# Weak and Strong Convergence Theorems for the Multiple-Set Split Equality Common Fixed-Point Problems of Demicontractive Mappings

Authors: Yaqin Wang; Tae-Hwa Kim; Xiaoli Fang
Journal: Journal of Function Spaces
 (2017)

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## Abstract

We consider mixed parallel and cyclic iterative algorithms in this paper to solve the multiple-set split equality common fixed-point problem which is a generalization of the split equality problem and the split feasibility problem for the demicontractive mappings without prior knowledge of operator norms in real Hilbert spaces. Some weak and strong convergence results are established. The results obtained in this paper generalize and improve the recent ones announced by many others.


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## Body

## 1. Introduction

For modeling inverse problems which arise from phase retrieval and in medical image reconstruction, in 1994, Censor and Elfving [1] firstly introduced the following split feasibility problem (SFP) in finite-dimensional Hilbert spaces.LetC and Q be nonempty closed convex sets of the Hilbert spaces H1 and H2, respectively, and let A:H1→H2 be a bounded linear operator. The split feasibility problem (SFP) is formulated to(1)findx∗∈CsuchthatAx∗∈Q. It has been found that the SFP can be used in many areas such as image restoration, computer tomograph, and radiation therapy treatment planning. Some methods have been proposed to solve split feasibility problems; see, for instance, [2–5].Assuming that the SFP (1) has a solution, it is not hard to see that (2)x∗=PCI-γA∗I-PQAx∗, where PC and PQ are the metric projections from H1 onto C and from H2 onto Q, respectively, γ is a positive constant, and A∗ denotes the adjoint of A. This implies that SFP can be solved by using fixed-point algorithms.In 2013, Moudafi and Al-Shemas [6] introduced the following new split feasibility problem, which is called the split equality fixed-point problem (SEFP). Let H1,H2,H3 be real Hilbert spaces, let A:H1→H3 and B:H2→H3 be two bounded linear operators, and let U:H1→H1 and T:H2→H2 be two firmly quasi-nonexpansive mappings. The SEFP in [6] is to(3)findx∗∈FU,y∗∈FTsuchthatAx∗=By∗, where FU and FT denote the sets of the fixed points of mappings U and T, respectively. The goal is to cover many situations, for instance, in decomposition methods for PDFs and applications in game theory and in intensity-modulated radiation therapy (IMRT).For solving the SEFP (3), Moudafi and Al-Shemas [6] introduced the following simultaneous iterative method: (4)xk+1=Uxk-γkA∗Axk-Byk,yk+1=Tyk+γkB∗Axk-Byk, for firmly quasi-nonexpansive mappings U and T, where γk∈ϵ,2/λA+λB-ϵandλAandλB stand for the spectral radii of A∗A and B∗B, respectively.In 2014, Zhao [7] introduced the following simultaneous Mann iterative algorithm:(5)uk=xk-γkA∗Axk-Byk,xk+1=1-αkuk+αkUuk,vk=yk+γkB∗Axk-Byk,yk+1=1-βkvk+βkTvk, where the step size γk does not depend on the operator norms A and B. And she proved the weak convergence of this algorithm (5) to solve SEFP (3) governed by quasi-nonexpansive operators U and T.Recently, the multiple-set split equality common fixed-point problem (MSECFP) studied by Zhao and Wang [8] is to(6)findx∗∈⋂i=1pFUi,y∗∈⋂j=1qFTjsuchthatAx∗=By∗,where p,q≥1 are integers. They introduced the following two mixed iterative algorithms for solving the MSECFP (6) of quasi-nonexpansive mappings:(7)uk=xk-γkA∗Axk-Byk,xk+1=αk0uk+αk1U1uk+⋯+αkpUpuk,vk=yk+γkB∗Axk-Byk,yk+1=βkvk+1-βkTjkvkand(8)uk=xk-γkA∗Axk-Byk,xk+1=αkuk+1-αkUikuk,vk=yk+γkB∗Axk-Byk,yk+1=βk0vk+βk1T1vk+⋯+βkqTqvk, where the step size γk does not depend on the operator norms A and B. And they proved the weak convergence of such algorithms.Very recently, Wang and Kim [9] introduced the following iterative scheme for solving SEFP (3) of demicontractive mappings,(9)∀x0∈H1,y0∈H2,uk=xk-γkA∗Axk-Byk,wk=1-tkuk,xk+1=αkwk+1-αkUwk,vk=yk+γkB∗Axk-Byk,pk=1-tkvk,yk+1=βkpk+1-βkTpk, and obtain a strong convergence result with no compactness assumptions on the spaces or the mappings and with no extra conditions on the fixed-point sets.Inspired and motivated by the works mentioned above, we consider the mixed parallel and cyclic iterative algorithms for MSECFP (6) of demicontractive mappings which are a generalization of quasi-nonexpansive mappings without prior knowledge of operator norms in Hilbert spaces. Under some mild assumptions, we prove weak and strong convergence results of such algorithms for solving MSECFP (6).


## 2. Preliminaries

Throughout this paper, we always assume thatH1,H2,H3 are real Hilbert spaces and let N and R be the set of positive integers and real numbers, respectively. In what follows, we denote strong and weak convergence in the space by “→" and “⇀", respectively, and the set of the fixed points of a mapping T by FT. Also, we use ωwxk=x:∃xkj⇀x to represent weak ω-limit set of xk.LetC be a nonempty closed convex subset of a Hilbert space H. The metric (or nearest point) projection PC from H onto C is defined as follows: given x∈H, the unique point PCx∈C satisfies the property (10)x-PCx=infy∈C⁡x-y. It is well known [10] that PC is a nonexpansive mapping and is characterized by the inequality(11)PCx∈C,x-PCx,y-PCx≤0,∀y∈C.Definition 1.
LetH be a real Hilbert space. A mapping T:H→H is said to be (i)
nonexpansive ifTx-Ty≤x-y,∀x,y∈H;(ii)
quasi-nonexpansive ifFT≠∅ and if Tx-q≤x-q,∀x∈H,q∈FT;(iii)
firmly nonexpansive if(12)Tx-Ty2≤x-y2-I-Tx-I-Ty2,∀x,y∈H, or equivalently (13)Tx-Ty2≤x-y,Tx-Ty,∀x,y∈H; (iv) firmly quasi-nonexpansive if FT≠∅ and (14)Tx-q2≤x-q2-x-Tx2,∀x∈H,q∈FT; (v) μ-demicontractive if FT≠∅ and there exists a constant μ∈-∞,1 such that (15)Tx-q2≤x-q2+μx-Tx2,∀x∈H,q∈FT.Remark 2.
Notice that every 0-demicontractive mapping is exactly quasi-nonexpansive. In particular, we say that it is quasi-strictly pseudocontractive [11] if 0≤μ<1. Moreover, if μ≤0, every μ-demicontractive mapping becomes quasi-nonexpansive. Therefore, it is sufficient to only take μ∈0,1 in (v) of Definition 1 in Hilbert spaces. However, as seen in (iv) of Definition 1, every firmly quasi-nonexpansive mapping (often called to be a directed operator [12])) is obviously -1-demicontractive.
It is worth noting that the class of demicontractive mappings is more general than the class of quasi-nonexpansive mappings and the class of firmly quasi-nonexpansive mappings.Definition 3.
LetH be a real Hilbert space. An operator T:H→H is called demiclosed at origin for any sequence xk which converges weakly to x, and if the sequence Txn converges strongly to 0, then Tx=0.As a special case of the demicloseness principle on uniformly convex Banach spaces given by [13], we know that if C is a nonempty closed convex subset of a Hilbert space H and T:C→H is a nonexpansive mapping, then the mapping I-T is demiclosed on C. Now, the following question naturally arises: If T:C→H is quasi-nonexpansive, is I-T still demiclosed on C? The answer is negative even at 0 as follows.Example 4 (see [9]; Example  2.11).
The mappingT  : 0,1→0,1 is defined by(16)Tx=x5,x∈0,12,xsin⁡πx,x∈12,1. Then, T is a quasi-nonexpansive mapping, but I-T is not demiclosed at 0.Definition 5.
LetS be a nonempty closed convex subset of a real Hilbert space H. A sequence xn⊂H isFejér-monotone relative to the target set S (simply S-Fejérian) if(17)xn+1-a≤xn-a,∀a∈S,n≥1.Lemma 6 (see [14]).
LetH be a real Hilbert space. Then, for all t∈0,1 and x,y∈H, (18)tx+1-ty2=tx2+1-ty2-t1-tx-y2.Since the square of the norm is convex, we have the following lemma.Lemma 7.
LetH be a real Hilbert space. Then, (19)α0x0+α1x1+⋯+αrxr2≤∑i=0rαixi2, for xi∈H,αi∈0,1,i=0,1,2,…,r, and ∑i=0rαi=1.Lemma 8 (see [15]).
LetT:H→H be μ-demicontractive with FT≠∅ and set Tα=1-αI+αT for α∈0,1. Then, Tα is quasi-nonexpansive for α∈0,1-μ and (20)Tαx-q2≤x-q2-α1-μ-αx-Tx2,∀x∈H,q∈FT.Lemma 9 (see [11]; Proposition 2.1).
Assume thatC is a closed convex subset of a Hilbert space H. Let T:C→C be a mapping. If T is a μ-demicontractive mapping (which is also called μ-quasi-strict pseudocontraction in [11]), then the fixed-point set FT is closed and convex.Lemma 10 (see [16]).
LetX and Y be Banach spaces and A be a continuous linear operator from X to Y. Then, A is weakly continuous.Lemma 11 (see [17, 18]).
LetS be a nonempty closed convex subset of a real Hilbert space H. Suppose that xn⊂H is S-Fejérian. Then, xn⇀x∈S if and only if ωwxn⊂S.Lemma 12 (see [19]; Lemma 3.1).
LetΓn be a sequence of real numbers that does not decrease at infinity in the sense that there exists a subsequence Γni of Γn which satisfies Γni<Γni+1 for all i∈N. Define the sequence τnn≥n0 of integers as follows: (21)τn=max⁡k≤n:Γk<Γk+1,where n0∈N such that {k≤n0:Γk<Γk+1}}≠∅. Then, there hold the following properties: (i)
τn0≤τn0+1≤⋯ and τn→∞;(ii)
Γτn≤Γτn+1 and Γn≤Γτn+1,∀n≥n0.Lemma 13 (see [20]).
Assume thatak is a sequence of nonnegative real numbers such that (22)ak+1≤1-λkak+λkδk,∀k≥0,where λk is a sequence in (0,1) and δk is a sequence in R such that(a)
∑k=0∞λk=∞ and limk→∞⁡λk=0;(b)
lim⁡supk→∞⁡δk≤0 or ∑k=0∞λkδk<∞.
Then,limk→∞⁡ak=0.


## 3. Main Results

In this section, we study two mixed parallel and cyclic iterative algorithms for MSECFP (6) of demicontractive mappings where the step sizes do not depend on the operator norms A and B, and we prove the weak and strong convergence of such algorithms. Denote the solution set of MSECFP (6) by Ω; that is,(23)Ω≔x,y∣x∈⋂i=1pFUi,y∈⋂j=1qFTjsuchthatAx=By.Given two positive integersp and q, the p-mod function i takes values in the set 1,2,…,p and the q-mod function j takes values in the set 1,2,…,q as(24)ik=r+1,0≤r≤p-1, for k=np+r for some integers n≥0 and 0≤r<p, and(25)jk=r+1,0≤r≤q-1, for k=nq+r for some integers n≥0 and 0≤r<q.PutH∗=H1×H2. Define the inner product on H∗ as follows: (26)x1,y1,x2,y2=x1,x2+y1,y2,∀x1,y1,x2,y2∈H∗. It is easy to see that H∗ is a real Hilbert space and (27)x,y=x2+y21/2,∀x,y∈H∗. Note also that if xk,yk is a sequence in H∗, there holds the following relation:(28)x∗,y∗∈ωwxk,yk⟹x∗∈ωwxk,y∗∈ωwyk.Algorithm 14.
Letx0∈H1,y0∈H2 be arbitrary and p,q≥1 be integers. Let αki⊂0,10≤i≤p such that ∑i=0pαki=1 and βk⊂0,1. Assume that the kth iterate xk,yk∈H∗ has been constructed and Axk-Byk≠0, and then we calculate the k+1th iterate xk+1,yk+1 via the formulas(29)uk=xk-γkA∗Axk-Byk,xk+1=αk0uk+αk1U1uk+⋯+αkpUpuk,vk=yk+γkB∗Axk-Byk,yk+1=1-βkvk+βkTjkvk. Assume the step size γk is chosen in such a way that(30)γk∈ε,min⁡η,2Axk-Byk2A∗Axk-Byk2+B∗Axk-Byk2-ε for all k∈Ω~ and some small enough ε>0, where the index set Ω~=k:Axk-Byk≠0, (31)η=2Axl-Byl2A∗Axl-Byl2+B∗Axl-Byl2,l=mink∈Ω~⁡k; otherwise, γk=γ(γ being any nonnegative value). If Axk=Byk=0, then we take uk=xk,vk=yk and(32)xk+1=αk0xk+αk1U1xk+⋯+αkpUpxk,yk+1=1-βkyk+βkTjkyk.Algorithm 15.
Letx0∈H1,y0∈H2 be arbitrary and p,q≥1 be integers. Let αki⊂0,10≤i≤p such that ∑i=0pαki=1 and βk⊂0,1,tk⊂0,1. Assume that the kth iterate xk,yk∈H∗ has been constructed and Axk-Byk≠0, and then we calculate the k+1th iterate xk+1,yk+1 via the formulas(33)uk=xk-γkA∗Axk-Byk,wk=1-tkuk,xk+1=αk0wk+αk1U1wk+⋯+αkpUpwk,vk=yk+γkB∗Axk-Byk,pk=1-tkvk,yk+1=1-βkpk+βkTjkpk. Assume that Ω~,η,l are the same as in Algorithm 14 and for k∈Ω~ the step size γk is chosen as (30); otherwise, γk=γ(γ being any nonnegative value). If Axk=Byk=0, then we take uk=xk,vk=yk and(34)wk=1-tkxk,xk+1=αk0wk+αk1U1wk+⋯+αkpUpwk,pk=1-tkyk,yk+1=1-βkpk+βkTjkpk.
Now, we will see from Lemma16 that γk is well defined and bounded. The proof of the following lemma is also added for completing the proof of Lemma 2.3 in [7] for the sake of convenience.Lemma 16.
Assume the solution setΩ of (6) is nonempty. Then, γk defined by (30) is well defined and bounded.Proof.
Takex,y∈Ω, that is, x∈⋂i=1pFUi, y∈⋂j=1qFTj, and Ax=By. We have (35)A∗Axk-Byk,xk-x=Axk-Byk,Axk-Ax;B∗Axk-Byk,y-yk=Axk-Byk,By-Byk. By adding the above two equalities and by taking Ax=By into account, we obtain (36)Axk-Byk2=A∗Axk-Byk,xk-x+B∗Axk-Byk,y-yk≤A∗Axk-Bykxk-x+B∗Axk-Byky-yk. Consequently, for k∈Ω, that is, Axk-Byk>0, we have A∗Axk-Byk≠0 or B∗Axk-Byk≠0. And since (37)A∗Axk-Byk2≤A∗2Axk-Byk2,(38)B∗Axk-Byk2≤B∗2Axk-Byk2, we have (39)2Axk-Byk2A∗Axk-Byk2+B∗Axk-Byk2≥2A∗2+B∗2=2A2+B2. Thus, we can choose small enough ε∈0,1/A2+B2. This causes γk to be well defined. From (30), we obtain that γk∈ε,η-ε and η is a fixed positive number, so γk is bounded.Assumption 17.
Assume that(i)
the solution setΩ of (6) is nonempty;(ii)
Ui:H1→H11≤i≤p and Tj:H2→H21≤j≤q are τi-demicontractive and μj-demicontractive, respectively;(iii)
I-Ui1≤i≤p and I-Tj1≤j≤q are demiclosed at origin.
Letτ=max1≤i≤p⁡τi and μ=max1≤j≤q⁡μj. Clearly, Ui is τ-demicontractive for all 1≤i≤p and Tj is μ-demicontractive for all 1≤j≤q.Lemma 18.
Given two bounded linear operatorsA:H1→H3 and B:H2→H3, let Ui:H1→H11≤i≤p and Tj:H2→H21≤j≤q be τi-demicontractive and μj-demicontractive, respectively. Assume that the solution set Ω of (6) is nonempty. Then, Ω is a nonempty closed convex set.Proof.
By Lemma9, we have that FTi1≤i≤p and FUj1≤j≤q are both closed convex subsets, and since A and B are both linear, it is easy to see that Ω is a closed convex subset in H∗.Lemma 19 (see [21]; Lemma 3.1).
Letuk be a bounded sequence of a Hilbert space H. Let q be a positive integer and J=1,2,…,q. If limk→∞⁡uk+1-uk=0 and x∗∈ωwuk, then, for any j∈J, there exists a subsequence ukm of uk depending on j such that jkm=j and ukm⇀x∗.Theorem 20.
Let Assumption17 be satisfied. Given two bounded linear operators A:H1→H3 and B:H2→H3, if the following conditions are satisfied:(a)
τ<lim⁡infk→∞⁡αk0 and lim⁡infk→∞⁡αki>01≤i≤p,(b)
0<lim⁡infk→∞⁡βk≤lim⁡supk→∞⁡βk<1-μ,
then the sequencexk,yk generated by Algorithm 14 converges weakly to a solution of (6).Proof.
First, we claim that the sequencexk,yk is Ω-Fejérian. Indeed, taking x∗,y∗∈Γ, that is, x∗∈⋂i=1pFUi, y∗∈⋂j=1qFTj, and Ax∗=By∗, by Algorithm 14, we obtain (40)uk-x∗2=xk-γkA∗Axk-Byk-x∗2=xk-x∗2-2γkxk-x∗,A∗Axk-Byk+γk2A∗Axk-Byk2=xk-x∗2-2γkAxk-Ax∗,Axk-Byk+γk2A∗Axk-Byk2.
Similarly, we have(41)vk-y∗2=yk+γkB∗Axk-Byk-y∗2=yk-y∗2+2γkyk-y∗,B∗Axk-Byk+γk2B∗Axk-Byk2=yk-y∗2+2γkByk-By∗,Axk-Byk+γk2B∗Axk-Byk2.
It follows from the above inequalities andAx∗=By∗ that(42)uk-x∗2+vk-y∗2=xk-x∗2+yk-y∗2-2γkAxk-Ax∗-Byk+By∗,Axk-Byk+γk2A∗Axk-Byk2+γk2B∗Axk-Byk2=xk-x∗2+yk-y∗2-γk2Axk-Byk2-γkA∗Axk-Byk2+B∗Axk-Byk2.
Takingωki=αki/1-αk01≤i≤p, we have ∑i=1pωki=1 for every k≥0 and 1-αk0∑i=1pωki=∑i=1pαki. It follows from (29) that(43)xk+1=αk0uk+1-αk0∑i=1pωkiUiuk=∑i=1pωkiαk0uk+1-αk0Uiuk.
Using Lemma8 for any i∈1,2,…,p, we have(44)αk0uk+1-αk0Uiuk-x∗2≤uk-x∗2-1-αk0αk0-τUiuk-uk2. Thus, by (43) and (44) and Lemma 7, we obtain(45)xk+1-x∗2≤∑i=1pωkiαk0uk+1-αk0Uiuk-x∗2≤∑i=1pωkiuk-x∗2-1-αk0αk0-τUiuk-uk2=uk-x∗2-1-αk0αk0-τ∑i=1pωkiUiuk-uk2. By (ii) of Assumption 17 and from (29) and Lemma 6, we also have(46)yk+1-y∗2=1-βkvk-y∗+βkTjkvk-y∗2=1-βkvk-y∗2+βkTjkvk-y∗2-βk1-βkTjkvk-vk2≤1-βkvk-y∗2+βkvk-y∗2+μβkTjkvk-vk2-βk1-βkTjkvk-vk2=vk-y∗2-βk1-βk-μvk-Tjkvk2. Then, it follows from (42), (45), and (46) that(47)xk+1,yk+1-x∗,y∗2=xk+1-x∗2+yk+1-y∗2≤uk-x∗2+vk-y∗2-1-αk0αk0-τ×∑i=1pωkiUiuk-uk2-βk1-βk-μvk-Tjkvk2≤xk-x∗2+yk-y∗2-γk2Axk-Byk2-γkA∗Axk-Byk2+B∗Axk-Byk2-αk0-τ×∑i=1pαkiUiuk-uk2-βk1-βk-μvk-Tjkvk2(48)≤xk-x∗2+yk-y∗2-γk2Axk-Byk2-γkA∗Axk-Byk2+B∗Axk-Byk2(49)≤xk,yk-x∗,y∗2-γkεA∗Axk-Byk2+B∗Axk-Byk2(50)≤xk,yk-x∗,y∗2. Hence, xk,yk is Ω-Fejérian.
Next, we show that, for alli∈1,2,…,p, limk→∞⁡uk-Uiuk=0, limk→∞⁡vk-Tjkvk=0, limk→∞⁡uk-xk=0, and limk→∞⁡vk+1-vk=0. In fact, from (50), we see that the sequence xk,yk-x∗,y∗2 is decreasing and bounded below by 0. Consequently, it converges to some finite limit. So, the sequences xk, yk, uk, and vk are also bounded. Then, from (49) and the assumption on γk in (30), it follows that (51)limk→∞⁡A∗Axk-Byk2+B∗Axk-Byk2=0, which together with (48) implies that(52)limk→∞⁡Axk-Byk=0. Also, from (47) and condition (a), it follows that (53)limk→∞⁡∑i=1pαkiUiuk-uk2=0. This, combined with condition (a), implies that, for any i∈1,2,…,p,(54)limk→∞⁡uk-Uiuk=0. It follows from (47) and condition (b) that(55)limk→∞⁡vk-Tjkvk=0. Since (56)uk-xk=γkA∗Axk-Byk≤γkA∗Axk-Byk, it follows from Lemma 16 that γk is bounded, so by (52) we have(57)limk→∞⁡uk-xk=0. Similarly, we have(58)limk→∞⁡vk-yk=0. It follows from (55) and (58) that(59)yk+1-yk≤1-βkvk-yk+βkvk-Tjkvk+βkvk-yk=vk-yk+βkvk-Tjkvk⟶0.
Finally, we claim thatωwxk,yk⊂Ω. To this end, let x~,y~∈ωwxk,yk. From (28), it follows that x~∈ωwxk, y~∈ωwyk. Since ωwxk=ωwuk by (57), we use (54) and Assumption 17(iii) to derive that x~∈⋂i=1pFUi. Given fixed j∈1,2,…,q, since yk+1-yk→0 and y~∈ωwyk, now we apply Lemma 19 to get a subsequence ykm of yk, depending on j, such that ykm⇀y~ and jkm=j for all m. Moreover, it turns out that vkm⇀y~ by (58) and(60)limm→∞⁡vkm-Tjvkm=limm→∞⁡vkm-Tjkmvkm=0by virtue of (55). Using Assumption 17(iii) yields Tjy~=y~ for any j∈1,2,…,q; that is, y~∈⋂j=1qFTj. On the other hand, by Lemma 10, we have Ax~-By~∈ωwAxk-Byk, which together with weakly lower semicontinuity of the norm implies that (61)Ax~-By~≤lim⁡infk→∞⁡Axk-Byk=0;that is, Ax~=By~; hence, x~,y~∈Ω. Since Ω is obviously a closed convex set by Lemma 18 and we have shown that xk,yk in H∗ is Ω-Fejérian and ωwxk,yk⊂Ω, by Lemma 11, we conclude that the sequence xk,yk generated by Algorithm 14 converges weakly to a point of Ω.LetPΩ be a metric projection from H∗ onto Ω. We denote the origins of H1 and H2 by θ1 and θ2, respectively.Theorem 21.
Let Assumption17 be satisfied. Given two bounded linear operators A:H1→H3 and B:H2→H3, if the following conditions are satisfied:(1)
τ<lim⁡infk→∞⁡αk0 and lim⁡infk→∞⁡αki>01≤i≤p;(2)
0<lim⁡infk→∞⁡βk≤lim⁡supk→∞⁡βk<1-μ;(3)
limk→∞⁡tk=0 and ∑k=1∞tk=∞,
then the sequencexk,yk generated by Algorithm 15 converges strongly to a solution PΩθ1,θ2 of (6).Proof.
Setx∗,y∗=PΩθ1,θ2. By (11), we readily see(62)-x∗,y∗,x,y-x∗,y∗≤0,∀x,y∈Ω.Since x∗,y∗∈Ω, we have x∗∈⋂i=1pFUi and y∗∈⋂j=1qFTj such that Ax∗=By∗. Taking δki=αki/1-αk01≤i≤p, we have ∑i=1pδki=1 for every k≥0 and 1-αk0∑i=1pδki=∑i=1pαki. Similar to the proof of (45) and (46), we obtain(63)xk+1-x∗2≤wk-x∗2-1-αk0αk0-τ∑i=1pδkiUiwk-wk2,(64)yk+1-y∗2≤pk-y∗2-βk1-βk-μpk-Tjkpk2.It follows from (33) and (42) that(65)wk-x∗2+pk-y∗2≤1-tkuk-x∗2+tkx∗2+1-tkvk-y∗2+tky∗2≤1-tkuk-x∗2+vk-y∗2+tkx∗2+y∗2≤1-tkxk-x∗2+yk-y∗2-γk2Axk-Byk2-γkA∗Axk-Byk2+B∗Axk-Byk2+tkx∗2+y∗2(66)≤1-tkxk-x∗2+yk-y∗2+tkx∗2+y∗2.By (63), (64), and (66), we have(67)xk+1-x∗2+yk+1-y∗2≤wk-x∗2+pk-y∗2≤1-tkxk-x∗2+yk-y∗2+tkx∗2+y∗2.Now, by setting Γk=xk-x∗2+yk-y∗2, from the above inequality, we obtain (68)Γk+1≤1-tkΓk+tkx∗2+y∗2.By induction, we have (69)Γk+1≤max⁡Γ1,x∗2+y∗2,which implies that Γk is bounded. Hence, xk,yk,uk,vk,wk, and pk are all bounded. By (63)–(65), we have(70)Γk+1≤wk-x∗2+pk-y∗2-1-αk0αk0-τ×∑i=1pδkiUiwk-wk2-βk1-βk-μpk-Tjkpk2≤1-tkΓk-1-tkγk2Axk-Byk2-γkA∗Axk-Byk2+B∗Axk-Byk2+tkx∗2+y∗2-αk0-τ∑i=1pαkiUiwk-wk2-βk1-βk-μpk-Tjkpk2.Then, it follows from (70) that(71)1-tkγk2Axk-Byk2-γkA∗Axk-Byk2+B∗Axk-Byk2+αk0-τ∑i=1pαkiUiwk-wk2+βk1-βk-μpk-Tjkpk2≤1-tkΓk-Γk+1+tkx∗2+y∗2.
Now, we will divide the proof into two cases to establish strong convergence.
Case 1. Suppose that there exists k0∈N such that Γk+1≤Γk for all k≥k0. In this case, limk→∞⁡Γk exists and then limk→∞⁡Γk+1-Γk=0. Hence, using (71) and the assumptions on αk0,βk,tk, and γk, we obtain(72)limk→∞⁡pk-Tjkpk=0,(73)limk→∞⁡Axk-Byk=0,(74)limk→∞⁡∑i=1pαkiUiwk-wk2=0. The last equality together with condition (1) implies that(75)limk→∞⁡Uiwk-wk=0,∀i∈1,2,…,p. From (33) and (73), we have(76)uk-xk=γkA∗Axk-Byk≤γkA∗Axk-Byk⟶0,(77)vk-yk=γkB∗Axk-Byk≤γkB∗Axk-Byk⟶0. Then, from (33), (75), and (76) and condition (3), we obtain (78)xk+1-xk≤αk0wk-xk+αk1U1wk-xk+⋯+αkpUpwk-xk≤wk-xk+αk1U1wk-wk+⋯+αkpUpwk-wk≤uk-xk+tkuk+αk1U1wk-wk+⋯+αkpUpwk-wk⟶0, which together with (75) implies that(79)xk-wk≤xk+1-xk+xk+1-wk≤xk+1-xk+αk1U1wk-wk+⋯+αkpUpwk-wk⟶0.It follows from (33), (72), and (77) that (80)yk+1-yk≤1-βkpk-yk+βkTjkpk-yk≤1-βkpk-yk+βkTjkpk-yk≤pk-yk+βkTjkpk-pk≤vk-yk+tkvk+βkTjkpk-pk⟶0. This, combined with (72), yields(81)yk-pk≤yk+1-yk+yk+1-pk≤yk+1-yk+βkTjkpk-pk⟶0. Since xk,yk is bounded in H∗, ωwxk,yk≠∅ and moreover(82)ωwxk,yk⊂Ω. Indeed, for any x^,y^∈ωwxk,yk, hence x^∈ωwxk, y^∈ωwyk by virtue of (28). Since x^∈ωwwk by (79), it immediately follows from (75) and Assumption 17(iii) that x^∈⋂i=1pFUi. On the other hand, since yk+1-yk→0, now we use Lemma 19 to derive that, for any j∈1,2,…,q, there exists a subsequence ykm⊂yk such that ykm⇀y^ and jkm=j for all m≥1. From (81), it results that pkm⇀y^. From (72), we have (83)limm→∞⁡pkm-Tjpkm=limm→∞⁡pkm-Tjkmpkm=0. Hence, it follows from (iii) of Assumption 17 that Tjy^=y^ for any j∈1,2,…,q; that is, y^∈⋂j=1qFTj. Mimicking the idea in the proof of Theorem 20, we conclude that Ax^=By^; hence, x^,y^∈Ω.
By (33), we observe(84)wk-x∗2=1-tkuk-x∗-tkx∗2=1-tk2uk-x∗2+tk2x∗2-2tk1-tkuk-x∗,x∗≤1-tkuk-x∗2+tk2x∗2-2tk1-tkuk-x∗,x∗. Similarly, we have(85)pk-y∗2≤1-tkvk-y∗2+tk2y∗2-2tk1-tkvk-y∗,y∗. Then, from (42), (63), (64), (84), and (85), it follows that(86)xk+1-x∗2+yk+1-y∗2≤wk-x∗2+pk-y∗2≤1-tkuk-x∗2+vk-y∗2+tktkx∗2+y∗2+21-tkuk-x∗,-x∗+vk-y∗,-y∗≤1-tkxk-x∗2+yk-y∗2+tktkx∗2+y∗2+21-tkuk-x∗,-x∗+vk-y∗,-y∗. By the boundedness of uk,vk in H∗, there exists a point p,q∈H∗ and a subsequence ukj,vkj of uk,vk in H∗ such that ukj,vkj⇀p,q and(87)lim⁡supk→∞⁡uk-x∗,-x∗+vk-y∗,-y∗=limj→∞⁡ukj-x∗,-x∗+vkj-y∗,-y∗. Then, it immediately follows from (76), (77), and (82) that p,q∈ωwuk,vk=ωwxk,yk⊂Ω. Thus, ukj,vkj⇀p,q and (87) and (62) yield(88)lim⁡supk→∞⁡uk-x∗,-x∗+vk-y∗,-y∗=limj→∞⁡ukj-x∗,-x∗+vki-y∗,-y∗=limj→∞⁡ukj,vkj-x∗,y∗,-x∗,y∗=p,q-x∗,y∗,-x∗,y∗≤0. Letting λk=tk, δk=21-tkuk-x∗,-x∗+vk-y∗,-y∗+tkx∗2+y∗2. From Lemma 13, (86) and (88), and the assumptions on tk, we conclude that the sequence xk,yk converges strongly to x∗,y∗.
Case
2. Suppose that there exists a subsequence Γki⊂Γk such that Γki<Γki+1 for all i∈N. In this case, let τ:N→N be a mapping for all n≥n0 (for some n0 large enough) by (89)τn=max⁡k≤n:Γk<Γk+1. Then, we have from Lemma 12 that Γτn≤Γτn+1 and, clearly, τn is a nondecreasing sequence such that τn→∞ as n→∞. From (71), we obtain (90)1-tτnγτn2Axτn-Byτn2-γτnA∗Axτn-Byτn2+B∗Axτn-Byτn2+ατn0-τ∑i=1pατniUiwτn-wτn2+βτn1-βτn-μpτn-Tjτnpτn2≤1-tτnΓτn-Γτn+1+tτnx∗2+y∗2≤tτnx∗2+y∗2⟶0.Conditions (1)–(3) imply(91)limk→∞⁡pτn-Tjτnpτn=limk→∞⁡Axτn-Byτn=0,limk→∞⁡Uiwτn-wτn=0,i=1,2,…,p. Now, using the sequence uτn,vτn in place of the bounded sequence uk,vk in H∗ and mimicking the proof of Case 1, we could obtain that(92)lim⁡supn→∞⁡uτn-x∗,-x∗+vτn-y∗,-y∗≤0. With the help of (86) and Γτn≤Γτn+1, we obtain(93)0≤Γτn+1-Γτn≤-tτnΓτn+tτntτnx∗2+y∗2+21-tτnuτn-x∗,-x∗+vτn-y∗,-y∗, which implies that(94)Γτn≤tτnx∗2+y∗2+21-tτnuτn-x∗,-x∗+vτn-y∗,-y∗. Therefore, from limn→∞⁡tτn=0, (92), and the above inequality, we have (95)0≤lim⁡infn→∞⁡Γτn≤lim⁡supn→∞⁡Γτn≤0⟹limn→∞⁡Γτn=0. In view of (93), it also results that (96)limn→∞⁡Γτn=limn→∞⁡Γτn+1=0. As a direct consequence of Lemma 12, we have (97)Γn≤Γτn+1⟶0. Since Γk=xk-x∗2+yk-y∗2, the sequence xk,yk generated by Algorithm 15 converges strongly to x∗,y∗, completing the proof.First, we shall give an example which satisfies all the conditions of the solution setΩ of MSECFP (6) and the mappings Uii=1p and Tii=1q in Theorems 20 and 21.Example 22.
LetH1=H2=H3=l2 and let i∈1,2,…,p and j∈1,2,…,q be arbitrarily fixed. Let Ui,Tj:l2→l2 be defined by Uix=-2ix and Tjx=-2j+1x for all x∈l2. Then, it is easy to see that ⋂i=1pFUi=0=⋂j=1qFTj and A0=0=B0. Thus, Ω=0,0≠∅. Also, Ui is τi-demicontractive and Tj is μj-demicontractive by Example 2.5 in [9], where τi=2i-1/2i+1 and μj=j/j+1; then, I-Ui and I-Tj are demiclosed at 0 by Remark 2.1 in [9].Next, we give an example which satisfies condition (a) in Theorem20 and condition (1) in Theorem 21.Example 23.
We can takeαk0=τ+(1-τ)/(p+1)+(1-τ)/p+2k+1, αk1=αk2=⋯=αkp = (1-τ)/(p+1)-(1-τ)/p(p+2)(k+1)k≥0.Remark 24.
The following two mistakes in the proof of Theorem3.2 in [9] should be modified as in the proof of Theorem 21:(i)
In Case1, the part “for any x^∈ωwxk and y^∈ωwyk” and in Case 2 the part “for any x¯∈ωwxτn and y¯∈ωwyτn” should be modified to “for any x^,y^∈ωwxk,yk” and “for any x¯,y¯∈ωwxτn,yτn,” respectively.(ii)
For proving(3.21) of Theorem 3.2 in [9], we should adopt the proving method of (87) and (88) in Theorem 21.Remark 25.
We can also consider the following two algorithms:(98)uk=xk-γkA∗Axk-Byk,xk+1=1-αkuk+αkUikuk,vk=yk+γkB∗Axk-Byk,yk+1=βk0vk+βk1T1vk+⋯+βkqTqvk
and(99)uk=xk-γkA∗Axk-Byk,wk=1-tkuk,xk+1=1-αkwk+αkUikwk,vk=yk+γkB∗Axk-Byk,pk=1-tkvk,yk+1=βk0pk+βk1T1pk+⋯+βkqTqpk. Under the suitable conditions, using methods similar to Theorems 20 and 21, we can prove the weak and strong convergence of the sequences xk,yk generated by these two algorithms, respectively.



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*Source: 5306802-2017-12-24.xml*