Dynamical Systems Method (DSM) for general nonlinear equations

If F:H→HF:H→H is a map in a Hilbert space HH, F∈Cloc2, and there exists yy such that F(y)=0F(y)=0, F′(y)≠0F′(y)≠0, then equation F(u)=0F(u)=0 can be solved by a DSM (dynamical systems method). This method yields also a convergent iterative method for finding yy, and this method converges at the rate of a geometric series. It is not assumed that yy is the only solution to F(u)=0F(u)=0. A stable approximation to a solution of the equation F(u)=fF(u)=f is constructed by a DSM when ff is unknown but fδfδ is known, where ‖fδ−f‖≤δ‖fδ−f‖≤δ.