numerical scheme is split into two parts: construction of the continuous process and numerical integration of the corresponding nonlinear operator differential equation. Convergence theorems for regularized continuous Newton-like methods are established in [@ars; @al; @r].

The goal of this paper is to develop a general approach to continuous analogs of discrete methods and to establish fairly general convergence theorem. This approach is based on an analysis of the solution to the Cauchy problem for a nonlinear differential equation in a Hilbert space. Such an analysis was done for well-posed problems in [@a], where it was based on a usage of an integral inequality. It is more difficult to study nonlinear ill-posed problems. In this case one has to use more complicated new integral inequality (Lemma \[ric\]). Based on Lemma \[ric\] the general theorem establishing convergence of a regularized continuous process is proved (Theorem \[t1\]). Applying this theorem to the regularized Newton’s and simple iteration methods (for monotone operators) and to Gauss-Newton-type methods (for non-monotone operators) convergence theorems are obtained under less restrictive conditions on the equation than the theorems known for the corresponding discrete methods. According to these theorems one can choose a regularizing operator depending on the “degree of 