On the stability of linear III-posed problems with a prescribed energy bound

In this paper we study the stability of linear operator equation A α u = ƒ under assumption of an a priori bound E(u)≤E, where α is a parameter in a metric space M and E(u) is a positive functional. Following[11] the problem A α u = ƒ,E(u)≤E is called stable in a Hilbert space H at a point α ∈ M if for any ƒ ∈ H, E,∈ > 0 there exists δ>0 such that for any functions satisfying , j = 1,2 we have H ≤∈ provided ρM(α j ,α)≤ δ , j=1,2. We show that if A α has a complete in H system of eigenvectors, and the eigenvectors and the eigenvalues depend continuously on α ∈ M then the problem is stable at α ∈ M if and only if 0∉σ p: (A α ).