Journal of Applied Mathematics

Volume 2013 (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

Shubo Cao1,2

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 [2–8]. 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 [13–17];(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 [19–23].

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 [30–34].

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:

References

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.

C. Byrne, “Iterative oblique projection onto convex sets and the split feasibility problem,” Inverse Problems, vol. 18, no. 2, pp. 441–453, 2002.

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.

Y. Censor, T. Bortfeld, B. Martin, and A. Trofimov, “Aunified approach for inversion problems in intensity modulated radiation therapy,” Physics in Medicine and Biology, vol. 51, no. 10, pp. 2353–2365, 2006.

Y. Censor, A. Motova, and A. Segal, “Perturbed projections and subgradient projections for the multiple-sets split feasibility problem,” Journal of Mathematical Analysis and Applications, vol. 327, no. 2, pp. 1244–1256, 2007.

G. Lopez, V. Martin, and H. K. Xu, “Iterative algorithms for the multiple-sets split feasibility problem,” in Biomedical Mathematics: Promising Directions in Imaging, Therapy Planning and Inverse Problems, Y. Censor, M. Jiang, and G. Wang, Eds., pp. 243–279, Medical Physics Publishing, Madison, Wis, USA, 2009.

F. Wang and H.-K. Xu, “Cyclic algorithms for split feasibility problems in Hilbert spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 12, pp. 4105–4111, 2011.

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.

L. Landweber, “An iteration formula for Fredholm integral equations of the first kind,” American Journal of Mathematics, vol. 73, pp. 615–624, 1951.

H.-K. Xu, “Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces,” Inverse Problems, vol. 26, no. 10, Article ID 105018, 17 pages, 2010.

A. Moudafi, “The split common fixed-point problem for demicontractive mappings,” Inverse Problems, vol. 26, no. 5, Article ID 055007, 6 pages, 2010.

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.

A. Moudafi, “A note on the split common fixed-point problem for quasi-nonexpansive operators,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 12, pp. 4083–4087, 2011.

J. Zhao and S. He, “Strong convergence of the viscosity approximation process for the split common fixed-point problem of quasi-nonexpansive mappings,” Journal of Applied Mathematics, vol. 2012, Article ID 438023, 12 pages, 2012.

B.-C. Deng, T. Chen, and B. Xin, “Parallel and cyclic algorithms for quasi-nonexpansives in Hilbert space,” Abstract and Applied Analysis, vol. 2012, Article ID 218341, 27 pages, 2012.

P.-E. Maingé, “The viscosity approximation process for quasi-nonexpansive mappings in Hilbert spaces,” Computers & Mathematics with Applications, vol. 59, no. 1, pp. 74–79, 2010.

I. Yamada and N. Ogura, “Hybrid steepest descent method for variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings,” Numerical Functional Analysis and Optimization, vol. 25, no. 7-8, pp. 619–655, 2004.

F. E. Browder and W. V. Petryshyn, “Construction of fixed points of nonlinear mappings in Hilbert space,” Journal of Mathematical Analysis and Applications, vol. 20, pp. 197–228, 1967.

K. Aoyama and F. Kohsaka, “Fixed point and mean convergence theorems for a family of -hybrid mappings,” Journal of Nonlinear Analysis and Optimization. Theory and Applications, vol. 2, no. 1, pp. 87–95, 2011.

K. Aoyama, “Halpern's iteration for a sequence of quasinonexpansive type mappings,” Non-Linear Mathematics for Uncertainty and Its Applications, vol. 100, pp. 387–394, 2011.

B.-C. Deng, T. Chen, and Z.-F. Li, “Cyclic iterative method for strictly pseudononspreading in Hilbert space,” Journal of Applied Mathematics, vol. 2012, Article ID 435676, 15 pages, 2012.

M. O. Osilike and F. O. Isiogugu, “Weak and strong convergence theorems for nonspreading-type mappings in Hilbert spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 5, pp. 1814–1822, 2011.

N. Petrot and R. Wangkeeree, “A general iterative scheme for strict pseudononspreading mapping related to optimization problem in Hilbert spaces,” Journal of Nonlinear Analysis and Optimization, vol. 2, no. 2, pp. 329–336, 2011.

M. Tian, “A general iterative algorithm for nonexpansive mappings in Hilbert spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 3, pp. 689–694, 2010.

P.-E. Maingé, “Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization,” Set-Valued Analysis, vol. 16, no. 7-8, pp. 899–912, 2008.

F. Kohsaka and W. Takahashi, “Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces,” Archiv der Mathematik, vol. 91, no. 2, pp. 166–177, 2008.

F. Kohsaka and W. Takahashi, “Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces,” SIAM Journal on Optimization, vol. 19, no. 2, pp. 824–835, 2008.

F. Wang and H.-K. Xu, “Cyclic algorithms for split feasibility problems in Hilbert spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 12, pp. 4105–4111, 2011.

S. Iemoto and W. Takahashi, “Approximating common fixed points of nonexpansive mappings and nonspreading mappings in a Hilbert space,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 12, pp. e2082–e2089, 2009.

M. Eslamian, “Convergence theorems for nonspreading mappings and nonexpansive multivalued mappings and equilibrium problems,” Optimization Letters, vol. 7, no. 3, pp. 547–557, 2013.

Y. Kurokawa and W. Takahashi, “Weak and strong convergence theorems for nonspreading mappings in Hilbert spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 6, pp. 1562–1568, 2010.

K. S. Kim, “Approximating common fixed points of nonspreading-type mappings and nonexpansive mappings in a Hilbert space,” Abstract and Applied Analysis, vol. 2012, Article ID 594218, 18 pages, 2012.

S. Plubtieng and S. Chornphrom, “Strong convergence theorem for equilibrium problems and fixed points of a nonspreading mapping in Hilbert spaces,” Fixed Point Theory and Applications, vol. 2010, Article ID 296759, 16 pages, 2010.

W. Takahashi, N.-C. Wong, and J.-C. Yao, “Fixed point theorems and convergence theorems for generalized nonspreading mappings in Banach spaces,” Journal of Fixed Point Theory and Applications, vol. 11, no. 1, pp. 159–183, 2012.