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^αu + G^α(u) = ? 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^αu + G^α(u) = ?, E^α(u) ? E is called stable in a Hilbert space H at a point α ? M if for any ??H, E, ? > 0 there exists δ > 0 such that for any functions u^α1, u^α2 satisfying A^αju^αj + G^αj(u^αj) = ?^αj, E^αj(u^αj) ? E, j = 1,2 we have ‖u^α1 ? u^α2_H ? ? provided ρ_M(α_j, α) ? δ, ‖?^αj ? ?‖_H ? δ, j = 1,2. In the present paper we obtain stability conditions for the problem A^αu + G^α(u) = ?, E^α(u) ? E.