Iterative methods for the split common fixed-point problem in Hilbert spaces

TL;DR

This paper introduces a new iterative algorithm for the split common fixed-point problem in Hilbert spaces that does not require prior knowledge of the operator norm, ensuring weak convergence under mild conditions.

Contribution

The paper proposes a novel algorithm for the split common fixed-point problem that removes the need for prior operator norm information and proves its weak convergence.

Findings

The new algorithm converges weakly under mild assumptions.

It does not require prior knowledge of the operator norm.

The algorithm is efficient for solving the split common fixed-point problem.

Abstract

The split common fixed-point problem is an inverse problem that consists in finding an element in a fixed-point set such that its image under a bounded linear operator belongs to another fixed-point set. Recently Censor and Segal proposed an efficient algorithm for solving such a problem. However, to employ their algorithm, one needs to know a prior information on the norm of the bounded linear operator. In this paper we propose a new algorithm that does not need any prior information of the operator norm, and we establish the weak convergence of the proposed algorithm under some mild assumptions.