Continuous methods for stable approximation of solutions to nonlinear equations in the Banach space based on the regularized Newton–Kantarovich scheme

We propose and study a class of methods for approximation of solutions to nonlinear equations with smooth operators in a Banach space, when the operators are approximately given and their derivatives are not regular. The construction of the presented methods is connected with the operator differential equation obtained by linearization of the original equation using the Newton–Kantorovich scheme and various ways of its regularization. When the initial discrepancy possesses a sourcewise representation, we establish estimates for the approximation errors.