1 Dynamical Systems Method (DSM) for solving nonlinear operator equations in Banach spaces

Let F (u) = h be a solvable operator equation in a Banach spaceX 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) u̇(t) = −A−1 a(t)(u(t))[F (u(t)) + a(t)u(t) − f], u(0) = u0, converges to y as t → +∞, for a(t) properly chosen. Here F (y) = f, and u ̇ denotes the time derivative