Shrinking Projection Methods for Split Common Fixed-Point Problems in Hilbert Spaces

and Applied Analysis 3 + γ 2 ⟨(I − T)Ah k , AA ∗ (I − T)Ah k ⟩ ≤ 󵄩󵄩󵄩󵄩hk − p 󵄩󵄩󵄩󵄩 2 + γ 2 λ AA ∗ 󵄩󵄩󵄩󵄩(I − T)Ahk 󵄩󵄩󵄩󵄩 2 + γ [ 󵄩󵄩󵄩󵄩TAhk − Ap 󵄩󵄩󵄩󵄩 2 − 󵄩󵄩󵄩󵄩TAhk − Ahk 󵄩󵄩󵄩󵄩 2 − 󵄩󵄩󵄩󵄩Ahk − Ap 󵄩󵄩󵄩󵄩 2 ] ≤ 󵄩󵄩󵄩󵄩hk − p 󵄩󵄩󵄩󵄩 2 + γ 2 λ AA ∗ 󵄩󵄩󵄩󵄩(I − T)Ahk 󵄩󵄩󵄩󵄩 2 − γ 󵄩󵄩󵄩󵄩TAhk − Ahk 󵄩󵄩󵄩󵄩 2 = 󵄩󵄩󵄩󵄩hk − p 󵄩󵄩󵄩󵄩 2 + γ (γλ AA ∗ − 1) 󵄩󵄩󵄩󵄩(I − T)Ahk 󵄩󵄩󵄩󵄩 2 ≤ 󵄩󵄩󵄩󵄩hk − p 󵄩󵄩󵄩󵄩 2 .


Introduction
Let  and  be nonempty closed convex sets in real Hilbert spaces  1 and  2 , respectively.The split feasibility problem (SFP) is to find  ∈ , such that  ∈ , where  :  1 →  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 [1].In 2010, Xu [2] 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][4][5][6]).Several iterative methods can be used to solve the SFP (1).Censor and Elfving [1] 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  0 be an arbitrary point in  1 : where  > 0 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 Fix() the fixed-point set of ; that is, Fix() = { ∈  :  = }.A mapping  :  →  is nonexpansive if ‖ − ‖ ≤ ‖ − ‖ forall ,  ∈ .A mapping  :  →  is quasinonexpansive if Fix() ̸ = 0 and ‖ − ‖ ≤ ‖ − ‖ forall  ∈  and  ∈ Fix().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][10][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 twooperator split common fixed-point problem (SCFP): where  :  1 →  2 is a bounded linear operator and  :  1 →  1 and  :  2 →  2 are two quasinonexpansive mappings with Fix() =  and Fix() = .The solution set of the SCFP (3) is denoted by As far as we know, the SCFP is introduced by Censor and Segal [12].By taking  =   and  =   , the SCFP reduces to the SFP.Hence, the SCFP is a generalization of the SFP.Moudafi [13] considered the following algorithm for the SCFP: let  0 ∈  1 be arbitrary,   =   −  * ( − )  and where  ∈ (0, 1),   ∈ (0, 1), and  ∈ (0, 1/), with  being the spectral radius of the operator  * .He obtained the weak convergence of the algorithm (5).
In 2008, Takahashi et al. [14] developed the shrinking projection method for the nonexpansive mapping.Let  be a nonexpansive mapping of  into itself such that Fix() ̸ = 0.
where 0 ≤   ≤  < 1.They proved that the sequence {  } converges strongly to  Fix()  0 .Motivated by the above results, especially by Moudafi [13] and Takahashi et al. [14], in this paper, we present the shrinking projection methods for the split common fixedpoint problems.As a special case, the split feasibility problem is also discussed.

Preliminaries
Throughout this paper, let N and R 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  = 0.It is well known that  −  is demiclosed whenever  is nonexpansive.In fact, this property is satisfied for more general mappings (see [15,16]).
Here are two useful lemmas.
Lemma 1.Let ,  ∈  and let  ∈ R. One has Lemma 2 (see [17]).Let  be a closed convex subset of a real Hilbert space  and let {  } be a sequence in  and  ∈ .

Shrinking Projection Methods
Now we are in a position to give the shrinking projection method for split common fixed-point problem (3).
Next we show that   is closed and convex for all  ∈ N. The set  1 =  1 is obviously closed and convex.Suppose that   is closed and convex.We see that  +1 is closed and convex since ‖  − ‖ ≤ ‖ℎ  − ‖ is equivalent to It follows that   is closed and convex for all  ∈ N. Therefore, we obtain that the sequence {ℎ  } is well defined.
From ℎ  =    , we have Recalling that Γ ⊂   , one has