Ill-posed problems with unbounded operators

Let A be a linear, closed, densely defined unbounded operator in a Hilbert space. Assume that A is not boundedly invertible. If Eq. (1) Au = f is solvable, and vertical bar vertical bar f(delta) - f vertical bar vertical bar <= delta, then the following results are provided: Problem F-delta (u) := vertical bar vertical bar Au - f(delta)vertical bar vertical bar(2) + alpha vertical bar vertical bar u vertical bar vertical bar(2) has a unique global minimizer u(alpha,delta) for any f(delta), u(alpha,delta) = A* (AA* + alpha 1)(-1)f(delta). There is a function alpha = alpha (delta), lims(delta -> 0)alpha(delta) = 0 such that lims(delta -> 0)vertical bar vertical bar u(alpha(delta).delta) - y vertical bar vertical bar = 0, where y is the unique minimal-norm solution to (1). A priori and a posteriori choices of alpha(delta) are given. Dynamical Systems Method (DSM) is justified for Eq. (1). (c) 2006 Elsevier Inc. All rights reserved.