  Let $F$ be a nonlinear map in a real Hilbert space $H$. Suppose that
$\sup_{u\in B(u_0,R)}$ $\|[F'(u)]^{-1}\|\leq m(R)$, where
$B(u_0,R)=\{u:\|u-u_0\|\leq R\}$, $R>0$ is arbitrary, $u_0\in H$ is an element.
If $\sup_{R>0}\frac{R}{m(R)}=\infty$, then $F$ is surjective. If
$\|[F'(u)]^{-1}\|\leq a\|u\|+b$, $a\geq 0$ and $b>0$ are constants independent
of $u$, then $F$ is a homeomorphism of $H$ onto $H$. The last result is known
as an Hadamard-type theorem, but we give a new simple proof of it based on the
DSM (dynamical systems method).
