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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 386930, 5 pages
Inertial Iteration for Split Common Fixed-Point Problem for Quasi-Nonexpansive Operators
1School of Management, University of Shanghai for Science and Technology, Shanghai 200093, China
2College of Computer Science and Technology, Henan Polytechnic University, Jiaozuo 454000, China
Received 14 March 2013; Accepted 6 May 2013
Academic Editor: Ru Dong Chen
Copyright © 2013 Yazheng Dang and Yan Gao. 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.
Inspired by the note on split common fixed-point problem for quasi-nonexpansive operators presented by Moudafi (2011), based on the very recent work by Dang et al. (2012), in this paper, we propose an inertial iterative algorithm for solving the split common fixed-point problem for quasi-nonexpansive operators in the Hilbert space. We also prove the asymptotical convergence of the algorithm under some suitable conditions. The results improve and develop previously discussed feasibility problems and related algorithms.
The convex feasibility problem (CFP), as an important optimization problem , is to find a common point in the intersection of finitely many convex sets. It has been applied to many areas, for instance, approximation theory , image reconstruction from projections [3, 4], control , and so on. When there are only two sets and constraints are imposed on the solutions in the domain of a linear operator as well as in this operator’s ranges, the problem is said to be a split feasibility problem (SFP) which has the following formula: finding a point satisfying where is a closed convex subset of a Hilbert space , is a closed convex subset of a Hilbert space , and is a bounded linear operator. The SFP was originally introduced in , and it has also broad applications in many fields, such as image reconstruction problem, signal processing, and radiation therapy. Many projection methods have also been developed for solving the SFP; see [7–9]. Denote by the orthogonal projection onto ; that is, , over all . Assuming that the SFP is consistent (i.e., (1) has a solution), it is not hard to see that solves (1) if and only if it solves the fixed-point equation: where is any positive constant and denotes the adjoint of .
The split common fixed-point problem (SCFP) is a generalization of the split feasibility problem (SFP) and the convex feasibility problem (CFP); see . Our main purpose here is to give an extension of the results developed in  to the split common fixed-point problem for quasi-nonexpansive operators, and we will introduce weak symposium convergence result of the algorithm under some suitable conditions. This will be done in the context of general Hilbert spaces.
Throughout the rest of the paper, denotes the identity operator and denotes the set of the fixed points of an operator , that is, .
Recall that a mapping is said to be quasi-nonexpansive () if A mapping is called nonexpansive ( if A mapping is called firmly nonexpansive () if A mapping is called firmly quasi-nonexpansive () 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 references therein).
Recently, Bauschke and Combettes  have considered a class of mappings satisfying the condition It can easily be seen that the class of mappings satisfying the latter condition coincides with that of firmly quasi-nonexpansive mappings.
Usually, the convergence of fixed-point algorithms requires some additional smoothness properties of the mapping such as demiclosedness.
Definition 1. A mapping is said to be demiclosed if for any sequence which weakly converges to and if the sequence strongly converges to , then .
In what follows, only the particular case of demiclosedness at zero will be used, which is the particular case when .
The following lemmas will be needed in the proof of the convergence of the algorithm.
Lemma 2. Let be a quasi-nonexpansive mapping. Set . Then, it is immediate that for all :(1) and ;(2);(3).
Lemma 3 (see ). Assume and satisfy(1), (2), (3), where .Then, the sequence is convergent with , where (for any ).
3. The Inertial Algorithm and Its Asymptotic Convergence
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 and are two quasi-nonexpansive operators with nonempty fixed-point sets and , and denote the solution set of the two-operator SCFP by
3.1. The Inertial Algorithm
To solve (9), Moudafi  proposed and proved, in finite-dimensional spaces, the convergence of the following algorithm: where , are relaxation parameters and . Inspired by the inertial technique, we introduce the following inertial method and then present its convergence analysis.
Algorithm 4. Initialization: Let be arbitrary. Iterative step: For , set , and let where , , and , with being the spectral radius of the operator , .
3.2. Asymptotic Convergence of the Inertial Algorithm
In this subsection, we establish the asymptotic convergence of Algorithm 4.
Lemma 5 (Opial ). Let be a Hilbert space and let be a sequence in such that there exists a nonempty set satisfying(1)for every , exists,(2)any weak cluster point of the sequence belongs to . Then, there exists such that weakly converges to .
Theorem 6. Given a bounded linear operator , let be a quasi-nonexpansive operator with nonempty and let be a quasi-nonexpansive operator with nonempty . Assume that and are demiclosed at . If , then any sequence generated by Algorithm 4 weakly converges to a split common fixed point, provided that we choose satisfying with , . and for a small enough .
Proof. Taking , and using (2) in Lemma 2, we obtain
On the other hand, we have
Now, by setting and using (1) of Lemma 2, we obtain
Combining the key inequality above with (15) yields
Define the auxiliary real sequence . Therefore, from (17), we have
By deducing, we have
It is easy to check that .
Hence, Putting (20) into (18), we get Since , according to , and (21), we derive Evidently, due to . Let in Lemma 3. We deduce that the sequence is convergent (hence, is bounded). By (23) and Lemma 3, we obtain . By reason of (21), we have Hence, By and the assumption on , we get Denoting by a weak-cluster point , let be a subsequence of . Obviously, Then, from (26) and the demiclosedness of at , we obtain it follows that .
Now, by setting , it follows that . By the demiclosedness of at , from (27), we have Hence, , and therefore .
Since there is no more than one weak-cluster point, the weak convergence of the whole sequence follows by applying Lemma 5 with .
Remark 7. Since the current value of is known before choosing the parameter , then is well-defined in Theorem 6. In fact, from the process of proof for Theorem 6, we can get the following assert: the convergence result of Theorem 6 always holds provided that we take , , with
To conclude, we have proposed an algorithm for solving the SCFP in the wide class of quasi-nonexpansive operators and proved its convergence in general Hilbert spaces. Next, we will improve the algorithm to assure the strong convergence in infinite Hilbert spaces.
This work was supported by the National Science Foundation of China (under Grant no. 11171221), Shanghai Municipal Committee of Science and Technology (under Grant no. 10550500800), Shanghai Leading Academic Discipline (under Grant no. XTKX 2012), Basic and Frontier Research Program of the Science and Technology Department of Henan Province (under Grant nos. 112300410277 and 082300440150), and China Coal Industry Association Scientific and Technical Guidance to Project (under Grant no. MTKJ-2011-403).
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