DSM for general nonlinear equations

Abstract

If F : H → H is a map in a Hilbert space H , F ∈ C l o c 2 , 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 f δ are known, where ‖ f δ − f ‖ ≤ δ .