Dynamical Systems Method (DSM) for general nonlinear equations

Abstract If F : H → H is a map in a Hilbert space H , F ∈ C loc 2 , and there exists y such that F ( y ) = 0 , F ′ ( y ) ≠ 0 , then equation F ( u ) = 0 can be solved by a DSM (dynamical systems method). This method yields also a convergent iterative method for finding y , and this method converges at the rate of a geometric series. It is not assumed that y is the only solution to F ( u ) = 0 . A stable approximation to a solution of the equation F ( u ) = f is constructed by a DSM when f is unknown but f δ is known, where ‖ f δ − f ‖ ≤ δ .