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 $0 \in {\rm Sp}\, A$. 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.