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Abstract and Applied Analysis

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

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

Iterative Algorithms for the Split Problem and Its Convergence Analysis

1School of Mathematics & Information Technology, Nanjing Xiaozhuang University, Nanjing 211171, China

2Department of Mathematics, Lahore Leads University, Lahore 54810, Pakistan

3Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea

4School 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 [13].

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 [613].

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.

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