Simultaneous iterative algorithms for the split common fixed-point problem of generalized asymptotically quasi-nonexpans
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RESEARCH
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Simultaneous iterative algorithms for the split common fixed-point problem of generalized asymptotically quasi-nonexpansive mappings without prior knowledge of operator norms Jing Zhao* and Songnian He *
Correspondence: [email protected] College of Science, Civil Aviation University of China, Tianjin, 300300, P.R. China
Abstract Let H1 , H2 , H3 be real Hilbert spaces, let A : H1 → H3 , B : H2 → H3 be two bounded linear operators. Moudafi introduced simultaneous iterative algorithms (Trans. Math. Program. Appl. 1:1-11, 2013) with weak convergence for the following split common fixed-point problem: 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). In this paper, we will continue to consider the split common fixed-point problem (1) governed by the general class of generalized asymptotically quasi-nonexpansive mappings. To estimate the norm of an operator is a very difficult, if it is not an impossible task. The purpose of this paper is to propose a simultaneous iterative algorithm with a way of selecting the stepsizes such that the implementation of the algorithm does not need any prior information as regards the operator norms. MSC: 47H09; 47H10; 47J05; 54H25 Keywords: split common fixed-point problem; generalized asymptotically quasi-nonexpansive mappings; weak convergence; simultaneous iterative algorithm; Hilbert space
1 Introduction and preliminaries Throughout this paper, we always assume that H is a real Hilbert space with the inner product ·, · and the norm · . Let T : H → H be a mapping. A point x ∈ H is said to be a fixed point of T provided Tx = x. In this paper, we use F(T) to denote the fixed point set and use → and to denote the strong convergence and weak convergence, respectively. We use ωw (xk ) = {x : ∃xkj x} stand for the weak ω-limit set of {xk }. © 2014 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 2014, 2014:73 http://www.fixedpointtheoryandapplications.com/content/2014/1/73
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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 []. Recently, it has been found that the SFP c
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