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

摘要

AbstractIn the present paper we discuss the stability of semilinear problems of the formAαu+Gα(u) = ƒ under assumption of ana prioribound for an energy functionalEα(u) ⩽E, where α is a parameter in a metric spaceM. Following 11 the problemAαu+Gα(u) = ƒ,Eα(u) ⩽Eis called stable in a Hilbert spaceHat a point α ϵMif for any ƒϵH,E, ϵ>0 there exists δ>0 such that for any functionsuα1,uα2satisfyingAαjuαj+Gαj(uαj) = ƒαj,Eαj(uαj) ⩽E,j= 1,2 we have ‖uα1−uα2H⩽ ϵ provided ρM(αj, α) ⩽ δ, ‖ƒαj− ƒ‖H⩽ δ,j= 1,2. In the present paper we obtain s