Let F(u)=h be an operator equation in a Banach space X,
∥F′(u)−F′(v)∥≤ω(∥u−v∥), where ω∈C([0,∞)),
ω(0)=0, ω(r)>0 if r>0, ω(r) is strictly growing on
[0,∞). Denote A(u):=F′(u), where F′(u) is the Fr\'{e}chet derivative
of F, and Aa​:=A+aI. Assume that (*) ∥Aa−1​(u)∥≤∣a∣bc1​​, ∣a∣>0, b>0, a∈L. Here a may be a complex number,
and L is a smooth path on the complex a-plane, joining the origin and some
point on the complex a−plane, 00 is a
small fixed number, such that for any a∈L estimate (*) holds. It is proved
that the DSM (Dynamical Systems Method) \bee
\dot{u}(t)=-A^{-1}_{a(t)}(u(t))[F(u(t))+a(t)u(t)-f],\quad u(0)=u_0,\
\dot{u}=\frac{d u}{dt}, \eee converges to y as t→+∞, where a(t)∈L,F(y)=f, r(t):=∣a(t)∣, and r(t)=c4​(t+c2​)−c3​, where cj​>0 are
some suitably chosen constants, j=2,3,4. Existence of a solution y to the
equation F(u)=f is assumed. It is also assumed that the equation
F(wa​)+awa​−f=0 is uniquely solvable for any f∈X, a∈L, and
$\lim_{|a|\to 0,a\in L}\|w_a-y\|=0.