Dynamical Systems Method 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) \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 $t\to +\infty$, for $a(t)$ properly chosen. Here $F(y)=f$, and $\dot{u}$ denotes the time derivative.