Shrinking Projection Methods for Split Common Fixed-Point Problems in Hilbert Spaces
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.
Let and be nonempty closed convex sets in real Hilbert spaces and , respectively. The split feasibility problem (SFP) is to find where is a bounded linear operator. We use 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 [3–6]). 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 be an arbitrary point in : where is a parameter and and are metric projections onto and , respectively.
Let be a nonempty closed convex subset of a real Hilbert space and let be a mapping. We denote by the fixed-point set of ; that is, . A mapping is nonexpansive if for all . A mapping is quasinonexpansive if and for all and . 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 [9–11]). 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): where is a bounded linear operator and and are two quasinonexpansive mappings with and . The solution set of the SCFP (3) is denoted by As far as we know, the SCFP is introduced by Censor and Segal . By taking and , the SCFP reduces to the SFP. Hence, the SCFP is a generalization of the SFP. Moudafi  considered the following algorithm for the SCFP: let be arbitrary, and where , , and , with being the spectral radius of the operator . He obtained the weak convergence of the algorithm (5).
In 2008, Takahashi et al.  developed the shrinking projection method for the nonexpansive mapping. Let be a nonexpansive mapping of into itself such that . Let , and ; where . They proved that the sequence converges strongly to .
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.
Throughout this paper, let and be the sets of positive integers and real numbers, respectively. For any , there exists a unique point such that where is a nonempty closed convex subset of a real Hilbert space . The mapping is called the metric projection of onto . Note that is a nonexpansive mapping. For and , we have
We say that a mapping is demiclosed at zero if for any sequence which converges weakly to , the strong convergence of the sequence to zero implies that . It is well known that is demiclosed whenever is nonexpansive. In fact, this property is satisfied for more general mappings (see [15, 16]).
We will use the following notations:(1) stands for the strong convergence of to ;(2) stands for the weak convergence of to ;(3) denotes the weak -limit set of .
Here are two useful lemmas.
Lemma 1. Let and let . One has
Lemma 2 (see ). Let be a closed convex subset of a real Hilbert space and let be a sequence in and . Let . If satisfies the following conditions:(1),(2) for all ,
then one has .
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 and be real Hilbert spaces and let be a bounded linear operator. Let and be two quasinonexpansive mappings with and . Suppose that and are demiclosed at zero and solution set of the SCFP (3) is nonempty. For chosen arbitrarily, , , define a sequence by the following algorithm: If the following are satisfied:(1) and (2), where denotes the spectral radius of the operator ,then the sequence converges strongly to .
Proof. We first show that for all . It is obvious that is contained in . Suppose that for some . We have, for any ,
It follows that . Thus, we get for all .
Next we show that is closed and convex for all . The set is obviously closed and convex. Suppose that is closed and convex. We see that is closed and convex since is equivalent to It follows that is closed and convex for all . Therefore, we obtain that the sequence is well defined.
From , we have Recalling that , one has Hence, This implies that which yields that is bounded.
From and , we get which gives that Hence,
It follows from (17) that Thus, we get The fact that gives The expressions (21) and (22) yield
We will prove that . Without loss of generality, we assume that . It follows from (11) that This together with (23) implies that We have since is demiclosed at zero.
Using (10) and (25), we get . For any , one has which implies that Therefore, one has It follows that since is demiclosed at zero. Thus, we have obtained . According to Lemma 2, we see that .
By Theorem 3, we immediately obtain the shrinking projection method for the split feasibility problem.
Theorem 4. Let and be real Hilbert spaces and let and be nonempty closed convex subsets of and , respectively. Let be a bounded linear operator. Suppose that the solution set of the SFP (1) is nonempty. For chosen arbitrarily, , , define a sequence by the following algorithm: If the following are satisfied: (1) and ,(2), where denotes the spectral radius of the operator , then the sequence converges strongly to .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank the referees and editors for their valuable comments and suggestions.
Y. Censor, T. Bortfeld, B. Martin, and A. Trofimov, “A unified approach for inversion problems in intensity-modulated radiation therapy,” Physics in Medicine and Biology, vol. 51, no. 10, pp. 2353–2365, 2006.View at: Google Scholar
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.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
P. L. Combettes and J. C. Pesquet, “Image restoration subject to a total variation constraint,” IEEE Transactions on Image Processing, vol. 13, no. 9, pp. 1213–1222, 2004.View at: Google Scholar
W. Takahashi, Y. Takeuchi, and R. Kubota, “Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 341, no. 1, pp. 276–286, 2008.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet