Let X, Y be Banach spaces, the closed ball with the centre and radius r in X. Suppose that are operators such that F is Fréchet differentiable on Int B(x, is continuously invertible and , where k(r) and (r) are nondecreasing functions on [0,R].
Put , ,
Theorem. Suppose that the function has a unique zero in [0,R] and (R). Then the equation has a solution in and this solution is unique in the ball . Moreover, the approximations are defined for all n, belong to and satisfy the estimates where is monotonically increasing sequence convergent to and defined by the formula These error estimates generalize the estimates of V. Pták [Comment. Math. Univ. Carol. 16, 699-705 (1975; Zbl 0314.65023), and Numer. Math. 25, 279-285 (1976; Zbl 0304.65037)] for the usual Newton-Kantorovich method.