Journal of Applied Mathematics

Volume 2012 (2012), Article ID 438121, 11 pages

doi:10.1155/2012/438121

Research Article

Strong Convergence Theorems for the Split Common Fixed Point Problem for Countable Family of Nonexpansive Operators

Cuijie Zhang and Songnian He

College of Science, Civil Aviation University of China, Tianjin 300300, China

Received 2 March 2012; Revised 15 May 2012; Accepted 21 May 2012

Academic Editor: Nazim I. Mahmudov

Copyright © 2012 Cuijie Zhang and Songnian He. 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.

We introduce a new iterative algorithm for solving the split common fixed point problem for countable family of nonexpansive operators. Under suitable assumptions, we prove that the iterative algorithm strongly converges to a solution of the problem.

1. Introduction

Let and be two real Hilbert spaces and let be a bounded linear operator. The split feasibility problem (SFP), see [1], is to find a point with the property:

where and are nonempty closed convex subsets of and , respectively. A more general form of the SFP is the so-called multiple-set split feasibility problem (MSSFP) which was recently introduced by Censor et al. [2]. Given integers , the MSSFP is to find a point with the property:

where and are nonempty closed convex subsets of and , respectively. The SFP (1.1) and the MSSFP (1.2) serve as a model for many inverse problems where constraints are imposed on the solutions in the domain of a linear operator as well as in this operator's ranges. Recently, the SFP (1.1) and the MSSFP (1.2) are widely applied in the image reconstructions [1, 3], the intensity-modulated radiation therapy [4, 5], and many other areas. The problems have been investigated by many researchers, for instance, [6–13]. The SFP (1.1) can be viewed as a special case of the convex feasibility problem (CFP) since the SFP (1.1) can be rewritten as

However, the methods for study the SFP (1.1) are actually different from those for the CFP in order to avoid the usage of the inverse . Byrne [6] introduced a so-called CQ algorithm:

where the operator is not relevant.

Censor and Segal in [14] firstly introduced the concept of the split common fixed point problem (SCFPP) in finite-dimensional Hilbert spaces. The SCFPP is a generalization of the convex feasibility problem (CFP) and the split feasibility problem (SFP). The SCFPP considers to find a common fixed point of a family of operators in such that its image under a linear transformation is a common fixed point of another family of operators in . That is, the SCFPP is to find a point with the property:

where and are nonlinear operators. If , the problem (1.5) deduces to the so-called two-set SCFPP, which is to find a point such that

where and are nonlinear operators.

Censor and Segal in [14] considered the following iterative algorithm for the SCFPP (1.6) for Class- operators in finite-dimensional Hilbert spaces:

where , and is the identity operator.

Recently, in the infinite-dimensional Hilbert space, Wang and Xu [15] studied the SCFPP (1.5) and introduced the following iterative algorithm for Class- operators:

where and . Under some mild conditions, they proved that converges weakly to a solution of the SCFPP (1.5), extended and improved Censor and Segal's results. Moreover, they proved that the SCFPP (1.5) for the Class- operators is equivalent to a common fixed point problem. This is also a classical method. Many problems eventually converted to a common fixed point problem, see [16–18]. Very recently, the split common fixed point problems for various types of operators were studied in [19–21].

The above-mentioned results are about a finite number of operators; that is, the constraints are finite imposed on the solutions. In this paper, we consider the constraints are infinite, but countable. That is, we consider the generalized case of SCFPP for two countable families of operators (denoted GSCFPP), which is to find a point such that

Of course, the GSCFPP is more general and widely used than the SCFPP. This is a novelty of this paper. At the same time, we consider the nonexpansive operator. The nonexpansive operator is important because it includes many types of nonlinear operator arising in applied mathematics. For instance, the projection and the identity operator are nonexpansive. We prove that the GSCFPP (1.9) for the nonexpansive operators is equivalent to a common fixed point problem. Very recently, Gu et al. [22] introduced a new iterative method for dealing with the countable family of operators. They studied the following iterative algorithm:

where and are nonexpansive, , is strictly decreasing sequence in , and is a sequence in . Under some certain conditions on parameters, they proved that the sequence converges strongly to . On the other hand, from weakly convergence to strongly convergence, the viscosity approximation method is also one of the classical methods, see [22–24].

Motivated and inspired by the above results, we introduce the following algorithm:

Under some certain conditions, we prove that the sequence generated by (1.11) converges strongly to the solution of the GSCFPP (1.9).

2. Preliminaries

Throughout this paper, we write and to indicate that converges weakly to and converges strongly to , respectively.

Let be a real Hilbert space. An operator is said to be nonexpansive if for all . The set of fixed points of is denoted by . It is known that is closed and convex, see [25]. An operator is called contraction if there exists a constant such that for all . Let be a nonempty closed convex subset of . For each , there exists a unique nearest point in , denoted by , such that for every . is called the metric projection of onto . It is known that, for each ,

for all .

In order to prove our main results, we collect the following lemmas in this section.

Lemma 2.1 (see [26]). Let be a Hilbert space, a closed convex subset of , and a nonexpansive operator with . If is a sequence in weakly converging to and converges strongly to , then . In particular, if , then .

Lemma 2.2 (see [23]). Assume is a sequence of nonnegative real numbers such that

where is a sequence in and is a sequence such that

(i),(ii).

Then .

3. Main Results

Now we state and prove our main results of this paper.

Theorem 3.1. Let and be sequences of nonexpansive operators on real Hilbert spaces and , respectively. Let be a contraction with coefficient . Suppose that the solution set of GSCFPP (1.9) is nonempty. Let and . Set , and let be a strictly decreasing sequence satisfying the following conditions:

(i); (ii); (iii).

Then the sequence generated by (1.11) converges strongly to , where .

Proof. We proceed with the following steps.

Step . First show that there exists such that .

In fact, since is a contraction with coefficient , we have

for every . Hence is also a contraction of into itself. Therefore, there exists a unique such that . At the same time, we note that .

Step . Now we show that is bounded.

For simplicity, we set . Then we can rewrite (1.11) to

Observe that

for all . Thus it follows that

For , we can immediately obtain that is a nonexpansive operator for every .

Let , then and for every . Thus , which implies that . Since , we have

Then it follows that

for every . This shows that and is bounded. Hence, is also bounded.

Step 3. We show .

From (3.2), we have

where is a constant such that

From (i), (ii), (iii), and Lemma 2.2, it follows that

Step . We show and for .

We first show for . Since , we note that

which implies that

Using (3.2) and (3.10), we deduce

Noting that and , then we immediately obtain

Since is strictly decreasing, it follows that

Next we show , for every . Note for every ,

which follows that

for every . From (3.2), we have

By (3.15), it follows that

Thus,

Using and , we have

By , there holds

Since is strictly decreasing, we obtain

Last we show for every . In fact, we note that for every ,

Then by (3.13) and (3.21), we obtain

Step 5. Show , where .

Since is bounded, there exist a point and a subsequence of such that

and . Since is a bounded linear operator, we have . Now applying (3.21), (3.23), and Lemma 2.1, we conclude that and for every . Hence, . Since is closed and convex, by (2.1), we get

Step 6. Show .

Since , we have and for every . It follows that . Using (3.2), we have

which implies that

for every . Consequently, according to (3.25), , and Lemma 2.2, we deduce that converges strongly to . This completes the proof.

Remark 3.2. If we set and for all , where is an arbitrary point in , it is easily seen that our conditions are satisfied.

Corollary 3.3. Let and be nonexpansive operators. Let be a contraction with coefficient . Suppose that the solution set of SCFPP (1.6) is nonempty. Let and define a sequence by the following algorithm:

where , and is a strictly decreasing sequence satisfying the following conditions:

(i); (ii); (iii).

Then converges strongly to , where .

Proof. Set and to be sequences of operators defined by and for all in Theorem 3.1. Then by Theorem 3.1 we obtain the desired result.

Remark 3.4. By adding more operators to the families and by setting for and for , the SCFPP (1.5) can be viewed as a special case of the GSCFPP (1.9).

Acknowledgment

This research is supported by the NSFC Tianyuan Youth Foundation of Mathematics of China (No. 11126136), the Fundamental Research Funds of Science for the Central Universities (Program No. ZXH2012K001), and the science research foundation program in Civil Aviation University of China (07kys09).

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