# Weak Convergence Theorems on the Split Common Fixed Point Problem for Demicontractive Continuous Mappings

Authors: Huanhuan Cui; Luchuan Ceng; Fenghui Wang
Journal: Journal of Function Spaces
 (2018)

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## Abstract

We are concerned with the split common fixed point problem in Hilbert spaces. We propose a new method for solving this problem and establish a weak convergence theorem whenever the involved mappings are demicontractive and Lipschitz continuous. As an application, we also obtain a new method for solving the split equality problem in Hilbert spaces.


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## Body

## 1. Introduction

The split common fixed point problem (SCFP) is an inverse problem that aims to find an element in a fixed point set such that its image under a linear transformation belongs to another fixed point set. More specifically, given two Hilbert spacesH1 and H2, the SCFP consists in finding x∈H1 such thatP1x∈FU,Ax∈FT,where A:H1→H2 is a bounded linear mapping and F(U) and F(T) are, respectively, the fixed point sets of nonlinear mappings U:H1→H1 and T:H2→H2. Particularly, if U and T are both metric projections, the SCFP is reduced to the well-known split feasibility problem (SFP). Actually, the SFP can be formulated as the problem of finding a point x∈H1 such that(1)x∈C,Ax∈Q,where C⊆H1 and Q⊆H2 are nonempty closed convex sets, and mapping A is as above. These two problems have been extensively investigated since they play an important role in various areas including signal processing and image reconstruction [1–5].We assume throughout the paper that problemP1 is consistent, which means that its solution set, denoted by S, is nonempty. Censor and Segal [6] studied the SCFP when U and T are firmly quasi-nonexpansive mappings, and proposed the following method:(2)xn+1=Uxn-τnA∗I-TAxn,where τn is a properly chosen stepsize. It is shown that if τn is chosen in (0,2/A2), then the sequence generated by method (2) converges weakly to a solution of problem P1. Subsequently, this result was extended to quasi-nonexpansive operators [7], demicontractive operators [8, 9], two groups of finitely many firmly quasi-nonexpansive mappings [10, 11], and the more general common null point problem [12]. Also, some variants of method (2) have been considered in [13–16].Since the choice of the stepsize is related toA, thus, to implement method (3), one has to compute (or at least estimate) the norm A, which is generally not easy in practice. A way to avoid this is to adopt variable stepsize which ultimately has no relation with A [8, 10, 17–19]. Wang [18] recently proposed a new method for solving the SCFP:(3)xn+1=xn-ρnI-Uxn+A∗I-TAxn,where {ρn}⊂(0,∞) is chosen such that(4)ρn=I-Uxn2+I-TAxn2I-Uxn+A∗I-TAxn2.Wang proved that if mappings U and T are firmly quasi-nonexpansive, then the sequence {xn} generated by (3)-(4) converges weakly to a solution of problem P1. Wang and Xu [19] recently proposed another choice of the stepsize:(5)ρn=τnI-Uxn+A∗I-TAxn,where {τn}⊂(0,∞) is chosen such that(6)∑n=0∞τn=∞,∑n=0∞τn2<∞.They proved that if mappings U and T are nonexpansive, then the sequence {xn} generated by (3) and (5)-(6) converges weakly to a solution of problem P1. It is clear that these choices of the stepsize do not rely on the norm A.In this paper, we first extend the above result for method (3) from nonexpansive mappings to demicontractive continuous mappings. By using properties of product spaces, we change the split equality problem into a special split common fixed point problem. As a result, based on our extension, we obtain a new method for solving the split equality problem in Hilbert spaces.


## 2. Preliminaries

Throughout the paper,Hi,i=1,2,3, are Hilbert spaces, I is the identity operator, “→" stands for strong convergence, and “⇀" stands for weak convergence. For a mapping W:H1→H1, F(W) is the set of the fixed points of W, W-1(0)={x∈H1:Wx=0}, and Wc≔I-W.Definition 1.
LetW:H1→H1 be a nonlinear mapping. (i)
W is calledfirmly nonexpansive, if (7)Wx-Wy2≤x-y2-Wcx-Wcy2,∀x,y∈H1.(ii)
W is callednonexpansive, if (8)Wx-Wy≤x-y,∀x,y∈H1.(iii)
W is calledstrictly pseudo-contractive, if there exists k<1 such that (9)Wx-Wy2≤x-y2+kWcx-Wcy2,∀x,y∈H1.(iv)
W is called L-Lipschitz continuous, if there exists L>0 such that (10)Wx-Wy≤Lx-y,∀x,y∈H1.Definition 2.
LetW:H1→H1 be a nonlinear mapping with F(W)≠∅. (i)
W is calledfirmly quasi-nonexpansive, if (11)Wx-z2≤x-z2-I-Wx2,∀x,z∈H1×FW.(ii)
W is calledquasi-nonexpansive, if (12)Wx-z≤x-z,∀x,z∈H1×FW.(iii)
W is calledk-demicontractive, if there exists k<1 such that(13)Wx-z2≤x-z2+kI-Wx2,∀x,z∈H1×FW.Note that the class of strictly pseudo-contractive mappings properly includes the class of nonexpansive mappings, while the class of nonexpansive mappings properly includes the class of firmly nonexpansive mappings. And the class of demicontractive mappings properly includes the class of quasi-nonexpansive mappings, while the class of quasi-nonexpansive mappings properly includes the class of firmly quasi-nonexpansive mappings.Definition 3 (demiclosedness property).
Let{xn} be a sequence in H1 and W:H1→H1 be a mapping. Then W is said to have demiclosedness property if the following implication holds: (14)I-Wxn⟶0xn⇀x⟹x∈FW.
It is known that strictly pseudo-contractive mappings possess the demiclosedness property [20]. In particular, both nonexpansive and firmly nonexpansive mappings possess such a property.Lemma 4 (see [20]).
LetW:H1→H1 be a k-strictly pseudo-contractive mapping. Then W is demicontractive and Lipschitz continuous and moreover has the demiclosedness property.The metric projectionPC from H1 onto a nonempty closed convex subset C⊆H1 is defined by (15)PCx≔argminy∈Cx-y,which is characterized by(16)x-PCx,z-PCx≤0,∀z∈C.It is well known that the metric projection is firmly nonexpansive.Definition 5.
LetC be a nonempty closed convex subset in H1. (i)
A sequence{xn} in H1 is Fejér-monotone with respect to C if (17)xn+1-z≤xn-z,∀n≥0,∀z∈C.(ii)
A sequence{xn} in H1 is quasi Fejér-monotone with respect to C if (18)xn+1-z2≤xn-z2+εn,∀n≥0,∀z∈C,where{εn}⊆(0,+∞) satisfies ∑n=1∞‍εn<∞.Lemma 6 (see [21]).
A quasi Fejér-monotone sequence{xn} (with respect to C) is weakly convergent to z∈C if and only if every weak cluster point of {xn} belongs to C.Lemma 7 (see [22]).
Let{εn} and {sn} be positive real sequences such that ∑n=0∞‍εn<∞. If sn+1≤(1+εn)sn, or sn+1≤sn+εn, then the limit of the sequence {sn} exists.


## 3. The Case for Demicontractive Continuous Mappings

In this section, we consider the SCFPP1 for demicontractive continuous mappings. Under this situation, we shall prove that the sequence {xn} generated by (3) and (5)-(6) still converges weakly to a solution of problem P1.Lemma 8.
Letk1,k2∈(-∞,1), L1,L2∈(0,+∞), and W=(I-U)+A∗(I-T)A, where U, T, and A are mappings defined in P1. Assume that U is k1-demicontractive and L1-Lipschitz continuous, T is k2-demicontractive and L2-Lipschitz continuous, and I-U and I-T are demiclosed at the origin. For any (x,z)∈H1×S, we have the following: (i)
S=W-1(0).(ii)
2Wx,x-z≥min1-k1/2,1-k2/2A2Wx2.(iii)
W is L-Lipschitz continuous with L=max⁡((L1+1),L2+1A2).(iv)
IfWxn→0 and xn⇀x as n→∞, then x∈S.Proof.
(i) It is readily seen thatS⊆W-1(0). To see the converse, let x∈W-1(0) and fix any z∈S. Since U and T are demicontractive, we have (19)1-k12I-Ux2≤I-Ux,x-z,1-k22I-TAx2≤I-TAx,Ax-Az.Adding up these two inequalities, we have(20)Wx,x-z≥1-k12I-Ux2+1-k22I-TAx2,which yields (I-U)x=(I-T)Ax=0, that is, x∈S. This implies S⊇W-1(0).
(ii) Let(x,z)∈H1×S. It follows from (20) that (21)Wx,x-z≥1-k12I-Ux2+1-k22I-TAx2=1-k12I-Ux2+1-k22A2A∗2I-TAx2≥1-k12I-Ux2+1-k22A2A∗I-TAx2≥min⁡1-k12,1-k22A2I-Ux2+A∗I-TAx2≥min⁡1-k14,1-k24A2I-U+A∗I-TAx2,which yields the desired inequality.
(iii) Letx,y∈H1. We have (22)Wx-Wy≤I-Ux-y+A∗I-TAx-y≤L1+1x-y+AI-TAx-y≤L1+1x-y+L2+1A2x-y≤max⁡L1+1,L2+1A2x-y.
(iv) We note that{xn} is bounded by its weak convergence. By inequality (20), we have(23)1-k12I-Uxn2+1-k22I-TAxn2≤Wxn,xn-z≤Wxnxn-z⟶0,which implies that (24)limn→∞I-Uxn=limn→∞I-TAxn=0.Since xn⇀x, this by the demiclosedness property implies x∈F(U). On the other hand, for any y∈H2, we have (25)limn→∞Axn,y=limn→∞xn,A∗y=x,A∗y=Ax,y. Hence Axn⇀Ax, which yields Ax∈F(T). Altogether, x∈S.Theorem 9.
Letk1,k2∈(-∞,1),L1,L2∈(0,+∞). Assume that U is k1-demicontractive and L1-Lipschitz continuous, T is k2-demicontractive and L2-Lipschitz continuous, and I-U and I-T are demiclosed at the origin. If condition (6) holds, then the sequence {xn}, generated by (3) and (5), converges weakly to a solution of problem P1.Proof.
Letz∈S, τ=min⁡((1-k1)/2,(1-k2)/2A2), and L=max⁡((L1+1),L2+1A2). It then follows from Lemma 8 that (26)xn+1-z2=xn-z2-2ρnWxn,xn-z+ρn2Wxn2≤xn-z2-τρnWxn2+ρn2Wxn2.By (5), we have(27)xn+1-z2≤xn-z2-ττnWxn+τn2;in particular, (28)xn+1-z2≤xn-z2+τn2.By our hypothesis (6), this implies that {xn} is quasi Fejér-monotone with respect to S.
Next, we deduce from (27) and the boundedness of {xn} (guaranteed by the quasi-Fejér-monotonicity) that (29)τnWxn≤1τxn-z2-xn+1-z2+τn2. Thus, we have(30)∑n=0∞τnWxn<∞.In view of (6), this implies(31)lim_n→∞Wxn=0.On the other hand, since (32)Wxn+1-Wxn≤Lxn-xn+1=LρnWxn,then we have (33)Wxn+12=Wxn2+Wxn+1-Wxn2+2Wxn,Wxn+1-Wxn≤Wxn2+Wxn+1-Wxn2+2WxnWxn+1-Wxn≤Wxn2+L2ρn2Wxn2+2LρnWxn2=Wxn2+L2τn2+2LτnWxn.In light of (30) and (6), we have ∑n‍(L2τn2+2LτnWxn)<∞. By Lemma 7, limnWxn exists, and further we have limnWxn=0 by (31). Hence, by Lemma 8, we conclude that every weak cluster point of {xn} belongs to S.
Finally, we deduce from Lemma6 that {xn} converges weakly to a solution of problem P1.Corollary 10.
Assume thatU and T are both strictly pseudo-contractive mappings. If condition (6) holds, then the sequence {xn}, generated by (3) and (5), converges weakly to a solution of problem P1.Proof.
It follows from Lemma4 and Theorem 9.Remark 11.
It is readily seen that the above result holds true for nonexpansive and firmly nonexpansive mappings. As a result, it extends the results in [19] from nonexpansive mappings to demicontractive continuous mappings.


## 4. New Methods for the Split Equality Problem

The split equality problem (SEP) is an inverse problem that requests findingP2x,y∈FU1×FU2s.t.A1x=A2y,where A1:H1→H3 and A2:H2→H3 are two bounded linear mappings, while U1:H1→H1 and U2:H2→H2 are two nonlinear mappings. The SEP was first introduced by Moudafi and Al-Shemas [23], and they proposed the following iterative method:(34)xn+1=U1xn-τnA∗Axn-Byn,yn+1=U2yn+τnB∗Axn-Byn. Under some certain conditions, they proved the weak convergence of the iterative sequence generated by method (34).Our method is actually motivated by (3), since problem P2 can be regarded as a special SCFP: find x=(x1,x2)∈H1×H2 such that(35)x∈FU,Ax∈FT,where Ux=(U1x1,U2x2), Ax=A1x1-A2x2, and T is the projection onto the set {0}. Motivated by (3), we now propose a new method for solving problem P2. For an arbitrary initial guess (x0,y0), define (xn,yn) recursively by(36)xn+1=xn-ρnI-U1xn+A1∗A1xn-A2yn,yn+1=yn-ρnI-U2yn-A2∗A1xn-A2yn,where {ρn} is chosen as(37)ρn=τnI-U1xn+A1∗A1xn-A2yn2+I-U2yn-A2∗A1xn-A2yn21/2.In what follows, we will show the SEP amounts to problem (35). Now consider the product space H1×H2, in which the inner product and the norm are, respectively, defined by (38)x,y=x1,y1+x2,y2,x=x12+x221/2, where x=(x1,x2), y=(y1,y2) with x1,y1∈H1, x2,y2∈H2.Lemma 12.
Letx1∈H1, x2∈H2, and A1:H1→H3 and A2:H2→H3 be as in problem P2. Define a mapping A:H1×H2→H3 by (39)Ax=Ax1,x2=A1x1-A2x2. Then we have the following: (i)
A is a bounded linear mapping.(ii)
A∗Ax=(A1∗(A1x1-A2x2),-A2∗(A1x1-A2x2)).Proof.
(i) Letα,β∈R. Since A1 and A2 are both linear, we have (40)Aαx+βy=Aαx1,x2+βy1,y2=Aαx1+βy1,αx2+βy2=A1αx1+βy1-A2αx2+βy2=αA1x1-A2x2+βA1y1-A2y2=αAx1,x2+βAy1,y2=αAx+βAy;on the other hand, we have (41)Ax=A1x1-A2x2≤A1x1+A2x2≤max⁡A1,A2x1+x2≤2max⁡A1,A2x12+x22=2max⁡A1,A2x,which implies A≤2max⁡A1,A2. Thus A is linear and bounded.
(ii) Forw∈H3, we have (42)Ax,w=Ax1,x2,w=A1x1-A2x2,w=x1,A1∗w+x2,-A2∗w=x1,x2,A1∗w,-A2∗w=x,A1∗w,-A2∗w.This gives A∗w=(A1∗w,-A2∗w), which implies that (43)A∗Ax=A∗A1x1-A2x2=A1∗A1x1-A2x2,-A2∗A1x1-A2x2. Hence the lemma is proved.Lemma 13.
Assume thatU1:H1→H1 is k1-demicontractive and L1-Lipschitz continuous, U2:H2→H2 is k2-demicontractive and L2-Lipschitz continuous, and I-U1 and I-U2 are demiclosed at the origin. Define a mapping U:H1×H2→H1×H2 by (44)Ux=U1x1,U2x2, where x=(x1,x2) is in H1×H2 with x1∈H1,x2∈H2. Then (i)
F(U)=F(U1)×F(U2);(ii)
U is demicontractive and Lipschitz continuous;(iii)
I-U is demiclosed at the origin.Proof.
It is easy to check (i). For (ii), fixz=(z1,z2)∈F(U). It follows that (45)Ux-z2=U1x1-z12+U2x2-z22=x1-z12+x2-z22+k1U1x1-x12+k2U2x2-x22≤x1-z12+x2-z22+max⁡k1,k2U1x1-x12+U2x2-x22=x1,x2-z1,z22+maxk1,k2I-Ux1,x22=x-z2+max⁡k1,k2I-Ux2,which implies that U is demicontractive. On the other hand, we have (46)Ux-Uy=U1x1-U1y1,U2x2-U2y2=U1x1-U1y12+U2x2-U2y221/2≤L12x1-y12+L22x2-y221/2≤max⁡L1,L2x1-y12+x2-y221/2=max⁡L1,L2x1,y1-x2,y1=max⁡L1,L2x-y,where y=(y1,y2) is in H1×H2 with y1∈H1, y2∈H2. This implies that U is Lipschitz continuous.
To show (iii), let{(xn,yn)} be a sequence such that it converges weakly to {(x,y)} and (I-U)(xn,yn) converges strongly to 0. This implies that xn⇀x and (I-U1)xn→0, which, by the demiclosedness of I-U1, gives x∈F(U1). Similarly, we have y∈F(U2), so that (x,y)∈F(U1)×F(U2)=F(U). So the lemma is proved.Theorem 14.
Assume thatU1:H1→H1 and U2:H2→H2 are two demicontractive and Lipschitz continuous mappings such that I-U1 and I-U2 are demiclosed at the origin. If condition (6) is fulfilled, then the sequence {(xn,yn)} generated by (36), (37), and (48) converges weakly to a solution of problem P2.Proof.
Letzn=(xn,yn), U, A be defined as in the previous lemmas, and T be the projection onto the set {0}. Then method (36) can be rewritten as (47)zn+1=zn-ρnI-Uzn+A∗I-TAzn,where(48)ρn=τnI-Uzn+A∗I-TAzn.Note that T is firmly nonexpansive. By Lemma 13, all assumptions in Theorem 9 are fulfilled. Hence, by applying Theorem 9, we conclude that {zn} converges weakly to some z=(x,y) such that z∈F(U)=F(U1)×F(U2) and Az∈F(T)={0}, which clearly yields x∈F(U1), y∈F(U2), and A1x=A2y. Hence the theorem is proved.


## 5. Conclusions

We studied the split common fixed point problem in Hilbert spaces. We proposed a new method for solving such a problem and established a weak convergence theorem whenever the involved mappings are demicontractive and continuous. We also obtained a new method for solving the split equality problem in Hilbert spaces.



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