Dynamical Systems Method ( Dsm ) and Nonlinear Problems

The dynamical systems method (DSM), for solving operator equations, especially nonlinear and ill-posed, is developed in this paper. Consider an operator equation F (u) = 0 in a Hilbert space H and assume that this equation is solvable. Let us call the problem of solving this equation illposed if the operator F ′(u) is not boundedly invertible, and well-posed otherwise. The DSM for solving linear and nonlinear ill-posed problems in H consists of the construction of a 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. The DSM is justified for: (a) an arbitrary linear solvable equations with bounded operator, (b) for well-posed solvable nonlinear equations with twice Fréchet differentiable operator F , (c) for ill-posed solvable nonlinear equations with monotone operators, (d) for ill-posed solvable nonlinear equations with non-monotone operators from a wide class of operators, (e) for ill-posed solvable nonlinear equations with operators F such that A := F ′(u) satisfies the spectral assumption of the type ‖(A+sI)−1‖ ≤ c/s, where c > 0 is a constant, and s ∈ (0, s0), s0 > 0 is a fixed number, arbitrarily small, c does not depend on s and u, and