# Parallel and Cyclic Algorithms for Quasi-Nonexpansives in Hilbert Space

Authors: Bin-Chao Deng; Tong Chen; Baogui Xin
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

Let{T}i=1N be N quasi-nonexpansive mappings defined on a closed
convex subset C of a real Hilbert space H. Consider the problem of finding a
common fixed point of these mappings and introduce the parallel and cyclic
algorithms for solving this problem. We will prove the strong convergence of
these algorithms.


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## Body

## 1. Introduction

Throughout this paper, we always assume thatC is a nonempty, closed, and convex subset of a real Hilbert space H. Let A:C→H be a nonlinear mapping. Recall the following definitions.(1)
Ais said to be monotone if
(1.1)〈Ax-Ay,x-y〉≥0,∀x,y∈C.(2)
A is said to be strongly positive if there exists a constant γ->0 such that
(1.2)〈Ax,x〉≥γ-∥x∥2,∀x∈C.(3)
A is said to be strongly monotone if there exists a constant α>0 such that
(1.3)〈Ax-Ay,x-y〉≥α∥x-y∥2,∀x,y∈C.For such a case, A is said to be α-strongly monotone.(4)
A is said to be inverse strongly if there exists a constant α>0 such that
(1.4)〈Ax-Ay,x-y〉≥α∥Ax-Ay∥2,∀x,y∈C.For such a case, A is said to be α-inverse-strongly-monotone (α-ism).AssumeA is strongly positive operator, that is, there is a constant γ- with the property
(1.5)〈Ax,x〉≥γ-∥x∥2,∀x∈H.Remark 1.1.
LetF=A-γf, where A is strongly positive operator, and f is contraction mapping with coefficient β∈(0,1). It is a simple matter to see that the operator F is (γ--γβ)-strongly monotone over C, that is,
(1.6)〈Fx-Fy,x-y〉≥(γ--γβ)∥x-y∥2,∀(x,y)∈C×C.The classical variational inequality which is denoted byVI(A,C) is to find x∈C such that
(1.7)〈Ax,y-x〉≥0,∀y∈C.The variational inequality has been extensively studied in literature; see, for example, [1, 2] and the reference therein. A mapping T:C→C is said to be a strict pseudocontraction [3] if there exists a constant 0≤k<1 such that
(1.8)∥Tx-Ty∥2≤∥x-y∥2+k∥(I-T)x-(I-T)y∥2,
for all x,y∈C (If (1.8) holds, we also say that T is a k-strict pseudo-contraction). These mappings are extensions of nonexpansive mappings which satisfy the inequality (1.8) with k=0. That is, T:C→C is nonexpansive if
(1.9)∥Tx-Ty∥≤∥x-y∥,∀x,y∈C.In [4], Xu proved that the sequence {xn} defined by the iterative method below with the initial guess x0∈H chosen arbitrarily,
(1.10)xn+1=αnb+(I-αnA)Txn,n≥0,
where the sequence {αn} satisfies certain conditions, he proved the sequence {xn} converges strongly to the unique solution of the following minimization problem:
(1.11)minx∈C12〈Ax,x〉-〈x,b〉.
In [5], Marino and Xu considered the following general iterative method:
(1.12)xn+1=αnγf(xn)+(I-αnA)Txn,n≥0,
they proved that if the sequence {αn} of parameters satisfies appropriate conditions, then the sequence {xn} generated by (1.12) converges strongly to the unique solution of the variational inequality
(1.13)〈(γf-A)x*,x-x*〉≤0,x∈C,
which is the optimality condition for the minimization problem
(1.14)minx∈C12〈Ax,x〉-h(x),
where h is a potential function for γf (i.e., h'(x)=γf(x) for x∈H). Some people also study the applications of the iterative method (1.12) [6, 7].Acedo and Xu [8] consider the following parallel and cyclic algorithms:Parallel Algorithm
The sequence{xn} was generated by
(1.15)xn+1=αnxn+(1-αn)∑i=1N‍λi(n)Tixn,
where {Ti}i=1N are N strict pseudocontractions defined on a closed convex subset C of a Hilbert space H. Under the following assumptions on the sequences of the weights {λi(n)}i=1N: (a1)
∑i=1N‍λi(n)=1 for all n and infn≥1λ(n)>0, for all 1≤i≤N,(a2)
∑i=1N‍∑i=1N‍|λi(n+1)-λi(n)|<∞.By (1.15), they will prove the weak convergence to a solution of the problem x∈⋂i=1NFix(Ti).Cyclic Algorithm
They define the sequence{xn} cyclically by
(1.16)x1=α0x0+(1-α0)T0x0;x2=α1x1+(1-α1)T1x1;⋮xN=αN-1xN-1+(1-αN-1)TN-1xN-1;xN+1=αNxN+(1-αN)T0xN;⋮
In a more compact form, they are rewritten xn+1 as
(1.17)xN+1=αNxN+(1-αN)TNxn,
where {Ti}i=1N are ki-strict pseudo-contractions and TN=Ti with i=n (mod N), 0≤i≤N-1. They show that this cyclic algorithm (1.17) is weakly convergent if the sequence {αn} of parameters is appropriately chosen. On the other hand, Osilike and Shehu [9] also consider the cyclic algorithm (1.17), under appropriate assumptions on the sequences of {αn}, some strong convergence theorems are proved.In this paper, we are concerned with the problem of finding a pointx such that
(1.18)x∈⋂i=1N‍Fix(Tωi),N≥1,
where Tωi=(1-ωi)I+ωiTi, {ωi}i=1N∈(0,1] and {Ti}i=1N are quasi-nonexpansive mappings defined on a closed convex subset C of a Hilbert space H. Here Fix(Tωi)={q∈C:Tωiq=q} is the set of fixed points of Ti, 1≤i≤N.LetT be defined by
(1.19)T=∑i=1N‍λiTωi,
where λi>0 for all i∈(0,1) such that ∑i=1N‍λi=1. Motivated and inspired by Acedo and Xu [8], we consider the following two general iterative algorithms for a family of quasi-nonexpansive mappings.Algorithm 1.2.
(1.20)
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In (1.20), the weights {λi}i=1N are constant in the sense that they are independent of n, the number of steps of the iteration process. In (1.21), we consider a more general case by allowing the weights {λi(n)}i=1N. Under appropriate assumptions on the sequences of the wights {λi(n)}i=1N,{λi}i=1N,{αn} and {βn}. From (1.20) and (1.21), we will prove some strong convergence to a solution of the problem (1.18). In addition, we can also know that the condition ∑i=1N‍∑i=1N‍|λi(n+1)-λi(n)|<∞ in [8] is superfluous.Another approach to the problem (1.18) is the cyclic algorithm (for convenience, we relabel the mappings {Tωi}i=1N as {Tωi}i=0N-1). This means that beginning with an x0∈C, we define the sequence {xn} cyclically by
(1.22)x1=α0γf(x0)+β0x0+((I-β0)I-α0A)Tω0x0,x2=α1γf(x1)+β1x1+((I-β1)I-α1A)Tω1x1,⋮xN=αN-1γf(xN-1)+βN-1xN-1+((I-βN-1)I-αN-1A)TωN-1xN-1,xN+1=αNγf(xN)+βNxN+((I-βN)I-αNA)TωNxN,(1.23)⋮
In a more compact from, xn+1 can be written as
(1.24)xn+1=αnγf(xn)+βnxn+((I-βn)I-αnA)T[n]xn,
where T[n]=Tωi, Tωi=(1-ωi)I+ωiTi, {ωi}i=1N∈(0,1], with i=n (mod N), 0≤i≤N-1.We will show that this cyclic algorithm (1.24) is also strongly convergent if the sequence {αn} of parameters is appropriately chosen.


## 2. Preliminaries

Throughout this paper, we writexn⇀x to indicate that the sequence {xn} converges weakly to x. xn→x implies that {xn} converges strongly to x. The following definitions and lemmas are useful for main results.Definition 2.1.
An operatorT:H→H is said to be quasi-nonexpansive if
(2.1)Fix(T)≠0andif∥Tx-z∥≤∥x-z∥,∀z∈Fix(T),∀x∈H.
Iterative methods for quasi-nonexpansive mappings have been extensively investigated; see [10, 11].Remark 2.2.
From the above definitions, It is easy to see that(i)
a nonexpansive mapping is a quasi-nonexpansive mapping;(ii)
the set of fixed points ofT is the set Fix(T)={x∈H:Tx=x}. We assume that Fix(T)≠∅, it is well know that Fix(T) is closed and convex.Remark 2.3 (see [10]).
LetTα=(1-α)I+αT, where T is a quasi-nonexpansive on H, Fix(T)≠∅ and α∈(0,1]. Then the following statements are reached: (i)
Fix(T)=Fix(Tα);(ii)
Tα is quasi-nonexpansive;(iii)
∥Tαx-q∥2≤∥x-q∥2-α(1-α)∥Tx-x∥2, for all (x,q)∈H×Fix(T);(iv)
〈x-Tαx,x-q〉≥(α/2)∥Tx-x∥2, for all (x,q)∈H×Fix(T).Example 2.4.
LetX=l2 with the norm ∥·∥ defined by
(2.2)∥X∥=∑i=1∞‍xi2,∀x=(x1,x2,…,xn,…)∈X,
and C={x=(x1,x2,…,xn,…)∣x1≤0,xi∈R1,i=2,3,…}. Then C is a nonempty subset of X.
Now, for anyx=(x1,x2,…,xn,…)∈C, define a mapping T:C→C as follows:
(2.3)T(x)=(0,4x1,0,…,0,…).
It is easy to see that T is a quasi-nonexpansive mapping. In fact, for any x=(x1,x2,…,xn,…)∈X, taking T(x)=x, that is,
(2.4)(0,4x1,0,…,0,…)=(x1,x2,…,xn,…),
we have F(T)={0} and
(2.5)∥T(x)-0∥=∥(0,4x1,0,…,0,…)-(0,0,0,…,0,…)∥=4|x1|≤4∑i=1∞‍xi2=∥(x1,x2,…,xn,…)-(0,0,0,…,0,…)∥=∥x-0∥.Lemma 2.5.
AssumeC is a closed convex subset of a Hilbert space H. (i)
Given an integerN≥1, assume, for all 1≤i≤N, Ti:C→C is a quasi-nonexpansive. Let {λi}i=1N be a positive sequence such that ∑i=1N‍λi=1. Then ∑i=1N‍λiTi is a quasi-nonexpansive.(ii)
Let{Ti}i=1N and ∑i=1N‍λi=1 be given as in (i) above. Suppose that {Ti}i=1N has a common fixed point. Then
(2.6)Fix(∑i=1N‍λiTi)=⋂i=1N‍Fix(Ti).(iii)
AssumeTi:C→C be quasi-nonexpansives, let Tαi=(1-αi)I+αiTi, 1≤i≤N. If ⋂i=1NFix(Ti)≠∅, then
(2.7)Fix(Tα1Tα2⋯TαN)=⋂i=1N‍Fix(Tαi).Proof.
To prove(i) we only need to consider the case of N=2 (the general case can be proved by induction). Set T=(1-λ)T1+λT2, where λ∈(0,1) and for i=1,2, Ti is a quasi-nonexpansive. We verify directly the following inequality: for all z∈Fix(T1)∩Fix(T2),
(2.8)∥Tx-z∥=∥((1-λ)T1+λT2)x-z∥≤(1-λ)∥T1x-z∥+λ∥T2x-z∥≤(1-λ)∥x-z∥+λ∥x-z∥≤∥x-z∥,
that is, T is a quasi-nonexpansive.
To prove(ii) again we can assume N=2. It suffices to prove that Fix(T)⊂Fix(T1)∩Fix(T2), where T=(1-λ)T1+λT2 with λ∈(0,1). Let x∈Fix(T).
Takingz∈Fix(T1)∩Fix(T2) to deduce that
(2.9)∥z-x∥=∥(1-λ)(z-T1x)+λ(z-T2x)∥≤(1-λ)∥z-T1x∥+λ∥z-T2x∥≤(1-λ)∥z-x∥+λ∥z-x∥≤∥z-x∥.
By the strict convexity of H, it follows that T1(x)-z=T2(x)-z=x-z; that is, T1(x)=T2(x)=x, hence x∈Fix(T1)∩Fix(T2). According to induction, we can easily claim that (2.6) is holds.
To prove(iii) by induction, for N=2, set Tαi=(1-αi)I+αiTi for all i=1,2. Obviously
(2.10)Fix(Tα1)⋂Fix(Tα2)⊂Fix(Tα1Tα2).
Now we prove
(2.11)Fix(Tα1Tα2)⊂Fix(Tα1)⋂Fix(Tα2).
For all q∈Fix(Tα1Tα2), Tα1Tα2q=q, if Tα2q=q, then Tα1q=q, the conclusion holds. In fact, we can claim that Tα2q=q. From Remark 2.3, we know that Tα2 is quasi-nonexpansive and Fix(Tα1)⋂Fix(Tα2)=Fix(T1)⋂Fix(T2)≠∅. Take p∈Fix(Tα1)⋂Fix(Tα2), then
(2.12)∥p-q∥2=∥p-Tα1Tα2q∥2=∥p-[(1-α1)Tα2q+α1T1Tα2q]∥2=∥(1-α1)(p-Tα2q)+α1(p-T1Tα2q)∥2=(1-α1)∥p-Tα2q∥2+α1∥p-T1Tα2q∥2-α1(1-α1)∥Tα2q-T1Tα2q∥2≤(1-α1)∥p-Tα2q∥2+α1∥p-Tα2q∥2-α1(1-α1)∥Tα2q-T1Tα2q∥2=∥p-Tα2q∥2-α1(1-α1)∥Tα2q-T1Tα2q∥2.
From (2.12), we have
(2.13)∥Tα2q-T1Tα2q∥2≤0,
namely, Tα2q=T1Tα2q, that is,
(2.14)Tα2q∈Fix(T1)=Fix(Tα1),Tα2q=Tα1Tα2q=q.
Suppose that the conclusion holds forN=k, we prove that
(2.15)Fix(Tα1Tα2⋯Tαk+1)=⋂i=1k+1‍Fix(Tαi).
It suffices to verify
(2.16)Fix(Tα1Tα2⋯Tαk+1)⊂⋂i=1k+1‍Fix(Tαi),
for all q∈Fix(Tα1Tα2⋯Tαk+1), that is, Tα1Tα2⋯Tαk+1q=q. Using Remark 2.3 again, take p∈⋂i=1k+1‍Fix(Tαi), we obtain
(2.17)∥p-q∥2=∥p-Tα1Tα2⋯Tαk+1q∥2=∥p-[(1-α1)Tα2⋯Tαk+1q-α1T1Tα2⋯Tαk+1q]∥2=∥[(1-α1)(p-Tα2⋯Tαk+1q)-α1(p-T1Tα2⋯Tαk+1q)]∥2=(1-α1)∥p-Tα2⋯Tαk+1q∥2+α1∥p-T1Tα2⋯Tαk+1q∥2-α1(1-α1)∥Tα2⋯Tαk+1q-T1Tα2⋯Tαk+1q∥2≤∥p-Tα2⋯Tαk+1q∥2-α1(1-α1)∥Tα2⋯Tαk+1q-T1Tα2⋯Tαk+1q∥2.
From (2.17), we obtain
(2.18)∥Tα2⋯Tαk+1q-T1Tα2⋯Tαk+1q∥2≤0,
this implies that
(2.19)Tα2⋯Tαk+1q∈Fix(T1)=Fix(Tα1),
namely,
(2.20)Tα2⋯Tαk+1q=Tα1Tα2⋯Tαk+1q=q.
From (2.20) and inductive assumption, we have
(2.21)q∈Fix(Tα1Tα2⋯Tαk+1)⊂⋂i=2k+1‍Fix(Tαi),
therefore
(2.22)Tαiq=q,i=2,3,…,k+1.
Substituting it into (2.20), we obtain Tα1q=q. Thus we assert that
(2.23)q∈⋂i=1k+1‍Fix(Tαi).Definition 2.6.
A mappingT is said to be demiclosed, if for any sequence {xn} weakly converges to y, and if the sequence {Txn} strongly converges to z, then T(y)=z.Lemma 2.7 (see [5]).
AssumeA is a strong positive linear bounded operator on a Hilbert space H with coefficient γ->0 and 0<ρ≤∥A∥-1, then ∥I-ρA∥≤1-ργ-.Lemma 2.8 (see [12]).
LetH be a Hilbert space, K a closed convex subset of H, and T:K→K a nonexpansive mapping with Fix(T)≠∅, if {xn} is a sequence in K weakly converging to x and if {(I-T)xn} converges strongly to y, then (I-T)x=y.Lemma 2.9 (see [13]).
Let{𝒯n} be a sequence of real numbers that does not decrease at infinity, in the sense that there exists a subsequence {𝒯nj}j≥0 of {𝒯n} which satisfies 𝒯nj<𝒯nj+1 for all j≥0. Also consider the sequence of integers {δ(n)}n≥n0 defined by
(2.24)δ(n)=max{k≤n∣𝒯k<𝒯k+1}.
Then {δ(n)}n≥n0 is a nondecreasing sequence verifying limn→∞δ(n)=∞, for all n≥n0, it holds that 𝒯δ(n)<𝒯δ(n)+1 and one has
(2.25)𝒯n<𝒯δ(n)+1.Lemma 2.10.
LetK be a closed convex subset of a real Hilbert space H, given x∈H and y∈K. Then y=PKx if and only if there holds the inequality
(2.26)〈x-y,y-z〉≥0,∀z∈K.


## 3. Parallel Algorithm

In this section, we discuss the parallel algorithm, respectively, for solving the variational inequality over the set of the common fixed points of finite quasi-nonexpansives.Before stating our main convergence result, we establish the boundedness of the iterates given by following algorithm:(3.1)T=∑i=1N‍λiTωi,xn+1=αnγf(xn)+βnxn+((I-βn)I-αnA)Txn.In (3.1), the weight {λi}i=1N are constant in the sense that they are independent of n, the number of steps of the iteration process. Below we consider a more general case by allowing the weights {λi}i=1N to be step dependent. That is, initializing with x0, we define {xn} by the algorithm
(3.2)xn+1=αnγf(xn)+βnxn+((I-βn)I-αnA)∑i=1N‍λi(n)Tωixn.
From (3.1) and (3.2), the sequence {xn} which converges strongly to the unique solution of variational inequality problem VI(γf-A,⋂i=1N‍Fix(Tωi)): find x* in ⋂i=1N‍Fix(Tωi) such that
(3.3)∀v∈⋂i=1N‍Fix(Tωi),〈(γf-A)x*,v-x*〉≤0,
or equivalently
(3.4)x*=(P⋂i=1N‍Fix(Tωi)·C)(x*),
where P⋂i=1N‍Fix(Tωi) denotes the metric projection from H onto ⋂i=1N‍Fix(Tωi) (see, [14] for more details on the metric projection).Lemma 3.1.
The sequence{xn} is generated by (3.2), where {αn} and {βn} are sequence in [0,1], and {Ti}i=1N is a quasi-nonexpansive mapping on H, is bounded and satisfies
(3.5)∥xn-v∥≤max{∥x1-v∥,∥γf(v)-Av∥γ--γβ},∀n≥1,
where v is any element in Fix(Ti), 1≤i≤N.Proof.
Sincelimn→∞αn=0, we shall assume that αn≤(1-βn)∥A∥-1 and 1-αn(γ--γβ)>0. Observe that if ∥u∥=1, then
(3.6)〈((I-βn)I-αnA)u,u〉=(1-βn)-αn〈Au,u〉≥(1-βn-αn∥A∥)≥0.
By Lemma 2.7, we obtain
(3.7)∥(I-βn)I-αnA∥≤1-βn-αnγ-.
Let Bn=∑i=1N‍λi(n)Tωi, for all n≥1. By Lemma 2.5, each Bn is a quasi-nonexpansive mapping on H, and in light of Remark 2.3. Taking v∈Fix(T), we have
(3.8)∥∑i=1N‍λi(n)Tωixn-v∥≤∥∑i=1N‍λi(n)Tωi(xn-v)∥≤∥∑i=1N‍λi(n)(xn-v)∥≤∥xn-v∥.
From (3.1), we have
(3.9)∥xn+1-v∥=∥αn(γf(xn)-Av)+βn(xn-v)+((I-βn)I-αnA)T(xn-v)∥≤αn∥γf(xn)-Av∥+βn∥xn-v∥+(1-βn-αnγ-)∥xn-v∥≤αnγ∥f(xn)-f(v)∥+αn∥γf(v)-Av∥+(1-αnγ-)∥xn-v∥=[1-αn(γ--γβ)]∥xn-v∥+αn∥γf(v)-Av∥.
By simple inductions, we obtain
(3.10)∥xn-v∥≤max{∥x1-v∥,∥γf(v)-Av∥γ--γβ},∀n≥1,
which gives that the sequence {xn} is bounded.Lemma 3.2.
Assume that{xn} is defined by (3.2), if x* is solution of (3.3) with T:C→C demiclosed and {yn}⊂H is a bounded sequence such that ∥Tyn-yn∥→0, then
(3.11)liminfn→∞〈(A-γf)x*,yn-x*〉≥0.Proof.
Clearly, by∥Tyn-yn∥→0 and T:H→H demi-closed, we know that any weak cluster point of {yn} belongs to Fix(T). It is also a simple matter to see that there exist y- and a subsequence {ynj} of {yn} such that limj→∞ynj⇀y- (hence y-∈Fix(T)) and such that
(3.12)liminfn→∞〈(A-γf)x*,yn-x*〉=limj→∞〈(A-γf)x*,ynj-x*〉,
it follows from (3.3), we can derive that
(3.13)liminfn→∞〈(A-γf)x*,yn-x*〉=〈(A-γf)x*,y--x*〉≥0,
that is the desired result.Theorem 3.3.
LetC be a closed convex subset of a Hilbert space H and let Ti:C→C be a quasi-nonexpansive for Tωi=(1-ωi)I+ωiTi, ωi∈(0,1), i∈(1,…,N) such that ⋂i=1N‍Fix(Tωi)≠∅, f be a contraction with coefficient β∈(0,1), and λi a positive constant such that ∑i=1N‍λi(n)=1 for all n and infn≥1λi(n)>0 for all i∈[1,N]. Let A be a strongly positive bounded linear operator with coefficient γ-. Given the initial guess x0∈H chosen arbitrarily and given sequences {αn} and {βn} in (0,1), satisfying the following conditions: (c1)
limn→∞αn=0 and ∑n=0∞‍αn=∞,(c2)
0<liminfn→∞βn≤limsupn→∞βn<1.
Let{xn} be the sequence generated by (3.2). Then {xn} converges strongly to the unique a element x*∈⋂i=1N‍Fix(Tωi),N≥1 verifying
(3.14)x*=(P⋂i=1N‍Fix(Tωi)·f)x*
which equivalently solves the following variational inequality problem:
(3.15)x*∈⋂i=1N‍Fix(Tωi),〈(γf-A)x*,x^-x*〉≤0,∀x^∈⋂i=1N‍Fix(Tωi).Proof.
TakingBn=∑i=1N‍λi(n)Tωi, for all n≥1. By Lemma 2.5(i), each Bn is a quasi-nonexpansive mapping on C, and (3.2) can be rewritten as
(3.16)xn+1=αnγf(xn)+βnxn+((I-βn)I-αnA)Bnxn.
Denote by Ω the common fixed point of the mappings {Tωi}i=1N (by Lemma 2.5(ii), we can easily know that Ω=⋂i=1NFix(Tωi)=⋂i=1NFix(Ti)) and take x*∈Ω and from (3.16) we deduce that
(3.17)xn+1-xn+αn(Axn-γf(xn))=(I-βn-αnA)(Bnxn-xn),
and hence
(3.18)〈xn+1-xn+αn(Axn-γf(xn)),xn-x*〉=〈(1-βn-αnA)Bnxn-xn,xn-x*〉=(1-βn-αn)〈Bnxn-xn,xn-x*〉+αn〈(I-A)(Bn-I)xn,xn-x*〉.
Moreover, by x*∈⋂i=1N‍Fix(Tωi) and using Remark 2.3(iv), we obtain
(3.19)〈xn-Bnxn,xn-x*〉≥〈xn-∑i=1N‍λi(n)Tωixn,xn-x*〉≥∑i=1N‍λi(n)〈xn-Tωixn,xn-x*〉≥∑i=1N‍λi(n)ωi2∥xn-Tixn∥2,
which combined with the (3.18) entails
(3.20)〈xn+1-xn+αn(A-γf)xn,xn-x*〉≤-(1-βn-αn)2∑i=1N‍(λi(n)ωi∥xn-Tixn∥2)+αn〈(I-A)(Bn-I)xn,xn-x*〉,
or equivalently
(3.21)-〈xn-xn+1,xn-x*〉≤-αn〈(A-γf)xn,xn-x*〉-(1-βn-αn)2∑i=1N‍(λi(n)ωi∥xn-Tixn∥2)+αn〈(I-A)(Bn-I)xn,xn-x*〉.
Furthermore, using the following classical equality:
(3.22)〈u,v〉=12∥u∥2-12∥u-v∥2+12∥v∥2,∀u,v∈C,
and setting 𝒯n=(1/2)∥xn-x*∥2, we have
(3.23)〈xn-xn+1,xn-x*〉=𝒯n-𝒯n+1+12∥xn-xn+1∥2.
So that (3.21) can be equivalently rewritten as
(3.24)𝒯n+1-𝒯n-12∥xn-xn+1∥2≤-αn〈(A-γf)xn,xn-x*〉-(1-βn-αn)2∑i=1N‍(λi(n)ωi∥xn-Tixn∥2)+αn〈(I-A)(Bn-I)xn,xn-x*〉.
Now using (3.16) again, we have
(3.25)∥xn+1-xn∥2=∥αn(γf(xn)-Axn)+(I-βn-αnA)(Bnxn-xn)∥2.
Since A:H→H is a strongly positive bounded linear operator with coefficient γ->0, hence it is a classical matter to see that
(3.26)∥xn+1-xn∥2≤2αn2∥γf(xn)-Axn∥2+2(1-βn-αnγ-)2∥Bnxn-xn∥2.
from
(3.27)∥Bnxn-xn∥2=∥∑i=1N‍λi(n)Tωixn-xn∥2=∥∑i=1N‍λi(n)(Tωixn-xn)∥2≤2∑i=1N‍(λi(n))2ωi2∥xn-Tixn∥2≤2∑i=1N‍λi(n)ωi∥xn-Tixn∥2
and (1-βn-αnγ-)2≤(1-βn-αnγ-) yields
(3.28)12∥xn+1-xn∥2≤αn2∥γf(xn)-Axn∥2+(1-βn-αnγ-)∑i=1N‍(λi(n)ωi∥xn-Tixn∥2).
Then from (3.24) and (3.28), we have
(3.29)𝒯n+1-𝒯n+[(1-βn-αn)2-(1-βn-αnγ-)]∑i=1N‍(λi(n)ωi∥xn-Tixn∥2)𝒯n+1≤αn(αn∥γf(xn)-Axn∥2-〈(A-γf)xn,xn-x*〉+〈(I-A)(Bn-I)xn,xn-x*〉).
The rest of the proof will be divided into two parts.
Case1. Suppose that there exists n0 such that {𝒯n}n≥n0 is nonincreasing. In this situation, {𝒯n} is then convergent because it is also nonnegative (hence it is bounded from below), so that limn→∞(𝒯n+1-𝒯n)=0; hence, in light of (3.29) together with limn→∞αn=0, 0<liminfn→∞βn≤liminfn→∞βn<1, and the boundedness of {xn}, we obtain
(3.30)limn→∞∑i=1N‍(λi(n)ωi∥xn-Tixn∥2)=0.
By (3.27) and (3.30), we can easily claim that
(3.31)limn→∞∥Bnxn-xn∥=0.
It also follows from (3.29) that
(3.32)𝒯n-𝒯n+1≥αn(-αn∥γf(xn)-Axn∥2+〈(A-γf)xn,xn-x*〉+〈(I-A)(Bn-I)xn,xn-x*〉).
Then, by ∑n=0∞‍αn=∞, we obviously deduce that
(3.33)liminfn→∞(-αn∥γf(xn)-Axn∥2+〈(A-γf)xn,xn-x*〉+〈(I-A)(Bn-I)xn,xn-x*〉)≤0.
Since {f(xn)} and {xn} are both bounded, limn→∞αn=0 and limn→∞∥Bnxn-xn∥=0, we obtain
(3.34)liminfn→∞〈(A-γf)xn,xn-x*〉≤0.
Moreover, by Remark 1.1, we have
(3.35)2(γ--γβ)𝒯n+〈(A-γf)x*,xn-x*〉≤〈(A-γf)xn,xn-x*〉,
which by (3.34) entails
(3.36)liminfn→∞(2(γ--γβ)𝒯n+〈(A-γf)x*,xn-x*〉)≤0,
hence, recalling that limn→∞𝒯n exists, we equivalently obtain
(3.37)2(γ--γβ)limn→∞𝒯n+liminfn→∞〈(A-γf)x*,xn-x*〉≤0,
namely,
(3.38)2(γ--γβ)limn→∞𝒯n≤-liminfn→∞〈(A-γf)x*,xn-x*〉.
From (3.30) and invoking Lemma 3.2, we obtain
(3.39)liminfn→∞〈(A-γf)x*,x^-x*〉≥0,x^∈⋂n=1N‍Fix(Tωi),
which by (3.38) yields limn→∞𝒯n=0, so that {xn} converges strongly to x*.
Case2. Suppose there exists a subsequence {𝒯nk}k≥0 of {𝒯n}n≥0 such that 𝒯nk≤𝒯nk+1 for all k≥0. In this situation, we consider the sequence of indices {δ(n)} as defined in Lemma 2.9. It follows that 𝒯δ(n+1)-𝒯δ(n)>0, which by (3.29) amounts to
(3.40)[(1-βδ(n)-αδ(n))2-(1-βδ(n)-αδ(n)γ-)]∑i=1N‍(λi(n)ωi∥xδ(n)-Tixδ(n)∥2)βδ(n)≤αδ(n)(αδ(n)∥γf(xδ(n))-Axδ(n)∥2-〈(A-γf)xδ(n),xn-x*〉),
hence, by the boundedness of {xn} and limn→∞αn=0, we immediately obtain
(3.41)limn→∞∑i=1N‍(λi(n)ωi∥xδ(n)-Tixδ(n)∥2)=0.
From (3.28) we have
(3.42)12∥xδ(n)+1-xδ(n)∥2xδ(n)≤αδ(n)2∥γf(xδ(n))-Axδ(n)∥2+(1-βδ(n)-αδ(n)γ-)∑i=1N‍(λi(n)ωi∥xδ(n)-Tixδ(n)∥2)xδ(n)≤αδ(n)∥γf(xδ(n))-Axδ(n)∥2+(1-βδ(n)-αδ(n)γ-)∑i=1N‍(λi(n)∥xδ(n)-Tixδ(n)∥2),
which together with (3.41), limn→∞αn=0 and 0<liminfn→∞βn≤limsupn→∞βn<1 yields
(3.43)limn→∞∥xδ(n)+1-xδ(n)∥=0.
Now by (3.40), we clearly have
(3.44)αδ(n)∥γf(xδ(n))-μBxδ(n)∥2≥〈(A-γf)xδ(n),xδ(n)-x*〉,
which in the light of (3.38) yields
(3.45)2(γ--γβ)𝒯δ(n)+〈(A-γf)x*,xδ(n)-x*〉≤αδ(n)∥γf(xδ(n))-Axδ(n)∥2,
hence (as limn→∞αδ(n)∥γf(xδ(n))-Axδ(n)∥2=0) it follows that
(3.46)2(γ--γβ)limsupn→∞𝒯δ(n)≤-liminfn→∞〈(A-γf)x*,xδ(n)-x*〉.
From (3.41) and invoking Lemma 3.2, we obtain
(3.47)limn→∞〈(A-γf)x*,x^-x*〉≥0,x^∈⋂n=1N‍Fix(Tωi),
which by (3.46) yields limsupn→∞𝒯δ(n)=0, so that limn→∞𝒯δ(n)=0. Combining (3.43), we have limn→∞𝒯δ(n)+1=0. Then, recalling that 𝒯n<𝒯δ(n)+1 (by Lemma 2.9), we get limn→∞𝒯n=0, so that xn→x* strongly. In addition, the variational inequality (3.39) and (3.47) can be written as
(3.48)〈(I-A+γf)x*-x*,x^-x*〉≤0,x^∈⋂n=1N‍Fix(Tωi).
So, by the Lemma 2.10, it is equivalent to the fixed point equation
(3.49)x*=P⋂n=1N‍Fix(Tωi)(I-A+γf)x*=(P⋂n=1N‍Fix(Tωi)·f)x*.If the sequences of the weights{λi(n)}i=1N={λi}i=1N in (3.2), according to the proof of Theorem 3.3, we can obtain the following corollary.Corollary 3.4.
LetC be a closed convex subset of a Hilbert space H and let Ti:C→C be a quasi-nonexpansive for Tωi=(1-ωi)I+ωiTi, ωi∈(0,1), i∈(1,…,N) such that ⋂i=1N‍Fix(Tωi)≠∅, f is a contraction with coefficient β∈(0,1) and λi is a positive constant such that ∑i=1N‍λi=1. Let A be a strongly positive bounded linear operator with coefficient γ-. Given the initial guess x0∈H chosen arbitrarily and given sequences {αn} and {βn} in (0,1), satisfying the following conditions: (c1)
limn→∞αn=0 and ∑n=0∞‍αn=∞;(c2)
0<liminfn→∞βn≤limsupn→∞βn<1.
Let{xn} be the sequence generated by (3.1). Then {xn} converges strongly to the unique a element x*∈⋂i=1N‍Fix(Tωi),N≥1 verifying
(3.50)x*=(P⋂i=1N‍Fix(Tωi)·f)x*,
which equivalently solves the following variational inequality problem:
(3.51)x*∈⋂i=1N‍Fix(Tωi),〈(γf-A)x*,x^-x*〉≤0,∀x^∈⋂i=1N‍Fix(Tωi).


## 4. Cyclic Algorithm

In this section, we discuss the cyclic algorithm, respectively, for solving the variational inequality over the set of the common fixed points of finite quasi-nonexpansives and introduce quasi-shrinking mapping and quoted its definition from [11]. Hereafter, for nonempty closed set S⊂H and r≥0, we use the notations dS:H∋u↦d(u,S):=infx∈S∥u-x∥, ⋄(S,r):={u∈H∣d(u,S)=r}, ⊲(S,r):={u∈H∣d(u,S)≤ r} and ⊳(S,r):={u∈H∣d(u,S)≥r}. In this case, by the upper semicontinuity of ds (see e.g., [14, Theorem 1.3.3]), ⊳(S,r) is closed. Moreover, for a nonempty bounded closed convex set C⊂H and r≥0, it is not hard to verify that (i) ⋄(C,r) and ⊲(C,r) are also closed; (ii) ⋄(C,r) and ⊲(C,r) are bounded; (iii) ⊲(C,r) is convex.Definition 4.1 (see [11]).
Suppose thatT:H→H is quasi-nonexpansive with Fix(T)⋂C≠∅ for some closed convex set C. Then T:H→H is called quasi-shrinking on C if
(4.1)D:r∈[0,∞)↦{infu∈⊳(Fix(T),r)∩Cd(u,Fix(T))-d(T(u),Fix(T)),ifu∈⊳(Fix(T),r)∩C≠∅,∞otherwise
satisfies D(r)=0⇔r=0. In particular, if T is quasi-shrinking on H, then T is just called quasi-shrinking.LetC be a closed convex subset of a Hilbert space H and let {Ti}i=1N-1 be quasi-nonexpansives defined on C such that the common fixed point set
(4.2)F:=⋂i=1N‍Fix(Tωi),N≥1,
where Tωi=(1-ωi)I+ωiTi, {ωi}i=1N∈(0,1). Let x0∈C, let {αn}n=0∞, and {βn}n=0∞ sequences in (0,1). The cyclic algorithm generates a sequence {xn}n=1∞ in the following way:
(4.3)x1=α0γf(x0)+β0x0+((I-β0)I-α0A)Tω0x0,x2=α1γf(x1)+β1x1+((I-β1)I-α1A)Tω1x1,⋮xN=αN-1γf(xN-1)+βN-1xN-1+((I-βN-1)I-αN-1A)TωN-1xN-1,xN+1=αN-1γf(xN-1)+βNxN+((I-βN)I-αNA)TωNxN,⋮
In general, xn+1 is defined by
(4.4)xn+1=αnγf(xn)+βnxn+((I-βn)I-αnA)T[n]xn,
where T[n]=Tωi=(1-ωi)I+ωiTi, with i=n (mod N), 0≤i≤N-1.Lemma 4.2 (see [11]).
Letφ(x):[0,∞)→[0,∞) satisfy (i)
x1>x2⇒φ(x1)>φ(x2),(ii)
φ(x)=0⇔x=0.
Let{zn}n≥1⊂[0,∞) satisfy limn→∞zn=0. Then any sequence {bn}n≥1⊂[0,∞) satisfying
(4.5)bn+1≤bn-φ(bn)+zn+1,n=0,1,2,…
converges to 0.Lemma 4.3 (see [15]).
Assume that{αn}n=0∞ is a sequence of nonnegative real numbers such that
(4.6)αn+1≤(1-γn)αn+γnδn,n≥0,
where {γn}n=0∞⊂[0,1] and {δn}n=0∞ satisfy the following conditions: (i)
∑n=0∞‍γn=∞ and limn→∞γn=0,(ii)
limsupn→∞δn≤0 or ∑n=0∞‍|γnδn|<∞.
Thenlimn→∞αn=0.Theorem 4.4.
LetC be a closed convex subset of a Hilbert space H and let Ti:H→H be quasi-nonexpansives for Tωi=(1-ωi)I+ωiTi, ωi∈(0,1), i∈(1,2,…,N) such that F:=⋂i=1N‍Fix(Tωi)≠∅ and f a contraction with coefficient β∈(0,1). Let A be a strongly positive bounded linear operator with coefficient γ-. Given the initial guess x0∈H chosen arbitrarily and given sequences {αn} and {βn} in (0,1), satisfying the following conditions: (4.1a)
limn→∞αn=0 and ∑n=0∞‍αn=∞;(4.1b)
∑n=0∞‍∥αn+1-αn∥<∞ or limn→∞αn/αn+1=1;(4.1c)
0<liminfn→∞βn≤limsupn→∞βn<1.
Let{xn} be the sequence generated by (4.4). Then {xn} converges strongly to the unique a element x*∈F:=⋂i=1N‍Fix(Tωi),N≥1 verifying
(4.7)x*=(P⋂i=1N‍Fix(Tωi)·f)x*,
which equivalently solves the following variational inequality problem:
(4.8)x*∈⋂i=1N‍Fix(Tωi),〈(γf-A)x*,x^-x*〉≤0,∀x^∈⋂i=1N‍Fix(Tωi).Proof.
Take ap∈F:=⋂i=1N‍Fix(Tωi). We break the proof process into several steps.
Step1.{xn} is bounded. In light of the Remark 2.3, we obtain
(4.9)∥T[n](xn-p)∥=∥Tωi(xn-p)∥≤∥xn-p∥.
From (4.4), we have
(4.10)∥xn+1-p∥=∥αn(γf(xn)-Ap)+βn(xn-p)+((I-βn)I-αnA)T[n](xn-p)∥≤αn∥γf(xn)-Ap∥+βn∥xn-p∥+(1-βn-αnγ-)∥xn-p∥≤αnγ∥f(xn)-f(p)∥+αn∥γf(p)-Ap∥+(1-αnγ-)∥xn-p∥=[1-αn(γ--γβ)]∥xn-p∥+αn∥γf(p)-Ap∥.
By simple inductions, we obtain
(4.11)∥xn-p∥≤max{∥x1-p∥,∥γf(p)-Ap∥γ--γβ},∀n≥1,
which gives that the sequence {xn} is bounded; we also know that {T[n]xn} and {f(xn)} are bounded.
Step2. Moreover if T[n]:H→H is quasi-shrinking on the set C, we obtain the following statements: (a)
limn→∞d(xn,F)=0;(b)
limn→∞∥T[n]xn-xn∥=0;(c)
limn→∞∥xn+1-xn∥=0.
By the boundedness of{xn}, {T[n]xn}, and {f(xn)}, there exists M>0 satisfying
(4.12)maxn≥0{∥xn∥,∥T[n]xn∥,∥f(xn)∥}≤M.
By a simple inspection, we deduce
(4.13)d(xn+1,F)≤∥xn+1-PF(T[n]xn)∥=∥αnγf(xn)+βnxn+((I-βn)I-αnA)T[n]xn-PF(T[n]xn)∥≤αn∥γf(xn)-AT[n]xn∥+βn∥xn-T[n]xn∥+∥T[n]xn-PF(T[n]xn)∥≤αn(γ∥f(xn)∥+∥AT[n]xn∥)+βn(∥xn∥+∥T[n]xn∥)+∥T[n]xn-PF(T[n]xn)∥≤d(T[n]xn,F)+2(αn+βn)M.
By {xn}n≥0⊂C, we can assume the boundedness of the sequence bn:=d(xn,F)≥0(n∈ℕ). Moreover, by Definition 4.1 and (4.13), it follows that
(4.14)D(bn)≤bn-d(T[n]xn,F)≤bn-bn+1+2(αn+βn)M,∀n≥0.
Now application of Lemma 4.2 to (4.14) yields limn→∞bn=0, hence (a) is proved.
The statements (b) and (c) are verified by(4.15)∥T[n]xn-xn∥=∥T[n]xn-PF(xn)+PF(xn)-xn∥≤∥T[n]xn-PF(xn)∥+∥PF(xn)-xn∥≤∥xn-PF(xn)∥+∥PF(xn)-xn∥=2d(xn,F)→0,(n→∞),(4.16)∥xn+1-xn∥=∥αnγf(xn)+βnxn+((I-βn)I-αnA)T[n]xn-xn∥≤αn∥γf(xn)-Axn∥+(1-βn-αnγ-)∥T[n]xn-xn∥→0,(n→∞).
Step3.limn→∞∥xn+N-xn∥=0. From (4.4) and (4.16), we obtain
(4.17)∥xn+N+1-xn+1∥=∥αn+Nγf(xn+N)+βn+Nxn+N+((I-βn+N)I-αn+NA)T[n+N]xn+N2-αnγf(xn)-βnxn-((I-βn)I-αnA)T[n]xn∥=∥αn+Nγf(xn+N)-αn+Nγf(xn)+βn+N(xn+N-T[n+N]xn+N)2+(I-αn+NA)T[n+N]xn+N-(I-αn+NA)T[n]xn2+(I-αn+NA)T[n]xn-(I-αnA)T[n]xn2+αn+Nγf(xn)-αnγf(xn)+βn(xn-T[n]xn)∥≤αn+Nγβ∥xn+N-xn∥+βn+N∥xn+N-T[n+N]xn+N∥+(1-αn+Nγ-)∥T[n]xn-T[n+N]xn+N∥+|αn+αn+N|γ-∥T[n]xn∥+|αn-αn+N|γβ∥xn∥-βn∥xn-T[n]xn∥=αn+Nγβ∥xn+N-xn∥+βn+N∥xn+N-T[n+N]xn+N∥+(1-αn+Nγ-)∑j=nN-1‍∥T[j]xj-T[j+1]xj+1∥+|αn-αn+N|γ-∥T[n]xn∥+|αn-αn+N|γβ∥xn∥-βn∥xn-T[n]xn∥≤αn+Nγβ∥xn+N-xn∥+βn+N∥xn+N-T[n+N]xn+N∥+(1-αn+Nγ-)∑j=nN-1‍(∥xj+1-T[j+1]xj+1∥+∥xj-xj+1∥+∥xj-T[j]xj∥)+|αn-αn+N|γ-∥T[n]xn∥+|αn-αn+N|γβ∥xn∥+βn∥xn-T[n]xn∥.
By conditions (4.1a), (4.1b), (4.1c), (4.15), and (4.16), {xn} and {T[n]xn} are bounded we obtain that
(4.18)limn→∞∥xn+N-xn∥=0.
Step4.limn→∞∥xn-T[n+N]⋯T[n+1]xn∥=0.
From (4.4), we observe that
(4.19)∥xn+1-T[n]xn∥=αnγ∥f(xn)-AT[n]xn∥+βn∥xn-T[n]xn∥.
It follows from the condition (4.1a), (4.1c), (4.15), and the boundedness of {f(xn)} and {T[n]xn} that
(4.20)∥xn+1-T[n]xn∥→0,(n→∞).
Recursively,
(4.21)∥xn+N-T[n+N]xn+N-1∥→0,(n→∞),∥xn+N-1-T[n+N-1]xn+N-2∥→0,(n→∞).
By Remark 2.3, T[n+N] is quasi-nonexpansive, we obtain
(4.22)∥T[n+N]xn+N-1-T[n+N]T[n+N-1]xn+N-2∥→0,(n→∞).
Proceeded accordingly, we obtain
(4.23)∥T[n+N]T[n+N-1]xn+N-2-T[n+N]T[n+N-1]T[n+N-2]xn+N-3∥→0,(n→∞),⋮∥T[n+N]⋯T[n+2]xn+1-T[n+N]⋯T[n+1]xn∥→0,(n→∞).
Note that
(4.24)∥x[n+N]-T[n+N]⋯T[n+1]xn∥≤∥xn+N-T[n+N]xn+N-1∥+∥T[n+N]xn+N-1-T[n+N]T[n+N-1]xn+N-2∥⋮+∥T[n+N]⋯T[n+2]xn+1-T[n+N]⋯T[n+1]xn∥.
From all the expressions above, we have
(4.25)∥x[n+N]-T[n+N]⋯T[n]xn∥→0,(n→∞).
Since
(4.26)∥xn-T[n+N]⋯T[n+1]xn∥≤∥xn+xn+N∥+∥xn+N-T[n+N]⋯T[n]xn∥,
it is concluded that
(4.27)limn→∞∥xn-T[n+N]⋯T[n+1]xn∥=0.
Step5.ωw(xn)⊂⋂n=1N‍Fix(T[n])=⋂n=1N‍Fix(Tωi),i=1,2,…,N. Assume that {xnj}⊂{xn} such that xnj⇀x^, we prove x^∈⋂n=1N‍Fix(T[n]). By the conclusion of step 4, we obtain
(4.28)limj→∞∥xnj-T[nj+N]⋯T[nj+1]xnj∥=0.
Observe that, for each nj, T[nj+N],…,T[nj+1] is some permutation of the mappings T[1],…,T[N], since T[1],…,T[N] are finite, all the full permutation are N!, there must be some permutation that appears infinite times. Without loss of generality, suppose that this permutation is T[1],…,T[N], we can take a subsequence {xnjk}⊂{xnj} such that
(4.29)limj→∞∥xnj-T[1]⋯T[N]xnj∥=0.
It is easy to prove that T[1],…,T[N] is quasi-nonexpansive. By Lemma 2.5, we have
(4.30)x^=T[1]⋯T[N]x^.
Using Remark 2.3 and Lemma 2.5, we obtain
(4.31)x^∈Fix(T[1]⋯T[N])=⋂n=1N‍Fix(T[n])=⋂n=1N‍Fix(Tn).
Step6.liminfn→∞〈(γf-A)x*,xn-x*〉≤0. Indeed, there exists a subsequence {xnj}⊂{xn} such that
(4.32)liminfn→∞〈(γf-A)x*,xn-x*〉=limj→∞〈(γf-A)x*,xnj-x*〉.
Without loss of generality, we may further assume that xnj⇀x^. It follows from (4.31) that x^∈⋂n=1N‍F(Tn). Since x* is the unique solution of (4.8), we have
(4.33)liminfn→∞〈(γf-A)x*,xn-x*〉=limj→∞〈(γf-A)x*,xnj-x*〉=〈(γf-A)x*,x^-x*〉≤0.
In addition, the variational inequality (4.33) can be written as
(4.34)〈(I-A+γf)x*-x*,x^-x*〉≤0,x^∈⋂n=1N‍Fix(Tωi).
So, by the Lemma 2.10, it is equivalent to the fixed point equation
(4.35)x*=P⋂n=1N‍Fix(Tωi)(I-A+γf)x*=(P⋂n=1N‍Fix(Tωi)·f)x*.
Step7.limn→∞∥xn-x*∥=0. From (4.4), we obtain
(4.36)∥xn+1-x*∥2=∥αnγf(xn)+βnxn+((I-βn)I-αnA)T[n]xn-x*∥2=〈αn(γf(xn)-Ax*)+βn(xn-x*)+((I-βn)I-αnA)(T[n]xn-x*),xn+1-x*〉≤αn〈γf(xn)-Ax*,xn+1-x*〉+βn〈xn-x*,xn+1-x*〉+(1-βn-αn)〈T[n]xn-x*,xn+1-x*〉+αn〈(I-A)(T[n]xn-x*),xn+1-x*〉≤αn(〈(I-A)(T[n]xn-x*),xn+1-x*〉〈γf(xn)-f(x*),xn+1-x*〉+〈γf(x*)-Ax*,xn+1-x*〉+〈(I-A)(T[n]xn-x*),xn+1-x*〉)+βn∥xn-x*∥∥xn+1-x*∥+(1-βn-αnγ-)∥T[n]xn-x*∥∥xn+1-x*∥≤αnγβ∥xn-x*∥∥xn+1-x*∥+αn〈γf(x*)-Ax*,xn+1-x*〉αn(1-γ-)∥T[n]xn-x*∥∥xn+1-x*∥+βn∥xn-x*∥∥xn+1-x*∥+(1-βn-αnγ-)∥xn-x*∥∥xn+1-x*∥≤1-(γ--γβ)2(∥xn-x*∥2+∥xn+1-x*∥2)+αnM≤1-(γ--γβ)2∥xn-x*∥2+12∥xn+1-x*∥2+αnM,
where M=(1-γ-)∥T[n]xn-x*∥∥xn+1-x*∥+〈γf(x*)-Ax*,xn+1-x*〉. It follows that
(4.37)∥xn+1-x*∥2≤(1-(γ--γβ))∥xn-x*∥2+2αnM.
By using Lemma 4.3, we can obtain the desired conclusion easily.


## 5. Application

In this section, we constructed a numerical example to compare the parallel algorithm and cyclic algorithm which is simple.Letx=(x1,x2)∈R2 and f(x)=(1/2)(sin(x1),cos(x2)) be a contraction mapping with coefficient 1/2. Let T1(x)=(0,4x1) and T2(x)=(4x2,0) be quasi-nonexpansive mappings. Let αn=βn=1/3, A=I and γ=λ1=λ2=1/2. According to (1.20) and (1.24), we can obtain the following parallel algorithm and cyclic algorithm:Parallel Algorithm
(5.1)
x
n
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1
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f
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x
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12
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.Cyclic Algorithm
(5.2)
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⋮
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T
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6
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T
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1
.
Form Theorems 3.3 and 4.4, we can easily know that parallel algorithm (5.1) and cyclic algorithm (5.2) are converge to the unique point in R2. Let x0=(5,2) and |xn+1-xn|2≤10-9, and let xP* and xX* be the fixed point of the parallel algorithm and cyclic algorithm. Using the software of MATLAB, we obtain xP*=x133=(0.6821,0.7080) and xX*=x166=(1.9325,0.8729). From the computed results of xP* and xX*, we can easily know that parallel algorithm (5.1) is simpler than cyclic algorithm (5.2). On the other hand, we need to explain that those algorithms do not converge a common fixed point, because parallel algorithm (5.1) and cyclic algorithm (5.2) have the different algorithm structure.



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*Source: 218341-2012-09-26.xml*