AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 817367 10.1155/2014/817367 817367 Research Article Shrinking Projection Methods for Split Common Fixed-Point Problems in Hilbert Spaces Wu Huan-chun Cheng Cao-zong Saejung Satit College of Applied Sciences Beijing University of Technology Beijing 100124 China bjpu.edu.cn 2014 1442014 2014 19 12 2013 02 04 2014 14 4 2014 2014 Copyright © 2014 Huan-chun Wu and Cao-zong Cheng. 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.

Inspired by Moudafi (2011) and Takahashi et al. (2008), we present the shrinking projection method for the split common fixed-point problem in Hilbert spaces, and we obtain the strong convergence theorem. As a special case, the split feasibility problem is also considered.

1. Introduction

Let C and Q be nonempty closed convex sets in real Hilbert spaces H 1 and H 2 , respectively. The split feasibility problem (SFP) is to find (1) x C , such    that    A x Q , where A : H 1 H 2 is a bounded linear operator. We use Φ to denote the solution set of the SFP (1). The SFP in finite-dimensional Hilbert space was first introduced by Censor and Elfving . In 2010, Xu  considered the SFP in the setting of infinite-dimensional Hilbert space. The SFP has received much attention due to its wide applications in signal processing, image reconstruction, intensity-modulated radiation therapy, and so on (see ). Several iterative methods can be used to solve the SFP (1). Censor and Elfving  constructed the iterative process which involves the computation of the inverse of a matrix. A more popular algorithm that solves the SFP is the CQ algorithm of Byrne [3, 4]; that is, let x 0 be an arbitrary point in H 1 : (2) x n + 1 = P C ( I - γ A * ( I - P Q ) A ) x n , where γ > 0 is a parameter and P C and P Q are metric projections onto C and Q , respectively.

Let K be a nonempty closed convex subset of a real Hilbert space H and let T : K K be a mapping. We denote by Fix ( T ) the fixed-point set of T ; that is, Fix ( T ) = { x K : T x = x } . A mapping T : K K is nonexpansive if T x - T y x - y for all x , y K . A mapping T : K K is quasinonexpansive if Fix ( T ) and T x - y x - y for all x K and y Fix ( T ) . It is known that the fixed-point set of a quasinonexpansive mapping is closed and convex (see [7, 8]). There are some quasinonexpansive mappings which are not nonexpansive (see ). For example, the level set of a continuous convex function is characterized as the fixed-point set of a nonlinear mapping called the subgradient projection, which is not nonexpansive but quasinonexpansive.

Now we focus our attention on the following two-operator split common fixed-point problem (SCFP): (3) find    x * C , such    that    A x * Q , where A : H 1 H 2 is a bounded linear operator and U : H 1 H 1 and T : H 2 H 2 are two quasinonexpansive mappings with Fix ( U ) = C and Fix ( T ) = Q . The solution set of the SCFP (3) is denoted by (4) Γ = { x * C : A x * Q } . As far as we know, the SCFP is introduced by Censor and Segal . By taking U = P C and T = P Q , the SCFP reduces to the SFP. Hence, the SCFP is a generalization of the SFP. Moudafi  considered the following algorithm for the SCFP: let x 0 H 1 be arbitrary, u k = x k - γ β A * ( I - T ) A x k and (5) x k + 1 = ( 1 - α k ) u k + α k U ( u k ) , where β ( 0,1 ) , α k ( 0,1 ) , and γ ( 0 , 1 / λ β ) , with λ being the spectral radius of the operator A * A . He obtained the weak convergence of the algorithm (5).

In 2008, Takahashi et al.  developed the shrinking projection method for the nonexpansive mapping. Let T be a nonexpansive mapping of K into itself such that Fix ( T ) . Let x 0 H , C 1 = K and u 1 = P C 1 x 0 ; (6) y n = α n u n + ( 1 - α n ) T u n , C n + 1 = { z C n : y n - z u n - z } , u n + 1 = P C n + 1 x 0 , where 0 α n a < 1 . They proved that the sequence { u n } converges strongly to P Fix ( T ) x 0 .

Motivated by the above results, especially by Moudafi  and Takahashi et al. , in this paper, we present the shrinking projection methods for the split common fixed-point problems. As a special case, the split feasibility problem is also discussed.

2. Preliminaries

Throughout this paper, let and be the sets of positive integers and real numbers, respectively. For any x H , there exists a unique point P K x K such that (7) x - P K x x - y y K , where K is a nonempty closed convex subset of a real Hilbert space H . The mapping P K is called the metric projection of H onto K . Note that P K is a nonexpansive mapping. For x H and z K , we have (8) z = P K x x - z , y - z 0 for    every    y K .

We say that a mapping T : K K is demiclosed at zero if for any sequence { x n } K which converges weakly to x , the strong convergence of the sequence { T x n } to zero implies that T x = 0 . It is well known that I - T is demiclosed whenever T is nonexpansive. In fact, this property is satisfied for more general mappings (see [15, 16]).

We will use the following notations:

x n x stands for the strong convergence of { x n } to x ;

x n x stands for the weak convergence of { x n } to x ;

ω ω ( x n ) = { x : x n j x } denotes the weak ω -limit set of { x n } .

Here are two useful lemmas.

Lemma 1.

Let x , y H and let λ . One has (9) λ x + ( 1 - λ ) y 2 = λ x 2 + ( 1 - λ ) y 2 - λ ( 1 - λ ) x - y 2 .

Lemma 2 (see [<xref ref-type="bibr" rid="B17">17</xref>]).

Let K be a closed convex subset of a real Hilbert space H and let { x n } be a sequence in H and u H . Let q = P K u . If { x n } satisfies the following conditions:

ω ω ( x n ) K ,

x n - u u - q for all n ,

then one has x n q .

3. Shrinking Projection Methods

Now we are in a position to give the shrinking projection method for split common fixed-point problem (3).

Theorem 3.

Let H 1 and H 2 be real Hilbert spaces and let A : H 1 H 2 be a bounded linear operator. Let U : H 1 H 1 and T : H 2 H 2 be two quasinonexpansive mappings with Fix ( U ) = C and Fix ( T ) = Q . Suppose that I - U and I - T are demiclosed at zero and solution set Γ of the SCFP (3) is nonempty. For u H 1 chosen arbitrarily, C 1 = H 1 , h 1 = P C 1 u , define a sequence { h n } by the following algorithm: (10) w n = h n - γ A * ( I - T ) A h n , y n = α n w n + ( 1 - α n ) U w n , C n + 1 = { z C n : y n - z h n - z } , h n + 1 = P C n + 1 u . If the following are satisfied:

{ α n } ( 0,1 ) and liminf n α n ( 1 - α n ) > 0 ,

0 < γ < ( 1 / λ A A * ) , where λ A A * denotes the spectral radius of the operator A A * ,

then the sequence { h n } converges strongly to P Γ u .

Proof.

We first show that Γ C n for all n . It is obvious that Γ is contained in C 1 = H 1 . Suppose that Γ C k for some k . We have, for any p Γ C k , (11) y k - p 2 = α k w k + ( 1 - α k ) U w k - p 2 α k w k - p 2 + ( 1 - α k ) w k - p 2 = h k - γ A * ( I - T ) A h k - p 2 = h k - p 2 - 2 γ A * ( I - T ) A h k , h k - p + γ 2 A * ( I - T ) A h k 2 = h k - p 2 + 2 γ T A h k - A h k , A h k - A p + γ 2 ( I - T ) A h k , A A * ( I - T ) A h k h k - p 2 + γ 2 λ A A * ( I - T ) A h k 2 + γ [ T A h k - A p 2 - T A h k - A h k 2 - A h k - A p 2 ] h k - p 2 + γ 2 λ A A * ( I - T ) A h k 2 - γ T A h k - A h k 2 = h k - p 2 + γ ( γ λ A A * - 1 ) ( I - T ) A h k 2 h k - p 2 . It follows that p C k + 1 . Thus, we get Γ C n for all n .

Next we show that C n is closed and convex for all n . The set C 1 = H 1 is obviously closed and convex. Suppose that C k is closed and convex. We see that C k + 1 is closed and convex since y n - z h n - z is equivalent to (12) y n 2 - h n 2 - 2 y n - h n , z 0 . It follows that C n is closed and convex for all n . Therefore, we obtain that the sequence { h n } is well defined.

From h n = P C n u , we have (13) u - h n , h n - y 0 y C n . Recalling that Γ C n , one has (14) u - h n , h n - p 0 p Γ . Hence, (15) 0 u - h n , h n - p = u - h n , h n - u + u - p - u - h n 2 + u - h n u - p . This implies that (16) u - h n u - p , which yields that { h n } is bounded.

From h n = P C n u and h n + 1 = P C n + 1 u C n + 1 C n , we get (17) 0 u - h n , h n - h n + 1 - u - h n 2 + u - h n u - h n + 1 , which gives that (18) u - h n u - h n + 1 . Hence, (19) the    limit    lim n u - h n exists .

It follows from (17) that (20) h n - h n + 1 2 = h n - u 2 + 2 h n - u , u - h n + 1 + u - h n + 1 2 = h n - u 2 + 2 h n - u , u - h n + h n - h n + 1 + u - h n + 1 2 = - h n - u 2 + 2 h n - u , h n - h n + 1 + u - h n + 1 2 - h n - u 2 + u - h n + 1 2 . Thus, we get (21) lim n h n - h n + 1 = 0 . The fact that h n + 1 = P C n + 1 u C n + 1 gives (22) y n - h n + 1 h n - h n + 1 0 . The expressions (21) and (22) yield (23) lim n y n - h n = 0 .

We will prove that ω ω ( h n ) Γ . Without loss of generality, we assume that h n h * . It follows from (11) that (24) γ ( 1 - γ λ A A * ) ( I - T ) A h n 2 h n - p 2 - y n - p 2 ( h n - p + y n - p ) h n - y n . This together with (23) implies that (25) lim n ( I - T ) A h n = 0 . We have A h * Fix ( T ) = Q since I - T is demiclosed at zero.

Using (10) and (25), we get w n h * . For any p Γ , one has (26) y n - p 2 = α n w n + ( 1 - α n ) U w n - p 2 = α n w n - p 2 + ( 1 - α n ) U w n - p 2 - α n ( 1 - α n ) U w n - w n 2 w n - p 2 - α n ( 1 - α n ) U w n - w n 2 h n - p 2 - α n ( 1 - α n ) U w n - w n 2 , which implies that (27) α n ( 1 - α n ) U w n - w n 2 h n - p 2 - y n - p 2 ( h n - p + y n - p ) h n - y n . Therefore, one has (28) lim n U w n - w n = 0 . It follows that h * Fix ( U ) = C since I - U is demiclosed at zero. Thus, we have obtained ω ω ( h n ) Γ . According to Lemma 2, we see that h n P Γ u .

By Theorem 3, we immediately obtain the shrinking projection method for the split feasibility problem.

Theorem 4.

Let H 1 and H 2 be real Hilbert spaces and let C and Q be nonempty closed convex subsets of H 1 and H 2 , respectively. Let A : H 1 H 2 be a bounded linear operator. Suppose that the solution set Φ of the SFP (1) is nonempty. For u H 1 chosen arbitrarily, C 1 = H 1 , h 1 = P C 1 u , define a sequence { h n } by the following algorithm: (29) w n = h n - γ A * ( I - P Q ) A h n , y n = α n w n + ( 1 - α n ) P C w n , C n + 1 = { z C n : y n - z h n - z } , h n + 1 = P C n + 1 u . If the following are satisfied:

{ α n } ( 0,1 ) and liminf n α n ( 1 - α n ) > 0 ,

0 < γ < 1 / λ A A * , where λ A A * denotes the spectral radius of the operator A A * ,

then the sequence { h n } converges strongly to P Φ u .

Remark 5.

Letting u = 0 in Theorems 3 and 4, we obtain the shrinking projection methods for minimum-norm solutions of corresponding problems.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the referees and editors for their valuable comments and suggestions.

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