solve nonlinear operator equation (\[noneq\]). Assume that $F$ is Fréchet differentiable and $$\label{frder}
(F'(h)\xi,\xi)\ge 0\q\t{ for all }\q h,\xi\in H.$$ Under this assumption the operator $F'(h)+\ep(t)I$ is boundedly invertible. Define $\P$ as follows: $$\label{rchn}
\P(h,t):=-[F'(h)+\ep(t)I]^{-1}[F(h)+\ep(t)(h-z_0)],$$ where $z_0\in H$ is an initial approximation point and $\ep(t)$ is some positive function on the interval $[0,\infty)$. Some restrictions on $\ep(t)$ will be stated in Theorem \[t3\].

An outline of the convergence proof is the following. First one considers an auxiliary well-posed problem: $$\label{pr1}
F_\ep(x):=F(x)+\ep(x-z_0)=0, \q \ep>0,$$ and shows that the difference between its solution $x(t)$ and the solution $z(t)$ to problem (\[caupr\]) tends to zero as $t\to +\infty$. On the other hand one shows that $x(t)$ converges to the exact solution $y$ of equation (\[noneq\]). Thus one proves the convergence of $z(t)$ to $y$ as $t\to +\infty$.

We recall first some definitions from nonlinear functional analysis which are used below. The most essential restrictions on the operator $F$ imposed in this section are (\[frder\]) and [*w*]{}-continuity of $F$. In particular they imply monotonicity and hemicontinuity of $F$.

A mapping $\p$ is monotone in a Hilbert space $H$ if $$(\p(x_1)-\p(x_2),x_1-x_2)\ge 0,\q \forall x_1,x_2\in H.$$

A mapping $\p$ is hemicontinuous at $x_0\in H$ if the 