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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 876069, 8 pages
http://dx.doi.org/10.1155/2012/876069
Research Article

A Modified Halpern's Iterative Scheme for Solving Split Feasibility Problems

Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi, Bangkok 10140, Thailand

Received 4 May 2012; Accepted 13 September 2012

Academic Editor: Hong-Kun Xu

Copyright © 2012 Jitsupa Deepho and Poom Kumam. 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

The purpose of this paper is to introduce and study a modified Halpern’s iterative scheme for solving the split feasibility problem (SFP) in the setting of infinite-dimensional Hilbert spaces. Under suitable conditions a strong convergence theorem is established. The main result presented in this paper improves and extends some recent results done by Xu (Iterative methods for the split feasibility problem in infinite-dimensional Hilbert space, Inverse Problem 26 (2010) 105018) and some others.

1. Introduction

Let and be nonempty-closed convex subsets of real Hilbert spaces and , respectively. Let be a linear-bounded operator from to . The split feasibility problem (SFP) is finding a point satisfying the following property:

The SFP was introduced by Censor and Elfving [1] for modeling inverse problems which arise from phase retrievals and medical image reconstruction [2], and very well-known iterative algorithms have been invented to solve it [2].

We use to denote the solution set of SFP: and assume that the SFP (1.1) is consistent (i.e., (1.1) has a solution) so that is closed, convex, and nonempty, it is not hard to see that solves (1.1) if and only if it solves the following fixed point equation; where and are the (orthogonal) projections onto and , respectively, is any positive constant and denotes the adjoint of . Moreover, for sufficiently small , the operator which defines the fixed point equation in (1.3) is nonexpansive.

To solve the SFP (1.1), Byrne [2] proposed his algorithm (see also [3]) which generates a sequence by where with being the spectral radius of the operator .

Very recently, Xu [4] has viewed the algorithm for averaged mappings and applied Mann's algorithm to solving the SFP, and he also proved that an averaged algorithm is weakly convergent to a solution of the SFP.

In this paper, we also regard the algorithm as a fixed point algorithm for averaged mappings and try to study the SFP by the following modified Halpern's iterative scheme; where , , and are three sequences in satisfying . Furthermore, our result extends and improves the result of Xu [4] from weak to strong convergence theorems.

2. Preliminaries

Throughout the paper, we adopt the following notation.

Let be a sequence and be a point in a normed space . We use and to denote strong and weak convergence to of the sequence , respectively. In addition, we use to denote the weak -limit set of the sequence ; namely,

Let be a real Hilbert space with inner product and norm , respectively, and let be a nonempty-closed convex subset of . For every point , there exists a unique nearest point in , denoted by , such that

is called the metric projection of onto . It is well known that is a nonexpansive mapping of onto and satisfies for every . Moreover, is characterized by the following properties: and for all .

Some important properties of projections are gathered in the following proposition.

Proposition 2.1. Given and . Then if and only if One also needs other sorts of nonlinear operators which are introduced below.
Let be the nonlinear operators. (1)is nonexpansive if for all , . (2) is firmly nonexpansive if is nonexpansive. Equivalent, , where is nonexpansive. Alternatively, is firmly nonexpansive if and only if (3) is averaged if , where and is nonexpansive. In this case, one also says that is -averaged. A firmly nonexpansive mapping is -averaged.(4) is monotone if for . (5) is -strongly monotone, with , if (6) is -inverse strongly monotone (-ism), with , if It is well known that both and are firmly nonexpansive and -ism.
Denote by the set of fixed points of a self-mapping defined on , (i.e., .

Proposition 2.2 (see [2, 5]). One has the following assertions.(1) is nonexpansive if and only if the complement is -ism. (2) If is -ism and , then is -ism. (3) is averaged if and only if the complement is -ism, for some .Indeed, for is -averaged if and only if is -ism.(4) If is -averaged and is -averaged, where , then the composite is -averaged, where .(5) If and are averaged and have a common fixed point, then .

Lemma 2.3 (see [6]). Let be a nonempty-closed convex subset of a real Hilbert space and be nonexpansive mapping on with . If is a sequence in which converges weakly to and if converges strongly to , then . In particular, if , then .

Lemma 2.4 (see [7]). Let be an inner product space. Then for all and with , one has

Lemma 2.5 (see [8]). Assume that is a sequence of nonnegative real numbers such that where is a sequence in and is a sequence such that (1); (2) or .
Then, .

3. Main Result

Let be a nonempty closed and convex subset of a Hilbert space . For any , we define the sequence by where , , and are three sequences in and satisfy .

Theorem 3.1. Suppose that the SFP is consistent and . Let be a sequence defined as in (3.1). If the following assumptions are satisfied:(C1) but , (C2), (C3) the sums , and are finite. Then converges strongly to a solution of the SFP (1.1).

Proof. We firstly show that the sequence is bounded. For our convenience, we take . Then, for any , we have . Now, we observe that Now, we note that the condition implies that the operator is averaged. Since is firmly nonexpansive mappings and so is -average, which is -ism. Also observe that is -ism so that is -ism. Further, from the fact that is -averaged and is -averaged, we may obtain that is -averaged, where
This implies that , where for some nonexpansive mappings . Note that is also nonexpansive mappings. Hence, we have From the inequalities (3.2) and (3.4), we have Continuing inductively, we may obtain that the inequality holds for all . So, is bounded so does .
Next, we will show that . Observe that Since and are bounded, there exists such that According to Lemma 2.5 and the condition (C3), we have .
We note that Consequently, by the condition (C1) and (C2), we also have . Next, we will show that
To show this, we can choose a subsequence of such that
As is bounded, there exists a subsequence which converges weakly to . We may assume without loss of generality that . Since , we obtain as . By Lemma 2.3, we obtain that .
Now from (2.4), observe that Therefore, we compute which implies that Finally, by (3.12), (3.14), and Lemma 2.5, we conclude that converges to . This completes the proof.

Letting of iterative scheme (3.1) in Theorem 3.1, then we obtain the following corollary.

Corollary 3.2. For any , one defines the sequence by where is a sequence in . Suppose that the SFP is consistent and .
Let be defined as in (3.15). If the following assumptions are satisfied: (C1) but , (C2). Then converges to a solution of the SFP (1.1).

Remark 3.3. Theorem 3.1 and Corollary 3.2 extend and improve the result of Xu [4] from weak to strong convergence theorems by using the modified Halpern's iterative scheme.

Acknowledgment

The first author was supported by the Thailand Research Fund through the Royal Golden Jubilee Ph.D. Program (Grant no. PHD/0033/2554).

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