Dynamical Systems Method for solving nonlinear operator equations in Banach spaces

TL;DR

This paper establishes conditions under which the Dynamical Systems Method (DSM) converges to a solution of nonlinear operator equations in Banach spaces with minimal smoothness assumptions.

Contribution

It provides new convergence results for DSM applied to nonlinear equations in Banach spaces under minimal smoothness conditions.

Findings

DSM converges to the solution as time approaches infinity.

Properly chosen regularization parameter ensures convergence.

Minimal smoothness assumptions are sufficient for convergence.

Abstract

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.