Alternating mann iterative algorithms for the split common fixed-point problem of quasi-nonexpansive mappings
- PDF / 345,174 Bytes
- 12 Pages / 595.276 x 793.701 pts Page_size
- 12 Downloads / 206 Views
RESEARCH
Open Access
Alternating mann iterative algorithms for the split common fixed-point problem of quasi-nonexpansive mappings Jing Zhao* and Songnian He * Correspondence: [email protected] College of Science, Civil Aviation University of China, Tianjin, 300300, P.R. China
Abstract Very recently, Moudafi (Alternating CQ-algorithms for convex feasibility and split fixed-point problems, J. Nonlinear Convex Anal. ) introduced an alternating CQ-algorithm with weak convergence for the following split common fixed-point problem. Let H1 , H2 , H3 be real Hilbert spaces, let A : H1 → H3 , B : H2 → H3 be two bounded linear operators. Find x ∈ F(U), y ∈ F(T) such that Ax = By,
()
where U : H1 → H1 and T : H2 → H2 are two firmly quasi-nonexpansive operators with nonempty fixed-point sets F(U) = {x ∈ H1 : Ux = x} and F(T) = {x ∈ H2 : Tx = x}. Note that by taking H2 = H3 and B = I, we recover the split common fixed-point problem originally introduced in Censor and Segal (J. Convex Anal. 16:587-600, 2009) and used to model many significant real-world inverse problems in sensor net-works and radiation therapy treatment planning. In this paper, we will continue to consider the split common fixed-point problem (1) governed by the general class of quasi-nonexpansive operators. We introduce two alternating Mann iterative algorithms and prove the weak convergence of algorithms. At last, we provide some applications. Our results improve and extend the corresponding results announced by many others. MSC: 47H09; 47H10; 47J05; 54H25 Keywords: split common fixed-point problem; quasi-nonexpansive mapping; weak convergence; Mann iterative algorithm; Hilbert space
1 Introduction Throughout this paper, we always assume that H is a real Hilbert space with the inner product ·, · and the norm · . Let I denote the identity operator on H. Let C and Q be nonempty closed convex subset of real Hilbert spaces H and H , respectively. The split feasibility problem (SFP) is to find a point x ∈ C such that Ax ∈ Q,
(.)
where A : H → H is a bounded linear operator. The SFP in finite-dimensional Hilbert spaces was first introduced by Censor and Elfving [] for modeling inverse problems which arise from phase retrievals and in medical image reconstruction []. The SFP attracts many ©2013 Zhao and He; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Zhao and He Fixed Point Theory and Applications 2013, 2013:288 http://www.fixedpointtheoryandapplications.com/content/2013/1/288
Page 2 of 12
authors’ attention due to its application in signal processing. Various algorithms have been invented to solve it (see [–] and references therein). Note that if the split feasibility problem (.) is consistent (i.e., (.) has a solution), then (.) can be formulated as a fixed point equation by using the fact
Data Loading...
