Dynamical systems method for solving operator equations.pdf

a r X i v : m a t h / 0 3 0 1 3 7 8 v 1 [ m a t h .D S ] 3 1 J a n 2 0 0 3 Dynamical systems method for solving operator equations ? A.G. Ramm Mathematics Department, Kansas State University, Manhattan, KS 66506-2602, USA E:mail: ramm@ /?ramm Abstract Consider an operator equation F (u) = 0 in a real Hilbert space. The problem of solving this equation is ill-posed if the operator F ′(u) is not boundedly invertible, and well-posed otherwise. A general method, dynamical systems method (DSM) for solving linear and nonlinear ill-posed problems in a Hilbert space is presented. This method consists of the construction of a nonlinear dynamical system, that is, a Cauchy problem, which has the following properties: 1) it has a global solution, 2) this solution tends to a limit as time tends to infinity, 3) the limit solves the original linear or non- linear problem. New convergence and discretization theorems are obtained. Examples of the applications of this approach are given. The method works for a wide range of well-posed problems as well. 1 Introduction This paper contains a recent development of the theory of DSM (dynamical systems method) earlier developed in papers [2]-[12]. DSM is a general method for solving op- erator equations, especially nonlinear, ill-posed, but also well-posed operator equations. The author hopes that DSM will demonstrate its practical efficiency and will allow one to solve ill-posed problems which cannot be solved by other methods. This paper is intended for a broad audience: the presentation is simplified considerably, and is non-technical in its present form. Most of the results are presented in a new way. Some of the results and/or proofs are new (Theorems 2.1, 3.1, 3.2, 4.2, 6.2, 7.1, 8.1, Remarks 4.4, 4.5, and the discussion of the stopping rules). We try to emphasize the basic ideas and methods of the proofs. What is the dynamical systems method (DSM) for solving operator equations? Consider an equation F (u) := B(u)? f = 0, f ∈