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

Abstract

In this paper we study the stability of linear operator equation A^αu = 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^αu = 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).

Keywords:

• Ill-posed problem
• Stability
• A priori energy bound

Additional information

Notes on contributors

A.A. Lyashenko