DSM of Newton type for solving operator equations F(u) = f with minimal smoothness assumptions on F.

Abstract

This paper is a review of the authors’ results on the DSM (Dynamical Systems Method) for solving operator equation () F(u) = f. It is assumed that () is solvable. The novel feature of the results is the minimal assumption on the smoothness of F. It is assumed that F is continuously Fr´echet differentiable, but no smoothness assumptions on F0(u) are imposed. The DSM for solving equation () is developed. Under weak assumptions global existence of the solution u(t) is proved, the existence of u(1) is established, and the relation F(u(1)) = f is obtained. The DSM is developed for a stable solution of equation () when noisy data f are given, kf − f k .