paper. Suppose that the operator f is defined on the closure B(x,, R)