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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 438023, 12 pages
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.
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 is a real Hilbert space with inner product and norm . Let denote the identity operator on . Let and be nonempty closed convex subset of real Hilbert spaces and , respectively. The split feasibility problem (SFP) is to find a point where 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 authors' attention due to its application in signal processing. Various algorithms have been invented to solve it (see [3–9] and references therein).
Note that the split feasibility problem (1.1) can be formulated as a fixed-point equation by using the fact that is, solves the SFP (1.1) if and only if solves the fixed point equation (1.2) (see  for the details). This implies that we can use fixed-point algorithms (see [11–13]) to solve SFP. A popular algorithm that solves the SFP (1.1) is due to Byrne's CQ algorithm  which is found to be a gradient-projection method (GPM) in convex minimization. Subsequently, Byrne  applied KM iteration to the CQ algorithm, and Zhao and Yang  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 . 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 such that 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 be a mapping. A point is said to be a fixed point of provided that . In this paper, we use to denote the fixed-point set and use and to denote the strong convergence and weak convergence, respectively. We use stand for the weak -limit set of .(i)A mapping belongs to the general class of (possibly discontinuous) quasi-nonexpansive mappings if (ii)A mapping belongs to the set of nonexpansive mappings if (iii)A mapping belongs to the set of firmly nonexpansive mappings if (iv)A mapping belongs to the set of firmly quasi-nonexpansive mappings if
It is easily observed that and that . Furthermore, is well known to include resolvents and projection operators, while contains subgradient projection operators (see, e.g.,  and the reference therein).
A mapping is called demiclosed at the origin if any sequence weakly converges to , and if the sequence strongly converges to 0, then . A mapping is called a contraction of modulus if
In what follows, we will focus our attention on the following general two-operator split common fixed-point problem: where is a bounded linear operator, and are two quasi-nonexpansive operators with nonempty fixed-point sets and , and denote the solution set of the two-operator SCFP by
Recall that and are nonempty closed convex subsets of and , respectively. If , we have which is close convex subset of . To solve (1.8), Censor and Segal  proposed and proved, in infinite-dimensional spaces, the convergence of the following algorithm:
where , with being the largest eigenvalue of the matrix ( stands for matrix transposition). Very recently, Moudafi  introduced the following relaxed algorithm:
where , , , and , with being the spectral radius of the operator . 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 be arbitrary.
Iterative step: Set . For , let where is a contraction of modulus , , with being the spectral radius of the operator , and .
This paper establishes the strong convergence of the sequence given by (1.12) to the unique solution of the variational inequality problem Now we give a series of preliminary results needed for the convergence analysis of algorithm (1.12).
Lemma 1.1. Let be a real Hilbert space and a quasi-nonexpansive mapping. Then, the following properties are reached: (i);(ii) and .
Remark 1.2. Let , where is the contraction defined in (1.7). It is a simple matter to see that the operator is strongly monotone over ; that is,
The next result is of fundamental importance for the techniques of analysis used in this paper. It was established in , and its proof is given for the sake of completeness.
Lemma 1.3 (see [18, Lemma 1.3]). Let be a sequence of real numbers that does not decrease at infinity, in the sense that there exists a subsequence of which satisfies for all . Also consider the sequence of integers defined by Then is a nondecreasing sequence verifying , and, for all , it holds that and one has
Proof. Clearly, we can see that is a well-defined sequence, and the fact that it is nondecreasing is obvious as well as and . Let us prove (1.16). It is easily observed that . Consequently, we prove (1.16) by distinguishing the three cases: (c1) ; (c2) ; (c3) . In the first case (i.e., ), (1.16) is immediately given by . In the second case (i.e., ), (1.16) becomes obvious. In the third case (i.e., ), by (1.15) and for any integer , we easily observe that for ; namely, which entails the desired result.
2. Main Results
Theorem 2.1. Given a bounded linear operator , let and be quasi-nonexpansive mappings with nonempty fixed-point set and . Assume that and are demiclosed at origin. Let be the sequence given by (1.12) with , such that and such that and . If , then the sequence strongly converges to a split common fixed-point , verifying which equivalently solves the following variational inequality problem:
Proof. Set . Then .
Firstly, we prove that is bounded. Taking , that is, , . We have From the definition of , we get On the other hand, we have From the definition of , it follows that Now, by using property (ii) of Lemma 1.1, we obtain Combining (2.4)–(2.6), we have From property (i) of Lemma 1.1, we have From (2.3) and (2.8), we have Combining (2.2), (2.3), and (2.9), it follows that It is obviously that and hence is bounded. Let . We have and hence By (2.9) we obtain that It follows from (2.13) that and hence Setting , we have so that (2.16) can be rewritten as Now using (2.12) again, we have which yields From (2.18) and (2.20), we obtain It follows from Remark 1.2 that and hence
The rest of the proof will be divided into two parts.
Case 1. Suppose that there exists such that is nonincreasing. In this situation, is convergent because it is nonnegative, so that ; hence, in light of (2.21) together with , the boundedness of , and , we obtain From (2.21) again, we have By , we deduce that and hence (as ) By (2.23) and (2.27), we have recalling that exists, we obtain Now we prove that It follows from (2.7) and (2.24) that and hence Taking , from the demiclosedness of at 0, we obtain Now, by setting , it follows that . On the other hand, which, combined with the demiclosedness of at 0, yields Hence, and . We can take subsequence of such that as and which leads to By (2.29), we have , and hence converges strongly to .
Case 2. Suppose there exists a subsequence of such that for all . In this situation, we consider the sequence of indices as defined in Lemma 1.3. It follows that , which by (2.21) amounts to By the boundedness of and , we immediately obtain Similar to Case 1, we have It follows from (2.38) that which in the light of (2.23) yields hence (as ) it follows that From (2.40) we have , so that , and hence . On the other hand, it follows that which, by (2.39), implies that So we have Then, recalling that (by Lemma 1.3), we get , so that the sequence converges strongly to .
Theorem 2.2. Given a bounded linear operator , let and be quasi-nonexpansive mappings with nonempty fixed-point set and . Assume that and are demiclosed at origin. Let be arbitrary and the sequence given by where , a contraction of modulus , , , and such that and . If , then the sequence strongly converges to a split common fixed-point , verifying which equivalently solves the following variational inequality problem:
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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