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

Abstract

Let \(F(u) = f\) 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 \[ \dot{u}(t) = -A^{-1}_{a(t)}(u(t))\left[F(u(t)) + a(t)u(t) - f\right], \quad u(0) = u_0, \] converges to \(y\) as \(t \to +\infty\), for properly chosen \(a(t)\). Here \(F(y) = f\), and \(\dot{u}\) denotes the time derivative.