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

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 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. MSC: 47J25; 47J20; 49N45; 65J15 Keywords: split common fixed point problem; directed operators; demicontractive operator; quasi-nonexpansive operator

Finding a solution of split null point of the sum of monotone operators without prior knowledge of operator norms in Hilbert spaces

“…This problem is called the split common fixed point problem (SCFP), and was studied by many authors, see [16,26,31,40] for example. We will denote by Γ V 1 ,V 2 for the solution set of problem (4.4) and introduce the following algorithm.…”

Strong convergence theorems for the general split common fixed point problem in Hilbert spaces

“…This algorithm was extended to the case of finitely many directed operators by Wang and Xu [14], quasi-nonexpansive operators by Moudafi [12],and demicontractive mappings by Moudafi [11]. Recently, Cui and Wang [9] proposed the following iterative method to solve the problem (1.1). For given x 0 ∈ H 1 and λ ∈ (0, 1 − τ), the iterative sequence {x n } is generated as follows:…”

“…Motivated and inspired by Cui and Wang's work [9], we consider the general split common fixed point problem GSCFP and construct an algorithm for demicontractive operators that produces sequences that always converge strongly to a solution of GSCFP and whose step size does not depend on the norm of the operator A. The constructed algorithm is Halpern type [10].…”

“…As we know, the SFP has received much attention due to its application in intensity-modulated radiation therapy, signal processing, and image reconstruction, for instance [2,3,5,6,8,9,11,12,14,16,17].…”

Hybrid iterative algorithms for the split common fixed point problems

“…They considered a parallel algorithm for solving the SCFPP (1.2) for a class of directed operators in finite dimensional spaces. Later, Ansari et al [2], Cui and Wang [13], Krailkaew and Saejung [17] and Moudafi [20,21] proposed different kinds of algorithms for solving SCFPP (1.2) in the Hilbert space setting. In this paper, motivated by the works [2,16], we present two iterative algorithms based on Yamada's the hybrid steepest descent method [28] for solving the SCFPP (1.2).…”