Journal of Mathematical Sciences, Vol. 243, No. 1, November, 2019 CONVERGENCE OF THE NEWTON–KURCHATOV METHOD UNDER WEAK CONDITIONS S. M. Shakhno and H. P. Yarmola UDC 519.6 We study the semilocal convergence of the combined Newton–Kurchatov method to a locally unique solution of the nonlinear equation under weak conditions imposed on the derivatives and first-order di- vided differences. The radius of the ball of convergence is established and the rate of convergence of the method is estimated. As a special case of these conditions, we consider the classical Lipschitz con- ditions. Introduction Consider an equation H ( x ) F ( x ) + G( x ) = 0 , (1) where F and G are nonlinear operators defined on an open convex set D of the Banach space X with values in the Banach space Y . Let F be a Frèchet differentiable operator and let G be a continuous operator whose differentiability is, generally speaking, not required. In view of the properties of the operator H , Eq. (1) cannot be solved by using the classical Newton meth- od. As a rule, for this purpose, the researchers use either a Newton-type method [8, 14] x n+1 = x n [ F ( x n )] 1 H ( x n ), n 0 , or difference methods, e.g., the method of chords (secants) [6, 9] x n+1 = x n [ H ( x n ; x n 1 )] 1 H ( x n ), n 0 , or the Kurchatov method of linear interpolation [1, 5, 12] x n+1 = x n [ H (2 x n x n 1 ; x n 1 )] 1 H ( x n ), n 0 , or the method developed in [11] x n+1 = x n [ F (2 x n x n 1 ; x n 1 ) + G( x n ; x n 1 )] 1 H ( x n ), n 0 , I. Franko Lviv National University, Lviv, Ukraine. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 60, No. 2, pp. 7–13, April–June, 2017. Original article sub- mitted March 9, 2017. 1072-3374/19/2431–0001 © 2019 Springer Science+Business Media, LLC 1 DOI 10.1007/s10958-019-04521-5