linear differential inequality has been used often in the literature by many authors.

[**Examples.**]{}

[**1.**]{} Let $\ep(t)=\ep_0(1+t)^{-\nu}$, $\ep_0$ and $\nu$ are positive constants. Then the assumptions of Theorem \[t5\] are satisfied if $\nu\in (0,1)$.

[**2.**]{} If $\ep(t)=\f{\ep_0}{\log(1+t)}$, then the assumptions of Theorem \[t5\] are satisfied.

If $\ep(t)=\ep_0e^{-\nu t}$ then condition 4 of Theorem \[t5\] is not satisfied.

Regularized Methods for Non-monotone Operators
==============================================

In this section we discuss two approaches to the regularization of the Gauss-Newton-type schemes for nonlinear equations with non-monotone operators. To describe the first one, assume that $F$ in (\[noneq\]) is compact and Fréchet differentiable. Denote: $$\label{T}
T(h):=F^{\pr *}(h)F'(h),\q T_{\ep}(h):=T(h)+\ep I.$$ Then $T(h)$ is a nonnegative self-adjoint compact operator. Such an operator cannot be boundedly invertible if $H$ is infinite-dimensional. One has: $$\label{ep}
|| T^{-1}_{\ep}(h)||\leq \f{1}{\ep}$$ for any $\ep >0$. Define $\P$: $$\label{P1}
\P(h,t):=(P_{\ep(t)}(\xi)-I)(h-z_0)-P_{\ep(t)}(\xi)T^{-1}_{\ep(t)}(h)F^{\pr 
*}(h)F(h),\q
\ep(t)>0.$$ Here $\xi\in H$ is a fixed element, which will be chosen so that inequality (\[sup\]) (see below) holds, and $$\label{proj}
P_{\ep}(\xi):=\il_{\ep}^{N_1^2}dE(s),$$ $E(s):=E(s,\xi)$ is the resolution of the identity of the self-adjoint operator $T(\xi)$, and $$\label{N12}
\sup_{h\in H}||F'(h)||\leq N_1,$$ so that $||T(\xi)||\le N_1^2$.

Assume that:

1.  problem (\[noneq\]) has a unique solution $y$;

2.  $\P$ is defined by (\[P1\]), $F$ is compact, 