# Internal Perturbation Projection Algorithm for the Extended Split Equality Problem and the Extended Split Equality Fixed Point Problem

Authors: Meixia Li; Xueling Zhou; Wenchao Wang
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this article, we study the extended split equality problem and extended split equality fixed point problem, which are extensions of the convex feasibility problem. For solving the extended split equality problem, we present two self-adaptive stepsize algorithms with internal perturbation projection and obtain the weak and the strong convergence of the algorithms, respectively. Furthermore, based on the operators being quasinonexpansive, we offer an iterative algorithm to solve the extended split equality fixed point problem. We introduce a way of selecting the stepsize which does not need any prior information about operator norms in the three algorithms. We apply our iterative algorithms to some convex and nonlinear problems. Finally, several numerical results are shown to confirm the feasibility and efficiency of the proposed algorithms.


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## Body

## 1. Introduction

LetH1,H2, and H3 be three real Hilbert spaces and C⊂H1 and Q⊂H2 be two nonempty, closed, and convex sets. The split feasibility problem (SFP) is formulated
(1)tofindx∈CsuchthatAx∈Q,where A:H1⟶H2 is a bounded linear operator. The SFP was first introduced by Censor and Elfving [1], which was used in modeling various inverse problems arising from phase retrievals and medical image reconstruction and further studied by many researchers. See, for instance, [2–10].Moudafi [11, 12] introduced the following split equality feasibility problem (SEFP), which is
(2)tofindx∈C,y∈QsuchthatAx=By,where A:H1⟶H3 and B:H2⟶H3 are two bounded linear operators. Obviously, if B=I and H3=H2, then (2) reduces to (1). The split equality feasibility problem (2) allows asymmetric and partial relations between the variables x and y. In order to solve SEFP, many researchers proposed their suggestions, such as [13–17] and references therein. Moudafi [11] introduced the following iterative method:
(3)xk+1=PCxk−λkA∗Axk−Byk,yk+1=PQyk+λkB∗Axk+1−Byk.Under some suitable conditions, he proved that the sequencexn,yn weakly converges to the solution of (2) in Hilbert spaces. In addition, Yu and Wang [18] proposed the following iterative algorithm:
(4)xk+1=xk−λxk−PCxk+A∗Axk−Byk,yk+1=yk−λyk−PCyk−B∗Axk+1−Byk,where 0<λ<1+c−1 with c=maxA2,B2. They studied the weak convergence of scheme (4).Recently, Che et al. [19] proposed the following extended split equality problem (ESEP) which is an extension of the convex feasibility problem. Let H be a real Hilbert space. For i=1,2,⋯,n, assume Ci are nonempty closed convex subsets of real Hilbert spaces Hi, respectively. The extended split equality problem is
(5)tofindx1∈C1,x2∈C2,⋯,xn∈CnsuchthatA1x1=A2x2=⋯=Anxn,where Ai:Hi⟶H are linear operators. They presented the following simultaneous iterative algorithm:
(6)ωk=∑i=1nAixikn,x1k+1=PC1x1k−λkA1∗A1x1k−ωk,x2k+1=PC2x2k−λkA2∗A2x2k−ωk,⋯xnk+1=PCnxnk−λkAn∗Anxnk−ωk.Under some suitable conditions, they obtained the weak convergence of (6).In order to avoid using the projection, Moudafi [11] introduced and studied the following problem. Let S:H1⟶H1 and T:H2⟶H2 be nonlinear operators such that FixS≠∅ and FixT≠∅, where FixS and FixT denote the sets of fixed points of S and T, respectively. If C=FixS and Q=FixT, then SEFP (2) reduces
(7)tofindx∈FixS,y∈FixTsuchthatAx=By,which is called the split equality fixed point problem (SEFPP). Many scholars have studied this issue, such as [20–22].To solve problem (7), Che and Li [23] proposed the following iterative algorithm:
(8)uk=xk−λkA∗Axk−Byk,xk+1=αkxk+1−αkSuk,vk=yk+λkB∗Axk−Byk,yk+1=αkyk+1−αkTvk.They established the weak convergence of scheme (8) under the conditions that the operators S and T are quasinonexpansive mappings.Similarly, Che et al. [19] proposed the following extended split equality fixed point problem (ESEFPP), which is
(9)tofindx1∈FixG1,x2∈FixG2,⋯,xn∈FixGnsuchthatA1x1=A2x2=⋯=Anxn,and presented the following simultaneous iterative algorithm:
(10)ωk=∑i=1nAixikn,u1k=x1k−λkA1∗A1x1k−ωk,x1k+1=αku1k+1−αkG1u1k,u2k=x2k−λkA2∗A2x2k−ωk,x2k+1=αku2k+1−αkG2u2k,⋯unk=xnk−λkAn∗Anxnk−ωk,xnk+1=αkunk+1−αkGnunk,where Gi are the G-mapping generated by Tii=1N which is a finite family of ki-strictly pseudononspreading. They obtained the weak convergence of (10).Motivated by the works mentioned above, we continue to study the ESEP (5) and ESEFPP (9) with internal perturbation projection and do not need any prior information about the operator norms. The paper is organized as follows. In Section 2, we introduce some preliminaries to be employed in the subsequent analysis. In Section 3, we present two simultaneous iterative algorithms to solve ESEP (5) and establish the weak and the strong convergence of the proposed algorithms, respectively. We propose a simultaneous iterative algorithm to solve ESEFPP (9) and obtain the weak convergence of the proposed algorithm in Section 4. In Section 5, we apply our iterative algorithms to some convex and nonlinear problems. In the concluding section, several numerical results are shown to confirm the effectiveness of our algorithms.


## 2. Preliminaries

In this paper, we use⟶ and ⇀ to denote the strong convergence and the weak convergence, respectively. We use ωωxik=x:∃xikj⇀x to stand for the weak ω-limit set of xik. For any x∈H, there exists a unique nearest point in C, denoted by PCx, such that
(11)x−PCx≤x−y,∀y∈C.It is well known thatPC is nonexpansive and firmly nonexpansive. And PC has the following well-known properties.Lemma 1.
LetC⊂H be a nonempty, closed, and convex set. The following conclusions hold:
(1)
x−PCx,z−PCx≤0,∀x∈H,∀z∈C(2)
PCx−PCy2≤PCx−PCy,x−y,∀x,y∈H(3)
PCx−x,y−x≥PCx−x2, ∀x∈H,∀y∈CDefinition 2.
A mappingT:C⟶C is said to be
(1)
nonexpansive ifTx−Ty≤x−y,∀x,y∈C(2)
quasinonexpansive ifTx−x∗≤x−x∗,∀x∈C,x∗∈FixT(3)
firmly nonexpansive ifTx−Ty2≤x−y2−I−Tx−I−Ty2,∀x,y∈C(4)
firmly quasinonexpansive ifTx−x∗2≤x−x∗2−I−Tx2,∀x∈C,x∗∈FixT(5)
k-strictly pseudononspreading if Tx−Ty2≤x−y2+kx−Tx−y−Ty2+2x−Tx,y−Ty,∃k∈0,1,∀x,y∈CLemma 3 [24].
In the real Hilbert spaceH, for ∀x,y∈H, the following relations hold:
(1)
x+y2≤x2+2y,x+y(2)
x+y2=x2+y2+2x,y(3)
αx+1−αy2=αx2+1−αy2−α1−αx−y2,α∈0,1(4)
∑i=1mαixi2=∑i=1mαixi2−∑i≠jαiαjxi−xj2, for ∑i=1mαi=1,αi∈0,1,i=1,2,⋯,mDefinition 4 [25].
A mappingT:C⟶C is said to be demiclosed at 0 if, for any sequence xn⊂C which converges weakly to x and with xn−Txn⟶0,Tx=x.Lemma 5 [26].
LetΓk be a sequence of real numbers that does not decrease at infinity in the sense that there exists a subsequence Γkj of Γk which satisfies Γkj≤Γkj+1 for all j∈N. Define the integer sequence τk, for k>k0 (such that Γk0≤Γk0+1) is as follows:
(12)τk=maxl∈N∣k0≤l<k,Γl≤Γl+1.
Then, there hold the following properties:(1)
τk0≤τk0+1≤⋯ and τk⟶∞(2)
Γτk≤Γτk+1 and Γk≤Γτk+1,∀k≥k0Lemma 6 [27].
Assume thatak is a sequence of nonnegative real numbers such that
(13)ak+1≤1−αkak+αkδk,∀k≥0,where αk is a sequence in 0,1 and δk is a sequence in R such that
(1)
∑k=1∞αk=∞ and limk→∞αk=0(2)
limsupk→∞δk≤0 or ∑k=1∞αk∣δk∣<∞
Then,limk→∞ak=0.Definition 7 [19].
LetC be a nonempty closed and convex subset of real Banach space. Let Tii=1N be a finite family of mappings of C into itself. For i=1,2,⋯,N, let πi=αi,βi,γi,δi, where αi,βi,γi,δi∈0,1 and αi+βi+γi+δi=1. We define a mapping G:C⟶C as follows:
(14)U0=I,U1=α1T12U0+β1T1U0+γ1U0+δ1I,U2=α2T22U1+β2T2U1+γ2U1+δ2I,U3=α3T32U2+β3T3U2+γ3U2+δ3I,⋯UN−1=αN−1TN−12UN−2+βN−1TN−1UN−2+γN−1UN−2+δN−1I,G=UN=αNTN2UN−1+βNTNUN−1+γNUN−1+δNI.
Such a mappingG is called the G-mapping generated by T1,T2,⋯,TN and π1,π2,⋯,πN.Lemma 8 [19].
LetC be a nonempty closed and convex subset of real Banach space. Let Tii=1N be a finite family of ρi-strictly pseudononspreading mappings of C into itself with ⋂i=1NFixTi≠∅ and ρ=maxi=1,2,⋯,Nρi. For i=1,2,⋯,N, let πi=αi,βi,γi,δi, where αi,βi,γi,δi∈0,1 and αi+βi+γi+δi=1. Assume that ρ≤βi for i=1,2,⋯,N, and αi+βiρ≤βiγi for i=2,3,⋯,N, and α1+β1ρ≤β1γ1+δ1. If G is the G-mapping generated by T1,T2,⋯,TN and π1,π2,⋯,πN, then FixG=⋂i=1NFixTi and G is a quasinonexpansive mapping.


## 3. Iterative Algorithms for ESEP

In this section, we introduce two simultaneous iterative algorithms with internal perturbation projection to solve ESEP (5) and define the solution set of ESEP (5) as Ω1.Algorithm 9.
Initialization: take x11,x21,⋯,xn1∈H1×H2×⋯×Hn arbitrary.
Iteration step: for a given current iterate x1k,x2k,⋯,xnk∈H1×H2×⋯×Hn, we calculate the next iterate x1k+1,x2k+1,⋯,xnk+1 by
(15)ωk=∑i=1nAixikn,x1k+1=x1k−λkx1k−PC1x1k+A1∗A1x1k−ωk,x2k+1=x2k−λkx2k−PC2x2k+A2∗A2x2k−ωk,⋯xnk+1=xnk−λkxnk−PCnxnk+An∗Anxnk−ωk,where the stepsize λk is chosen in such a way that if k∈Ω, then
(16)λk∈ε,ρk2∑i=1nxik−PCixik2+∑i=1nAixik−ωk2∑i=1nxik−PCixik+Ai∗Aixik−ωk2,where the index set Ω=k:Aixik−ωk≠0,∃i∈1,2,⋯,n, small enough ε>0 and 0<δ1<ρk<δ2<1, set k≔k+1 and go to (15). Otherwise, Aixik−ωk=0, for i=1,2,⋯,n, the iteration stops.Remark 10.
Note that in (16), the choice of the stepsize λk is independent of the norm Ai, for i=1,2,⋯,n. Furthermore, we will show from Lemma 11 that λk is well defined.Lemma 11.
Assume the solution setΩ1 of ESEP (5) is nonempty, then λk defined by (16) is well defined.Proof.
Letz1,z2,⋯,zn∈Ω1, then A1z1=A2z2=⋯=Anzn. Noting that for i=1,2,⋯,n, we have
(17)xik−PCixik+Ai∗Aixik−ωk,xik−zi=xik−PCixik,xik−zi+Aixik−ωk,Aixik−Aizi=xik−PCixik,xik−zi+Aixik−ωk,Aixik−ωk+Aixik−ωk,ωk−Aizi.
As a result,(18)x1k−PC1x1k+A1∗A1x1k−ωk,x1k−z1=x1k−PC1x1k,x1k−z1+A1x1k−ωk,A1x1k−ωk+A1x1k−ωk,ωk−A1z1,x2k−PC2x2k+A2∗A2x2k−ωk,x2k−z2=x2k−PC2x2k,x2k−z2+A2x2k−ωk,A2x2k−ωk+A2x2k−ωk,ωk−A2z2,xnk−PCnxnk+An∗Anxnk−ωk,xnk−zn=xnk−PCnxnk,xnk−zn+Anxnk−ωk,Anxnk−ωk+Anxnk−ωk,ωk−Anzn.
Summing the above equalities and applying Lemma1 (3) as well as the condition A1z1=A2z2=⋯=Anzn, one has
(19)∑i=1nxik−PCixik2+∑i=1nAixik−ωk2≤∑i=1nxik−PCixik+Ai∗Aixik−ωk,xik−zi≤∑i=1nxik−PCixik+Ai∗Aixik−ωkxik−zi.
Consequently, fork∈Ω, we have ∑i=1nAixik−ωk2≠0, then ∑i=1nxik−PCixik+Ai∗Aixik−ωk≠0, which leads that λk is well defined.Theorem 12.
LetH be a real Hilbert space. For i=1,2,⋯,n, assume that Ci are nonempty closed convex subsets of real Hilbert spaces Hi, and Ai:Hi⟶H are bounded linear operators with their adjoint operators Ai∗. Then, the sequence x1k,x2k,⋯,xnk generated by Algorithm 9 converges weakly to a solution of ESEP (5). Furthermore, for i=1,2,⋯,n,xik−PCixik⟶0,Aixik−ωk⟶0 and xik+1−xik⟶0 as k⟶∞.Proof.
Letz1,z2,⋯,zn∈Ω1 and
(20)z=A1z1=A2z2=⋯=Anzn.
Fori=1,2,⋯,n, from (15) and Lemma 3, we have
(21)xik+1−zi2=xik−λkxik−PCixik+Ai∗Aixik−ωk−zi2=xik−zi2+λk2xik−PCixik+Ai∗Aixik−ωk2−2λkxik−zi,xik−PCixik+Ai∗Aixik−ωk=xik−zi2+λk2xik−PCixik+Ai∗Aixik−ωk2−2λkxik−zi,xik−PCixik−2λkxik−zi,Ai∗Aixik−ωk.
According to Lemma1 and
(22)−2xik−zi,Ai∗Aixik−ωk=−2Aixik−Aizi,Aixik−ωk=−Aixik−Aizi2−Aixik−ωk2+Aizi−ωk2,
(21) can be written as
(23)xik+1−zi2≤xik−zi2+λk2xik−PCixik+Ai∗Aixik−ωk2−2λkxik−PCixik2−λkAixik−Aizi2−λkAixik−ωk2+λkAizi−ωk2.
Note that(24)Aizi−ωk2=Aizi−∑i=1nAixikn2=∑i=1nAizi−Aixikn2≤∑i=1nAizi−Aixik2n.
Summing (23) for i from 1 to n, we can obtain
(25)∑i=1nxik+1−zi2≤∑i=1nxik−zi2+λk2∑i=1nxik−PCixik+Ai∗Aixik−ωk2−2λk∑i=1nxik−PCixik2−λk∑i=1nAixik−ωk2≤∑i=1nxik−zi2+λkρk2∑i=1nxik−PCixik2+∑i=1nAixik−ωk2−λk2∑i=1nxik−PCixik2+∑i=1nAixik−ωk2=∑i=1nxik−zi2−λk1−ρk2∑i=1nxik−PCixik2+∑i=1nAixik−ωk2.
LetΓkz1,z2,⋯,zn=∑i=1nxik−zi2, we have
(26)Γk+1z1,z2,⋯,zn≤Γkz1,z2,⋯,zn.
Therefore, the sequenceΓkz1,z2,⋯,zn is nonincreasing and lower bounded by 0. Hence, Γkz1,z2,⋯,zn converges to some finite limit, suppose as lz1,z2,⋯,zn, and sequence xik is bounded. Letting k⟶∞ and taking the limit in the two sides of (25), for 0<δ1<ρk<δ2<1, we can obtain that
(27)λk1−ρk2∑i=1nxik−PCixik2+∑i=1nAixik−ωk2⟶0,which implies
(28)2λk∑i=1nxik−PCixik2⟶0,λk∑i=1nAixik−ωk2⟶0.
Sinceλk>ε, then
(29)xik−PCixik⟶0,Aixik−ωk⟶0.
From the definition ofλk, we have
(30)λk2∑i=1nxik−PCixik+Ai∗Aixik−ωk2≤λkρk2∑i=1nxik−PCixik2+∑i=1nAixik−ωk2⟶0,then,
(31)λk2∑i=1nxik−PCixik+Ai∗Aixik−ωk2⟶0,which implies
(32)λkxik−PCixik+Ai∗Aixik−ωk⟶0.
From (32), for k⟶∞, we can get
(33)xik+1−xik=λkxik−PCixik+Ai∗Aixik−ωk=λkxik−PCixik+Ai∗Aixik−ωk⟶0.
Consequently, the sequencexik is asymptotically regular.
Letx¯1,x¯2,⋯,x¯n∈ωωx1k,x2k,⋯,xnk, then for i=1,2,⋯,n, there exists a subsequence of xik which converges weakly to x¯i. From (29) and the lower semicontinuity of the squared norm, we have
(34)0≤x¯i−PCix¯i≤liminfj→∞xikj−PCixikj=limk→∞xik−PCixik=0,which implies x¯i∈Ci.
Furthermore, it follows from (29) and the lower semicontinuity of the squared norm
(35)Aix¯i−ω¯2≤liminfk→∞Aixik−ωk2=0,where ω¯=∑i=1nAix¯i/n, which means that Aix¯i−ω¯=0. That is,
(36)A2x¯2+A3x¯3+⋯+Anx¯n=n−1A1x¯1,A1x¯1+A3x¯3+⋯+Anx¯n=n−1A2x¯2,⋯A1x¯1+A2x¯2+⋯+An−1x¯n−1=n−1Anx¯n.
Obviously,(37)A1x¯1=A2x¯2=⋯=Anx¯n.
Hence,x¯1,x¯2,⋯,x¯n∈Ω1.
In the sequel, we will show the uniqueness of the weak cluster points ofxik. Assume that xi∗ is another weak cluster point of xik. It follows from the definition of Γk that
(38)Γkx¯1,x¯2,⋯,x¯n=∑i=1nxik−x¯i2=∑i=1nxik−xi∗2+∑i=1nxi∗−x¯i2+2∑i=1nxik−xi∗,xi∗−x¯i=Γkx1∗,x2∗,⋯,xn∗+∑i=1nxi∗−x¯i2+2∑i=1nxik−xi∗,xi∗−x¯i.
Without loss of generality, we can supposexik⇀xi∗, and then
(39)lx¯1,x¯2,⋯x¯n=lx1∗,x2∗,⋯,xn∗+∑i=1nxi∗−x¯i2.
Reversing the role ofx¯1,x¯2,⋯x¯n and x1∗,x2∗,⋯,xn∗, we can obtain
(40)lx1∗,x2∗,⋯,xn∗=lx¯1,x¯2,⋯x¯n+∑i=1nxi∗−x¯i2.
Adding (39) and (40), we deduce
(41)∑i=1nxi∗−x¯i2=0,which yields that xi∗=x¯i. Hence, the sequence xik weakly converges to a solution of ESEP (5), which completes the proof.In the following, we introduce another simultaneous iterative algorithm with internal perturbation projection to solve ESEP (5) and prove the strong convergence of the algorithm.Algorithm 13.
Initialization: let x11,x21,⋯,xn1∈H1×H2×⋯×Hn arbitrary.
Iteration step: for a given current iterate x1k,x2k,⋯,xnk∈H1×H2×⋯×Hn, we calculate the next iterate x1k+1,x2k+1,⋯,xnk+1 by
(42)ωk=∑i=1nAixikn,u1k=x1k−λkx1k−PC1x1k+A1∗A1x1k−ωk,x1k+1=αkv1+1−αku1k,u2k=x2k−λkx2k−PC2x2k+A2∗A2x2k−ωk,x2k+1=αkv2+1−αku2k,⋯unk=xnk−λkxnk−PCnxnk+An∗Anxnk−ωk,xnk+1=αkvn+1−αkunk,where λk is the same as Algorithm 9, v1,v2,⋯,vn∈H1×H2×⋯×Hn.Theorem 14.
LetH be a real Hilbert space. For i=1,2,⋯,n, let Ci be nonempty closed convex subsets of real Hilbert spaces Hi, respectively. Ai:Hi⟶H are bounded linear operators with their adjoint operators Ai∗, respectively. If αk∈0,1, limk→∞αk=0 and ∑k=1∞αk=∞, then the sequence x1k,x2k,⋯,xnk generated by Algorithm 13 converges strongly to a solution of ESEP (5) denoted by z1,z2,⋯,zn=PΩ1v1,v2,⋯vn. Furthermore, for i=1,2,⋯,n,xik−PCixik⟶0,Aixik−ωk⟶0 and xik+1−xik⟶0 as k⟶∞.Proof.
Letz1,z2,⋯,zn=PΩ1v1,v2,⋯vn, which means z1,z2,⋯,zn∈Ω1. Let
(43)z=A1z1=A2z2=⋯=Anzn.
Similar to the proof of the Theorem12, we have the following inequality
(44)∑i=1nuik−zi2≤∑i=1nxik−zi2−λk1−ρk2∑i=1nxik−PCixik2+∑i=1nAixik−ωk2.
From Lemma3, we have
(45)∑i=1nxik+1−zi2=∑i=1nαkvi+1−αkuik−zi2≤αk∑i=1nvi−zi2+1−αk∑i=1nuik−zi2≤αk∑i=1nvi−zi2+1−αk∑i=1nxik−zi2−λk1−ρk2∑i=1nxik−PCixik2+∑i=1nAixik−ωk2≤αk∑i=1nvi−zi2+1−αk∑i=1nxik−zi2≤max∑i=1nvi−zi2,∑i=1nxik−zi2.
By induction, we can get(46)∑i=1nxik+1−zi2≤max∑i=1nvi−zi2,∑i=1nxi1−zi2,which implies that sequence Γk is bounded by setting Γk=∑i=1nxik−zi2. In the sequel, we divide the proof into two cases.Case 1.
Assume that there exists integerk0>0 such that Γk is decreasing sequence for all k>k0, which implies limk→∞Γk exists. Consequently, (45) can be rewritten as
(47)λk1−ρk2∑i=1nxik−PCixik2+∑i=1nAixik−ωk2≤−∑i=1nxik+1−zi2+1−αk∑i=1nxik−zi2+αk∑i=1nvi−zi2.
Similar to the proof of Theorem12, taking limit in the two sides of (47) for k⟶∞, i=1,2,⋯n, we can obtain
(48)xik−PCixik⟶0;Aixik−ωk⟶0,λkxik−PCixik+A∗Aixik−ωk⟶0.
For another, sincexik is bounded, xik has a convergent subsequence. Without loss of generality, we have x¯1,x¯2,⋯,x¯n∈ωωx1k,x2k,⋯,xnk. From (42), for k⟶∞, we have
(49)uik−xik=λkxik−PCixik+Ai∗Aixik−ωk=λkxik−PCixik+Ai∗Aixik−ωk⟶0,which implies uik is bounded. Since limk→∞αk=0, for k⟶∞, we can infer
(50)xik+1−uik=αkvi+1−αkuik−uik=αkvi−αkuik=αkvi−uik⟶0.
Combining (49) and (50), we have
(51)xik+1−xik≤xik+1−uik+uik−xik⟶0.
Similar to the proof of Theorem12, we can get x¯1,x¯2,⋯,x¯n∈Ω1. Since xik is asymptotically regular, we can get x¯1,x¯2,⋯,x¯n∈ωωx1k+1,x2k+1,⋯,xnk+1. Depending on the property of the projection, we have
(52)limsupk→∞vi−zi,xik+1−zi=limj→∞vi−zi,xikj+1−zi=vi−zi,x¯i−zi≤0.(53)∑i=1nxik+1−zi2=∑i=1nαkvi+1−αkuik−zi2≤1−αk2∑i=1nuik−zi2+2αk∑i=1nvi−zi,xik+1−zi≤1−αk∑i=1nxik−zi2+2αk∑i=1nvi−zi,xik+1−zi,which combining with Lemma 6 and (52), we conclude the sequence x1k,x2k,⋯,xnk converges strongly to z1,z2,⋯,zn.Case 2.
AssumeΓk is not a decreasing sequence. Then suppose that there exists a subsequence Γkj of Γk, for j≥0, such that Γkj≤Γkj+1. For all k>k0 (such that Γk0≤Γk0+1), define a integer sequence as follows:
(54)τk=maxl∈N∣k0≤l<k,Γl≤Γl+1.
It is clear thatτk is a nondecreasing sequence satisfying limk→∞τk=∞, and Γτk≤Γτk+1.
Similar to the proof of Case1, we can obtain
(55)λτk1−ρτk2∑i=1nxiτk−PCixiτk2+∑i=1nAixiτk−ωτk2≤−Γτk+1+1−ατkΓτk+ατk∑i=1nvi−zi2.
Taking the limit in the two sides of (55) fork⟶∞, according to Case 1, we have
(56)xiτk−PCixiτk⟶0,Aixiτk−ωτk⟶0,λτkxiτk−PCixiτk+A∗Aixiτk−ωτk⟶0,xiτk+1−xiτk⟶0.
That impliesx¯1,x¯2,⋯,x¯n∈ωωx1τk+1,x2τk+1,⋯,xnτk+1. Depending on the property of the projection, we have
(57)limsupτk→∞vi−zi,xiτk+1−zi≤0.
From (53), we can get
(58)Γτk+1≤1−ατkΓτk+2ατk∑i=1nvi−zi,xiτk+1−zi.
SinceΓτk≤Γτk+1, we have
(59)Γτk≤2∑i=1nvi−zi,xiτk+1−zi,which implies limsupτk→∞Γτk≤0, that is,
(60)limτk→∞Γτk=0.
By (58), we obtain
(61)limsupτk→∞Γτk+1≤limsupτk→∞Γτk≤0,which implies
(62)limτk→∞Γτk+1=0.
By Lemma5, we have Γk≤Γτk+1, which means the sequence x1k,xik,⋯,xnk converges strongly to z1,z2,⋯,zn. Thus, we complete the proof.


## 4. Iterative Algorithm for ESEFPP

In this section, we introduce a simultaneous iterative algorithm with internal perturbation projection to solve ESEFPP (9) and define the solution set of ESEFPP (9) as Ω2.Algorithm 15.
Initialization: let 0<a<αk<b<1, x11,x21,⋯,xn1∈H1×H2×⋯×Hn arbitrary.
Iteration step: for a given current iterate x1k,x2k,⋯,xnk∈H1×H2×⋯×Hn, we calculate the next iterate x1k+1,x2k+1,⋯,xnk+1 by
(63)ωk=∑i=1nAixikn,u1k=x1k−λkx1k−G1x1k+A1∗A1x1k−ωk,x1k+1=αku1k+1−αkG1u1k,u2k=x2k−λkx2k−G2x2k+A2∗A2x2k−ωk,x2k+1=αku2k+1−αkG2u2k,⋯unk=xnk−λkxnk−Gnxnk+An∗Anxnk−ωk,xnk+1=αkunk+1−αkGnunk,where the stepsize λk is chosen in such a way that if k∈Ω, then
(64)λk∈ε,ρk∑i=1nxik−Gixik2+∑i=1nAixik−ωk2∑i=1nxik−Gixik+Ai∗Aixik−ωk2,where the index set Ω=k:Aixik−ωk≠0,∃i∈1,2,⋯,n, small enough ε>0 and 0<δ1<ρk<δ2<1, set k≔k+1 and go to (63). Otherwise, Aixik−ωk=0, for i=1,2,⋯,n, the iteration stops.Theorem 16.
LetH be a real Hilbert space. For i=1,2,⋯,n, let Ai:Hi⟶H be bounded linear operators with their adjoint operators Ai∗, respectively. And Gi:Hi⟶Hi are quasinonexpansive mappings and demiclosed at 0. Then, the sequence x1k,x2k,⋯,xnk generated by Algorithm 15 weakly converges to a solution of ESEFPP (9). Furthermore, for i=1,2,⋯,n,xik−Gixik⟶0,Aixik−ωk⟶0, Giuik−uik⟶0 and xik+1−xik⟶0 as k⟶∞.Proof.
Letz1,z2,⋯,zn∈Ω2 and
(65)z=A1z1=A2z2=⋯=Anzn.
Fori=1,2,⋯,n, from (63) and Lemma 3 (2), we have
(66)uik−zi2=xik−λkxik−Gixik+Ai∗Aixik−ωk−zi2=xik−zi2+λk2xik−Gixik+Ai∗Aixik−ωk2−2λkxik−zi,xik−Gixik+Ai∗Aixik−ωk=xik−zi2+λk2xik−Gixik+Ai∗Aixik−ωk2−2λkxik−zi,xik−Gixik−2λkxik−zi,Ai∗Aixik−ωk.
It follows from(67)−2xik−zi,xik−Gixik=−2xik−zi2−2xik−zi,zi−Gixik=−2xik−zi2+xik−zi2+Gixik−zi2−xik−Gixik2≤−xik−Gixik2,−2xik−zi,Ai∗Aixik−ωk=−2Aixik−Aizi,Aixik−ωk=Aizi−ωk2−Aixik−Aizi2−Aixik−ωk2,then (66) can be rewritten as
(68)uik−zi2≤xik−zi2+λk2xik−Gixik+Ai∗Aixik−ωk2−λkxik−Gixik2+λkAizi−ωk2−λkAixik−Aizi2−λkAixik−ωk2.
SinceGi are quasinonexpansive mappings, by (68), we have
(69)xik+1−zi2=αkuik+1−αkGiuik−zi2=αkuik−zi2+1−αkGiuik−zi2−αk1−αkGiuik−uik2≤uik−zi2−αk1−αkGiuik−uik2≤xik−zi2+λk2xik−Gixik+Ai∗Aixik−ωk2−λkxik−Gixik2+λkAizi−ωk2−λkAixik−Aizi2−λkAixik−ωk2−αk1−αkGiuik−uik2.
Note that(70)Aizi−ωk2=Aizi−∑i=1nAixikn2=∑i=1nAizi−Aixikn2≤∑i=1nAizi−Aixik2n.
Summing (69) for i from 1 to n, we can obtain
(71)∑i=1nxik+1−zi2≤∑i=1nxik−zi2+λk2∑i=1nxik−Gixik+Ai∗Aixik−ωk2−λk∑i=1nxik−Gixik2−λk∑i=1nAixik−ωk2−αk1−αk∑i=1nGiuik−uik2≤∑i=1nxik−zi2+λkρk∑i=1nxik−Gixik2+∑i=1nAixik−ωk2−λk∑i=1nxik−Gixik2+∑i=1nAixik−ωk2−αk1−αk∑i=1nGiuik−uik2=∑i=1n∥xik−zi∥2−λk1−ρk∑i=1nxik−Gixik2+∑i=1nAixik−ωk2−αk1−αk∑i=1nGiuik−uik2.
LetΓkz1,z2,⋯,zn=∑i=1nxik−zi2. According to the definition of λk, we have
(72)Γk+1z1,z2,⋯,zn≤Γkz1,z2,⋯,zn.
Therefore, the sequenceΓkz1,z2,⋯,zn is a nonincreasing sequence and lower bounded by 0. As a result Γkz1,z2,⋯,zn converges to some finite limit, suppose as lz1,z2,⋯,zn, and sequence xik is bounded. Letting n⟶∞ and taking limit in the two sides of (71) for k⟶∞, we obtain that
(73)limk→∞λk∑i=1nxik−Gixik2=0,limk→∞λk∑i=1nAixik−ωk2=0,limk→∞Giuik−uik=0.
Sinceλk>ε, we have
(74)limk→∞xik−Gixik=0,limk→∞Aixik−ωk=0.
From the definition ofλk, we have
(75)λk2∑i=1nxik−Gixik+Ai∗Aixik−ωk2≤λkρk∑i=1nxik−Gixik2+∑i=1nAixik−ωk2⟶0,which implies
(76)λkxik−Gixik+Ai∗Aixik−ωk⟶0.
Now, let us prove thatxik is asymptotically regular. Indeed, from
(77)uik−xik=xik−λkxik−Gixik+Ai∗Aixik−ωk−xik=λkxik−Gixik+Ai∗Aixik−ωk,we have
(78)limk→∞uik−xik=0.
Consequently,(79)limk→∞xik+1−xik=limk→∞αkuik+1−αkGiuik−xik≤limk→∞αkuik−xik+1−αkGiuik−xik≤limk→∞αkuik−xik+1−αkGiuik−uik+1−αkuik−xik=0,which yields xik is asymptotically regular.
Letx¯1,x¯2,⋯,x¯n∈ωωx1k,x2k,⋯,xnk, then there exists a subsequence of xik which weakly converges to x¯i. For limk→∞uik−xik=0, we can obtain x¯i∈ωωuik. Due to the demiclosednesses of Gi at 0 and limk→∞Giuik−uik=0, it yields Gix¯i=x¯i, which implies x¯i∈FixGi. Furthermore, it follows from (74) and the lower semicontinuity of the squared norm, we have
(80)Aix¯i−ω¯2≤liminfk→∞Aixik−ωk2=0,where ω¯=∑i=1nAix¯i/n, which means that Aix¯i−ω¯=0. That is,
(81)A2x¯2+A3x¯3+⋯+Anx¯n=n−1A1x¯1,A1x¯1+A3x¯3+⋯+Anx¯n=n−1A2x¯2,⋯A1x¯1+A2x¯2+⋯+An−1x¯n−1=n−1Anx¯n.
Obviously,(82)A1x¯1=A2x¯2=⋯=Anx¯n,x¯1,x¯2,⋯,x¯n∈Ω2.
Finally, following the proof of Theorem12, we can prove the uniqueness of the weak cluster points of xik. Thus, we complete the proof.Similar to the proof of Theorem16, from Definition 7 and Lemma 8, we have the following corollary.Corollary 17.
LetH be a real Hilbert space. For i=1,2,⋯,n, let Ai:Hi⟶H be bounded linear operators with their adjoint operators Ai∗ and Ci be nonempty closed convex subsets of real Hilbert spaces Hi. Let Tjii=1N be a finite family of ρji-strictly pseudononspreading mappings of Ci into itself with ⋂i=1NFixTji≠∅ and ρi=maxj=1,2,⋯,Nρji. For i=1,2,⋯,N, let πji=αji,βji,γji,δji, where αji,βji,γji,δji∈0,1 and αji+βji+γji+δji=1. Assume that ρi≤βji for j=1,2,⋯N, and αji+βjiρi≤βjiγji for j=2,3,⋯,N, and α1i+β1iρi≤β1iγ1i+δ1i. If Gi are the G-mapping generated by T1i,T2i,⋯,TNi and π1i,π2i,⋯,πNi and demiclosed at 0, then the sequence x1k,x2k,⋯,xnk generated by Algorithm 15 weakly converges to a solution of ESEFPP (9). Furthermore, for i=1,2,⋯,n,xik−Gixik⟶0,Aixik−ωk⟶0, Giuik−uik⟶0, and xik+1−xik⟶0 as k⟶∞.


## 5. Application

In this section, we apply our iterative algorithms to some convex and nonlinear problems, for example, [11, 28].


### 5.1. Split Equality Feasibility Problem

Note that ESEP (5) reduces to SEFP (2) as n=2. Then, we can obtain the following algorithm to study SEFP (2) from Algorithm 9.Algorithm 18.
Initialization: let x11,x21∈H1×H2 arbitrary.
Iteration step: for a given current iterate x1k,x2k∈H1×H2, we calculate the next iterate x1k+1,x2k+1 by
(83)ωk=A1x1k+A2x2k2,x1k+1=x1k−λkx1k−PC1x1k+A1∗A1x1k−ωk,x2k+1=x2k−λkx2k−PC2x2k+A2∗A2x2k−ωk,where the stepsize λk is chosen as Algorithm 9. Set k≔k+1 and go to (83). Otherwise, Aixik−ωk=0, for i=1,2, iteration stops.


### 5.2. Split Equality Fixed Point Problem

Note that ESEFPP (9) reduces to SEFPP (7) as n=2. Then, we can obtain the following algorithm to study SEFPP (7) from Algorithm 15.Algorithm 19.
Initialization: let 0<a<αk<b<1, x11,x21∈H1×H2 arbitrary.
Iteration step: for a given current iterate x1k,x2k∈H1×H2, we calculate the next iterate x1k+1,x2k+1 by
(84)ωk=A1x1k+A2x2k2,u1k=x1k−λkx1k−G1x1k+A1∗A1x1k−ωk,x1k+1=αku1k+1−αkG1u1k,u2k=x2k−λkx2k−G2x2k+A2∗A2x2k−ωk,x2k+1=αku2k+1−αkG2u2k,where the stepsize λk is chosen as Algorithm 15. Set k≔k+1 and go to (84). Otherwise, Aixik−ωk=0, for i=1,2, iteration stops.


### 5.3. Extend Split Equation Problem

Similar to Algorithm15, we have another algorithm to solve ESEP (5).Algorithm 20.
Initialization: let 0<a<αk<b<1, x11,x21,⋯,xn1∈H1×H2×⋯×Hn arbitrary.
Iteration step: for a given current iterate x1k,x2k,⋯,xnk∈H1×H2×⋯×Hn, we calculate the next iterate x1k+1,x2k+1,⋯,xnk+1 by
(85)ωk=∑i=1nAixikn,u1k=x1k−λkx1k−PC1x1k+A1∗A1x1k−ωk,x1k+1=αku1k+1−αkPC1u1k,u2k=x2k−λkx2k−PC2x2k+A2∗A2x2k−ωk,x2k+1=αku2k+1−αkPC2u2k,⋯unk=xnk−λkxnk−PCnxnk+An∗Anxnk−ωk,xnk+1=αkunk+1−αkPCnunk,where the stepsize λk is chosen as Algorithm 9. Set k≔k+1 and go to (85). Otherwise, Aixik−ωk=0, for i=1,2,⋯,n, iteration stops.


## 5.1. Split Equality Feasibility Problem

Note that ESEP (5) reduces to SEFP (2) as n=2. Then, we can obtain the following algorithm to study SEFP (2) from Algorithm 9.Algorithm 18.
Initialization: let x11,x21∈H1×H2 arbitrary.
Iteration step: for a given current iterate x1k,x2k∈H1×H2, we calculate the next iterate x1k+1,x2k+1 by
(83)ωk=A1x1k+A2x2k2,x1k+1=x1k−λkx1k−PC1x1k+A1∗A1x1k−ωk,x2k+1=x2k−λkx2k−PC2x2k+A2∗A2x2k−ωk,where the stepsize λk is chosen as Algorithm 9. Set k≔k+1 and go to (83). Otherwise, Aixik−ωk=0, for i=1,2, iteration stops.


## 5.2. Split Equality Fixed Point Problem

Note that ESEFPP (9) reduces to SEFPP (7) as n=2. Then, we can obtain the following algorithm to study SEFPP (7) from Algorithm 15.Algorithm 19.
Initialization: let 0<a<αk<b<1, x11,x21∈H1×H2 arbitrary.
Iteration step: for a given current iterate x1k,x2k∈H1×H2, we calculate the next iterate x1k+1,x2k+1 by
(84)ωk=A1x1k+A2x2k2,u1k=x1k−λkx1k−G1x1k+A1∗A1x1k−ωk,x1k+1=αku1k+1−αkG1u1k,u2k=x2k−λkx2k−G2x2k+A2∗A2x2k−ωk,x2k+1=αku2k+1−αkG2u2k,where the stepsize λk is chosen as Algorithm 15. Set k≔k+1 and go to (84). Otherwise, Aixik−ωk=0, for i=1,2, iteration stops.


## 5.3. Extend Split Equation Problem

Similar to Algorithm15, we have another algorithm to solve ESEP (5).Algorithm 20.
Initialization: let 0<a<αk<b<1, x11,x21,⋯,xn1∈H1×H2×⋯×Hn arbitrary.
Iteration step: for a given current iterate x1k,x2k,⋯,xnk∈H1×H2×⋯×Hn, we calculate the next iterate x1k+1,x2k+1,⋯,xnk+1 by
(85)ωk=∑i=1nAixikn,u1k=x1k−λkx1k−PC1x1k+A1∗A1x1k−ωk,x1k+1=αku1k+1−αkPC1u1k,u2k=x2k−λkx2k−PC2x2k+A2∗A2x2k−ωk,x2k+1=αku2k+1−αkPC2u2k,⋯unk=xnk−λkxnk−PCnxnk+An∗Anxnk−ωk,xnk+1=αkunk+1−αkPCnunk,where the stepsize λk is chosen as Algorithm 9. Set k≔k+1 and go to (85). Otherwise, Aixik−ωk=0, for i=1,2,⋯,n, iteration stops.


## 6. Numerical Examples

We are in a position to show some numerical examples to demonstrate the performance and convergence of Algorithms9 and 13. The whole programs are written in MATLAB 2017b. All the numerical results are carried out on a personal Lenovo computer with Intel(R) Core(TM) i7-7500U CPU 2.70 GHz RAM 4.00 GB. We denote the vector with all elements 1 by e in what follows.Example 21.
Let(86)A=0.6948286229758170.9502220488383550.4387443596563980.3170994800608610.03444608050290880.381558457093008,B=0.7655167881490020.1868726045543790.7951999011370630.489764395788231,C=0.4455862007109000.7093648308580730.2760250769985780.6550980039738410.6463130101112650.7546866819823610.6797026768536750.162611735194631.
C1=x1∈R3:x1≤1,C2=x2∈R2:−3e≤x2≤e,andC3=x3∈R4:−2e≤x3≤2e, to find x,y,z∈C1×C2×C3 such that Ax=By=Cz.Example 22.
LetA1=aijP×M,A2=bijP×N,A3=cijP×Q,C1=x1∈RM:x1≤1,C2=x2∈RN:x2≤2,andC3=x3∈Q:x3≤3, where aij,bij,cij∈0,1 are all generated randomly.In the Example22, we consider P=10,M=20,N=9,Q=25;P=25,M=35,N=30,Q=35;P=50,M=30,N=40,Q=50, and four initial values.Case 1.
x=onesM,1,y=onesN,1,andz=10∗onesQ,1.Case 2.
x=−10∗onesM,1,y=0∗onesN,1,andz=10∗onesQ,1.Case 3.
x=100∗onesN,1,y=onesM,1,andz=−10∗onesQ,1.Case 4.
x=10∗onesN,1,y=−10∗onesM,1,andz=onesQ,1.We takeαk=5/6k, ρk=1/2+1/10k, v1,v2,⋯,vn=0,0,⋯,0 in Algorithms 9 and 13. In the following tables and figures, we denote Algorithm 5.4 in reference [19] by ECCL. And we set “n” and “s” to express the number of iteration and CPU time in seconds, respectively. We use ∑i=13Aixik−ωk≤10−4 as the stop criterion, where ωk=∑i=13Aixik/3. The numerical results can be seen from Table 1 and Figures 1–5.Table 1
The numerical results of Example22.
PMNQAlgorithm9Algorithm13ECCLnsnsnsCase110209251910.0166571440.0126852060.027401253530354750.0198022480.0239215550.0445545030405016840.2013848570.10387326750.20225610209251840.0198021130.0096222300.026396Case2253530353670.0487781460.00179664330.0223745030405021470.2648697580.08914925330.17985910209252300.0194261700.0151822320.011593Case3253530353470.0382272690.0285354330.0393845030405019350.23599214460.17538720870.13164610209251920.0168251210.0090102130.014185Case4253530353610.0472101670.0180084670.0401005030405020650.2560915620.06653323070.180630Figure 1
The behaviors ofxik,ωk, and Aixik−ωk of Algorithm 9 for Example 21.Figure 2
The behaviors ofxik,ωk, and Aixik−ωk of Algorithm 13 for Example 21.Figure 3
The behaviors ofxik,ωk, and Aixik−ωk of ECCL for Example 21.Figure 4
The iterative number of Algorithm13 and ECCL for Case A.Figure 5
The iterative number of Algorithm13 and ECCL for Case B.We take the initial pointx=ones3,1, y=ones2,1, and z=ones4,1 in the Example 21. Figures 1–3 express the behaviors of xik,ωk, and Aixik−ωk, for i=1,2,3, of Algorithms 9 and 13 and ECCl for Example 21, respectively.Furthermore, for testing the stationary property of iterative number, we carry 500 experiments for different initial points which are presented randomly, such as(87)CaseAx=rand3,1,y=rand2,1,z=rand4,1,CaseBx=1000∗rand3,1,y=1000∗rand2,1,z=rand4,1,in Example 21 for Algorithm 13 and ECCL, the results can be found in Figures 4 and 5, respectively.Table1 shows the numerical results of Example 22 when Algorithms 9 and 13 and ECCL select different initial points and matrix dimensions.From Figures1–3 and Table 1, we can see that the iterative numbers of Algorithms 9 and 13 are less than ECCL. From Figures 4 and 5 and Table 1, we can see that the stationary property of the iterative number and CPU time of Algorithm 13 are exceeded by ECCL. And Algorithm 2 is better than Algorithm 13 in these aspects.



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*Source: 6034754-2020-06-03.xml*