Stability of semilinear ill-posed problems with a prescribed energy bound

✍ Scribed by A. A. Lyashenko

• Publisher: John Wiley and Sons
• Year: 1995
• Tongue: English
• Weight: 714 KB
• Volume: 18
• Category: Article
• ISSN: 0170-4214
• DOI: 10.1002/mma.1670180705

✦ Synopsis

Abstract

In the present paper we discuss the stability of semilinear problems of the form A^α^u + G^α^(u) = ƒ under assumption of an a priori bound for an energy functional E^α^(u) ⩽ E, where α is a parameter in a metric space M. Following [11] the problem A^α^u + G^α^(u) = ƒ, E^α^(u) ⩽ E is called stable in a Hilbert space H at a point α ϵ M if for any ƒϵ__H__, E, ϵ > 0 there exists δ > 0 such that for any functions u^α1^, u^α2^ satisfying A^α__j__^u^α__j__^ + G^α__j__^(u^α__j__^) = ƒ^α__j__^, E^α__j__^(u^α__j__^) ⩽ E, j = 1,2 we have ‖u^α1^ − u^α2^~H~ ⩽ ϵ provided ρ~M~(α~j~, α) ⩽ δ, ‖ƒ^α__j__^ − ƒ‖~H~ ⩽ δ, j = 1,2. In the present paper we obtain stability conditions for the problem A^α^u + G^α^(u) = ƒ, E^α^(u) ⩽ E.