DSM for general nonlinear equations

If F:H→H is a map in a Hilbert space H, F∈Cloc2, and there exists a solution y, possibly non-unique, such that F(y)=0, F′(y)≠0, then equation F(u)=0 can be solved by a DSM (Dynamical Systems Method) and the rate of convergence of the DSM is given provided that a source-type assumption holds. A discrete version of the DSM yields also a convergent iterative method for finding y. This method converges at the rate of a geometric series. Stable approximation to a solution of the equation F(u)=f is constructed by a DSM when f is unknown but the noisy data fδ are known, where ‖fδ−f‖≤δ.