\dot{z}(t)=-[F'(z(t))+\ep(t)I]^{-1}[F(z(t))+\ep(t)(z(t)-z_0)],\q z(0)=z_0\in H,$$

-   regularized Gauss-Newton’s methods: equation (\[caupr\]) with the function $\P$ defined in (\[P1\]) or in (\[P2\]),

with an appropriate choice of the function $\ep(t)$ and the point $z_0$. Here $I$ is the identity operator.

The goal of this paper is to develop a uniform approach to such regularized methods. Let us consider the Cauchy problem: $$\label{caupr}
\dot{z}(t)=\P(z(t),t),\q z(0)=z_0\in H,$$ with an operator $\P:H\times[0,\infty)\to H$. The choice of $\P$ yields the corresponding continuous process.

In this section a general convergence theorem (Theorem \[t1\]) is established. In the next two sections the convergence theorems for the processes mentioned above are derived from this general theorem. In the proof of the general theorem the technique of integral inequalities is used.

The following lemma is known. It is a version of some results concerning integral inequalities (see e.g. Theorem 22.1 in [@JS]). For convenience of the reader and to make the presentation essentially self-contained we include a proof.

Let $f(t,w)$, $g(t,u)$ be continuous on region $[0,T)\times D$ ($D\subset
R$, $T\le\infty$) and $f(t,w)\leq g(t,u)$ if $w\leq u$, $t\in (0,T)$, $w,u\in D$. Assume that $g(t,u)$ is such that the Cauchy problem $$\label{Cauchy}
\dot u = g(t,u), \q u(0)=u_0,\q u_0\in D$$ has a unique 