General Source Conditions for Nonlinear Ill-Posed Equations

Abstract

In order to derive error estimates for regularized approximate solutions of ill-posed equations F(x) = y, where F : D(F) ⊆ X → Y is a nonlinear operator between Hilbert spaces X and Y, certain source conditions are to be imposed on the unknown solution . For linear ill-posed equations of the form Ax = y, where A:X → Y is a bounded linear operator, Nair, Schock, and Tautenhahn considered a general source condition, namely, ∈ M ρ, ϕ := {[ϕ(A*A)]1/2 v: ‖v‖ ≤ ρ}, where the function ϕ is general enough to include some of the well-studied special cases. We extend such procedure to the nonlinear ill-posed equations F(x) = y, so that it also includes the results of Tautenhahn where the special case of ϕ(λ) := λν/2 is considered.