twice Fréchet differentiable and inequalities (\[N\]) hold;

3.  there exist $\xi\in H$ and $ \ep_0>0$ such that $$\label{sup}
    C:=\sup_{\ep\in (0,\ep_0]} ||P_{\ep}(\xi)T^{-1}_{\ep}(\xi)
    \{T(y)-T(\xi)\}||<\f{1}{2};$$

4.  $0<\ep(t)\leq \ep_0$.

Then there exist positive functions $\a(t)$, $\g(t)$ and $\s(t)$, $t\in [0,+\infty)$, such that estimate (\[semon\]) holds with $x(t)$ replaced by $y$. \[l41\]

Condition (\[sup\]) contains a priori information about a nonlinear operator $F$. This condition allows one to get a convergence rate for ill-posed problem (\[noneq\]). It is always satisfied in a well-posed case (for a boundedly invertible operator $T$) if $\xi$ is sufficiently close to $y$. However it is not clear yet how restrictive this condition is, and how it is related to other conditions that one has to use in order to prove the convergence of the process in ill-posed cases. \[wrem\]

[**Proof**]{} [**of Lemma \[l41\]**]{} Using the polar decomposition $F'(h)=U(F^{\pr *}(h)F'(h))^{\f{1}{2}}$, where $U$ is a partial isometry, one gets $F^{\pr *}(h)=T^{\f{1}{2}}U^*$, and, since $||U^*||=1$, one obtains: $$\label{polar}
||T^{-1}_{\ep(t)}(h)F^{\pr *}(h)||\leq||T^{-1}_{\ep(t)}(h)T^{\f{1}{2}}(h)||\leq
\max_{0\le s<+\infty}\f{\sqrt{s}}{s+\ep(t)}=\f{1}{2\sqrt{\ep(t)}}.$$ Using (\[P1\]) and the relation $$\label{47a}
F(h)=F(h)-F(y)=F'(h)(h-y)+R(y,h),$$ where $||R(y,h)||\le\f{N_2}{2}||h-y||^2$, one gets $$(\P(h,t),h-y)=((P_{\ep(t)}(\xi)-I)(h-z_0),h-y) 
-(P_{\ep(t)}(\xi)T^{-1}_{\ep(t)}(h)T(h)(h-y),h-y)$$ $$+\f{N_2}{4\sqrt{\ep(t)}}||h-y||^3\leq -||h-y||^2+||(P_{\ep(t)}(\xi)-I)(y-z_0)||
||h-y||$$ $$\label{4.7}
+\ep(t)(P_{\ep(t)}(\xi)T^{-1}_{\ep(t)}(h)(h-y),h-y)+\f{N_2}{4\sqrt{\ep(t)}}||h-y||^3.$$ Also one has the following estimates: $$\label{2ep}
||P_{\ep(t)}(\xi)T^{-1}_{\ep(t)}(\xi)||\le\max_{s\ge\ep(t)}\f{1}{s+\ep(t)}=
\f{1}{2\ep(t)},$$ and $$||T(h)-T(y)||\le ||F^{\pr *}(h)[F'(h)-F'(y)]+[F^{\pr *}(h)-F^{\pr *}(y)]F'(y)||$$ $$\label{TT}
\leq 2N_1N_2||h-y||.$$ Thus, using the identity $A^{-1}-B^{-1}=-B^{-1}(A-B)A^{-1}$ 