`Journal of Applied MathematicsVolume 2012, Article ID 575014, 13 pageshttp://dx.doi.org/10.1155/2012/575014`
Research Article

Strong Convergence Theorems for the Generalized Split Common Fixed Point Problem

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

Received 9 January 2012; Accepted 17 February 2012

Copyright © 2012 Cuijie Zhang. 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 the generalized split common fixed point problem (GSCFPP) and show that the GSCFPP for nonexpansive operators is equivalent to the common fixed point problem. Moreover, we introduce a new iterative algorithm for finding a solution of the GSCFPP and obtain some strong convergence theorems under suitable assumptions.

1. Introduction

Let and be real Hilbert spaces and let be a bounded linear operator. Given intergers , let us recall that the multiple-set split feasibility problem (MSSFP) was recently introduced  and is to find a point: where and are nonempty closed convex subsets of and , respectively. If , the MSSFP (1.1) becomes the so-called split feasibility problem (SFP)  which is to find a point: where and are nonempty closed convex subsets of and , respectively. Recently, the SFP (1.2) and MSSFP (1.1) have been investigated by many researchers; see, .

Since every closed convex subset in a Hilbert space is looked as the fixed point set of its associating projection, the MSSFP (1.1) becomes a special case of the split common fixed point problem (SCFPP), which is to find a point: where and are nonlinear operators. If , the problem (1.3) reduces to the so-called two-set SCFPP, which is to find a point:

Censor and Segal in  firstly introduced the concept of SCFPP in finite-dimensional Hilbert spaces and considered the following iterative algorithm for the two-set SCFPP for Class- operators: where , and is the identity operator. They proved the convergence of the algorithm (1.5) to a solution of problem (1.4). Moreover, they introduced a parallel iterative algorithm, which converges to a solution of the SCFPP (1.3). However, the parallel iterative algorithm does not include the algorithm (1.5) as a special case.

Very recently, Wang and Xu in  considered the SCFPP (1.3) for Class- operators and introduced the following iterative algorithm for solving the SCFPP (1.3): Under some mild conditions, they proved some weak and strong convergence theorems. Their iterative algorithm (1.6) includes Censor and Segal’s algorithm (1.5) as a special case for the two-set SCFPP (1.4). Moreover, they prove that the SCFPP (1.3) 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 .

Motivated and inspired by the aforementioned research works, we introduce a generalized split common fixed point problem (GSCFPP) which is to find a point: Then, we show that the GSCFPP (1.7) for nonexpansive operators is equivalent to the following common fixed point problem: where for every . Moreover, we give a new iterative algorithm for solving the GSCFPP (1.7) for nonexpansive operators and obtain some strong convergence theorems.

2. Preliminaries

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

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. 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 a metric projection of onto . It is known that for each , for all .

Let be a sequence of operators of into itself. The set of common fixed points of is denoted by , that is, . A sequence is said to be strongly nonexpansive if each is nonexpansive and whenever and are sequences in such that is bounded and ; see [16, 17]. A sequence in is said to be an approximate fixed point sequence of if . The set of all bounded approximate fixed point sequences of is denoted by ; see [16, 17]. We know that if has a common fixed point, then is nonempty; that is, every bounded sequence in the common fixed point set is an approximate fixed point sequence. A sequence with a common fixed point is said to satisfy the condition () if every weak cluster point of is a common fixed point whenever . A sequence of nonexpansive mappings of into itself is said to satisfy the condition () if for every nonempty bounded subset of ; see .

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

Lemma 2.1 (see ). Let be a nonempty subset of a Hilbert space . Let be a sequence of nonexpansive mappings of into . Let be a sequence in such that . Let be a sequence of mappings of into defined by for , where is the identity mapping on . Then is a strongly nonexpansive sequence.

Lemma 2.2 (see ). Let be a Hilbert space, a nonempty subset of , and and sequences of nonexpansive self-mappings of . Suppose that or is a strongly nonexpansive sequence and is nonempty. Then .

Lemma 2.3 (see ). Let be a Hilbert space, and a nonempty subset of . Both and satisfy the condition and is bounded for any bounded subset of . Then satisfies the condition .

Lemma 2.4 (see ). Let and be bounded sequences in a Banach space and let be a sequence in with . Suppose for all integers and Then .

Lemma 2.5 (see ). Assume that is a sequence of nonnegative real numbers such that where is a sequence in and is a sequence such that(i),(ii) or . Then .

3. Main Results

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

Lemma 3.1. Let be a given bounded linear operator and let be a sequence of nonexpansive operators. Assume For each constant , is defined by the following: Then . Moreover, for , is nonexpansive on for .

Proof. Since the inclusion is evident, now we only need to show the converse inclusion. If , then we have . Since , we take an arbitrary . Hence It follows that , then for every , hence . Next we turn to show that is a nonexpansive operator for . Since is nonexpansive, we have Hence For , we can immediately obtain that is a nonexpansive operator for every .

From Lemma 3.1, we can obtain that the solution set of GSCFPP (1.7) is identical to the solution set of problem (1.8).

Theorem 3.2. Let and be sequences of nonexpansive operators on Hilbert space . Both and satisfy the conditions and . Let be a contraction with coefficient . Suppose . Take an initial guess and define a sequence by the following algorithm: where , , , and are sequences in . If the following conditions are satisfied:(i);(ii)  and   ;(iii);(iv);(v), then converges strongly to where .

Proof. We proceed with the following steps.
Step 1. First show that there exists such that .
In fact, since is a contraction with coefficient , we have for every , . Hence is also a contraction. Therefore, there exists a unique such that .
Step 2. Now we show that is bounded.
Let , then and . Hence Then By induction on , for every . This shows that and are bounded, and hence, , , and are also bounded.
Step 3. We claim that and , where .
We first show the former equality. Let be a bounded sequence in . If , then Hence . On the other hand, if , combining (3.11) and , we obtain that . Hence . Therefore, .
Next, we show the latter equality. Using Lemma 2.1, we know that is a strongly nonexpansive sequence. Thus, since , from Lemma 2.2 we have
Step 4. satisfies the condition , where .
Let be a nonempty bounded subset of . From the definition of , we have, for all , It follows that Since satisfies the condition and , we have that is, satisfies the condition . Since is bounded for any bounded subset of , by using Lemma 2.3, we have that satisfies the condition , that is, satisfies the condition .
Step 5. We show .
We can write (3.6) as where . It follows that From Step 2, we may assume that , where is a bounded set of . Then from (3.16), we obtain It follows that Since satisfies the condition , combining as , we have Hence by Lemma 2.4, we get as . Consequently,
Step 6. We claim that .
From (3.6), we have and hence Since , and , we derive Thus (3.23) and Steps 2 and 3 imply that
Step 7. Show , where .
Since is bounded, there exist a point and a subsequence of such that and . Since and satisfy the condition , from Step 6, we have . Using (2.1), we get
Step 8. Show .
Since , using (3.8), we have which implies that for every . Consequently, according to Step 7, , and Lemma 2.5, we deduce that converges strongly to . This completes the proof.

Combining Lemma 3.1 and Theorem 3.2, we can obtain the following strong convergence theorem for solving the GSCFPP (1.7).

Theorem 3.3. Let and be sequences of nonexpansive operators on Hilbert space and , respectively. Both and satisfy the conditions and . Let be a contraction with coefficient . Suppose that the solution set of GSCFPP (1.7) is nonempty. Take an initial guess and define a sequence by the following algorithm: where , and , , , are sequences in . If the following conditions are satisfied:(i);(ii)  and  ;(iii);(iv);(v), then converges strongly to where .

Proof. Set . By Lemma 3.1, is a nonexpansive operator for every . We can rewrite (3.29) as
We only need to prove that satisfies the conditions and . Assume that is a nonempty bounded subset of . For every , we have Since satisfies the condition , and is bounded, it follows from (3.31) that Therefore, satisfies the condition .
Assume that and ; we next show that . By using , we have . Since , we choose an arbitrary point ; then for every , Hence Then we get . Since satisfies the condition and , we have . From Lemma 3.1, we have .

Let be a nonexpansive mapping with a fixed point, and define for all . Then satisfies the conditions and . Thus, one obtains the algorithm for solving the two-set SCFPP (1.4).

Corollary 3.4. Let and be nonexpansive operators on Hilbert space and , respectively. Let be a contraction with coefficient . Suppose that the solution set of SCFPP (1.4) is nonempty. Take an initial guess and define a sequence by the following algorithm in (3.29), where , and , , , are sequences in . If the following conditions are satisfied:(i);(ii) and ;(iii);(iv);(v). Then converges strongly to where .

Remark 3.5. By adding more operators to the families and by setting for and for , the SCFPP (1.3) can be viewed as a special case of the GSCFPP (1.7).

Acknowledgment

This research is supported by the science research foundation program in Civil Aviation University of China (07kys09), the Fundamental Research Funds for the Central Universities (Program No. ZXH2011D005), and the NSFC Tianyuan Youth Foundation of Mathematics of China (No. 11126136).

References

1. Y. Censor, T. Elfving, N. Kopf, and T. Bortfeld, “The multiple-sets split feasibility problem and its applications for inverse problems,” Inverse Problems, vol. 21, no. 6, pp. 2071–2084, 2005.
2. 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.
3. C. Byrne, “Iterative oblique projection onto convex sets and the split feasibility problem,” Inverse Problems, vol. 18, no. 2, pp. 441–453, 2002.
4. C. Byrne, “A unified treatment of some iterative algorithms in signal processing and image reconstruction,” Inverse Problems, vol. 20, no. 1, pp. 103–120, 2004.
5. 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, Article ID 102085, 13 pages, 2010.
6. F. Wang and H. K. Xu, “Strongly convergent iterative algorithms for solving a class of variational inequalities,” Journal of Nonlinear and Convex Analysis, vol. 11, no. 3, pp. 407–421, 2010.
7. H. K. Xu, “A variable Krasnoselskii-Mann algorithm and the multiple-set split feasibility problem,” Inverse Problems, vol. 22, no. 6, pp. 2021–2034, 2006.
8. H. K. Xu, “Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces,” Inverse Problems, vol. 26, no. 10, Article ID 105018, p. 17, 2010.
9. Y. Yao, W. Jigang, and Y.-C. Liou, “Regularized methods for the split feasibility problem,” Abstract and Applied Analysis, vol. 2012, Article ID 140679, 13 pages, 2012.
10. A. Moudafi, “A note on the split common fixed-point problem for quasi-nonexpansive operators,” Nonlinear Analysis, vol. 74, no. 12, pp. 4083–4087, 2011.
11. 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.
12. F. Wang and H. K. Xu, “Cyclic algorithms for split feasibility problems in Hilbert spaces,” Nonlinear Analysis, vol. 74, no. 12, pp. 4105–4111, 2011.
13. Y. Yao and N. Shahzad, “Strong convergence of a proximal point algorithm with general errors,” Optimization Letters, vol. 6, no. 4, pp. 621–628, 2012.
14. Y. Yao, R. Chen, and Y.-C. Liou, “A unified implicit algorithm for solving the triple-hierarchical constrained optimization problem,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 1506–1515, 2012.
15. Y. Yao, Y. Je Cho, and Y.-C. Liou, “Hierarchical convergence of an implicit double-net algorithm for nonexpansive semigroups and variational inequalities,” Fixed Point Theory and Applications, vol. 2011, article 101, 2011.
16. K. Aoyama, Y. Kimura, W. Takahashi, and M. Toyoda, “On a strongly nonexpansive sequence in Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol. 8, no. 3, pp. 471–489, 2007.
17. K. Aoyama and Y. Kimura, “Strong convergence theorems for strongly nonexpansive sequences,” Applied Mathematics and Computation, vol. 217, no. 19, pp. 7537–7545, 2011.
18. K. Aoyama, “An iterative method for fixed point problems for sequences of nonexpansive mappings,” in Fixed Point Theory and Applications, pp. 1–7, Yokohama Publication, Yokohama, Japan, 2010.
19. T. Suzuki, “Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without bochner integrals,” Journal of Mathematical Analysis and Applications, vol. 305, no. 1, pp. 227–239, 2005.
20. H. K. Xu, “Viscosity approximation methods for nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 279–291, 2004.