M. A. Krasnosel'skii Theorem and Iterative Methods for Solving Ill-Posed Linear Problems with a Self-Adjoint Operator

Abstract

Abstract The paper deals with iterative methods for solving linear operator equations x = B x + f ${x = Bx + f}$ and A x = f ${Ax = f}$ with self-adjoint operators in Hilbert space X in the critical case when ρ ( B ) = 1 ${\rho (B) = 1}$ and 0 ∈ Sp A ${0 \in \operatorname{Sp} A}$ . The results obtained are based on a theorem by M. A. Krasnosel'skii on the convergence of the successive approximations, their modifications and refinements.