Approximation Method for Solving Equilibrium Problems and Fixed Point Problems of Infinitely Many Nonexpansive Mappings

Volume 2009, Article ID 374815,32pages doi:10.1155/2009/374815

Research Article

A Hybrid Extragradient Viscosity

Approximation Method for Solving Equilibrium Problems and Fixed Point Problems of Infinitely Many Nonexpansive Mappings

Chaichana Jaiboon and Poom Kumam

Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi, Bangkok 10140, Thailand

Correspondence should be addressed to Poom Kumam,poom.kum@kmutt.ac.th Received 25 December 2008; Accepted 4 May 2009

Recommended by Wataru Takahashi

We introduce a new hybrid extragradient viscosity approximation method for finding the common element of the set of equilibrium problems, the set of solutions of fixed points of an infinitely many nonexpansive mappings, and the set of solutions of the variational inequality problems for β-inverse-strongly monotone mapping in Hilbert spaces. Then, we prove the strong convergence of the proposed iterative scheme to the unique solution of variational inequality, which is the optimality condition for a minimization problem. Results obtained in this paper improve the previously known results in this area.

Copyrightq2009 C. Jaiboon and P. Kumam. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

LetH be a real Hilbert space, and let Cbe a nonempty closed convex subset ofH. Recall that a mappingT of H into itself is called nonexpansivesee1if Tx−Ty ≤ x−y for allx, y ∈H. We denote byFT {x ∈ C: Tx x}the set of fixed points ofT. Recall also that a self-mappingf :H → His a contraction if there exists a constantα∈0,1such thatfx−fy ≤αx−y, for all x, y ∈H.In addition, letB:C → Hbe a nonlinear mapping. LetP_C be the projection ofHontoC. The classical variational inequality which is denoted byV IC, Bis to findu∈Csuch that

Bu, v−u ≥0, ∀v∈C. 1.1

For a givenz∈H,u∈Csatisfies the inequality

u−z, v−u ≥0, ∀v∈C, 1.2 if and only ifuPCz. It is well known thatPCis a nonexpansive mapping ofHontoCand satisfies

x−y, P_Cx−P_Cy

≥P_Cx−P_Cy², ∀x, y∈H. 1.3 Moreover,P_Cxis characterized by the following properties:P_Cx∈Cand for allx∈H, y∈C,

x−P_Cx, y−P_Cx

≤0, 1.4

x−y²≥ x−P_Cx²y−P_Cx². 1.5

It is easy to see that the following is true:

u∈V IC, B⇐⇒uP_Cu−λBu, λ >0. 1.6 One can see that the variational inequality1.1is equivalent to a fixed point problem.

The variational inequality has been extensively studied in literature; see, for instance, 2–

6. This alternative equivalent formulation has played a significant role in the studies of the variational inequalities and related optimization problems. Recall the following.

1A mappingBofCintoHis called monotone if Bx−By, x−y

≥0, ∀x, y∈C. 1.7

2A mappingB is calledβ-strongly monotonesee7,8if there exists a constant β >0 such that

Bx−By, x−y

≥βx−y², ∀x, y∈C. 1.8 3A mappingBis calledk-Lipschitz continuous if there exists a positive real number

ksuch that

Bx−By≤kx−y, ∀x, y∈C. 1.9 4A mapping Bis called β-inverse-strongly monotonesee 7, 8 if there exists a

constantβ >0 such that Bx−By, x−y

≥βBx−By², ∀x, y∈C. 1.10

Remark 1.1. It is obvious that anyβ-inverse-strongly monotone mappingBis monotone and 1/β-Lipschitz continuous.

5An operatorAis strongly positive onH if there exists a constantγ > 0 with the property

Ax, x ≥γx², ∀x∈H. 1.11

6A set-valued mappingT :H → 2^His called monotone if for allx, y∈H,f ∈Tx, andg ∈Tyimplyx−y, f−g ≥0. A monotone mappingT :H → 2^His maximal if the graph ofGT ofT is not properly contained in the graph of any other monotone mapping. It is known that a monotone mappingTis maximal if and only if forx, f∈H×H,x−y, f−g ≥ 0 for everyy, g∈GTimpliesf∈Tx. LetBbe a monotone map ofCintoH,and letNCv be the normal cone toCatv∈C, that is,N_Cv{w∈H:u−v, w ≥0, for allu∈C},.

Tv

⎧⎨

⎩

BvN_Cv, v∈ C,

∅, v /∈ C. 1.12

ThenT is the maximal monotone and 0∈Tvif and only ifv∈V IC, B; see9.

7 Let F be a bifunction ofC×Cinto R, whereR is the set of real numbers. The equilibrium problem forF:C×C → Ris to findx∈Csuch that

F

x, y ≥0, ∀y∈C. 1.13

The set of solutions of 1.13 is denoted by EPF. Given a mapping T : C → H, let Fx, y Tx, y− x for all x, y ∈ C. Then, z ∈ EPF if and only if Tz, y−z ≥ 0 for all y ∈ C.Numerous problems in physics, saddle point problem, fixed point problem, variational inequality problems, optimization, and economics are reduced to find a solution of 1.13. Some methods have been proposed to solve the equilibrium problem; see, for instance, 10–16. Recently, Combettes and Hirstoaga 17introduced an iterative scheme of finding the best approximation to the initial data whenEPFis nonempty and proved a strong convergence theorem.

In 1976, Korpelevich18introduced the following so-called extragradient method:

x₀x∈C,

y_nP_Cx_n−λBx_n, x_n1P_C

x_n−λBy_n

1.14

for alln≥0,whereλ∈0,1/k, Cis a closed convex subset ofRⁿ,andBis a monotone and k-Lipschitz continuous mapping ofCintoRⁿ. He proved that ifV IC, Bis nonempty, then the sequences{xn}and{yn}, generated by1.14, converge to the same pointz∈V IC, B.

For finding a common element of the set of fixed points of a nonexpansive mapping and

the set of solution of variational inequalities forβ-inverse-strongly monotone, Takahashi and Toyoda19introduced the following iterative scheme:

x₀∈C chosen arbitrary,

x_n1αnxn 1−αnSPCxn−λnBxn, ∀n≥0, 1.15 whereBisβ-inverse-strongly monotone,{αn}is a sequence in0, 1, and{λn}is a sequence in0,2β. They showed that ifFS∩V IC, Bis nonempty, then the sequence{xn}generated by1.15converges weakly to somez∈FS∩V IC, B. Recently, Iiduka and Takahashi20 proposed a new iterative scheme as follows:

x₀x∈C chosen arbitrary,

x_n1α_nx 1−α_nSP_Cx_n−λ_nBx_n, ∀n≥0, 1.16 whereBisβ-inverse-strongly monotone,{αn}is a sequence in0, 1, and{λn}is a sequence in0,2β. They showed that ifFS∩V IC, Bis nonempty, then the sequence{xn}generated by1.16converges strongly to somez∈FS∩V IC, B.

Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example,21–24and the references therein. Convex minimization problems have a great impact and influence in the development of almost all branches of pure and applied sciences. A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert spaceH:

minx∈C

1

2Ax, x − x, b, 1.17

whereAis a linear bounded operator,Cis the fixed point set of a nonexpansive mapping SonH, andbis a given point inH. Moreover, it is shown in25that the sequence {xn} defined by the scheme

x_{n1 n}γfxn 1−_nASxn 1.18 converges strongly to z P_FSI − A γfz. Recently, Plubtieng and Punpaeng 26 proposed the following iterative algorithm:

F

u_n, y 1 r_n

y−u_n, u_n−x_n

≥0, ∀y∈H, x_n1nγfxn I−_nASun.

1.19

They prove that if the sequences{n}and{rn}of parameters satisfy appropriate condition, then the sequences{xn}and{un}both converge to the unique solutionzof the variational inequality

A−γf q, q−p ≥0, p∈FS∩EPF, 1.20

which is the optimality condition for the minimization problem

x∈FS∩EPFmin 1

2Ax, x −hx, 1.21 wherehis a potential function forγfi.e.,hx γfxforx∈H.

Furthermore, for finding approximate common fixed points of an infinite countable family of nonexpansive mappings {Tn} under very mild conditions on the parameters.

Wangkeeree27introduced an iterative scheme for finding a common element of the set of solutions of the equilibrium problem1.13and the set of common fixed points of a countable family of nonexpansive mappings onC. Starting with an arbitrary initialx₁ ∈ C, define a sequence{xn}recursively by

F

un, y 1 r_n

y−un, un−xn

≥0, ∀y∈C, ynPCun−λnBun,

x_n1 αnfxn βnxnγnSnPC

un−λnByn , ∀n≥1,

1.22

where{αn},{βn},and{γn}are sequences in0,1. It is proved that under certain appropriate conditions imposed on {αn},{βn},{γn}, and {rn}, the sequence {xn} generated by 1.22 strongly converges to the unique solution q ∈ ∩^∞_n1FSn∩V IC, B∩EPF, where p P_∩^∞_n1FSn∩V IC,B∩EPFfqwhich extend and improve the result of Kumam14.

Definition 1.2see21. Let{Tn}be a sequence of nonexpansive mappings ofCinto itself, and let{μn}be a sequence of nonnegative numbers in0,1. For eachn≥1, define a mapping WnofCinto itself as follows:

U_n,n1I,

Un,nμnTnU_n,n1

1−μn I, U_n,n−1μ_n−1T_n−1Un,n

1−μ_n−1 I, ...

Un,kμkTkU_n,k1

1−μk I, U_n,k−1μ_k−1T_k−1Un,k

1−μ_k−1 I, ...

Un,2μ₂T₂Un,3

1−μ₂ I, WnUn,1μ1T1Un,2

1−μ1 I.

1.23

Such a mappingWnis nonexpansive fromCtoC,and it is called theW-mapping generated byT₁, T₂, . . . , T_nandμ₁, μ₂, . . . , μ_n.

On the other hand, Colao et al.28introduced and considered an iterative scheme for finding a common element of the set of solutions of the equilibrium problem1.13and the set of common fixed points of infinitely many nonexpansive mappings onC. Starting with an arbitrary initialx0∈C, define a sequence{xn}recursively by

F

u_n, y 1

rny−u_n, u_n−x_n ≥0, ∀y∈H, x_n1nγfxn βxn

1−β I−nA Wnun,

1.24

where{n}is a sequence in0,1. It is proved28that under certain appropriate conditions imposed on{n}and{rn}, the sequence{xn}generated by1.24strongly converges toz ∈

∩^∞_n1FTn∩EPF, wherezis an equilibrium point forF and is the unique solution of the variational inequality1.20, that is,zP_∩^∞_n1FTn∩EPFI−A−γfz.

In this paper, motivated by Wangkeeree27, Plubtieng and Punpaeng26, Marino and Xu25, and Colao, et al.28, we introduce a new iterative scheme in a Hilbert spaceH which is mixed by the iterative schemes of1.18,1.19,1.22, and1.24as follows.

Letfbe a contraction ofHinto itself,Aa strongly positive bounded linear operator on Hwith coefficientγ >0,andBaβ-inverse-strongly monotone mapping ofCintoH; define sequences{xn},{yn},{kn},and{un}recursively by

x₁ x∈C chosen arbitrary, F

un, y 1

r_ny−un, un−xn ≥0, ∀y∈C, y_nP_Cun−λ_nBu_n,

knαnun 1−αnPC

un−λnByn , x_n1nγfxn β_nx_n

1−β_n I−_nA W_nk_n, ∀n≥1,

1.25

where {Wn}is the sequence generated by 1.23,{n},{αn},and {βn} ⊂ 0,1 and {rn} ⊂ 0,∞satisfying appropriate conditions. We prove that the sequences{xn},{yn},{kn}and {un}generated by the above iterative scheme1.25converge strongly to a common element of the set of solutions of the equilibrium problem1.13, the set of common fixed points of infinitely family nonexpansive mappings, and the set of solutions of variational inequality 1.1for a β-inverse-strongly monotone mapping in Hilbert spaces. The results obtained in this paper improve and extend the recent ones announced by Wangkeeree 27, Plubtieng and Punpaeng26, Marino and Xu25, Colao, et al.28, and many others.

2. Preliminaries

We now recall some well-known concepts and results.

LetHbe a real Hilbert space, whose inner product and norm are denoted by·,·and · , respectively. We denote weak convergence and strong convergence by notationsand

→, respectively.

A spaceHis said to satisfy Opial’s condition29if for each sequence{xn}inHwhich converges weakly to pointx∈H, we have

lim inf

n→ ∞ xn−x<lim inf

n→ ∞ x_n−y, ∀y∈H, y /x. 2.1 Lemma 2.1see25. LetCbe a nonempty closed convex subset ofH, letf be a contraction of H into itself withα ∈ 0,1, and letAbe a strongly positive linear bounded operator on Hwith coefficientγ >0. Then , for 0< γ < γ/α,

x−y,

A−γf x−

A−γf y

≥

γ−αγ x−y², x, y∈H. 2.2 That is,A−γfis strongly monotone with coefficientγ−γα.

Lemma 2.2see25. Assume thatA is a strongly positive linear bounded operator onH with coefficientγ >0 and 0< ρ≤ A⁻¹. ThenI−ρA ≤1−ργ.

For solving the equilibrium problem for a bifunctionF :C×C → R, let us assume thatFsatisfies the following conditions:

A1Fx, x 0 for allx∈C;

A2Fis monotone, that is,Fx, y Fy, x≤0 for allx, y∈C;

A3for eachx, y, z∈C,lim_t↓0Ftz 1−tx, y≤Fx, y;

A4for eachx∈C, y→Fx, yis convex and lower semicontinuous.

The following lemma appears implicitly in30.

Lemma 2.3see30. LetCbe a nonempty closed convex subset ofHand letFbe a bifunction of C×CintoRsatisfying (A1)–(A4). Letr >0 andx∈H. Then, there existsz∈Csuch that

F

z, y 1 r

y−z, z−x

≥0 ∀y∈C. 2.3

The following lemma was also given in17.

Lemma 2.4see17. Assume thatF :C×C → Rsatisfies (A1)–(A4). Forr > 0 andx ∈H, define a mappingTr :H → Cas follows:

T_rx

z∈C:F

z, y 1 r

y−z, z−x

≥0, ∀y∈C

2.4

for allz∈H. Then, the following holds:

1Tr is single-valued;

2T_r is firmly nonexpansive, that is, for anyx, y∈H, Trx−Try²≤

Trx−Try, x−y

; 2.5

3FTr EPF;

4EPFis closed and convex.

For eachn, k ∈ N, let the mappingUn,kbe defined by1.23. Then we can have the following crucial conclusions concerningWn. You can find them in31. Now we only need the following similar version in Hilbert spaces.

Lemma 2.5see31. LetC be a nonempty closed convex subset of a real Hilbert spaceH. Let T1, T2, . . . be nonexpansive mappings of C into itself such that ∩^∞_n1FTn is nonempty, and let μ₁, μ₂, . . .be real numbers such that 0 ≤ μ_n ≤ b < 1 for every n ≥ 1. Then, for everyx ∈ Cand k∈N, the limit limn→ ∞Un,kxexists.

UsingLemma 2.5, one can define a mappingWofCinto itself as follows:

Wx lim

n→ ∞Wnx lim

n→ ∞Un,1x 2.6

for everyx ∈ C. Such aW is called the W-mapping generated byT₁, T₂, . . . and μ₁, μ₂, . . ..

Throughout this paper, we will assume that 0≤μ_n≤b <1 for everyn≥1. Then, we have the following results.

Lemma 2.6see31. LetC be a nonempty closed convex subset of a real Hilbert spaceH. Let T₁, T₂, . . . be nonexpansive mappings of C into itself such that ∩^∞_n1FTn is nonempty, and let μ₁, μ₂, . . .be real numbers such that 0≤μn≤b <1 for everyn≥1. Then,FW ∩^∞_n1FTn. Lemma 2.7see32. If{xn}is a bounded sequence inC, then lim_{n→ ∞}Wxn−W_nx_n0.

Lemma 2.8see33. Let{xn}and{zn}be bounded sequences in a Banach spaceX,and let{βn}be a sequence in0,1with 0<lim infn→ ∞βn≤lim sup_{n→ ∞}βn<1.Supposex_n1 1−βnznβnxn

for all integersn≥0 and lim sup_{n→ ∞}y_n1−z_n − x_n1−x_n≤0.Then, lim_{n→ ∞}zn−x_n0.

Lemma 2.9see34. Assume that{an}is a sequence of nonnegative real numbers such that

a_n1≤1−l_nanσ_n, n≥0, 2.7

where{ln}is a sequence in0,1and{σn}is a sequence inRsuch that 1_∞

n1l_n∞;

2lim sup_{n→ ∞}σn/ln≤0 or_∞

n1|σn|<∞.

Then lim_{n→ ∞}a_n0.

Lemma 2.10. LetHbe a real Hilbert space. Then for allx, y∈H, 1xy²≤ x²2y, xy;

2xy²≥ x²2y, x.

3. Main Results

In this section, we prove the strong convergence theorem for infinitely many nonexpansive mappings in a real Hilbert space.

Theorem 3.1. LetCbe a nonempty closed convex subset of a real Hilbert spaceH, letFbe a bifunction fromC×CtoRsatisfying (A1)–(A4), let{Tn}be an infinitely many nonexpansive ofCinto itself, and let B be an β-inverse-strongly monotone mapping ofC into H such that Θ : ∩^∞_n1FTn∩ EPF∩V IC, B/∅. Letfbe a contraction ofHinto itself withα∈0,1,and letAbe a strongly positive linear bounded operator onH with coefficient γ > 0 and 0 < γ < γ/α. Let {xn},{yn}, {kn},and{un}be sequences generated by1.25, where{Wn}is the sequence generated by1.23, {n},{αn},and{βn}are three sequences in0,1,and{rn}is a real sequence in0,∞satisfying the following conditions:

ilimn→ ∞n0,_∞

n1n∞;

iilim_{n→ ∞}α_n0 and_∞

n1α_n∞;

iii0<lim infn→ ∞βn≤lim sup_{n→ ∞}βn<1;

ivlim inf_{n→ ∞}r_n>0 and lim_{n→ ∞}|r_n1−r_n|0;

v{λn/β} ⊂τ,1−δfor someτ, δ∈0,1and limn→ ∞λn0.

Then,{xn}and{un}converge strongly to a pointz∈Θwhich is the unique solution of the variational inequality

A−γf z, z−x

≥0, ∀x∈Θ. 3.1

Equivalently, one haszP_ΘI−Aγfz.

Proof. Note that from the conditioni, we may assume, without loss of generality, that_n ≤ 1−β_nA⁻¹for alln∈N. FromLemma 2.2, we know that if 0≤ρ≤ A⁻¹, thenI−ρA ≤ 1−ργ. We will assume thatI−A ≤1−γ. First, we show thatI−λnBis nonexpansive. Indeed, from theβ-inverse-strongly monotone mapping definition onBand conditionv, we have

I−λnBx−I−λnBy²x−y−λnBx−By² x−y²−2λ_n

x−y, Bx−By

λ²_nBx−By²

≤x−y²−2λnβBx−By²λ²_nBx−By² x−y²λ_n

λ_n−2β Bx−By²

≤x−y²,

3.2

which implies that the mapping I−λ_nB is nonexpansive. On the other hand, sinceAis a strongly positive bounded linear operator on H, we have

Asup{|Ax, x|:x∈H,x1}. 3.3

Observe that

1−βn I−nA x, x

1−βn−nAx, x

≥1−β_n−_nA

≥0,

3.4

and this show that1−βnI−nAis positive. It follows that

1−βn I−nAsup1−βn I−nA x, x:x∈H,x1 sup

1−β_n−_nAx, x:x∈H,x1

≤1−βn−nγ.

3.5

LetQP_Θ, whereΘ:∩^∞_n1FTn∩EPF∩V IC, B. Note thatf is a contraction ofHinto itself withα∈0,1. Then, we have

Q

I−Aγf x−Q

I−Aγf y P_Θ

I−Aγf x−P_Θ

I−Aγf y

≤I−Aγf x−

I−Aγf y

≤ I−Ax−yγfx−f y

≤

1−γ x−yγαx−y

1−γγα x−y

1−

γ−γα x−y, ∀x, y∈H.

3.6

Since 0 < 1−γ −γα < 1, it follows that QI−Aγf is a contraction ofH into itself.

Therefore by the Banach Contraction Mapping Principle, which implies that there exists a unique elementz∈Hsuch thatzQI−Aγfz P_ΘI−Aγfz.

We will divide the proof into five steps.

Step 1. We claim that{xn}is bounded. Indeed, pick anyp∈Θ. From the definition ofT_r, we note thatunTrnxn. If follows that

un−pTrnxn−Trnp≤xn−p. 3.7 SinceI−λnBis nonexpansive andpPCp−λnBpfrom1.6, we have

yn−pPCun−λnBun−PC

p−λnBp

≤u_n−λ_nAu_n−

p−λ_nBp I−λ_nAun−I−λ_nBp

≤un−p≤xn−p.

3.8

Putvn PCun−λnByn. Sincep ∈ V IC, B, we havep PCp−λnBp. Substitutingx u_n−λ_nAy_nandypin1.5, we can write

vn−p² ≤un−λnByn−p²−un−λnByn−vn² u_n−p²−2λ_n

By_n, u_n−p

λ²_nBy_n²

− un−vn²2λn

Byn, un−vn

−λ²_nByn² u_n−p²− u_n−v_n²2λ_n

By_n, p−v_n u_n−p²− u_n−v_n²2λ_n

By_n−Bp, p−y_n 2λ_n

Bp, p−y_n 2λ_n

By_n, y_n−v_n .

3.9

Using the fact that B is β-inverse-strongly monotone mapping, and p is a solution of the variational inequality problemV IC, B, we also have

Byn−Bp, p−yn

≤0, Bp, p−yn ≤0. 3.10

It follows from3.9and3.10that

v_n−p²≤u_n−p²− u_n−v_n²2λ_nByn, y_n−v_n u_n−p²−un−y_n yn−v_n²2λ_n

By_n, y_n−v_n

≤un−p²−un−yn²−yn−vn²

−2

un−yn, yn−vn 2λn

Byn, yn−vn un−p²−un−yn²−yn−vn²2

un−λnByn−yn, vn−yn .

3.11

Substitutingxbyun−λnBunandyvnin1.4, we obtain u_n−λ_nBu_n−y_n, v_n−y_n

≤0. 3.12

It follows that

un−λnByn−yn, vn−yn

un−λnBun−yn, vn−yn

λnBun−λnByn, vn−yn

≤

λ_nBu_n−λ_nBy_n, v_n−y_n

≤λnBun−Bynvn−yn

≤ λn

β un−ynvn−yn.

3.13

Substituting3.13into3.11, we have

v_n−p²≤u_n−p²−u_n−y_n²−y_n−v_n²2

u_n−λ_nBy_n−y_n, v_n−y_n

≤u_n−p²−u_n−y_n²−y_n−v_n²2λ_n

β u_n−y_nv_n−y_n

≤u_n−p²−u_n−y_n²−y_n−v_n²λ²_n

β²u_n−y_n²v_n−y_n² un−p²−un−yn²λ²_n

β²un−yn² u_n−p²

λ²_n β² −1

u_n−y_n²

≤u_n−p²≤x_n−p².

3.14

Settingk_nα_nu_n 1−α_nvn, we can calculate x_n1−pn

γfxn−Ap βn

xn−p

1−βn I−nA Wnkn−p

≤

1−β_n−_nγ k_n−pβ_nx_n−p_nγfxn−Ap

≤

1−βn−nγ αnun−p 1−αnvn−p βnxn−pnγfxn−Ap

≤

1−β_n−_nγ α_nx_n−p 1−α_nx_n−p βnxn−pnγfxn−Ap

1−β_n−_nγ x_n−pβ_nx_n−p_nγfx_n−Ap

1−_nγ x_n−p_nγfxn−f

p _nγf

p −Ap

≤

1−nγ xn−pnγαxn−pnγf

p −Ap

1−

γ−γα n xn−p

γ−γα n

γf

p −Ap γ−γα .

3.15

By induction,

xn−p≤max

x1−p,γf

p −Ap γ−γα

, n∈N. 3.16

Hence,{xn}is bounded, so are{un},{vn},{Wnkn},{fxn},{Bun},{yn},and{Byn}.

Step 2. We claim that lim_{n→ ∞}x_n1−x_n0.

Observing thatunTrnxnandu_n1Tr_n1x_n1,we get

F

u_n, y 1 rn

y−u_n, u_n−x_n

≥0 ∀y∈H 3.17 F

u_n1, y 1 r_n1

y−u_n1, u_n1−x_n1

≥0 ∀y∈H. 3.18

Puttingyu_n1in3.17andyunin3.18, we have

Fun, u_n1 1

rnun1−un, un−xn ≥0 Fu_n1, u_n 1

r_n1un−u_n1, u_n1−x_n1 ≥0.

3.19

So, fromA2we have

u_n1−u_n,u_n−x_n

r_n −u_n1−x_n1 r_n1

≥0, 3.20

and hence

u_n1−un, un−u_n1u_n1−xn− rn

r_n1un1−x_n1

≥0. 3.21

Without loss of generality, let us assume that there exists a real numbercsuch thatr_n> c >0 for alln∈N.Then, we have

u_n1−u_n²≤

u_n1−u_n, x_n1−x_n

1− r_n r_n1

u_n1−x_n1

≤ u_n1−un

xn1−xn 1− rn

r_n1

un1−x_n1

,

3.22

and hence

u_n1−u_n ≤ x_n1−x_n 1

r_n1|r_n1−r_n|u_n1−x_n1

≤ x_n1−xnM1

c |rn1−rn|,

3.23

whereM1sup{un−xn:n∈N}. Note that v_n1−v_n ≤P_C

u_n1−λ_n1By_n1 −P_C

u_n−λ_nBy_n

≤u_n1−λ_n1By_n1−

u_n−λ_nBy_n u_n1−λ_n1Bu_n1−u_n−λ_n1Bu_n

λ_n1

Bu_n1−By_n1−Bun λnByn

≤ un1−λ_n1Bu_n1−un−λ_n1Bun λ_n1

Bun1By_n1Bun λnByn

≤ u_n1−unλ_n1

Bun1By_n1Bun λnByn, kn1−knαn1u_n1 1−α_n1vn1−αnun−1−αnvn

α_n1u_n1−u_n α_n1−α_nun

1−α_n1v_n1−v_n αn−α_n1vn

≤α_n1u_n1−u_n 1−α_n1v_n1−v_n|αn−α_n1|unv_n α_n1u_n1−u_n 1−α_n1

×

un1−unλ_n1

Bun1By_n1Bun λnByn

|αn−α_n1|unvn un1−un 1−α_n1λn1

Bun1By_n1Bun 1−α_n1λnByn|αn−α_n1|unvn

≤ x_n1−xnM₁

c |rn1−rn| 1−α_n1λn1

×

Bu_n1By_n1Bu_n

1−α_n1λnByn|αn−α_n1|unvn.

3.24

Setting

zn x_n1−βnxn

1−β_{n n}γfx_n

1−β_n I−_nA W_nk_n

1−β_n , 3.25

we havex_n1 1−βnznβnxn, n≥1. It follows that z_n1−z_{n n1}γfx_n1

1−β_n1 I−_n1A W_n1k_n1 1−β_n1

− _nγfxn

1−β_n I−_nA W_nk_n 1−βn

_n1

1−β_n1γfx_n1− _n

1−β_nγfxn W_n1k_n1−W_nk_n n

1−βnAWnkn− _n1

1−β_n1AW_n1k_{n1 n1}

1−β_n1

γfx_n1−AW_n1k_{n1 n} 1−β_n

AW_nk_n−γfxn

W_n1k_n1−W_n1knW_n1kn−Wnkn.

3.26

It follows from3.24and3.26that zn1−zn − xn1−xn ≤ _n1

1−β_n1γfxn1AWn1k_{n1 n}

1−βn

AWnk_nγfxn W_n1k_n1−W_n1k_n

Wn1kn−Wnkn − xn1−xn

≤ _n1

1−β_n1γfx_n1AW_n1k_n1 n

1−β_n

AWnknγfxn kn1−kn

W_n1k_n−W_nk_n − x_n1−x_n

≤ _n1

1−β_n1γfx_n1AW_n1k_n1 n

1−β_n

AWnk_nγfxn M₁

c |r_n1−r_n| 1−α_n1λn1

Bun1By_n1Bun 1−α_n1λnByn|αn−α_n1|unvn Wn1kn−Wnkn.

3.27

SinceTiandUn,iare nonexpansive, we have

Wn1kn−Wnknμ₁T₁U_n1,2kn−μ₁T₁Un,2kn

≤μ1Un1,2kn−Un,2kn

μ₁μ₂T₂U_n1,3k_n−μ₂T₂U_n,3k_n

≤μ₁μ₂Un1,3kn−Un,3kn ...

≤μ₁μ₂· · ·μ_nU_n1,n1k_n−U_n,n1k_n

≤M₂ⁿ

i1

μ_i,

3.28

whereM2≥0 is a constant such thatUn1,n1kn−U_n,n1kn ≤M2for alln≥0.

Combining3.27and3.28, we have zn1−zn − xn1−xn ≤ _n1

1−β_n1γfxn1AWn1k_{n1 n}

1−βn

AWnk_nγfxn M₁

c |r_n1−r_n| 1−α_n1λ_n1

Bu_n1By_n1Bun 1−α_n1λnByn|αn−α_n1|unvn M₂ⁿ

i1

μ_i,

3.29

which implies thatnoting thati,ii,iii,iv,v, and 0< μ_i≤b <1,for alli≥1 lim sup

n→ ∞ zn1−zn − xn1−xn≤0. 3.30

Hence, byLemma 2.8, we obtain

n→ ∞limzn−x_n0. 3.31

It follows that

nlim→ ∞xn1−xn lim

n→ ∞

1−βn zn−xn0. 3.32

Applying3.32andii,iv, andvto3.23and3.24, we obtain that

nlim→ ∞un1−un lim

n→ ∞kn1−kn0. 3.33

Sincex_n1nγfxn βnxn 1−βnI−nAWnkn, we have xn−W_nk_n ≤ xn−x_n1x_n1−W_nk_n

xn−x_n1nγfxn βnxn

1−βn I−nA Wnkn−Wnkn xn−x_n1n

γfxn−AW_nk_n β_nxn−W_nk_n

≤ xn−x_n1nγfxnAWnkn βnxn−Wnkn,

3.34

that is

xn−Wnkn ≤ 1

1−βnxn−x_n1 n

1−βn

γfxnAWnkn . 3.35

Byi,iii, and3.32it follows that

n→ ∞limWnkn−xn0. 3.36

Step 3. We claim that the following statements hold:

ilim_{n→ ∞}un−k_n0;

iilim_{n→ ∞}xn−u_n0.

For anyp∈Θ:∩^∞_n1FTn∩EPF∩V IC, Band3.14, we have k_n−p²α_nun−p 1−α_nvn−p²

≤αnun−p² 1−αnvn−p²

≤αnun−p² 1−αn

un−p² λ²_n

β² −1

un−yn² un−p² 1−αn

λ²_n β² −1

un−yn²

≤xn−p² 1−αn λ²_n

β² −1

un−yn².

3.37