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

This paper is a review of the authors’ results on the Dynamical Systems Method (DSM) 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 Fréchet differentiable, but no smoothness assumptions on F ′(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(∞) is established, and the relation F (u(∞)) = f is obtained. The DSM is developed for a stable solution of equation (*) when noisy data fδ are given, ‖f − fδ‖ ≤ δ.