Dynamical systems method (DSM) for solving nonlinear operator equations in Banach spaces

Let $F(u)=h$ be a solvable operator equation in a Banach space $X$ with a Gateaux differentiable norm. Under minimal smoothness assumptions on $F$, sufficient conditions are given for the validity of the Dynamical Systems Method (DSM) for solving the above operator equation. It is proved that the DSM (Dynamical Systems Method)
$$ \dot u(t)=-A_{a(t)}^{-1}(u(t))[F(u(t))+a(t)u(t)-f)],\quad u(0)=u_0, $$
converges to $y$ as $t\to+\infty$, for $a(t)$ properly chosen. Here $F(y)=f$, and $\dot u$ denotes the time derivative.