# Strong Convergence Theorems for an Implicit Iterative Algorithm for the Split Common Fixed Point Problem

Authors: Huimin He; Sanyang Liu; Rudong Chen
Journal: Journal of Function Spaces
 (2016)

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## Abstract

The aim of this paper is to construct a novel implicit iterative algorithm for the split common fixed point problem for the demicontractive operatorsU, T, and  xn=αnfxn+1-αnUλxn-ρnA*I-TAxn, n≥0, where Uλ=(1-λ)I+λU, and we obtain the sequence which strongly converges to a solution x^ of this problem, and the solution x^ satisfies the variational inequality. 〈x^-f(x^),x^-z〉≤0, ∀z∈S, where S denotes the set of all solutions of the split common fixed point problem.


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## Body

## 1. Introduction

The split feasibility problem (SFP) is to find a point(1)x∈Csuch  thatAx∈Q,where C is a nonempty closed convex subset of a Hilbert space H1, Q is a nonempty closed convex subset of a Hilbert space H2, and A:H1→H2 is a bounded linear operator.This problem was proposed by Censor and Elfving [1] in 1994.Since the SFP can extensively be applied in fields such as intensity-modulated radiation therapy, signal processing, and image reconstruction, then the SFP has received so much attention by so many scholars; see [2–23].In 1994, Censor and Elfving [1] proposed the original algorithm in Rn,(2)xn+1=A-1PQPACAxn,where C and Q are nonempty closed convex subsets of Rn, A in the finite-dimensional Rn is a n×n matrix, and PQ is the projection operator from H2 onto Q.As we know, the computation of the inverseA-1 is not easy if the inverse of A existed. So, the algorithm (2) does not become popular.In 2002 and 2004, Byrne [2, 3] gave the so-called CQ algorithm as follows:(3)xn+1=PCxn-γA∗I-PQAxn,n≥0,where 0<γ<2/ρ with ρ taken as the largest eigenvalue of the operator A∗A and PC and PQ denote the projection operators from H1 and H2 onto the sets C, Q, respectively.For the stepsize of algorithm (3) is fixed and closely related to spectral radius of A∗A, then the projection operators PC and PQ are not easily calculated usually.The split common fixed point problem (SCFP) is to find a point(4)x∈Fix⁡Usuch  thatAx∈Fix⁡T,where U:H1→H1 and T:H2→H2, and Fix(U) and Fix(T) denote the fixed point sets of U and T.This problem was proposed by Censor and Segal [12] in 2009. Note that the SCFP is closely related to SFP and it is a particular case of SFP.In 2009, Censor and Segal [12] introduced the original algorithm for directed operators as follows:(5)xn+1=Uxn-ρA∗I-TAxn,n≥0,where the step size ρ satisfies 0<ρ<2/A2, and they obtained that {xn} weakly converges to a solution of the SCFP (4) if the solution of SCFP exists. But it is obvious that the choice of the step size ρ depends on the norm of operator, A, which is the disadvantage of this algorithm.The next two years, some extension results on the operators are obtained, such as Moudafi (2010) [24], Moudafi (2011) [25], and Wang and Xu (2011) [14].In order to overcome this disadvantage, Cui and Wang [26] proposed the following algorithm in 2014:(6)xn+1=Uλxn-ρnA∗I-TAxn,n≥0,where Uλ=(1-λ)I+λU and the step size ρn is chosen by the following way:(7)ρn=1-τI-TAxn22A∗I-TAxn2,Axn≠TAxn,0,otherwiseand they proved that the sequence {xn} converges weakly to a solution of the SCFP (4). Note that the advantage of this algorithm is that the step size ρn searches automatically and does not depend on the norm of operator A.Recently, Byrne et al. [27] introduced the split common null point problem (SCNPP) for set-valued maximal monotone mappings in Hilbert spaces. Given set-valued mappings Bi:H1→2H1, 1≤i≤p, and Fj:H2→2H2, 1≤j≤r, respectively, and the bounded linear operators Aj:H1→H2, 1≤j≤r, the SCNPP is formulated as follows:(8)findx∈H1such  that0∈⋂i=1pBixsuch  thatyj=Ajx∈H2solve0∈⋂j=1rFJyj.As we know, the SCNPP generalizes the split common fixed point problem and the split variational inequality problem [28, 29].Motivated by the viscosity idea of [30], in this paper, we construct a novel algorithm for demicontractive operators to approximate the solution of the SCFP (4), that is, the following implicit iterative algorithm:(9)xn=αnfxn+1-αnUλxn-ρnA∗I-TAxn,n≥0,where Uλ=(1-λ)I+λU and the step size ρn is also chosen as (7).The research highlight of this paper is that the strong convergence of the SCFP (4) is constructed; that is to say the sequence {xn} generated by (9) converges strongly to a solution of the SCFP.


## 2. Preliminaries

Throughout this paper, we denote the set of all solutions of the SCFP (4) by S. We use xn⇀x to indicate that {xn} converges weakly to x. Similarly, xn→x symbolizes the sequence {xn} which converges strongly to x.LetH, H1, and H2 be Hilbert spaces endowed with the inner product ·,· and norm ·, and C and Q are nonempty closed convex subsets of H1 and H2, respectively.Some concepts and lemmas are given in the following and they are useful in proving our main results.Definition 1.
A operatorT:H→H is said to be
(i) nonexpansive if(10)Tx-Ty≤x-z,∀x,y∈H
(ii) quasi-nonexpansive if(11)Tx-z≤x-z,∀x∈H,∀z∈Fix⁡T
(iii) directed if(12)z-Tx,x-Tx≤0,∀x∈H,∀z∈Fix⁡T
(iv)τ-demicontractive with τ<1 if(13)Tx-z2≤x-z2+τx-Tx2,∀x∈H,∀z∈Fix⁡TNote that (12) is equivalent to (14)z-Tx2+x-Tx2-x-z2≤0,∀x∈H,∀z∈Fix⁡TDefinition 2.
LetT:H→H be an operator, then I-T is said to be demiclosed at zero, if for any {xn} in H, the following implication holds(15)xn⇀xI-Txn⟶0⇓x=TxAs we know, the nonexpansive mappings are demiclosed at zero [31].Definition 3.
LetC be a nonempty closed convex subset of a Hilbert space H, the metric (nearest point) projection PC from H to C is defined as follows: Given x∈H, PCx is the only point in C with the property (16)x-PCx=inf⁡x-y:y∈C.Lemma 4 (see [32]).
LetC be a nonempty closed convex subset of a Hilbert space H, PC is a nonexpansive mapping from H onto C and is characterized as: Given x∈H, there holds the inequality (17)x-PCx,y-PCx≤0,∀y∈C.Lemma 5 (see [32]).
LetH be a Hilbert space, then the following inequality holds,(18)x+y2≤x2+2y,x+y,∀x,y∈H.Lemma 6 (Cui and Wang [26]).
LetA:H1→H2 be a bounded linear operator and T:H2→H2 a τ-demicontractive operator with τ<1. If A-1Fix(T)≠∅, then(a)
(I-T)Ax=0⇔A∗(I-T)Ax=0, ∀x∈H1.(b)
In addition, forz∈A-1Fix(T)(19)x-ρA∗I-TAx-z2≤x-z2-1-τ24I-TAx4A∗I-TAx2wherex∈H1, Ax≠T(Ax) and (20)ρ≔1-τ2I-TAx2A∗I-TAx2.Lemma 7 (Cui and Wang [26]).
LetU:H1→H1 be a k-demicontractive operator with k<1. Denote Uλ≔(1-λ)I+λU for λ∈(0,1-k). Then for any x∈H1 and z∈Fix(U),(21)Uλx-z2≤x-z2-λ1-k-λx-Ux2.


## 3. Main Results

Proposition 8.
Based on the definitions in preliminaries, the classes ofk-demicontractive operators, directed operators, quasi-nonexpansive operators, and nonexpansive operators have close relations. We can visually use the following Venn diagram (Figure 1) to denote their relations.Figure 1
The relations ofk-demicontractive operator, directed operator, quasi-nonexpansive operator, and nonexpansive operator.Proof.
From Definition1, the following conclusion is obtained easily.(i)
The nonexpansive operator is quasi-nonexpansive operator.(ii)
The quasi-nonexpansive operator is0-demicontractive operator.(iii)
The directed operator is-1-demicontractive operator.Next, we give the novel implicit algorithm to solve the SCFP (4) for demicontractive operators. In the sequel, the assumptions are given as follows:(i)
U:H1→H1 is a k-demicontractive operator with k<1.(ii)
T:H2→H2 is a τ-demicontractive operator with τ<1.(iii)
BothI-U and I-T are demiclosed at zero.Algorithm 9.
Choose an initial guessx0∈H1 arbitrarily. Let f be a fixed contraction on Fix(U) with coefficient α (0<α<1), λ∈(0,1-τ). Assume that the nth iteration xn has been constructed. Then the (n+1)th iteration is via the following formula:(22)xn=αnfxn+1-αnUλxn-ρnA∗I-TAxn,n≥0,where A∗ is the adjoint of bounded linear operator A and the step size ρn is chosen in the following way:(23)ρn=1-τI-TAxn22A∗I-TAxn2,Axn≠TAxn0,otherwise.Theorem 10.
Assume the solution set of the SCFP (4) S≠∅. If αn∈(0,1) satisfies limn→∞⁡αn=0 and ∑n=0∞αn=∞, then the sequence {xn} generated by implicit algorithm (22) converges strongly to a point x^∈S, and x^=PSf(x^); that is, x^ satisfies the following variational inequality:(24)x^-fx^,x^-z≤0,∀z∈S.Proof.
The proof is divided into three steps.
Step 1. We show that {xn} is bounded.
Denoteyn≔xn-ρnA∗(I-T)Axn and take z∈S; it follows from (22) that(25)xn-z=αnfxn-z+1-αnUλyn-z≤αnfxn-fz+1-αnUλyn-z+αnfz-z≤ααnxn-z+1-αnUλyn-z+αnfz-z.
(i) If ρ n = 0. Then yn=xn; from (21), we get (26)Uλxn-z≤xn-z.Thus (27)xn-z≤ααnxn-z+1-αnxn-z+αnfz-z.Hence(28)xn-z≤11-αfz-z.So, {xn} is bounded, so is {f(xn)}.
(ii) If ρ n ≠ 0. It follows from (19) and (21) that we get(29)Uλyn-z2≤yn-z2-λ1-λ-kyn-Uyn2=xn-ρnA∗I-TAxn-z2-λ1-λ-kyn-Uyn2≤xn-z2-1-τ24I-TAxn4A∗I-TAxn2-λ1-λ-kyn-Uyn2.Thus,(30)Uλyn-z≤xn-z.Combining with (30) and (25), we get (28). So, {xn} is bounded, so is {f(xn)}.
Step 2. We show that there exists a subsequence {xnj}⊆{xn} such that xnj→x^ as j→∞, and x^∈S solves the variational inequality (24).
By the reflexivity of Hilbert spaceH1 and the boundedness of {xn}, there exists a weakly convergence subsequence {xnj}⊆{xn} such that xnj⇀x^, as j→∞.
First, we show thatxnj→x^, as j→∞.
Next, we denotexnj by xj.
(i) If ρ n j = 0. From (18) and (21), we have(31)xj-x^2≤1-αjUλxj-x^2+2αjfxj-x^,xj-x^≤1-αjxj-x^2-λ1-k-λxj-Uxj2+2αjfxj-x^,xj-x^≤1-αjxj-x^2+2αjfxj-x^,xj-x^.Hence(32)xj-x^2≤2fxj-x^,xj-x^≤2fxj-fx^,xj-x^+2fx^-x^,xj-x^≤2αxj-x^2+2fx^-x^,xj-x^.So(33)xj-x^2≤21-2αfx^-x^,xj-x^.
For{xj}⇀x^ as j→∞, the above inequality implies that(34)xj⟶x^asj⟶∞.
(ii) If ρ n j ≠ 0. From (18), (19) and (21), we have(35)xj-x^2≤1-αjUλyj-x^2+2αjfxj-x^,xn+1-x^≤1-αjyj-x^2-λ1-k-λyj-Uyj2+2αjfxj-x^,xn+1-x^≤1-αjxj-x^2-1-τ24I-TAxj4A∗I-TAxj2-1-αjλ1-k-λyj-Uyj2+2αjfxj-x^,xj-x^≤1-αjxj-x^2+2αjfxj-x^,xj-x^.
Then, (33) is obtained. By the similar proofs of the case of ρnj=0, we conclude that (36)xj⟶x^asj⟶∞.
Second, we show thatx^∈S.
(i) If ρ n j = 0. From (31), we get(37)λ1-k-λxj-Uxj2≤αjfxj-x^,xj-x^-αjxj-x^2≤αjfxj-x^,xj-x^.Hence(38)xj-Uxj⟶0asj⟶∞.
For the caseρnj=0, then it is clear we obtain(39)I-TAxj⟶0asj⟶∞.From (37) and the demiclosedness of I-U at zero, then(40)x^∈Fix⁡U.
SinceA is bounded linear operator, then A is weak continuity; then(41)xj⇀x^⟹Axj⇀Ax^,asj⟶∞.
From (39) and the demiclosedness of I-T at zero, then(42)Ax^∈Fix⁡T.
Hence,x^∈S by (40) and (42).
(ii) If ρ n j ≠ 0. From (35), we get(43)λ1-k-λyj-Uyj2+1-τ24I-TAxj4A∗I-TAxj2≤2αj1-αjfxj-x^,xj-x^-αjxj-x^2≤2αj1-αjfxj-x^,xj-x^.So, we have (44)0≤λ1-k-λyj-Uyj2≤2αj1-αjfxj-x^,xj-x^,1-τ24I-TAxj4A∗I-TAxj2≤2αj1-αjfxj-x^,xj-x^.Take j→∞, we have(45)yj-Uyj⟶0asj⟶∞,(46)I-TAxj2A∗I-TAxj⟶0asj⟶∞.
Moreover,(47)I-TAxj=A·I-TAxjA=A·I-TAxjI-TAxjAI-TAxj≤A·I-TAxjI-TAxjA∗I-TAxj=AI-TAxj2A∗I-TAxj.Hence, from (46)(48)I-TAxj⟶0,asj⟶∞.
Byyj≔xj-ρjA∗(I-T)Axj, we have (49)xj-yj=ρjA∗I-TAxj=1-τ2I-TAxj2A∗I-TAxj.So,(50)xj-yj⟶0,asj⟶∞.
Forxj→x^, then yj→x^ by (50).
From (45) and the demiclosedness of I-U at zero, then(51)x^∈Fix⁡U.
From (48) and the demiclosedness of I-T at zero, then(52)Ax^∈Fix⁡T.So, x^∈S by (51) and (52).
Third, we show thatx^∈S solves the variational inequality (24).
Indeed, from (22), we get(53)I-fxj=-1-αjαjxj-Uλxj-ρnA∗I-TAxj=-1-αjαjxj-Uλyj.The above equality and (30) imply that (54)I-fxj,xj-z=-1-αjαjxj-Uλyj,xj-z=-1-αjαjxj-z-Uλyj-z,xj-z≤0.Since (55)xj-z-Uλyj-z,xj-z=xj-z2-Uλyj-z,xj-z≥xj-z2-Uλyj-z·xj-z≥xj-z2-xj-z·xj-z=0,
take the limit throughj→∞ and we obtain(56)x^-fx^,x^-z≤0,∀z∈S.
Step 3. We show that xn→x^ as n→∞.
To show thatxn→x^ as n→∞, we only need to show that any subsequence of {xn} converges strongly to x^.
Assuming the above conclusion does not hold, that is to say, there exists another subsequence{xnk}⊆{xn}, which converges strongly to x~≠x^ as k→∞. Similarly, we know x~∈S solves the variational inequality(57)x~-fx~,x~-z≤0,∀z∈S.
Replacingz∈S with x~∈S in (56) and replacing z∈S with x^∈S in (57), we obtain (58)x^-fx^,x^-x~≤0,x~-fx~,x~-x^≤0.
Adding up the above variational inequality yields(59)1-αx^-x~2≤x^-x~,I-fx^-I-fx~≤0.
Thusx^=x~. This is contradicting with the assumption x^≠x~, so {xn} converges strongly to x^.
The proof is completed.


## 4. Applications

In this section, we consider some special cases as the applications of Theorem10.Based on the relations ofk-demicontractive operators, directed operators, and quasi-nonexpansive operators (Proposition 8), the following corollaries are obtained easily.Corollary 11.
LetU:H1→H1 and T:H2→H2 be quasi-nonexpansive operators and I-U and I-T be demiclosed at zero. Assume the SCFP (4) is consistent (S≠∅). If αn∈(0,1) satisfies limn→∞⁡αn=0 and ∑n=0∞αn=∞, then the sequence {xn} generated by implicit algorithm (22) converges strongly to a point x^∈S, and x^=PSf(x^); that is, x^ satisfies the following variational inequality (24).Corollary 12.
LetU:H1→H1 and T:H2→H2 be directed operators and I-U and I-T be demiclosed at zero. Assume the SCFP (4) is consistent (S≠∅). If αn∈(0,1) satisfies limn→∞⁡αn=0 and ∑n=0∞αn=∞, then the sequence {xn} generated by implicit algorithm (22) converges strongly to a point x^∈S, and x^=PSf(x^); that is, x^ satisfies the following variational inequality (24).Corollary 13.
LetU:H1→H1 be a directed operator, T:H2→H2 be a quasi-nonexpansive operator, and I-U and I-T be demiclosed at zero. Assume the SCFP (4) is consistent (S≠∅). If αn∈(0,1) satisfies limn→∞⁡αn=0 and ∑n=0∞αn=∞, then the sequence {xn} generated by implicit algorithm (22) converges strongly to a point x^∈S, and x^=PSf(x^); that is, x^ satisfies the following variational inequality (24).Corollary 14.
LetU:H1→H1 be a directed operator, T:H2→H2 be a τ-demicontractive operator, and I-U and I-T be demiclosed at zero. Assume the SCFP (4) is consistent (S≠∅). If αn∈(0,1) satisfies limn→∞⁡αn=0 and ∑n=0∞αn=∞, then the sequence {xn} generated by implicit algorithm (22) converges strongly to a point x^∈S, and x^=PSf(x^); that is, x^ satisfies the following variational inequality (24).Corollary 15.
LetU:H1→H1 be a quasi-nonexpansive operator, T:H2→H2 be a τ-demicontractive operator, and I-U and I-T be demiclosed at zero. Assume the SCFP (4) is consistent (S≠∅). If αn∈(0,1) satisfies limn→∞⁡αn=0 and ∑n=0∞αn=∞, then the sequence {xn} generated by implicit algorithm (22) converges strongly to a point x^∈S, and x^=PSf(x^); that is, x^ satisfies the following variational inequality (24).


## 5. Conclusions

In this paper, the research highlights that the strong convergence of the SCFP (4) is constructed. We construct a novel implicit algorithm for demicontractive operator to solve the split common fixed points problem SCFP, and we prove that the sequence {xn} strongly converges to a solution of the SCFP. These results further complete the theory of the SCFP, and some relevant work can be extended in the future.



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