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

TL;DR

This paper explores iterative methods for solving ill-posed linear equations with self-adjoint operators in Hilbert spaces, focusing on convergence issues when spectral radius conditions are critical, using Krasnoselskii's theorem and its variants.

Contribution

It extends Krasnoselskii's convergence theorem to critical cases in ill-posed problems involving self-adjoint operators, providing new insights into iterative solution methods.

Findings

Convergence of iterative methods is established under critical spectral conditions.

Modified Krasnoselskii theorems improve understanding of solution behavior.

Results apply to ill-posed linear operator equations with self-adjoint operators.

Abstract

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.