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Journal of Applied Mathematics
Volume 2013, Article ID 241789, 9 pages
http://dx.doi.org/10.1155/2013/241789
Research Article

The Split Common Fixed Point Problem for -Strictly Pseudononspreading Mappings

1School of Management, Tianjin University, Tianjin 300072, China
2School of Economics and Management, Civil Aviation University of China, Tianjin 300300, China

Received 10 January 2013; Revised 12 March 2013; Accepted 11 April 2013

Academic Editor: Filomena Cianciaruso

Copyright © 2013 Shubo Cao. 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 and analyze the viscosity approximation algorithm for solving the split common fixed point problem for the strictly pseudononspreading mappings in Hilbert spaces. Our results improve and develop previously discussed feasibility problems and related results.

1. Introduction

Throughout this paper, we always assume that is a real Hilbert space with inner product and norm . Let denote the identity operator on . Let and be two real Hilbert spaces and let be a bounded linear operator. Given closed convex subsets and of and , respectively.

The split feasibility problem (SFP) (Censor and Elfving 1994 [1]), modeling phase retrieval problems, is to find a point with the property Recently, it has been found that the SFP can also be used to model the intensity-modulated radiation therapy [28]. A special case of the SFP (1) is the convexly constrained linear problem: This problem, due to its applications in many applied disciplines, has extensively been investigated in the literature ever since Landweber [9] introduced his iterative method in 1951.

Note that the split feasibility problem (1) can be formulated as fixed point equation by using the fact where and are the projections onto and , respectively, is any positive constant, and denotes the adjoint of ; that is, solves the SFP (1) if and only if solves the fixed point equation (3) (see [10] for more details). This implies that we can use fixed point algorithms to solve SFP.

In 2002, Byrne [2] proposed his CQ algorithm to solve (1). The sequence is generated by the following iteration scheme: where , with being the spectral radius of the operator .

The CQ algorithm (4) is a special case of the K-M algorithm. Due to the fixed point formulation (2) of the SFP, Moudafi [11] applied the K-M algorithm to the operator to obtain a sequence given by where , with being the spectral radius of the operator , and the sequence satisfies the condition ; he proved weak convergence result of the algorithm (5) in Hilbert spaces.

In 2009, Censor and Segal [12] considered the following algorithm to be solved (1).

Algorithm 1. Initialization: let be arbitrary.
Iterative step: for let where with being the spectral radius of the operator and be a single pair of directed operators.

In 2010, Moudafi [13] extended the Algorithm 1 and introduced the following algorithm with weak convergence for the split common fixed point problem.

Algorithm 2. Initialization: let be arbitrary.
Iterative step: for let where , , and with being the spectral radius of the operator and be a pair of quasi-nonexpansive operators.

In 2012, Zhao and He [14] continue to consider the split common fixed point problem with quasi-nonexpansive operators and to use the following algorithm to obtain the strong convergence of the viscosity method for solving the split common fixed point problem.

Algorithm 3. Initialization: let be arbitrary.
Iterative step: for let where is a contractive mapping with constant , , and with being the spectral radius of the operator and be a pair of quasi-nonexpansive operators.

Motivated and inspired by Censor and Segal [12], Moudafi [11], and Zhao and He [14], we introduce the following relaxed algorithm.

Algorithm 4. Initialization: let be arbitrary.
Iterative step: for let where is a contractive mapping with constant , is -strongly monotone and boundedly Lipschitzian, , and with being the spectral radius of the operator and be a pair of -strictly pseudononspreading mappings .

This paper establishes the strong convergence of the sequence given by (9) to the unique solution of solving the split common fixed point problem and the following variational inequality problem :

2. Preliminaries

In this section, we introduce the concepts of contraction mappings, nonexpansive mappings, quasi-nonexpansive mappings, and -strictly pseudononspreading mappings and some Lemmas.

Assume that is a nonempty closed and convex subset of Hilbert space . Recall that the (nearest point or metric) projection from onto , that denoted , assigns, to each , the unique point with the property

Definition 5. A mapping is said to be (1)contraction, if , and ;(2)nonexpansive, if , ;(3)quasi-nonexpansive, , .

Remark 6. From the Definition 5, It is easy to see that(i)iterative methods for quasi-nonexpansive mappings have been extensively investigated; see [1317];(ii)a nonexpansive mapping is a quasi-nonexpansive mapping.

Following the terminology of Browder and Petryshyn [18], we obtain the following definitions.

Definition 7. A mapping is -strictly pseudononspreading if there exists such that for all .

Iterative methods for strictly pseudononspreading mapping have been extensively investigated; see [1923].

Lemma 8 (see [24]). Let be a Hilbert spaces, and is a contractive mapping with constant . is -Lipschitzian and -strongly monotone operator with , . Then for , That is, is strongly monotone with coefficient .

Lemma 9. Let be a real Hilbert space. Then the following well-known results hold: for all and (i); (ii); (iii).

Lemma 10. Let be a -strictly pseudononspreading mapping with , and set , . The following properties are reached for each : (1) and ; (2); (3).

Proof. Note that property (1) is easily deduced from the Lemma 8(iii) and the fact that is -strictly pseudononspreading mapping, we obtain
Property (2) is obtained from property (1) and by
Property (3) is given by and property (1).

Lemma 11 (see [25]). Let be a sequence of real numbers that does not decrease at infinity, in the sense that there exists a subsequence of which satisfies for all . Also consider the sequence of integers defined by Then is a nondecreasing sequence verifying , for all ; it holds that and one has

Lemma 12. Let be a closed convex subset of a real Hilbert space , given and . Then if and only if there holds the inequality

3. Main Results

In what follows, we will focus our attention on the following general two operator split common fixed point problem in real Hilbert space : where is a bounded linear operator, and are two -strictly pseudononspreading mappings with nonempty fixed point sets and , and denote the solution set of the two-operator SCFP by On the other hand, is also unique solution of solving the variational inequality problem : where is -strongly monotone and -Lipschitzian on with , . Let , .

Before stating our main convergence result, we establish the boundedness of the iterates given by (9).

Lemma 13. The sequence is generated by (9), and let and be two -strictly pseudononspreading mappings on , , and is a contractive mapping with constant , and , . Then is bounded.

Proof. Set . Then .
Taking , that is, and . We obtain From the definition of , we have On the other hand, we obtain According to the definition of , we have Now, by using property (1) of Lemma 9, we obtain Combining (24)–(26), we obtain From property (i) of Lemma 9 and (23), we get Combining (22), (23), and (28), we have It follows from (29) and induction that and hence is bounded.

Now we are in position to claim the main convergence result.

Theorem 14. Given a bounded linear operator , let and be two -strictly pseudononspreading mappings, and with fixed point and . Assume that and are demiclosed at origin. Let be -strongly monotone and -Lipschitzian on with , , and is a contractive mapping with constant . Assume that is the sequence given by Algorithm 4 with , , and such that and such that and . If , then the sequence strongly converges to a split common fixed point , verifying which equivalently solves the following variational inequality problem:

Proof. Let be the solution of (31). From (9) we obtain that hence By (28), we obtain that It follows from (33) that or equivalently Furthermore, using the classical equality (iii) in Lemma 10 and setting , we have So that (36) can be equivalently rewritten as Now using (32) again, we have Since is -strongly monotone and -Lipschitzian on , hence it is a classical matter to see that which by yields Then from (38) and (41), we have The rest of the proof will be divided into two parts.
Case  1.  Suppose that there exists such that is nonincreasing. In this situation, is then convergent because it is also nonnegative (hence it is bounded from below), so that ; hence, in light of (42) together with , the boundedness of and , we obtain It also follows from (42) that Then, by , we obviously deduce that or equivalently (as and ). From (45), we get Moreover, by Lemma 8, we have which by (46) entails Hence, recalling that exists, we equivalently obtain Namely, Now we prove that It follows from (27) and (43) that and hence Taking , from the demiclosedness of at , we have Now, by setting , it follows that . On the other hand, which, combined with the demiclosedness of at , yields Hence, and . We can take subsequence of such that and which leads to By (50), we have , and hence converges strongly to .
Case  2.  Suppose that there exists a subsequence of such that for all . In this situation, we consider the sequence of indices as defined in Lemma 11. It follows that , which by (42) amounts to By the boundedness of and , we immediately obtain Using (9), we have which together with (60) and yields Similar to Case 1, we have Now by (59) we clearly have which in the light of (47) yields Hence (as and ) it follows that From (59) and (63), we obtain which by (60) yields , so that . Combining (62), we have . Then, recalling that (by Lemma 11), we get , so that strongly.
In addition, the variational inequality (50) and (67) can be written as So, by the Lemma 12, it is equivalent to the fixed point equation

4. Application in Other Nonlinear Operators

In order to define our motivations, we recall some definitions of classed of operators as follows

Definition 15. is said to be (1)nonspreading in [26, 27], if (2)demicontractive in [28], if there exists a constant such that

Remark 16. Iemoto and Takahashi [29] proved that (70) is equivalent to Iterative methods for nonspreading mapping have been extensively investigated; see [3034].

Remark 17. From the Definition 5 (3), Definition 7, and Definition 15, we have the following facts. (i)Observe that every nonspreading mapping is 0-strictly pseudononspreading.(ii)If is nonspreading mapping and the set of fixed point is nonempty, then is quasi-nonexpansive mapping.(iii)Every pseudononspreading mapping with a nonempty fixed point set is demicontractive (see [28]).

Corollary 18. Given a bounded linear operator , let and be two nonspreading mappings with fixed point and . Assume that and are demiclosed at origin. Let be -strongly monotone and -Lipschitzian on with , , and let be a contractive mapping with constant . Let be the sequence given by (9) with , , and such that and such that and . If , then the sequence strongly converges to a split common fixed point , verifying which equivalently solves the following variational inequality problem:

Proof. Form the proof of the Theorem 14, we can easily certify this theorem by nonspreading mapping (i.e., nonspreading is 0-strictly pseudononspreading).

From the Remark 17(ii) and the Corollary 18, we have the following corollary.

Corollary 19. Given a bounded linear operator , let and be two quasi-nonexpansives with fixed point and . Assume that and are demiclosed at origin. Let be -strongly monotone and -Lipschitzian on with , , and let be a contractive mapping with constant . Let be the sequence given by (9) with , such that and such that and . If , then the sequence strongly converges to a split common fixed point , verifying which equivalently solves the following variational inequality problem:

If and in (9), thus and , and then we obtain (8) and the following corollary. On the other hand, this corollary was proven by Zhao and He [14].

Corollary 20. Given a bounded linear operator , let and be two quasi-nonexpansives with fixed point and . Assume that and are demiclosed at origin. Let be a contractive mapping with constant . Let be the sequence given by (8) with , such that , and such that and . If, then the sequence strongly converges to a split common fixed point , verifying which equivalently solves the following variational inequality problem:

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