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

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 .