- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Abstract and Applied Analysis

Volume 2013 (2013), Article ID 386930, 5 pages

http://dx.doi.org/10.1155/2013/386930

## Inertial Iteration for Split Common Fixed-Point Problem for Quasi-Nonexpansive Operators

^{1}School of Management, University of Shanghai for Science and Technology, Shanghai 200093, China^{2}College 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.

#### Abstract

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.

#### 1. Introduction

The convex feasibility problem (CFP), as an important optimization problem [1], 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 [2], image reconstruction from projections [3, 4], control [5], 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 [6], 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 .

To solve (2), in [10], Byrne introduced the so-called CQ algorithm, which generates a sequence by where and is the spectral radius of .

The split common fixed-point problem (SCFP) is a generalization of the split feasibility problem (SFP) and the convex feasibility problem (CFP); see [11]. Our main purpose here is to give an extension of the results developed in [12] 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.

The paper is organized as follows. In Section 2, we recall some preliminaries. In Section 3, we present an inertial CQ algorithm and show its convergence.

#### 2. Preliminaries

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., [13], and the references therein).

Recently, Bauschke and Combettes [14] 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 [8]). *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 [15] 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 [16]). *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
that is,
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.

#### Acknowledgments

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).

#### References

- J. W. Chinneck, “The constraint consensus method for finding approximately feasible points in nonlinear programs,”
*INFORMS Journal on Computing*, vol. 16, no. 3, pp. 255–265, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Deutsch, “The method of alternating orthogonal projections,” in
*Approximation Theory, Spline Functions and Applications*, vol. 356 of*NATO Advanced Science Institutes Series C*, pp. 105–121, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992. View at Zentralblatt MATH · View at MathSciNet - Y. Censor, “Parallel application of block-iterative methods in medical imaging and radiation therapy,”
*Mathematical Programming*, vol. 42, no. 2, pp. 307–325, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. T. Herman,
*Image Reconstruction from Projections: The Fundamentals of Computerized Tomography*, Academic Press, New York, NY, USA, 1980. View at MathSciNet - Y. Gao, “Determining the viability for a affine nonlinear control system,”
*Journal of Control Theory & Applications*, vol. 26, no. 6, pp. 654–656, 2009 (Chinese). View at Scopus - Y. Censor and T. Elfving, “A multiprojection algorithm using Bregman projections in a product space,”
*Numerical Algorithms*, vol. 8, no. 2–4, pp. 221–239, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Dang and Y. Gao, “The strong convergence of a KM-CQ-like algorithm for a split feasibility problem,”
*Inverse Problems*, vol. 27, no. 1, Article ID 015007, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P.-E. Maingé, “Convergence theorems for inertial KM-type algorithms,”
*Journal of Computational and Applied Mathematics*, vol. 219, no. 1, pp. 223–236, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. Qu and N. Xiu, “A note on the $CQ$ algorithm for the split feasibility problem,”
*Inverse Problems*, vol. 21, no. 5, pp. 1655–1665, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Byrne, “Iterative oblique projection onto convex sets and the split feasibility problem,”
*Inverse Problems*, vol. 18, no. 2, pp. 441–453, 2002. View at Publisher · View at Google Scholar · View at MathSciNet - Y. Censor and A. Segal, “The split common fixed point problem for directed operators,”
*Journal of Convex Analysis*, vol. 16, no. 2, pp. 587–600, 2009. View at Zentralblatt MATH · View at MathSciNet - Y. Dang, Y. Gao, and Y. Han, “A perturbed projection algorithm with inertial technique for split feasibility problem,”
*Journal of Applied Mathematics*, vol. 2012, Article ID 207323, 10 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Măruşter and C. Popirlan, “On the Mann-type iteration and the convex feasibility problem,”
*Journal of Computational and Applied Mathematics*, vol. 212, no. 2, pp. 390–396, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. H. Bauschke and P. L. Combettes, “A weak-to-strong convergence principle for Fejér-monotone methods in Hilbert spaces,”
*Mathematics of Operations Research*, vol. 26, no. 2, pp. 248–264, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Moudafi, “A note on the split common fixed-point problem for quasi-nonexpansive operators,”
*Nonlinear Analysis*, vol. 74, no. 12, pp. 4083–4087, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,”
*Bulletin of the American Mathematical Society*, vol. 73, pp. 591–597, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet