  Let $F:X\to X$ be a $C^2_\loc$ map in a Banach space $X$, and $A$ be its
Fr\`echet derivative at the element $w:=w_\ve$, which solves the problem
$(\ast) \dotw=-A^{-1}_\ve(F(w)+\ve w)$, $w(0)=w_0$, where $A_\ve:=A+\ve I$.
Assume that $\|A^{-1}_\ve\|\leq c \ve^{-k}$, $0<k\leq 1$, $0<\ve>\ve_0$. Then
$(\ast)$ has a unique global solution, $w(t)$, there exists $w(\infty)$, and
$(\ast\ast) F(w(\infty))+\ve w(\infty)=0$. Thus the DSM (Dynamical Systems
Method) is justified for equation $(\ast\ast)$. The limit of $w_\ve$ as $\ve\to
0$ is studied.
