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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 438023, 12 pages
http://dx.doi.org/10.1155/2012/438023
Research Article

Strong Convergence of the Viscosity Approximation Process for the Split Common Fixed-Point Problem of Quasi-Nonexpansive Mappings

College of Science, Civil Aviation University of China, Tianjin 300300, China

Received 14 December 2011; Accepted 11 January 2012

Academic Editor: Yonghong Yao

Copyright © 2012 Jing Zhao and Songnian He. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Very recently, Moudafi (2011) introduced an algorithm with weak convergence for the split common fixed-point problem. In this paper, we will continue to consider the split common fixed-point problem. We discuss the strong convergence of the viscosity approximation method for solving the split common fixed-point problem for the class of quasi-nonexpansive mappings in Hilbert spaces. Our results improve and extend the corresponding results announced by many others.

1. Introduction and Preliminary

Throughout this paper, we always assume that 𝐻 is a real Hilbert space with inner product , and norm . Let 𝐼 denote the identity operator on 𝐻. Let 𝐶 and 𝑄 be nonempty closed convex subset of real Hilbert spaces 𝐻1 and 𝐻2, respectively. The split feasibility problem (SFP) is to find a point 𝑥𝐶suchthat𝐴𝑥𝑄,(1.1) where 𝐴𝐻1𝐻2 is a bounded linear operator. The SFP in finite-dimensional Hilbert spaces was first introduced by Censor and Elfving [1] for modeling inverse problems which arise from phase retrievals and in medical image reconstruction [2]. The SFP attracts many authors' attention due to its application in signal processing. Various algorithms have been invented to solve it (see [39] and references therein).

Note that the split feasibility problem (1.1) can be formulated as a fixed-point equation by using the fact 𝑃𝐶𝐼𝛾𝐴𝐼𝑃𝑄𝐴𝑥=𝑥;(1.2) that is, 𝑥 solves the SFP (1.1) if and only if 𝑥 solves the fixed point equation (1.2) (see [10] for the details). This implies that we can use fixed-point algorithms (see [1113]) to solve SFP. A popular algorithm that solves the SFP (1.1) is due to Byrne's CQ algorithm [2] which is found to be a gradient-projection method (GPM) in convex minimization. Subsequently, Byrne [3] applied KM iteration to the CQ algorithm, and Zhao and Yang [14] applied KM iteration to the perturbed CQ algorithm to solve the SFP. It is well known that the CQ algorithm and the KM algorithm for a split feasibility problem do not necessarily converge strongly in the infinite-dimensional Hilbert spaces.

The split common fixed-point problem (SCFP) is a generalization of the split feasibility problem (SFP) and the convex feasibility problem (CFP); see [15]. In this paper, we introduce and study the convergence properties of a viscosity approximation algorithm for solving the SCFP for the class of quasi-nonexpansive operators 𝑆 such that 𝐼𝑆 is demiclosed at the origin.

Now let us first recall the definition of quasi-nonexpansive operators which appear naturally when using subgradient projection operator techniques in solving some feasibility problems, and also some definitions of classes of operators often used in fixed-point theory and which are commonly encountered in the literature.

Let 𝑇𝐻𝐻 be a mapping. A point 𝑥𝐻 is said to be a fixed point of 𝑇 provided that 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)={𝑥𝑥𝑘𝑗𝑥} stand for the weak 𝜔-limit set of {𝑥𝑘}.(i)A mapping 𝑇𝐻𝐻 belongs to the general class Φ𝑄 of (possibly discontinuous) quasi-nonexpansive mappings if 𝑇𝑥𝑞𝑥𝑞,(𝑥,𝑞)𝐻×𝐹(𝑇).(1.3)(ii)A mapping 𝑇𝐻𝐻 belongs to the set Φ𝑁 of nonexpansive mappings if 𝑇𝑥𝑇𝑦𝑥𝑦,(𝑥,𝑦)𝐻×𝐻.(1.4)(iii)A mapping 𝑇𝐻𝐻 belongs to the set ΦFN of firmly nonexpansive mappings if 𝑇𝑥𝑇𝑦2𝑥𝑦2(𝑥𝑦)(𝑇𝑥𝑇𝑦)2,(𝑥,𝑦)𝐻×𝐻.(1.5)(iv)A mapping 𝑇𝐻𝐻 belongs to the set ΦFQ of firmly quasi-nonexpansive mappings if 𝑇𝑥𝑞2𝑥𝑞2𝑥𝑇𝑥2,(𝑥,𝑞)𝐻×𝐹(𝑇).(1.6)

It is easily observed that ΦFNΦ𝑁Φ𝑄 and that ΦFNΦFQΦ𝑄. Furthermore, ΦFN is well known to include resolvents and projection operators, while ΦFQ contains subgradient projection operators (see, e.g., [16] and the reference therein).

A mapping 𝑇𝐻𝐻 is called demiclosed at the origin if any sequence {𝑥𝑛} weakly converges to 𝑥, and if the sequence {𝑇𝑥𝑛} strongly converges to 0, then 𝑇𝑥=0. A mapping 𝑓𝐻𝐻 is called a contraction of modulus 𝜌[0,1) if 𝑓𝑥𝑓𝑦𝜌𝑥𝑦,(𝑥,𝑦)𝐻×𝐻.(1.7)

In what follows, we will focus our attention on the following general two-operator split common fixed-point problem: nd𝑥𝐶suchthat𝐴𝑥𝑄,(1.8) where 𝐴𝐻1𝐻2 is a bounded linear operator, 𝑈𝐻1𝐻1 and 𝑆𝐻2𝐻2 are two quasi-nonexpansive operators with nonempty fixed-point sets 𝐹(𝑈)=𝐶 and 𝐹(𝑆)=𝑄, and denote the solution set of the two-operator SCFP by Γ={𝑦𝐶;𝐴𝑦𝑄}.(1.9)

Recall that 𝐹(𝑈) and 𝐹(𝑆) are nonempty closed convex subsets of 𝐻1 and 𝐻2, respectively. If Γ, we have Γ which is close convex subset of 𝐻1. To solve (1.8), Censor and Segal [15] proposed and proved, in infinite-dimensional spaces, the convergence of the following algorithm: 𝑥𝑘+1𝑥=𝑈𝑘+𝛾𝐴𝑡(𝑆𝐼)𝐴𝑥𝑘,𝑘𝑁,(1.10)

where 𝛾(0,2/𝜆), with 𝜆 being the largest eigenvalue of the matrix 𝐴𝑡𝐴 (𝐴𝑡 stands for matrix transposition). Very recently, Moudafi [17] introduced the following relaxed algorithm: 𝑥𝑘+1=1𝛼𝑘𝑢𝑘+𝛼𝑘𝑈𝑢𝑘,𝑘𝑁,(1.11)

where 𝑢𝑘=𝑥𝑘+𝛾𝛽𝐴(𝑆𝐼)𝐴𝑥𝑘, 𝛽(0,1), 𝛼𝑘(0,1), and 𝛾(0,1/𝜆𝛽), with 𝜆 being the spectral radius of the operator 𝐴𝐴. Moudafi proved weak convergence result of the algorithm in Hilbert spaces.

Inspired by their work, we introduce the following viscosity approximation algorithm.

Algorithm 1. Initialization: Let 𝑥0𝐻 be arbitrary.
Iterative step: Set 𝑇=𝑈(𝐼+𝛾𝐴(𝑆𝐼)𝐴). For 𝑘𝑁, let 𝑥𝑘+1=𝛼𝑘𝑓𝑥𝑘+1𝛼𝑘1𝜔𝑘𝑥𝑘+𝜔𝑘𝑇𝑥𝑘,(1.12) where 𝑓𝐻𝐻 is a contraction of modulus 𝜌, 𝜔𝑘(0,1/2), 𝛾(0,1/𝜆) with 𝜆 being the spectral radius of the operator 𝐴𝐴, and 𝛼𝑘(0,1).
This paper establishes the strong convergence of the sequence given by (1.12) to the unique solution of the variational inequality problem VIP(𝐼𝑓,Γ)nd𝑥Γsuchthat(𝐼𝑓)𝑥,𝑣𝑥0,𝑣Γ.(1.13) Now we give a series of preliminary results needed for the convergence analysis of algorithm (1.12).

Lemma 1.1. Let 𝐻 be a real Hilbert space and 𝑇𝐻𝐻 a quasi-nonexpansive mapping. Then, the following properties are reached: (i)1𝑥,𝑦=2𝑥𝑦2+12𝑥2+12𝑦2,(𝑥,𝑦)𝐻×𝐻;(ii)1𝑥𝑇𝑥,𝑥𝑞2𝑥𝑇𝑥2 and 1𝑥𝑇𝑥,𝑞𝑇𝑥2𝑥𝑇𝑥2,(𝑥,𝑞)𝐻×𝐹(𝑇).

Remark 1.2. Let 𝐹=𝐼𝑓, where 𝑓 is the contraction defined in (1.7). It is a simple matter to see that the operator 𝐹 is (1𝜌) strongly monotone over 𝐻; that is,𝐹𝑥𝐹𝑦,𝑥𝑦(1𝜌)𝑥𝑦2,(𝑥,𝑦)𝐻×𝐻.(1.14)

The next result is of fundamental importance for the techniques of analysis used in this paper. It was established in [18], and its proof is given for the sake of completeness.

Lemma 1.3 (see [18, Lemma 1.3]). Let {𝛿𝑛} be a sequence of real numbers that does not decrease at infinity, in the sense that there exists a subsequence {𝛿𝑛𝑗}𝑗0 of {𝛿𝑛} which satisfies 𝛿𝑛𝑗<𝛿𝑛𝑗+1 for all 𝑗0. Also consider the sequence of integers {𝜏(𝑛)}𝑛𝑛0 defined by 𝜏(𝑛)=max𝑘𝑛𝛿𝑘<𝛿𝑘+1.(1.15) Then {𝜏(𝑛)}𝑛𝑛0 is a nondecreasing sequence verifying lim𝑛𝜏(𝑛)=, and, for all 𝑛𝑛0, it holds that 𝛿𝜏(𝑛)𝛿𝜏(𝑛)+1 and one has 𝛿𝑛𝛿𝜏(𝑛)+1.(1.16)

Proof. Clearly, we can see that {𝜏(𝑛)} is a well-defined sequence, and the fact that it is nondecreasing is obvious as well as lim𝑛𝜏(𝑛)= and 𝛿𝜏(𝑛)𝛿𝜏(𝑛)+1. Let us prove (1.16). It is easily observed that 𝜏(𝑛)𝑛. Consequently, we prove (1.16) by distinguishing the three cases: (c1) 𝜏(𝑛)=𝑛; (c2) 𝜏(𝑛)=𝑛1; (c3) 𝜏(𝑛)<𝑛1. In the first case (i.e., 𝜏(𝑛)=𝑛), (1.16) is immediately given by 𝛿𝜏(𝑛)𝛿𝜏(𝑛)+1. In the second case (i.e., 𝜏(𝑛)=𝑛1), (1.16) becomes obvious. In the third case (i.e., 𝜏(𝑛)𝑛2), by (1.15) and for any integer 𝑛𝑛0, we easily observe that 𝛿𝑗𝛿𝑗+1 for 𝜏(𝑛)+1𝑗𝑛1; namely, 𝛿𝜏(𝑛)+1𝛿𝜏(𝑛)+2𝛿𝑛1𝛿𝑛,(1.17) which entails the desired result.

2. Main Results

Theorem 2.1. Given a bounded linear operator 𝐴𝐻1𝐻2, let 𝑈𝐻1𝐻1 and 𝑆𝐻2𝐻2 be quasi-nonexpansive mappings with nonempty fixed-point set 𝐹(𝑈)=𝐶 and 𝐹(𝑆)=𝑄. Assume that 𝑈𝐼 and 𝑆𝐼 are demiclosed at origin. Let {𝑥𝑘} be the sequence given by (1.12) with 𝛾(0,1/𝜆), 𝜔𝑘(0,1/2) such that 0<liminf𝑘𝜔𝑘limsup𝑘𝜔𝑘<1/2 and {𝛼𝑘}(0,1) such that lim𝑘𝛼𝑘=0 and 𝑘𝛼𝑘=. If Γ, then the sequence {𝑥𝑘} strongly converges to a split common fixed-point 𝑥Γ, verifying 𝑥=𝑃Γ𝑓(𝑥) which equivalently solves the following variational inequality problem: 𝑥Γ,(𝐼𝑓)𝑥,𝑣𝑥0,𝑣Γ.(2.1)

Proof. Set 𝑇𝜔𝑘=(1𝜔𝑘)𝐼+𝜔𝑘𝑇. Then 𝑥𝑘+1=𝛼𝑘𝑓(𝑥𝑘)+(1𝛼𝑘)𝑇𝜔𝑘𝑥𝑘.
Firstly, we prove that {𝑥𝑘} is bounded. Taking 𝑦Γ, that is, 𝑦𝐹(𝑈), 𝐴𝑦𝐹(𝑆). We have 𝑥𝑘+1=𝛼𝑦𝑘𝑓𝑥𝑘𝑓(𝑦)+𝛼𝑘(𝑓(𝑦)𝑦)+1𝛼𝑘𝑇𝜔𝑘𝑥𝑘𝑦𝛼𝑘𝑓𝑥𝑘𝑓(𝑦)+𝛼𝑘𝑓(𝑦)𝑦+1𝛼𝑘𝑇𝜔𝑘𝑥𝑘𝑦𝛼𝑘𝜌𝑥𝑘𝑦+𝛼𝑘𝑓(𝑦)𝑦+1𝛼𝑘𝑇𝜔𝑘𝑥𝑘.𝑦(2.2) From the definition of 𝑇𝜔𝑘, we get 𝑇𝜔𝑘𝑥𝑘𝑦2=1𝜔𝑘𝑥𝑘+𝜔𝑘𝑇𝑥𝑘𝑦2=𝑥𝑘𝑦+𝜔𝑘𝑇𝑥𝑘𝑥𝑘2=𝑥𝑘𝑦22𝜔𝑘𝑥𝑘𝑦,𝑥𝑘𝑇𝑥𝑘+𝜔2𝑘𝑇𝑥𝑘𝑥𝑘2.(2.3) On the other hand, we have 𝑇𝑥𝑘𝑦2=𝑈𝐼+𝛾𝐴𝑥(𝑆𝐼)𝐴𝑘𝑦2𝐼+𝛾𝐴𝑥(𝑆𝐼)𝐴𝑘𝑦2=𝑥𝑘𝑦2+𝛾2𝐴(𝑆𝐼)𝐴𝑥𝑘2+2𝛾𝑥𝑘𝑦,𝐴(𝑆𝐼)𝐴𝑥𝑘=𝑥𝑘𝑦2+𝛾2(𝑆𝐼)𝐴𝑥𝑘,𝐴𝐴(𝑆𝐼)𝐴𝑥𝑘+2𝛾𝑥𝑘𝑦,𝐴(𝑆𝐼)𝐴𝑥𝑘.(2.4) From the definition of 𝜆, it follows that 𝛾2(𝑆𝐼)𝐴𝑥𝑘,𝐴𝐴(𝑆𝐼)𝐴𝑥𝑘𝜆𝛾2(𝑆𝐼)𝐴𝑥𝑘,(𝑆𝐼)𝐴𝑥𝑘=𝜆𝛾2(𝑆𝐼)𝐴𝑥𝑘2.(2.5) Now, by using property (ii) of Lemma 1.1, we obtain 2𝛾𝑥𝑘𝑦,𝐴(𝑆𝐼)𝐴𝑥𝑘𝐴𝑥=2𝛾𝑘,𝑦(𝑆𝐼)𝐴𝑥𝑘𝐴𝑥=2𝛾𝑘𝑦+(𝑆𝐼)𝐴𝑥𝑘(𝑆𝐼)𝐴𝑥𝑘,(𝑆𝐼)𝐴𝑥𝑘𝑆=2𝛾𝐴𝑥𝑘𝐴𝑦,(𝑆𝐼)𝐴𝑥𝑘(𝑆𝐼)𝐴𝑥𝑘212𝛾2(𝑆𝐼)𝐴𝑥𝑘2(𝑆𝐼)𝐴𝑥𝑘2=𝛾(𝑆𝐼)𝐴𝑥𝑘2.(2.6) Combining (2.4)–(2.6), we have 𝑇𝑥𝑘𝑦2𝑥𝑘𝑦2+𝜆𝛾2(𝑆𝐼)𝐴𝑥𝑘2𝛾(𝑆𝐼)𝐴𝑥𝑘2=𝑥𝑘𝑦2𝛾(1𝜆𝛾)(𝑆𝐼)𝐴𝑥𝑘2𝑥𝑘𝑦2.(2.7) From property (i) of Lemma 1.1, we have 𝑥𝑘𝑦,𝑥𝑘𝑇𝑥𝑘1=2𝑇𝑥𝑘𝑦2+12𝑥𝑘𝑦2+12𝑥𝑘𝑇𝑥𝑘212𝑥𝑘𝑇𝑥𝑘2.(2.8) From (2.3) and (2.8), we have 𝑇𝜔𝑘𝑥𝑘𝑦2𝑥𝑘𝑦2𝜔𝑘𝑥𝑘𝑇𝑥𝑘2+𝜔2𝑘𝑥𝑘𝑇𝑥𝑘2=𝑥𝑘𝑦2𝜔𝑘1𝜔𝑘𝑥𝑘𝑇𝑥𝑘2𝑥𝑘𝑦2,(2.9) Combining (2.2), (2.3), and (2.9), it follows that 𝑥𝑘+1𝑦𝛼𝑘𝜌𝑥𝑘𝑦+𝛼𝑘𝑓(𝑦)𝑦+1𝛼𝑘𝑥𝑘=𝑦1𝛼𝑘(𝑥1𝜌)𝑘𝑦+𝛼𝑘(𝑥𝑓𝑦)𝑦max𝑘,1𝑦.1𝜌𝑓(𝑦)𝑦(2.10)It is obviously that 𝑥𝑘𝑥𝑦max0,1𝑦1𝜌𝑓(𝑦)𝑦,(2.11) and hence {𝑥𝑘} is bounded. Let 𝑥=𝑃Γ𝑓(𝑥). We have 𝑥𝑘+1𝑥𝑘+𝛼𝑘𝑥𝑘𝑥𝑓𝑘=1𝛼𝑘𝑇𝜔𝑘𝑥𝑘𝑥𝑘,(2.12) and hence 𝑥𝑘+1𝑥𝑘+𝛼𝑘(𝐼𝑓)𝑥𝑘,𝑥𝑘𝑥=1𝛼𝑘𝑥𝑘𝑇𝜔𝑘𝑥𝑘,𝑥𝑘𝑥.(2.13) By (2.9) we obtain that 𝑥𝑘𝑇𝜔𝑘𝑥𝑘,𝑥𝑘𝑥=12𝑥𝑘𝑇𝜔𝑘𝑥𝑘2+12𝑥𝑘𝑥212𝑇𝜔𝑘𝑥𝑘𝑥2𝜔2𝑘2𝑥𝑘𝑇𝑥𝑘2+12𝑥𝑘𝑥212𝑥𝑘𝑥2+𝜔𝑘21𝜔𝑘𝑥𝑘𝑇𝑥𝑘2=𝜔𝑘2𝑥𝑘𝑇𝑥𝑘2.(2.14) It follows from (2.13) that 𝑥𝑘+1𝑥𝑘+𝛼𝑘(𝐼𝑓)𝑥𝑘,𝑥𝑘𝑥𝜔𝑘21𝛼𝑘𝑥𝑘𝑇𝑥𝑘2,(2.15) and hence 𝑥𝑘𝑥𝑘+1,𝑥𝑘𝑥𝛼𝑘(𝐼𝑓)𝑥𝑘,𝑥𝑘𝑥𝜔𝑘21𝛼𝑘𝑥𝑘𝑇𝑥𝑘2.(2.16) Setting 𝛿𝑘=12𝑥𝑘𝑥2, we have 𝑥𝑘𝑥𝑘+1,𝑥𝑘𝑥1=2𝑥𝑘+1𝑥2+12𝑥𝑘𝑥2+12𝑥𝑘𝑥𝑘+12=𝛿𝑘+1+𝛿𝑘+12𝑥𝑘𝑥𝑘+12,(2.17) so that (2.16) can be rewritten as 𝛿𝑘+1𝛿𝑘12𝑥𝑘𝑥𝑘+12𝛼𝑘(𝐼𝑓)𝑥𝑘,𝑥𝑘𝑥𝜔𝑘21𝛼𝑘𝑥𝑘𝑇𝑥𝑘2.(2.18) Now using (2.12) again, we have 𝑥𝑘+1𝑥𝑘2=𝛼𝑘𝑓𝑥𝑘𝑥𝑘+1𝛼𝑘(𝑇𝜔𝑘𝑥𝑘𝑥𝑘2𝛼𝑘𝑓𝑥𝑘𝑥𝑘+1𝛼𝑘𝑇𝜔𝑘𝑥𝑘𝑥𝑘22𝛼2𝑘𝑓𝑥𝑘𝑥𝑘2+21𝛼𝑘2𝑇𝜔𝑘𝑥𝑘𝑥𝑘22𝛼2𝑘𝑓𝑥𝑘𝑥𝑘2+21𝛼𝑘𝜔2𝑘𝑇𝑥𝑘𝑥𝑘2,(2.19) which yields 12𝑥𝑘+1𝑥𝑘2𝛼2𝑘𝑓𝑥𝑘𝑥𝑘2+1𝛼𝑘𝜔2𝑘𝑇𝑥𝑘𝑥𝑘2.(2.20) From (2.18) and (2.20), we obtain 𝛿𝑘+1𝛿𝑘+𝜔𝑘1𝛼𝑘12𝜔𝑘𝑇𝑥𝑘𝑥𝑘2𝛼𝑘𝛼𝑘𝑓𝑥𝑘𝑥𝑘2(𝐼𝑓)𝑥𝑘,𝑥𝑘𝑥.(2.21) It follows from Remark 1.2 that (𝐼𝑓)𝑥𝑘(𝐼𝑓)𝑥,𝑥𝑘𝑥𝑥(1𝜌)𝑘𝑥2=2(1𝜌)𝛿𝑘,(2.22) and hence 2(1𝜌)𝛿𝑘+(𝐼𝑓)𝑥,𝑥𝑘𝑥(𝐼𝑓)𝑥𝑘,𝑥𝑘𝑥.(2.23)
The rest of the proof will be divided into two parts.
Case 1. Suppose that there exists 𝑘0 such that {𝛿𝑘}𝑘𝑘0 is nonincreasing. In this situation, {𝛿𝑘} is convergent because it is nonnegative, so that lim𝑘(𝛿𝑘+1𝛿𝑘)=0; hence, in light of (2.21) together with 𝛼𝑘0, the boundedness of {𝑥𝑘}, and 0<liminf𝑘𝜔𝑘limsup𝑘𝜔𝑘<1/2, we obtain lim𝑘𝑥𝑘𝑇𝑥𝑘=0.(2.24) From (2.21) again, we have 𝛼𝑘𝛼𝑘𝑓𝑥𝑘𝑥𝑘2+(𝐼𝑓)𝑥𝑘,𝑥𝑘𝑥𝛿𝑘𝛿𝑘+1.(2.25) By 𝑘𝛼𝑘=, we deduce that liminf𝑘𝛼𝑘𝑓𝑥𝑘𝑥𝑘2+(𝐼𝑓)𝑥𝑘,𝑥𝑘𝑥0(2.26) and hence (as 𝛼𝑘𝑓(𝑥𝑘)𝑥𝑘20) liminf𝑘(𝐼𝑓)𝑥𝑘,𝑥𝑘𝑥0.(2.27) By (2.23) and (2.27), we have liminf𝑘2(1𝜌)𝛿𝑘+(𝐼𝑓)𝑥,𝑥𝑘𝑥0;(2.28) recalling that lim𝑘𝛿𝑘 exists, we obtain 2(1𝜌)lim𝑘𝛿𝑘+liminf𝑘(𝐼𝑓)𝑥,𝑥𝑘𝑥0.(2.29) Now we prove that liminf𝑘(𝐼𝑓)𝑥,𝑥𝑘𝑥0.(2.30) It follows from (2.7) and (2.24) that 𝛾(1𝜆𝛾)(𝑆𝐼)𝐴𝑥𝑘2𝑥𝑘𝑦2𝑇𝑥𝑘𝑦2=𝑥𝑘𝑦𝑇𝑥𝑘𝑥𝑦𝑘+𝑦𝑇𝑥𝑘𝑥𝑦𝑘𝑇𝑥𝑘𝑥𝑘+𝑦𝑇𝑥𝑘𝑦0(𝑘),(2.31) and hence lim𝑘(𝑆𝐼)𝐴𝑥𝑘=0.(2.32) Taking 𝑦𝜔𝑤(𝑥𝑘), from the demiclosedness of 𝑆𝐼 at 0, we obtain 𝑆(𝐴𝑦)=𝐴𝑦.(2.33) Now, by setting 𝑢𝑘=𝑥𝑘+𝛾𝐴(𝑆𝐼)𝐴𝑥𝑘, it follows that 𝑦𝜔𝑤(𝑢𝑘). On the other hand, 𝑈𝑢𝑘𝑢𝑘=𝑇𝑥𝑘𝑥𝑘𝛾𝐴(𝑆𝐼)𝐴𝑥𝑘𝑇𝑥𝑘𝑥𝑘+𝛾𝐴(𝑆𝐼)𝐴𝑥𝑘0,(2.34) which, combined with the demiclosedness of 𝑈𝐼 at 0, yields 𝑈𝑦=𝑦.(2.35) Hence, 𝑦𝐶 and 𝑦Γ. We can take subsequence {𝑥𝑘𝑗} of {𝑥𝑘} such that 𝑥𝑘𝑗𝑦 as j and liminf𝑘(𝐼𝑓)𝑥,𝑥𝑘𝑥=lim𝑗(𝐼𝑓)𝑥,𝑥𝑘𝑗𝑥,(2.36) which leads to liminf𝑘(𝐼𝑓)𝑥,𝑥𝑘𝑥=(𝐼𝑓)𝑥,𝑦𝑥0.(2.37) By (2.29), we have lim𝑘𝛿𝑘=0, and hence {𝑥𝑘} converges strongly to 𝑥.
Case 2. Suppose there exists a subsequence {𝛿𝑘𝑗}𝑗0 of {𝛿𝑘} such that 𝛿𝑘𝑗<𝛿𝑘𝑗+1 for all 𝑗0. In this situation, we consider the sequence of indices {𝜏(𝑘)} as defined in Lemma 1.3. It follows that 𝛿𝜏(𝑘)+1𝛿𝜏(𝑘)>0, which by (2.21) amounts to 𝜔𝑘1𝛼𝜏(𝑘)12𝜔𝑘𝑇𝑥𝜏(𝑘)𝑥𝜏(𝑘)2𝛼𝜏(𝑘)𝛼𝜏(𝑘)𝑓𝑥𝜏(𝑘)𝑥𝜏(𝑘)2(𝐼𝑓)𝑥𝜏(𝑘),𝑥𝜏(𝑘)𝑥.(2.38) By the boundedness of {𝑥𝑘} and 𝛼𝑘0, we immediately obtain lim𝑘𝑇𝑥𝜏(𝑘)𝑥𝜏(𝑘)=0.(2.39) Similar to Case 1, we have liminf𝑘(𝐼𝑓)𝑥,𝑥𝜏(𝑘)𝑥0.(2.40) It follows from (2.38) that (𝐼𝑓)𝑥𝜏(𝑘),𝑥𝜏(𝑘)𝑥𝛼𝜏(𝑘)𝑓𝑥𝜏(𝑘)𝑥𝜏(𝑘)2,(2.41) which in the light of (2.23) yields 2(1𝜌)𝛿𝜏(𝑘)+(𝐼𝑓)𝑥,𝑥𝜏(𝑘)𝑥𝛼𝜏(𝑘)𝑓𝑥𝜏(𝑘)𝑥𝜏(𝑘)2;(2.42) hence (as 𝛼𝜏(𝑘)𝑓(𝑥𝜏(𝑘))𝑥𝜏(𝑘)20) it follows that 2(1𝜌)limsup𝑘𝛿𝜏(𝑘)liminf𝑘(𝐼𝑓)𝑥,𝑥𝜏(𝑘)𝑥.(2.43) From (2.40) we have limsup𝑘𝛿𝜏(𝑘)=0, so that lim𝑘𝛿𝜏(𝑘)=0, and hence lim𝑘𝑥𝜏(𝑘)𝑥=0. On the other hand, it follows that 𝑥𝜏(𝑘)+1𝑥𝜏(𝑘)=𝛼𝜏(𝑘)𝑓𝑥𝜏(𝑘)𝑥𝜏(𝑘)+1𝛼𝜏(𝑘)𝑇𝜔𝑘𝑥𝜏(𝑘)𝑥𝜏(𝑘)𝛼𝜏(𝑘)𝑓𝑥𝜏(𝑘)𝑥𝜏(𝑘)+1𝛼𝜏(𝑘)𝜔𝑘𝑇𝑥𝜏(𝑘)𝑥𝜏(𝑘),(2.44) which, by (2.39), implies that lim𝑘𝑥𝜏(𝑘)+1𝑥𝜏(𝑘)=0.(2.45) So we have lim𝑘𝛿𝜏(𝑘)+1=12𝑥𝜏(𝑘)+1𝑥=0.(2.46) Then, recalling that 𝛿𝑘𝛿𝜏(𝑘)+1 (by Lemma 1.3), we get lim𝑘𝛿𝑘=0, so that the sequence {𝑥𝑘} converges strongly to 𝑥.

Theorem 2.2. Given a bounded linear operator 𝐴𝐻1𝐻2, let 𝑈𝐻1𝐻1 and 𝑆𝐻2𝐻2 be quasi-nonexpansive mappings with nonempty fixed-point set 𝐹(𝑈)=𝐶 and 𝐹(𝑆)=𝑄. Assume that 𝑈𝐼 and 𝑆𝐼 are demiclosed at origin. Let 𝑥0𝐻 be arbitrary and {𝑥𝑘} the sequence given by 𝑥𝑘+1=𝛼𝑘𝑓𝑥𝑘+1𝛼𝑘(1𝜔)𝑥𝑘+𝜔𝑇𝑥𝑘,(2.47) where 𝑇=𝑈(𝐼+𝛾𝐴(𝑆𝐼)𝐴), 𝑓𝐻𝐻 a contraction of modulus 𝜌, 𝛾(0,1/𝜆), 𝜔(0,1/2), and {𝛼𝑘}(0,1) such that lim𝑘𝛼𝑘=0 and 𝑘𝛼𝑘=. If Γ, then the sequence {𝑥𝑘} strongly converges to a split common fixed-point 𝑥Γ, verifying 𝑥=𝑃Γ𝑓(𝑥) which equivalently solves the following variational inequality problem: 𝑥Γ,(𝐼𝑓)𝑥,𝑣𝑥0,𝑣Γ.(2.48)

Acknowledgments

The research was supported by Fundamental Research Funds for the Central Universities (Program No. ZXH2011D005); it was also supported by science research foundation program in Civil Aviation University of China (2011kys02).

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