Dynamical systems method (DSM) for selfadjoint operators

Let A be a selfadjoint linear operator in a Hilbert space H. The DSM (dynamical systems method) for solving equation Av = f consists of solving the Cauchy problem ˙u = Φ(t,u), u(0) = u0, where Φ is a suitable operator, and proving that (i) ∃u(t) ∀t > 0, (ii) ∃u(∞), and (iii) A(u(∞)) = f . It is proved that if equation Av = f is solvable and u solves the problem ˙u = i(A + ia)u − if , u(0) = u0, where a > 0 is a parameter and u0 is arbitrary, then lima→0 limt→∞u(t, a) = y, where y is the unique minimal-norm solution of the equation Av = f . Stable solution of the equation Av = f is constructed when the data are noisy, i.e., fδ is given in place of f , fδ −f δ. The case when a = a(t) > 0, ∞0 a(t) dt =∞, a(t) 0 as t→∞is considered. It is proved that in this case limt→∞u(t) = y and if fδ is given in place of f , then limt→∞u(tδ) = y, where tδ is properly chosen. © 2006 Elsevier Inc. All rights reserved