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 .