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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 491760, 12 pages
doi:10.1155/2012/491760
Research Article

Multiple-Set Split Feasibility Problems for Asymptotically Strict Pseudocontractions

Shih-Sen Chang,1 Yeol Je Cho,2 J. K. Kim,3 W. B. Zhang,1 and L. Yang4

1College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China
2Department of Mathematics Education and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea
3Department of Mathematics Education, Kyungnam University, Masan, Kyungnam 631-701, Republic of Korea
4Department of Mathematics, South West University of Science and Technology, Mianyang Sichuan 621010, China

Received 5 October 2011; Revised 5 December 2011; Accepted 7 December 2011

Academic Editor: Khalida Inayat Noor

Copyright © 2012 Shih-Sen Chang 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

In this paper, we introduce an iterative method for solving the multiple-set split feasibility problems for asymptotically strict pseudocontractions in infinite-dimensional Hilbert spaces, and, by using the proposed iterative method, we improve and extend some recent results given by some authors.

1. Introduction

The split feasibility problem (SFP) in finite dimensional spaces was first introduced by Censor and Elfving [1] for modeling inverse problems which arise from phase retrievals and in medical image reconstruction [2]. Recently, it has been found that the SFP can also be used in various disciplines such as image restoration, computer tomograph, and radiation therapy treatment planning [3–5].

The split feasibility problem in an infinite dimensional Hilbert space can be found in [2, 4, 6–8].

Throughout this paper, we always assume that are real Hilbert spaces, “ ”, “ ” are denoted by strong and weak convergence, respectively.

The purpose of this paper is to introduce and study the following multiple-set split feasibility problem for asymptotically strict pseudocontraction (MSSFP) in the framework of infinite-dimensional Hilbert spaces. Find such that where is a bounded linear operator, and , , are the families of mappings and , respectively, and , where and denote the sets of fixed points of and , respectively. In the sequel, we use to denote the set of solutions of the problem (MSSFP), that is,

2. Preliminaries

We first recall some definitions, notations, and conclusions which will be needed in proving our main results.

Let be a Banach space. A mapping is said to be demiclosed at origin if, for any sequence with and , we have . A Banach space is said to have Opial’s property if, for any sequence with , we have for all with .

Remark 2.1. It is well known that each Hilbert space possesses Opial’s property.

Definition 2.2. Let be a real Hilbert space.
(1)A mapping is called a -asymptotically strict pseudocontraction if there exists a constant and a sequence with such that Especially, if for each in (2.2) and there exists such that then is called a -strict pseudocontraction.(2)A mapping is said to be uniformly -Lipschitzian if there exists a constant such that (3)A mapping is said to be semicompact if, for any bounded sequence with , there exists a subsequence such that converges strongly to a point .

Now, we give one example of the ( )-asymptotically strict pseudocontraction mapping.

Example 2.3. Let be the unit ball in a Hilbert space , and define a mapping by where is a sequence in such that . It is proved in Goebel and Kirk [9] that(a) for all ,(b) for all and .Denote by , and . Then, we have and so the mapping is a ( )-asymptotically strict pseudocontraction.

Remark 2.4. (1) If we put in (2.2), then the mapping is asymptotically nonexpansive.
(2) If we put in (2.3), then the mapping is nonexpansive.
(3) Each ( )-asymptotically strict pseudocontraction and each -strictly pseudocontraction both are demiclosed at origin [10].

Proposition 2.5. Let be a ( )- asymptotically strict pseudocontraction. If , then, for any and , the following inequalities hold and they are equivalent:

Lemma 2.6 (see [11]). Let , , and be sequences of nonnegative real numbers satisfying If and , then the limit exists.

3. Multiple-Set Split Feasibility Problem

For solving the multiple-set split feasibility problem (1.1), let us assume that the following conditions are satisfied:(C1) and are two real Hilbert spaces, is a bounded linear operator;(C2) , , is a uniformly -Lipschitzian and -asymptotically strict pseudocontraction, and , , is a uniformly -Lipschitzian and -asymptotically strict pseudocontraction satisfying the following conditions:(a) and ,(b) and ,(c) and ,(d) and .

We are now in a position to give the following result.

Theorem 3.1. Let , , , , , , , , , , , and be the same as above. Let be the sequence generated by where , for all , is a sequence in , and is a constant satisfying the following conditions.(e) for all and , where is a positive constant.(1) If , then the sequence converges weakly to a point .(2) In addition, if there exists a positive integer such that is semicompact, then the sequences and both converge strongly to a point .

Proof. (1) The proof is divided into 5 steps as follows.
Step 1. We first prove that, for any , the limit exists. In fact, since , , and . From (3.1) and (2.8), it follows that
On the other hand, since Further, letting , , , in (2.9) and noting , it follows that Substituting (3.7) into (3.6) and simplifying it, we have Substituting (3.5) and (3.8) into (3.4) and simplifying it, we have Again, substituting (3.9) into (3.3) and simplifying it, we have By the condition (e), we have where By the condition (d), ; hence, from Lemma 2.6, we know that the following limit exists:
Step 2. We will now prove that, for each , the limit exists. In fact, from (3.10) and (3.13), it follows that This together with the condition (e) implies that Therefore, it follows from (3.4), (3.13), and (3.17) that the limit exists.
Step 3. Now, we prove that
In fact, it follows from (3.1) that In view of (3.16) and (3.17), we have
Similarly, it follows from (3.1), (3.17), and (3.20) that The conclusion (3.18) is proved.
Step 4. Next, we prove that, for each , In fact, from (3.16), it follows that Since is uniformly -Lipschitzian continuous, it follows from (3.18) and (3.23) that
Similarly, for each , it follows from (3.17) that Since is uniformly -Lipschitzian continuous, by the same way as above, from (3.18) and (3.25), we can also prove that
Step 5. Finally, we prove that and , which is a solution of the problem (MSSFP). In fact, since is bounded, there exists a subsequence such that . Hence, for any positive integer , there exists a subsequence with such that . Again, from (3.22), it follows that Since is demiclosed at zero (see Remark 2.4), it follows that . By the arbitrariness of , we have .
Moreover, it follows from (3.1) and (3.17) that Since is a linear bounded operator, it follows that . For any positive integer , there exists a subsequence with such that . In view of (3.22), we have Since is demiclosed at zero, we have . By the arbitrariness of , it follows that . This together with shows that , that is, is a solution to the problem (MSSFP).
Now, we prove that and . In fact, assume that there exists another subsequence such that with . Consequently, by virtue of (3.2) and Opial’s property of Hilbert space, we have This is a contradiction. Therefore, . By using (3.1) and (3.17), we have Therefore, the conclusion (I) follows.
(2) Without loss of generality, we can assume that is semicompact. It follows from (3.27) that Therefore, there exists a subsequence of (for the sake of convenience, we still denote it by ) such that . Since , and so . By virtue of (3.2), we know that that is, and both converge strongly to the point . This completes the proof.

If we put in Theorem 3.1, we can get the following.

Corollary 3.2. Let , , and be the same as above and a family of asymptotically nonexpansive mappings. Let be the sequence generated by where for all and is a sequence in satisfying the following conditions.(e) for all , where is a positive constant.(1) If , then the sequence converges weakly to a point .(2) In addition, if there exists a positive integer such that is semicompact, then the sequence converges strongly to a point .

The following theorem can be obtained from Theorem 3.1 immediately.

Theorem 3.3. Let and be two real Hilbert spaces, a bounded linear operator, , , a uniformly -Lipschitzian and -strict pseudocontraction, and , a uniformly -Lipschitzian and -strict pseudocontraction satisfying the following conditions: (a) and ,(b) and .Let be the sequence generated by where , , is a sequence in , and is a constant. If and the following condition is satisfied:(c) for all and , where is a constant,then the sequence converges weakly to a point . In addition, if there exists a positive integer such that is semicompact, then the sequences and both converge strongly to the point .

Proof. By the same way as given in the proof of Theorem 3.1 and using the case of strict pseudocontraction with the sequence , we can prove that, for each , the limits and exist,
In addition, if there exists a positive integer such that is semicompact, we can also prove that and both converge strongly to the point . This completes the proof.

If you put or (: the identity mapping) for each in Theorem 3.3, then we have the following.

Corollary 3.4. Let be a real Hilbert space and , , a uniformly -Lipschitzian and -strict pseudocontraction satisfying the following conditions: (a) ,(b) . Let be the sequence generated by where and is a sequence in . If and the following condition is satisfied:(c) for all , where is a constant,then the sequence converges weakly to a point . In addition, if there exists a positive integer such that is semicompact, then the sequences converges strongly to the point .

Remark 3.5. Theorems 3.1 and 3.3 improve and extend the corresponding results of Censor et al. [1, 4, 5], Byrne [2], Yang [7], Moudafi [12], Xu [13], Censor and Segal [14], Masad and Reich [15], and others in the following aspects:(1)for the framework of spaces, we extend the space from finite dimension Hilbert space to infinite dimension Hilbert space;(2)for the mappings, we extend the mappings from nonexpansive mappings, quasi-nonexpansive mapping or demicontractive mappings to finite families of asymptotically strictly pseudocontractions;(3)for the algorithms, we propose some new hybrid iterative algorithms which are different from ones given in [1, 2, 4, 5, 7, 14, 15]. And, under suitable conditions, some weak and strong convergences for the algorithms are proved.

Acknowledgments

The authors would like to express their thanks to the referees for their helpful suggestions and comments. This work was supported by the Natural Science Foundation of Yunnan University of Economics and Finance and the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant no. 2011–0021821).

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