z_0-y=F'(y)v,\q ||v||<\f{2}{N_2}.$$ Then for the solution $z(t)$ to problem (\[caupr\]) the following convergence rate estimate holds: $$\label{pr22}
||z(t)-y||\le \left [\f{1-C_\ep}{N_2}+\f{4||v||}{2-N_2||v||}\right ]\ep(t).$$* ]{} \[prop2\]

[**Proof**]{} From (\[pr1\]) for an arbitrary $\ep>0$ one gets $$F(x)-F(y)+\ep(x-y)=\ep(z_0-y).$$ Therefore by the Lagrange formula one has: $$\label{pr23}
\left\{\il_0^1(F'(y+s(x-y))ds+\ep I\right\}(x-y)=\ep(z_0-y).$$ Introduce the notation $Q_\ep(x):=\il_0^1(F'(y+s(x-y))ds+\ep I$. From (\[pr23\]) it follows that $$||x-y||=\ep||Q_\ep^{-1}(x)Q_0(y)v||\le \ep||Q_\ep^{-1}(x)(Q_0(y)-Q_\ep(x))v||
+\ep||Q_\ep^{-1}(x)Q_\ep(x)v||.$$ Since $Q_\ep(x)=Q_0(x)+\ep I$, one obtains $$||x-y||\le  \ep||Q_\ep^{-1}(x)(Q_0(y)-Q_0(x))v||+\ep||Q_\ep^{-1}(x)\ep v||
+\ep||v||$$ $$\label{pr24} 
\le\f{N_2}{2}||x-y||\,||v||+2\ep ||v||.$$ So, from (\[lap\]), (\[pr21\]) and (\[pr24\]) for $\ep=\ep(t)$ and correspondingly $x=x(t)$ satisfying the assumptions of Theorem  \[t3\] one concludes that estimate (\[pr22\]) holds. $\Box$

Now we describe the simple iteration scheme for solving nonlinear equation (\[noneq\]). Define: $$\label{4.1}
\P(h,t):=-[F(h)+\ep(t)(h-z_0)],\q \ep(t)>0.$$

Assume that $F$ is monotone, $\P$ is defined by (\[4.1\]), and $x(t)$ is a solution to problem (\[pr1\]) with $\ep=\ep(t)>0,$ $t\in [0,+\infty)$. Then for the positive function $\g(t):=\ep(t)$ and for $\s(t)=\a(t)\equiv 0$ estimate (\[semon\]) holds. \[l9\]

[**Proof**]{} Since $x(t)$ is a solution to problem (\[pr1\]), by the monotonicity of $F$ one has: $$(\P(h,t),h-x(t))=-(F(h)-F(x(t),h-x(t))-\ep(t)(h-x(t),h-x(t))$$ $$\label{star1}
\le -\ep(t)||h-x(t)||^2.$$ Lemma \[l9\] is proved. $\Box$

Lemma \[l9\] together with Lemma \[stupid\] presented below allow one to formulate the convergence result concerning the simple iteration procedure (see Theorem  \[t5\]).

Let $\nu(t)$ be integrable on $[0,+\infty)$. Suppose that 