On a new notion of the solution to an ill-posed problem

A new understanding of the notion of the stable solution to ill-posed problems is proposed. The new notion is more realistic than the old one and better fits the practical computational needs. A method for constructing stable solutions in the new sense is proposed and justified. The basic point is: in the traditional definition of the stable solution to an ill-posed problem $Au=f$, where $A$ is a linear or nonlinear operator in a Hilbert space $H$, it is assumed that the noisy data $\{f_{\delta },\delta \}$ are given, $\mathrm{\mid }\mathrm{\mid }f-f_{\delta }\mathrm{\mid }\mathrm{\mid }\le \delta$, and a stable solution $u_{\text{d}}:=R_{\text{d}}{f}_{\text{d}}$ is defined by the relation ${\lim }_{\text{d}\to 0}\mathrm{\mid }\mathrm{\mid }{R}_{\text{d}}{f}_{\text{d}}-y\mathrm{\mid }\mathrm{\mid }=0$, where $y$ solves the equation $Au=f$, i.e., $Ay=f$. In this definition $y$ and $f$ are unknown. Any $f\in B(f_{\text{d}},\text{d})$ can be the exact data, where $B(f_{\text{d}},\text{d}):=\{f:\mathrm{\mid }\mathrm{\mid }f-f_{\delta }\mathrm{\mid }\mathrm{\mid }\le \delta \}$.The new notion of the stable solution excludes the unknown $y$ and $f$ from the definition of the solution.