Abstract and Applied Analysis

Volume 2014 (2014), Article ID 897370, 6 pages

http://dx.doi.org/10.1155/2014/897370

## Iterative Algorithms for the Split Problem and Its Convergence Analysis

^{1}School of Mathematics & Information Technology, Nanjing Xiaozhuang University, Nanjing 211171, China^{2}Department of Mathematics, Lahore Leads University, Lahore 54810, Pakistan^{3}Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea^{4}School of Mathematics and Information Science, Beifang University of Nationalities, Yinchuan 750021, China

Received 12 May 2014; Accepted 18 May 2014; Published 28 May 2014

Academic Editor: Jen-Chih Yao

Copyright © 2014 Zhangsong Yao et al. 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

Now, it is known that the split common fixed point problem is a generalization of the split feasibility problem and of the convex feasibility problem. In this paper, the split common fixed point problem associated with the pseudocontractions is studied. An iterative algorithm has been presented for solving the split common fixed point problem. Strong convergence result is obtained.

#### 1. Introduction

Now, we know that the convex feasibility problem can be formulated as finding a point such that where is a closed convex subset of a Hilbert space . The convex feasibility problem has extensive applications in many applied disciplines such as signal processing, biomedical engineering, and communications. For related works, please see [1–3].

If in (1), then a special case of (1) is the following split feasibility problem.

*Problem 1. *The split feasibility problem: let and be two Hilbert spaces. Let and be two nonempty closed convex sets. Let be a bounded linear operator. The split feasibility problem is

Such problem arises in the intensity-modulated radiation therapy. In the finite-dimensional space, Censor and Elfving [4] firstly constructed the following iterative algorithm to solve (2): where and are closed convex sets and is an matrix.

However, we note that calculating inverse is very time-consuming, if the dimension is large. For overcoming this problem, Byrne [5] introduced the following more popular algorithm: where denotes the transposition of . Consequently, (4) and its variant have been studied extensively. For related results, please refer to [6–13].

In the case where and in (2) are the fixed point sets of nonlinear operators, problem (2) is called by Censor and Segal [14] the split common fixed point problem.

*Problem 2. *The split common fixed point problem: this problem is to find a fixed point of an operator in the space such that its image under a linear transformation is a fixed point of another operator in the image space . Namely, find a vector such that
where and denote the fixed point sets of nonlinear operators and , respectively.

A natural idea is to apply (4) to the split common fixed point problem (5). That is, taking and in (4), we get However, and are generally not easy to calculate. Thus, (6) may fail. We have to find new algorithm to solve (5). In this respect, Censor and Segal [14] proposed the following iterative method: for any initial guess , define a sequence by where and are directed operators. Moudafi [15] relaxed (7) to the following form: where and are demicontractive operators.

Note that (7) and (8) have weak convergence. Some strong convergence results have been given in the literature; see, for instance, [16, 17]. In the present paper, we consider an interesting respect: could we extend the classes of directed and demicontractive operators to the class of pseudocontractive mappings?

Our main purpose of this paper is to solve the above problem. We construct an iterative algorithm in which the involved operators are pseudocontractions and show its strong convergence.

#### 2. Definitions and Lemmas

In this section, we collect some definitions and lemmas. Let be a real Hilbert space with inner product and norm , respectively. Let be a closed convex set.

*Definition 3. *An operator is called Lipschitzian, if
for some and all , .

In this case, we call *-*Lipschitzian continuous. If in (9), we call nonexpansive*.*

*Definition 4. *An operator is called a directed operator, if
equivalently,
for all and , the fixed points set of .

*Definition 5. *An operator is called a demicontractive operator, if
where

From the above definitions, we note that the class of demicontractive operators contains important operators such as the directed operators and the nonexpansive operators with fixed points. Such a class of operators is fundamental because they include many types of nonlinear operators arising in applied mathematics and optimization; see, for example, [18] and references therein.

*Definition 6. *An operator is called pseudocontractive, if
equivalently,
for all , .

It is obvious that the class of pseudocontractive mappings with fixed points includes the class of demicontractive mappings.

Lemma 7 (see [19]). *Let be a real Hilbert space; let be a closed convex set. Let be a continuous pseudocontractive mapping. Then*(i)* is a closed convex subset of ;*(ii)* is demiclosed at zero.*

#### 3. Main Results

Let and be two real Hilbert spaces and let be a bounded linear operator with its adjoint . This section is devoted to study problem (5), where and are two -Lipschitzian pseudocontractive mappings. We denote the solution set of problem (5) by In the sequel, we assume .

In order to solve problem (5), we present the following iterative algorithm.

*Algorithm 8. *Let , and be five constants. For , arbitrarily, we define the following iterative manner:

Theorem 9. *Assume that , , , , and satisfy the following assumptions: , , and . Then, , , and defined by (16) converge strongly to .*

*Proof. *The outline of our proof details is as follows:(i), for all ;(ii) is closed and convex, for all ;(iii);(iv).*
Proof of (i).* We show this by induction. is obvious.

Suppose that for some . For any , we have, from (14), that
Observing that in any Hilbert space, we have
Set , for all . By (17) and (18), we obtain
Since is -Lipschitzian and , we have
Since , we deduce
This together with (20) implies that
Hence,
Noting that , we deduce
Similarly, we also have
In Hilbert spaces, there holds
With the help of (25) and (26), we get
By (24) and (27), we have
This shows that . Thus, we get , for all . *Proof of (ii).* It is easy to verify that is closed, for all . Next, we only need to verify that is convex, for all . In fact, let ; for each , we have
namely,
this shows and is a convex set, for all .*Proof of (iii).* Since and , we obtain
It follows that is bounded.

It is known that the metric projection can be characterized by
equivalently,
With the help of (33), we have
which implies that
It follows that
Since is bounded, we get
This together with (35) implies that
The fact that gives
By (38) and (39), we derive
Next we show that is a Cauchy sequence. As a matter of fact, for any with , we have
It follows that
Note that . Therefore, . So, is a Cauchy sequence and hence .*Proof of (iv).* From (24), (27), and (40), we have
Hence,
It follows that
Next, we firstly show that
As a matter of fact, is obvious. We only need to show that .

Take any . We have . Set . We have . Write . Then, . Next, we show . In fact,
Since , we deduce . Thus, . Hence, . Therefore, .

With (46) in hand, we prove that is demiclosed at . Let the sequence satisfying and . Now, we show that .

Since is -Lipschitzian, we have
It follows that
Hence,
Since is demiclosed at by Lemma 7, we immediately deduce . Therefore, is demiclosed at . Since is a bounded linear operator, we get . From (45), we deduce .

By (16), (40), and (44), we deduce
So,
Similarly, we can show that . To this end, we have proven that and . Therefore, . This completes the proof.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The third author was supported by the fund of the Research Promotion Program, Gyeongsang National University, 2013 (RPP-2013-023). The fourth author was supported in part by the NNSF of China (61362033) and NZ13087.

#### References

- H. Stark, Ed.,
*Image Recovery Theory and Applications*, Academic Press, Orlando, Fla, USA, 1987. - P. L. Combettes, “The convex feasibility problem in image recovery,” in
*Advances in Imaging and Electron Physics*, P. Hawkes, Ed., vol. 95, pp. 155–270, Academic Press, New York, NY, USA, 1996. View at Google Scholar - H. H. Bauschke and J. M. Borwein, “On projection algorithms for solving convex feasibility problems,”
*SIAM Review*, vol. 38, no. 3, pp. 367–426, 1996. View at Publisher · View at Google Scholar · View at MathSciNet - 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 MathSciNet - C. Byrne, “Iterative oblique projection onto convex sets and the split feasibility problem,”
*Inverse Problems. An International Journal on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data*, vol. 18, no. 2, pp. 441–453, 2002. View at Publisher · View at Google Scholar · View at MathSciNet - H.-K. Xu, “Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces,”
*Inverse Problems. An International Journal on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data*, vol. 26, no. 10, 17 pages, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - J. Zhao and Q. Yang, “Several solution methods for the split feasibility problem,”
*Inverse Problems. An International Journal on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data*, vol. 21, no. 5, pp. 1791–1799, 2005. View at Publisher · View at Google Scholar · View at MathSciNet - Y. Dang and Y. Gao, “The strong convergence of a KM-CQ-like algorithm for a split feasibility problem,”
- F. Wang and H.-K. Xu, “Approximating curve and strong convergence of the $CQ$ algorithm for the split feasibility problem,”
*Journal of Inequalities and Applications*, vol. 2010, Article ID 102085, 13 pages, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - Y. Yao, W. Jigang, and Y.-C. Liou, “Regularized methods for the split feasibility problem,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 140679, 15 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - L.-C. Ceng, Q. H. Ansari, and J.-C. Yao, “Relaxed extragradient methods for finding minimum-norm solutions of the split feasibility problem,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 75, no. 4, pp. 2116–2125, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - Y. Yao, T. H. Kim, S. Chebbi, and H.-K. Xu, “A modified extragradient method for the split feasibility and fixed point problems,”
*Journal of Nonlinear and Convex Analysis*, vol. 13, no. 3, pp. 383–396, 2012. View at Google Scholar · View at MathSciNet - Y. Yao, M. Postolache, and Y.-C. Liou, “Strong convergence of a self-adaptive method for the split feasibility problem,”
*Fixed Point Theory and Applications*, vol. 2013, 13 pages, 2013. 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 Google Scholar · View at MathSciNet - A. Moudafi, “The split common fixed-point problem for demicontractive mappings,”
- Z. H. He and W. S. Du, “Nonlinear algorithms approach to split common solution problems,”
*Fixed Point Theory and Applications*, vol. 130, 14 pages, 2012. View at Publisher · View at Google Scholar - H. Cui, M. Su, and F. Wang, “Damped projection method for split common fixed point problems,”
*Journal of Inequalities and Applications*, vol. 2013, 10 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - S. Maruster 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 MathSciNet - H. Zhou, “Strong convergence of an explicit iterative algorithm for continuous pseudo-contractions in Banach spaces,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 70, no. 11, pp. 4039–4046, 2009. View at Publisher · View at Google Scholar · View at MathSciNet