General Iterative Methods for

Volume 2012, Article ID 602513,17pages doi:10.1155/2012/602513

Research Article

General Iterative Methods for

Equilibrium Problems and Infinitely Many Strict Pseudocontractions in Hilbert Spaces

Peichao Duan and Aihong Wang

College of Science, Civil Aviation University of China, Tianjin 300300, China

Correspondence should be addressed to Peichao Duan,[email protected] Received 11 January 2012; Revised 24 February 2012; Accepted 25 February 2012 Academic Editor: Rudong Chen

Copyrightq2012 P. Duan and A. Wang. 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.

We propose an implicit iterative scheme and an explicit iterative scheme for finding a common element of the set of fixed point of infinitely many strict pseudocontractive mappings and the set of solutions of an equilibrium problem by the general iterative methods. In the setting of real Hilbert spaces, strong convergence theorems are proved. Our results improve and extend the corresponding results reported by many others.

1. Introduction

LetHbe a real Hilbert space and letCbe a nonempty closed convex subset ofH. LetFbe a bifunction fromC×CtoR, whereRis the set of real numbers.

The equilibrium problem forF:C×C → Ris to findx∈Csuch that F

x, y

≥0 1.1

for ally∈C. The set of such solutions is denoted by EPF.

A mappingSofCis said to be aκ-strict pseudocontraction if there exists a constant κ∈0,1such that

Sx−Sy²≤x−y²κI−Sx−I−Sy² 1.2 for allx, y ∈C; see1. We denote the set of fixed points ofSbyFS i.e.,FS {x ∈C: Sxx}.

Note that the class of strict pseudocontractions strictly includes the class of nonex- pansive mappings which are mappingS on Csuch that

Sx−Sy≤x−y 1.3 for allx, y∈C. That is,Sis nonexpansive if and only ifSis a 0-strict pseudocontraction.

Numerous problems in physics, optimization, and economics reduce to finding a solution of the equilibrium problem. Some methods have been proposed to solve the equilibrium problem1.1; see, for instance,2–4. In particular, Combettes and Hirstoaga5 proposed several methods for solving the equilibrium problem. On the other hand, Mann 6, Shimoji and Takahashi 7 considered iterative schemes for finding a fixed point of a nonexpansive mapping. Further, Acedo and Xu 8 projected new iterative methods for finding a fixed point of strict pseudocontractions.

In 2006, Marino and Xu3introduced the general iterative method and proved that the algorithm converged strongly. Recently, Liu 2 considered a general iterative method for equilibrium problems and strict pseudocontractions. Tian 9 proposed a new general iterative algorithm combining an L-Lipschitzian and η-strong monotone operator. Very recently, Wang10considered a general composite iterative method for infinite family strict pseudocontractions.

In this paper, motivated by the above facts, we introduce two iterative schemes and obtain strong convergence theorems for finding a common element of the set of fixed points of a infinite family of strict pseudocontractions and the set of solutions of the equilibrium problem1.1.

2. Preliminaries

Throughout this paper, we always write for weak convergence and → for strong convergence. We need some facts and tools in a real Hilbert space H which are listed as below.

Lemma 2.1. LetHbe a real Hilbert space. There hold the following identities:

ix−y²x²− y²−2x−y, y, ∀x, y∈H;

iitx 1−ty²tx² 1−ty²−t1−tx−y², ∀t∈0,1, ∀x, y∈H.

Lemma 2.2see11. Assume that{αn}is a sequence of nonnegative real numbers such that

α_n1≤ 1−γ_n

α_nδ_n, 2.1

where{γ_n}is a sequence in0,1and{δ_n}is a sequence such that i_∞

n1γ_n∞;

iilim_{n→ ∞}supδ_n/γ_n≤0 or _∞

n1|δ_n|<∞.

Then, lim_{n→ ∞}α_n0.

Recall that given a nonempty closed convex subsetCof a real Hilbert spaceH, for anyx∈H, there exists a unique nearest point inC, denoted byP_Cx, such that

x−PCx ≤x−y 2.2

for ally∈C. Such aP_Cis called the metric (or the nearest point) projection ofHontoC. As known, yPCxif and only if there holds the relation:

x−y, y−z

≥0 ∀z∈C. 2.3

Lemma 2.3see10. LetA:H → Hbe aL-Lipschitzian andη-strongly monotone operator on a Hilbert spaceHwithL >0,η >0, 0< μ <2η/L², and 0< t <1. Then,S I−tμA:H → His a contraction with contractive coefficient 1−tτandτ 1/2μ2η−μL².

Lemma 2.4see1. LetS : C → Cbe aκ-strict pseudocontraction. DefineT : C → Cby Tx λx 1−λSxfor eachx ∈C. Then, asλ ∈κ,1,T is a nonexpansive mapping such that

FT FS.

Lemma 2.5see9. LetHbe a Hilbert space andf :H → Hbe a contraction with coefficient 0 < α <1, andA:H → HanL-Lipschitzian continuous operator andη-strongly monotone with L >0,η >0. Then for 0< γ < μη/α:

x−y,

μA−γf x−

μA−γf y

≥

μη−γαx−y², x, y∈H. 2.4 That is,μA−γfis strongly monotone with coefficientμη−γα.

Let{S_n}be a sequence ofκ_n-strict pseudo-contractions. DefineS_n θ_nI 1−θ_nS_n, θ_n ∈ κn,1. Then, byLemma 2.4,S_nis nonexpansive. In this paper, consider the mappingWndefined by

U_n,n1I,

U_n,nt_nS_nU_n,n1 1−t_nI, Un,n−1tn−1S_n−1Un,n 1−tn−1I,

. . . ,

Un,itiS_iUn,i1 1−tiI, . . . ,

U_n,2 t2S₂U_n,3 1−t2I, WnUn,1t1S₁Un,2 1−t1I,

2.5

wheret1, t2, . . .are real numbers such that 0 ≤t_n < 1. Such a mappingW_nis called aW-mapping generated byS₁, S₂, . . .andt1, t2, . . .. It is easy to seeW_nis nonexpansive.

Lemma 2.6see7. LetCbe a nonempty closed convex subset of a strictly convex Banach space E, letS₁, S₂, . . .be nonexpansive mappings ofCinto itself such that∩^∞_i1FS_i/∅and lett1, t2, . . .be

real numbers such that 0< ti ≤b <1, for everyi1,2, . . .. Then, for anyx∈Candk∈N, the limit lim_{n→ ∞}U_n,kxexists.

UsingLemma 2.6, one can define the mappingWofCinto itself as follows:

Wx: lim

n→ ∞W_nx lim

n→ ∞U_n,1x, x∈C. 2.6

Lemma 2.7see7. LetCbe a nonempty closed convex subset of a strictly convex Banach spaceE.

LetS₁, S₂, . . .be nonexpansive mappings ofCinto itself such that∩^∞_i1FS_i/∅and lett1, t2, . . .be real numbers such that 0< ti≤b <1, for alli≥1. IfKis any bounded subset ofC, then

nlim→ ∞sup

x∈KWx−Wnx0. 2.7

Lemma 2.8see12. LetCbe a nonempty closed convex subset of a Hilbert spaceH, let{S_i:C → C}be a family of infinite nonexpansive mappings with∩^∞_i1FS_i/∅, lett1, t2, . . .be real numbers such that 0< ti≤b <1, for everyi1,2, . . .. ThenFW ∩^∞_i1FS_i.

For solving the equilibrium problem, assume that the bifunction F satisfies the following conditions:

A1Fx, x 0 for allx∈C;

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

A3for eachx, y, z∈C, lim sup_t→0Ftz 1−tx, y≤Fx, y;

A4Fx,·is convex and lower semicontinuous for eachx∈C.

Recall some lemmas which will be needed in the rest of this paper.

Lemma 2.9see13. LetCbe a nonempty closed convex subset of H, letF be bifunction from C×CtoRsatisfying (A1)–(A4), and letr >0 andx∈H. Then, there existsz∈Csuch that

F z, y

1 r

y−z, z−x

≥0, ∀y∈C. 2.8

Lemma 2.10see5. Forr >0, x∈H, define a mappingT_r :H → Cas follows:

T_rx

z∈C|F z, y

1 r

y−z, z−x

≥0, ∀y∈C 2.9

for allx∈H. Then, the following statements hold:

iTr is single-valued;

iiT_r is firmly nonexpansive, that is, for anyx, y∈H, T_rx−T_ry²≤

T_rx−T_ry, x−y

; 2.10

iiiFTr EPF;

ivEPFis closed and convex.

Lemma 2.11see14. Let{xn}and{zn}be bounded sequences in a Banach space and let{βn}be a sequence of real numbers such that 0<lim inf_{n→ ∞}β_n ≤lim sup_{n→ ∞}β_n <1 for alln0,1,2, . . ..

Suppose thatxn1 1−βnznβnxnfor alln0,1,2, . . .and lim sup_{n→ ∞}zn1−zn−xn1−xn ≤ 0. Then lim_{n→ ∞}zn−xn0.

Lemma 2.12see4. LetC, H, F, andT_rxbe as inLemma 2.10. Then, the following holds:

Tsx−Ttx²≤ s−t

s Tsx−Ttx, Tsx−x 2.11 for alls, t >0 andx∈H.

Lemma 2.13see10. LetHbe a Hilbert space and letCbe a nonempty closed convex subset ofH, andT :C → Ca nonexpansive mapping withFT/∅. If{x_n}is a sequence inCweakly converging toxand if{I−Tx_n}converges strongly toy, thenI−Txy.

3. Main Result

Throughout the rest of this paper, we always assume thatf is a contraction ofHinto itself with coefficient α ∈ 0,1, and A is a L-Lipschitzian continuous operator and η-strongly monotone onHwithL >0,η >0. Assume that 0< μ <2η/L²and 0< γ < μη−μL²/2/α τ/α.

Define a mappingVnβnI 1−βnWnTrn. Since bothWnandTrn are nonexpansive, it is easy to getVnis also nonexpansive. Consider the following mappingGnonHdefined by

G_nxα_nγfx

I−α_nμA

V_nx, ∀x∈H, n∈N, 3.1

whereα_n∈0,1. By Lemmas2.3and2.10, we have G_nx−G_ny≤α_nγfx−f

y 1−α_nτV_nx−V_ny

≤αnγαx−y 1−αnτx−y

1−α_n

τ−γαx−y.

3.2

Since 0 < 1−αnτ −γα < 1, it follows thatGn is a contraction. Therefore, by the Banach contraction principle,Gnhas a unique fixed pointedx_n^f ∈Hsuch that

x^f_nαnγf x^f_n

I−αnμA

Vnx^f_n. 3.3

For simplicity, we will writex_nforx^f_n provided no confusion occurs. Next we prove the sequences{xn} converges strongly to ax^∗ ∈ Ω ∩^∞_i1FSi∩EPF which solves the variational inequality:

γf−μA

x^∗, p−x^∗

≤0, ∀p∈Ω. 3.4

Equivalently,x^∗P_ΩI−μAγfx^∗.

Theorem 3.1. LetCbe a nonempty closed convex subset of a real Hilbert spaceHandFa bifunction fromC×CtoRsatisfying (A1)–(A4). LetS_i :C → Cbe a familyκ_i-strict pseudocontractions for some 0≤κ_i <1. Assume the setΩ ∩^∞_i1FS_i∩EPF/∅. Letfbe a contraction ofHinto itself with α ∈ 0,1and letA be aL-Lipschitzian continuous operator and η-strongly monotone with L >0, η >0, 0< μ <2η/L²and 0< γ < μη−μL²/2/ατ/α. For everyn∈N, letW_nbe the mapping generated byS_iandtias in2.5. Let{xn}and{un}be sequences generated by the following algorithm:

u_n T_rnx_n, y_nβ_nx_n

1−β_n W_nu_n, x_nα_nγfx_n

I−μα_nA y_n.

3.5

If{α_n}, {β_n}, and{r_n}satisfy the following conditions:

i{αn} ⊂0,1, limn→ ∞αn0;

ii0<lim inf_{n→ ∞}β_n≤lim sup_{n→ ∞}β_n<1;

iii{rn} ⊂0,∞, lim infn→ ∞rn>0.

Then,{xn}converges strongly to a pointx^∗∈Ω, which solves the variational inequality3.4.

Proof. The proof is divided into several steps.

Step 1. Show first that{xn}is bounded.

Take anyp∈Ω, by3.5andLemma 2.3, we derive that xn−pαn

γfxn−μAp

I−μαnA yn−

I−μαnA p

≤αnαγxn−pαnγf p

−μAp 1−αnτyn−p

≤ 1−αn

τ−γαxn−pαnγf p

−μAp.

3.6

It follows thatxn−p ≤γfp−μAp/τ−γα.

Hence,{x_n}is bounded, so are{u_n}and{y_n}. It follows from the Lipschitz continuity ofAthat{Axn}and{Aun}are also bounded. From the nonexpansivity offandWn, it follows that{fx_n}and{W_nx_n}are also bounded.

Step 2. Show that

n→ ∞limun−xn0, lim

n→ ∞un−yn0. 3.7

Notice that

un−yn≤ un−xnxn−ynun−xnαnγfxn−μAyn. 3.8

ByLemma 2.10, we have u_n−p²T_rnx_n−T_rnp²≤

x_n−p, u_n−p 1

2

u_n−p²x_n−p²− xn−u_n² . 3.9 It follows that

un−p²≤xn−p²− xn−un². 3.10 Thus, fromLemma 2.1and3.10, we get

xn−p² α_n

γfx_n−μAp

I−μα_nA y_n−

I−μα_nA p²

≤1−α_nτ²y_n−p²2α_n

γfx_n−γf p

γf p

−μAp, x_n−p

≤1−αnτ²un−p²2αn

γfxn−γf p

γf p

−μAp, xn−p

≤1−α_nτ²x_n−p²− xn−u_n²

2α_nγαx_n−p²2α_nγf p

−μApx_n−p

1−2αn τ−γα

αnτ²xn−p²−1−αnτ²xn−un²2αnγf p

−μApxn−p

≤xn−p² αnτ²xn−p²−1−αnτ²xn−un²2αnγf p

−μApxn−p. 3.11

It follows that

1−αnτ²xn−un²≤αnτ²xn−p²2αnγf p

−μApxn−p. 3.12 Sinceαn → 0, we have

nlim→ ∞u_n−x_n0. 3.13

From3.8, it is easy to get

nlim→ ∞un−yn0. 3.14

Step 3. Show that

nlim→ ∞un−Wun0, 3.15

u_n−W_nu_n ≤u_n−y_ny_n−W_nu_nu_n−y_nβ_nx_n−u_nu_n−W_nu_n. 3.16

This implies that

1−β_n

u_n−W_nu_n ≤u_n−y_nβ_nx_n−u_n. 3.17

From conditionii,3.13, and3.14, we have

u_n−W_nu_n −→0. 3.18

Notice that

un−Wun ≤ un−WnunWnun−Wun. 3.19

ByLemma 2.7and3.18, we get3.15.

Since{u_n}is bounded, so there exists a subsequence{u_nj}which converges weakly to x^∗.

Step 4. Show thatx^∗∈Ω.

SinceCis closed and convex,Cis weakly closed. So, we havex^∗∈C.

From3.15, we obtainWunj x^∗. From Lemmas2.8,2.4, and 2.13, we havex^∗ ∈ FW ∩^∞_i1FS_i ∩^∞_i1FSi.

Byu_nT_rnx_n, for alln≥1, we have F

u_n, y 1

rn

y−u_n, u_n−x_n

≥0, ∀y∈C. 3.20

It follows fromA2that 1 rn

y−un, un−xn

≥F y, un

, ∀y∈C. 3.21

Hence, we get

1 rnj

y−u_nj, u_nj−x_nj

≥F y, u_nj

, ∀y∈C. 3.22

It follows from conditioniii,3.13, andA4that 0≥F

y, x^∗

, ∀y∈C. 3.23

Forswith 0 < s≤ 1 andy ∈C, letys sy 1−sx^∗. Sincey ∈Candx^∗ ∈C, we obtain y_s∈Cand henceFy_s, x^∗≤0. So, we have

0f ys, ys

≤sF ys, y

1−sF ys, x^∗

≤sF ys, y

. 3.24

Dividing bys, we get

F ys,y

≥0, ∀y∈C. 3.25

Lettings → 0 and fromA3, we get F

x^∗, y

≥0 3.26

for ally∈Candx^∗∈EPF.Hencex^∗∈Ω.

Step 5. Show thatxn → x^∗,wherex^∗PΩI−μAγfx^∗: xn−x^∗αn

γfxn−μAx^∗

I−μα_nA yn−

I−μα_nA

x^∗. 3.27

Hence, we obtain x_n−x^∗2α_n

γfx_n−μAx^∗, x_n−x^∗

I−μα_nA y_n−

I−μα_nA

x^∗, x_n−x^∗

;

≤αn

γfxn−μAx^∗, xn−x^∗

1−αnτxn−x^∗2. 3.28

It follows that

xn−x^∗2≤ 1 τ

γfxn−μAx^∗, xn−x^∗

1 τ

γ

fx_n−fx^∗, x_n−x^∗

γfx^∗−μAx^∗, x_n−x^∗

≤ 1 τ

γαxn−x^∗2

γfx^∗−μAx^∗, xn−x^∗ .

3.29

This implies that

x_n−x^∗2≤

γfx^∗−μAx^∗, xn−x^∗

τ−γα . 3.30

In particular,

xnj−x^∗2≤

γfx^∗−μAx^∗, x_nj−x^∗

τ−γα . 3.31

Sincex_nj x^∗, it follows from3.31thatx_nj → x^∗asj → ∞. Next, we show thatx^∗ solves the variational inequality3.4.

By the iterative algorithm3.5, we have xnαnγfxn

I−μαnA

ynαnγfxn

I−μαnA

Vnxn. 3.32

Therefore, we have

μα_nAx_n−α_nγfx_n μα_nAx_n−x_n

I−μα_nA

V_nx_n, 3.33

that is,

μA−γf

x_n− 1 α_n

I−V_nx_n−μα_nAx_n−AV_nx_n

. 3.34

Hence, forp∈Ω, μA−γf

x_n, x_n−p −1

αn

I−V_nx_n−μα_nAx_n−AV_nx_n, x_n−p

−1 α_n

I−Vnxn−I−Vnp, xn−p μ

Axn−AVnxn, xn−p

≤μ

Ax_n−AV_nx_n, x_n−p .

3.35 SinceI−V_nis monotonei.e.,x−y,I−V_nx−I−V_ny ≥0, for allx, y∈H. This is due to the nonexpansivity ofVn.

Now replacingnin3.35withn_jand lettingj → ∞, we obtain μA−γf

x^∗, x^∗−p lim

j→ ∞

μA−γf

xnj, xnj−p

≤ lim

j→ ∞μ

Axnj−AVnxnj, xnj−p 0.

3.36 That is,x^∗∈Ωis a solution of3.4. To show that the sequence{x_n}converges strongly tox^∗, we assume thatxnk → x. By the same processing as the proof above, we derive x∈Ω.

Moreover, it follows from the inequality3.36that μA−γf

x^∗, x^∗−x

≤0. 3.37

Interchangingx^∗andx, we get

μA−γf

x,x−x^∗

≤0. 3.38

ByLemma 2.5, adding up3.37and3.38yields μη−γα

x^∗−x ²≤

μA−γf x^∗−

μA−γf

x, x^∗−x

≤0. 3.39 Hencex^∗xand, therefore,xn → x^∗asn → ∞,

I−μAγf

x^∗−x^∗, x^∗−p

≥0, ∀p∈Ω. 3.40

This is equivalent to the fixed point equation:

PΩ

I−μAγf

x^∗x^∗. 3.41

Theorem 3.2. LetCbe a nonempty closed convex subset of a real Hilbert spaceHandFa bifunction fromC×CtoRsatisfying (A1)–(A4). LetSi :C → Cbe a familyκi-strict pseudocontractions for some 0≤κ_i <1. Assume the setΩ ∩^∞_i1FS_i∩EPF/∅. Letfbe a contraction ofHinto itself with α ∈ 0,1and letA be aL-Lipschitzian continuous operator and η-strongly monotone with L >0,η >0,0< μ <2η/L², and 0< γ < μη−μL²/2/ατ/α. For everyn∈N, letWnbe the mapping generated byS_iand 0< t_i≤b <1. Givenx1∈H, let{x_n}and{u_n}be sequences generated by the following algorithm:

u_nT_rnx_n, y_nβ_nx_n

1−β_n W_nu_n, xn1αnγfxn

I−μαnA yn.

3.42

If{αn},{βn}and{rn}satisfy the following conditions:

i{αn} ⊂0,1, limn→ ∞αn0 and_∞

n1αn ∞;

ii0<lim inf_{n→ ∞}βn≤lim sup_{n→ ∞}βn<1;

iii{rn} ⊂0,∞, lim infn→ ∞rn>0 and lim_{n→ ∞}|rn1−rn|0.

Then,{x_n}converges strongly tox^∗∈Ω, which solves the variational inequality3.4.

Proof. The proof is divided into several steps.

Step 1. Show first that{xn}is bounded.

Taking anyp∈Ω, we have xn1−pαn

γfxn−μAp

I−μαnA yn−

I−μαnA p

≤αnγfxn−γf

pγf p

−μAp 1−αnτyn−p

≤α_nαγx_n−pα_nγf p

−μAp 1−α_nτy_n−p

1−αn

τ−αγxn−pαn

τ−αγγf p

−μAp τ−αγ

≤max

xn−p,γf p

−μAp τ−αγ

.

3.43

By induction, we obtainx_n−p ≤max{x1−p,γfp−μAp/τ−αγ}, n≥1.Hence,{x_n} is bounded, so are{un}and{yn}. It follows from the Lipschitz continuity ofAthat{Axn}and {Au_n}are also bounded. From the nonexpansivity off andW_n, it follows that{fx_n}and {W_nx_n}are also bounded.

Step 2. Show that

xn1−xn −→0. 3.44

Observe that

un1−unTrn1xn1−Trnxn

≤ Trn1xn1−Trn1xnTrn1xn−Trnxn

≤ xn1−xnTrn1xn−Trnxn,

3.45

and from2.5, we have

W_n1u_n−W_nu_nt1S₁U_n1,2u_n−t1S₁U_n,2u_n

≤t1Un1,2un−Un,2un

t1t2S₂U_n1,3u_n−t2S₂U_n,3u_n

≤t1t2Un1,3un−Un,3un

≤ · · ·

≤ⁿ

i1

tiUn1,n1un−Un,n1un

≤M1

n i1

t_i,

3.46

whereM1sup_n{U_n1,n1u_n−U_n,n1u_n}.

Supposexn1βnxn 1−βnzn, thenzn xn1−βnxn/1−βn αnγfxn I− μα_nAy_n−β_nx_n/1−β_n.

Hence, we have z_n1−z_n αn1γfxn1

I−μαn1A

yn1−βn1xn1

1−β_n1 − αnγfxn

I−μαnA

yn−βnxn

1−β_n α_n1

γfx_n1−μAy_n1

1−β_n1 yn1−βn1xn1

1−β_n1 −α_n

γfx_n−μAy_n

1−β_n −yn−βnxn

1−β_n αn1

γfxn1−μAyn1

1−βn1 βn1xn1

1−βn1

Wn1un1−βn1xn1

1−βn1

−αn

γfxn−μAyn

1−β_n −βnxn 1−βn

Wnun−βnxn

1−β_n

≤ α_n1

γfx_n1−μAy_n1 1−βn1 −α_n

γfx_n−μAy_n

1−βn Wn1un1−Wnun.

3.47

It follows from3.45,3.46, and the above result that zn1−zn

≤ αn1

1−β_n1γfxn1μAyn1 αn

1−β_nγfxnμAynWn1un1−Wnun

≤

αn1

1−βn1 αn

1−βn

M2Wn1un1−Wn1unWn1un−Wnun

≤

α_n1

1−βn1 α_n 1−βn

M2u_n1−u_nW_n1u_n−W_nu_n

≤ xn1−x_nT_rn1x_n−T_rnx_n

α_n1

1−βn1 α_n 1−βn

M2M1

n i1

t_i,

3.48

whereM2sup_n{γfx_nμAy_n}. Hence, we get z_n1−z_n − x_n1−x_n ≤ T_rn1x_n−T_rnx_n

α_n1

1−βn1 α_n 1−βn

M2M1

n i1

t_i. 3.49

From conditioni,iii, 0< t_n≤b <1, andLemma 2.12, we obtain lim sup

n→ ∞ z_n1−z_n − x_n1−x_n≤0. 3.50

ByLemma 2.11,we have lim_{n→ ∞}z_n−x_n0. Thus,

nlim→ ∞xn1−xn lim

n→ ∞

1−βn

zn−xn0. 3.51

ByLemma 2.12,3.45and3.44, we obtain

u_n1−u_n −→0. 3.52

Step 3. Show that

xn−Wxn −→0. 3.53

Observe that

x_n−W_nx_n ≤ x_n−W_nu_nW_nu_n−W_nx_n ≤ x_n−W_nu_nu_n−x_n, xn−Wnun ≤ xn−xn1xn1−ynyn−Wnunxn−xn1

xn1−ynβnun−xnxn−Wnun.

3.54

From conditioniand3.5,we can obtain 1−β_n

x_n−W_nu_n ≤ x_n−x_n1x_n1−y_nβ_nu_n−x_n

≤ xn−xn1αnγfxn−μAynβnun−xn. 3.55 ByLemma 2.10, we get

u_n−p²T_rnx_n−T_rnp² ≤

T_rnx_n−T_rnp, x_n−p 1

2

u_n−p²x_n−p²x_n−u_n² . 3.56 This implies that

un−p²≤xn−p²− xn−un². 3.57 By nonexpansivity ofW_n, we have

yn−p²≤βnxn−p²

1−βnun−p²≤xn−p²− 1−βn

xn−un². 3.58

It follows from3.42that x_n1−p²α_n

γfx_n−p

I−μα_nA y_n−

I−μα_nA pα_n

p−μAp²

≤αnγfxn−p² 1−αnτyn−p²αnp−μAp²

≤α_nγfx_n−p² 1−α_nτx_n−p²− 1−β_n

x_n−u_n²

α_np−μAp²

≤α_nγfx_n−p²x_n−p²− 1−β_n

x_n−u_n²α_np−μAp².

3.59 This implies that

1−β_n

x_n−u_n²≤α_nγfx_n−p²p−μAp²

x_n−p²−x_n1−p²

≤α_nγfx_n−p²p−μAp²

x_n−px_n1−px_n1−x_n. 3.60

From conditioni,ii, and3.44, we have

xn−un −→0. 3.61

Further we havex_n−W_nu_n → 0. Thus we get

xn−Wnxn −→0. 3.62

On the other hand, we have

xn−Wxn ≤ xn−WnxnWnxn−Wxn ≤ xn−Wnxnsup

xn∈CWnxn−Wxn. 3.63 Combining3.62, the last inequality, andLemma 2.7, we obtain3.53.

Step 4. Show that

lim sup

n→ ∞

γf−μA

x^∗, x_n−x^∗

≤0, 3.64

wherex^∗PΩI−μAγfx^∗is a unique solution of the variational inequality3.4. Indeed, take a subsequence{x_nj}of{x_n}such that

lim sup

n→ ∞

γf−μA

x^∗, xn−x^∗ lim

j→ ∞

γf−μA

x^∗, xnj−x^∗

. 3.65

Since{x_nj} is bounded, there exists a subsequence{x_njk}of {x_nj} which converges weakly to q. Without loss of generality, we can assumexnj q. From 3.53, we obtain Wx_nj q.

By the same argument as in the proof of Theorem 3.1, we have q ∈ Ω. Since x^∗ PΩI−μAγfx^∗, it follows that

lim sup

n→ ∞

γf−μA

x^∗, xn−x^∗ lim

j→ ∞

γf−μA

x^∗, xnj−x^∗

γf−μA

x^∗, q−x^∗

≤0.

3.66

Step 5. Show that

x_n−→x^∗. 3.67

Since

γf−μA

x^∗, x_n1−x^∗

γf−μA

x^∗, x_n1−x_n

γf−μA

x^∗, x_n−x^∗

≤γf−μA

x^∗x_n1−x_n

γf−μA

x^∗, x_n−x^∗

. 3.68

It follows from3.44and3.66that lim sup

n→ ∞

γf−μA

x^∗, xn1−x^∗

≤0.

x_n1−x^∗2 α_nγfx_n

I−μα_nA

y_n−x^∗2 I−μα_nA

y_n−

I−μα_nA

x^∗α_n

γfx_n−μAx^∗2

≤I−μα_nA y_n−

I−μα_nA

x^∗22α_n

γfx_n−μAx^∗, x_n1−x^∗

≤1−α_nτ²y_n−x^∗22α_n

γfx_n−γfx^∗, x_n1−x^∗

2α_n

γf−μA

x^∗, x_n1−x^∗

≤1−α_nτ²x_n−x^∗2α_nαγ

x_n−x^∗2x_n1−x^∗2

2α_n

γf−μA

x^∗, x_n1−x^∗ . 3.69

This implies that xn1−x^∗2

≤ 1−αnτ²αnαγ

1−α_nαγ xn−x^∗2 2αn

1−α_nαγ

γf−μA

x^∗, xn1−x^∗

≤

1−2α_n τ−αγ 1−α_nαγ

xn−x^∗2 2αn

1−α_nαγ

γf−μA

x^∗, xn1−x^∗

αnτ² 1−α_nαγM3,

3.70 whereM3sup_nxn−x^∗2, n≥1. It is easily to see thatγn2αnτ−αγ/1−αnαγ. Hence, byLemma 2.2, the sequence{xn}converges strongly tox^∗.

Remark 3.3. IfF≡0, thenTheorem 3.2reduces toTheorem 3.1of Wang10.

Acknowledgments

The authors would like to thank the referee for valuable suggestions to improve the manuscript NSFC Tianyuan Youth Foundation of Mathematics of Chinano. 11126136, and the Fundamental Research Funds for the Central UniversitiesGRANT: ZXH2011C002.

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