C_\ep:=\max_{t\in[0,\infty)}\f{\ep(0)|\dot{\ep}(t)|}{\ep^2(t)}<1;$$

4.  $$\label{rho}
    \ep(0)>\f{N_2||z_0-y||}{1-C_\ep}
    \max\left\{1,\f{2C_\ep}{1-C_\ep}\right\};$$

5.  $\P$ is defined by (\[rchn\]).

Then Cauchy problem (\[caupr\]) has a unique solution $z(t)$ for $t\in [0,+\infty)$ and $$\label{lap}
||z(t)-x(t)||\le\f{1-C_\ep}{N_2}\ep(t),\q \lim_{t\to+\infty}||z(t)-y||=0.$$ \[t3\]

First notice that Theorem \[t3\] establishes convergence for any initial approximation point $z_0$ if $\ep(t)$ is appropriately chosen. To make an appropriate choice of $\ep(t)$ one has to choose some function $\ep(t)$ satisfying condition (\[rhoc\]). Examples of such functions $\ep(t)$ are given below. One can observe that condition (\[rhoc\]) is invariant with respect to a multiplication $\ep(t)$ by a constant. Therefore one can choose $\ep(t)$ satisfying condition (\[rho\]) by a multiplication of the original $\ep(t)$ by a sufficiently large constant. If $\f{|\dot{\ep}(t)|}{\ep^2(t)}$ is not increasing, then in condition (\[rho\]) $C_\ep:=\max_{t\in[0,\infty)}\f{\ep(0)|\dot{\ep}(t)|}{\ep^2(t)}$ can be replaced by $C_\ep:=\f{|\dot{\ep}(0)|}{\ep(0)}$. \[noincon\]

In order to get an estimate of the convergence rate for $||x(t)-y||$ one has to make some additional assumptions either on $F(x)$ or on the choice of the initial approximation $z_0$. Without such assumptions one cannot give an estimate of the convergence rate. Indeed, as a simple example consider the scalar equation $F(x):=x^m=0$. Then one gets the following algebraic equation for $x(\ep)$: $$\label{algeq} 
F_\ep(x):=x^m+\ep(x-z_0)=0.$$ Assume $m$ is a positive integer and $z_0>0$. It is known that 