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 fδ − f δ, then the following results are provided: Problem Fδ(u) := Au − fδ 2 + α u 2 has a unique global minimizer uα,δ for any fδ, uα,δ = A∗(AA∗ + αI)−1fδ. There is a function α = α(δ), limδ→0 α(δ) = 0 such that limδ→0 uα(δ),δ − y = 0, where y is the unique minimal-norm solution to (1). A priori and a posteriori choices of α(δ) are given. Dynamical Systems Method (DSM) is justified for Eq. (1). © 2006 Elsevier Inc. All rights reserved.