M.A. Krasnoselskii theorem and iteration methods of solving ill-posed linear problems with a self-adjoint operator

The article deals with iterative methods of solving linear operator equations $x=Bx+f$ and $Ax=f$ with self-adjoint operators in Hilbert space $X$ in critical case when $\rho (B)=1$ and $0\in \mathrm{S}\mathrm{p}A$. The main results are based on the use of M.A. Krasnosel’ski\u{i} theorem about the convergence of the successive approximations and some its modifications and refinements.