The construction of methods for solving applied problems is of несомненную актуальность In this case, the following requirements for the method are of particular importance: algorithmic simplicity and speed, estimation of the accuracy of an approximate solution; minimum a priori information about the desired solution; certain universality of the numerical algorithm. Bearing in mind a large number of specific mathematical problems (integral equations, boundary value problems for differential equations, etc.), it is convenient to study approximate methods immediately for some classes of equations, that is, in the form of operator equations. This work is a direct generalization and development of the method of connected differential descent [1,2,3] for solving systems of finite-dimensional equations as applied to solving operator equations considered in separable Banach spaces. The differential descent method based on the solution of the Cauchy problem was considered in [4]. S. M. Gerashenko [5] investigated the possibility of improving the convergence of differential descent methods. To dampen oscillations near the extremum point, an additional coefficient is introduced into the right side of the system of differential equations in order to increase the roughness of the system with respect to the calculation error. The so-called sliding mode is introduced into the search algorithm. The study of sliding mode differential descent methods was continued in [4,6]. In [6], the rate of convergence of such methods is studied. In articles by B. A. Galanov [1,2], S. I. Alyber and Ya. I. Alyber [7], the differential descent method was applied to solve a system of equations. The method makes it possible to obtain an exact solution.