Dynamical systems method for solving operator equations

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 for solving linear and non-linear ill-posed problems in a Hilbert space is presented. This method consists of the construction of a non-linear 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.

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