degeneracy” of the original nonlinear problem and estimate the rate of convergence of the regularized process.

The paper is organized as follows. In Section 2 Lemma \[ric\] about the solution of a new integral inequality and Theorem \[t1\] about the convergence of a regularized continuous process are formulated and proved. In Section 3 this theorem is applied to continuous Newton’s and simple iteration methods and in Section 4 to the Gauss-Newton-type methods. A practically interesting numerical example is considered in Section 5. A lemma about nonlinear differential inequality is proved in the Appendix.

Regularization procedure
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In the well-posed case (the Fréchet derivative $F'$ of the operator $F$ is a bijection in a neighborhood of the solution of equation (\[noneq\])) in order to solve equation (\[noneq\]) one can use the following continuous processes:

-   simple iteration method: $$\label{esim2}
    \dot{z}(t)=-F(z(t)),\q z(0)=z_0\in H,$$

-   Newton’s method: $$\label{enm2}
    \dot{z}(t)=-[F'(z(t))]^{-1}F(z(t)),\q z(0)=z_0\in H,$$

-   Gauss-Newton’s method: $$\label{egnm}
    \dot{z}(t)=-[F^{\pr *}(z(t))F^{\pr}(z(t))]^{-1}F^{\pr *}(z(t))F(z(t)),\q
    z(0)=z_0\in H,$$

or some of their modifications (see [@a; @g]).

However if $F'$ is not continuously invertible (ill-posed case) one has to replace equations (\[esim2\]) -(\[egnm\]) by the corresponding regularized equations:

-   regularized simple iteration method: $$\label{esim1}
    \dot{z}(t)=-[F(z(t))+\ep(t)(z(t)-z_0)],\q z(0)=z_0\in H,$$

-   regularized Newton’s method: $$\label{enm1} 
    