Strong Convergence of a New Iterative Method for Infinite Family of Generalized Equilibrium and Fixed-Point Problems of Nonexpansive Mappings in Hilbert Spaces

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Volume 2011, Article ID 392741,24pages doi:10.1155/2011/392741

Research Article

Strong Convergence of a New Iterative Method

for Infinite Family of Generalized Equilibrium and

Fixed-Point Problems of Nonexpansive Mappings

in Hilbert Spaces

Shenghua Wang

1, 2

_{and Baohua Guo}

1, 2

1_{National Engineering Laboratory for Biomass Power Generation Equipment,}

North China Electric Power University, Baoding 071003, China

2_{Department of Mathematics and Physics, North China Electric Power University, Baoding 071003, China}

Correspondence should be addressed to Shenghua Wang,[email protected]

Received 15 October 2010; Accepted 18 November 2010

Academic Editor: Qamrul Hasan Ansari

Copyrightq2011 S. Wang and B. Guo. 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 introduce an iterative algorithm for finding a common element of the set of solutions of an infinite family of equilibrium problems and the set of fixed points of a finite family of nonexpansive mappings in a Hilbert space. We prove some strong convergence theorems for the proposed iterative scheme to a fixed point of the family of nonexpansive mappings, which is the unique solution of a variational inequality. As an application, we use the result of this paper to solve a multiobjective optimization problem. Our result extends and improves the ones of Colao et al.

2008and some others.

1. Introduction

LetHbe a real Hilbert space andTbe a mapping ofHinto itself.Tis said to be nonexpansive if

_Tx₋_Ty_≤_x₋_y_, _∀_{x, y}_∈_H. _1.1

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Letf:H → Hbe a mapping. If there exists a constant 0≤κ <1 such that

_fx₋_fy_≤_κ_x₋_y_, _∀_{x, y}_∈_H, _1.2

thenfis called a contraction with the constantκ. Recall that an operatorA:H → His called to be strongly positive with coeﬃcientγ >0 if

Ax, x ≥γ x 2, ∀x∈H. 1.3

Let u ∈ H be a fixed point, A be a strongly positive linear bounded operator on

H and {T}N_n₁ be a finite family of nonexpansive mappings of H into itself such thatF _N

n1FTn/∅.

In 2003, Xu2introduced the following iterative scheme:

xn1 I−n1ATn1xnn1u, ∀n≥1, 1.4

where I is the identical mapping on H and Tn TnmodN, and proved some strong convergence theorems for the iterative scheme to the solution of the quadratic minimization problem

min x∈F

1

2Ax, x − x, u 1.5

under suitable hypotheses onnand the additional hypothesis:

FFT1T2· · ·TN FTNT1· · ·TN−1 · · ·FT2T3· · ·TNT1. 1.6

Recently, Marino and Xu3introduced a new iterative scheme from an arbitrary point

x0∈Hby the viscosity approximation method as follows:

xn1nγfxn I−nATxn, ∀n≥1, 1.7

and prove that the scheme strongly converges to the unique solution x∗ of the variational inequality:

A−γfx∗, x−x∗≥0, ∀x∈FT, 1.8

which is the optimality condition for the minimization problem:

min x∈F

1

2Ax, x −hx, 1.9

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Let{Tn}Nn1be a finite family of nonexpansive mappings ofHinto itself. In 2007, Yao

4defined the mappings

Un,1 λn,1T1 1−λn,1I,

Un,2 λn,2T2Un,1 1−λn,2I,

.. .

Un,N−1 λn,N−1TN−1Un,N−2 1−λn,N−1I,

Wn ≡Un,Nλn,NTNUn,N−1 1−λn,NI

1.10

and, by extending1.10, proposed the iterative scheme:

xn1nγfxn βxn

1−βI−nA

Wnxn, ∀n≥1. 1.11

Then he proved that the iterative scheme1.10strongly converges to the unique solutionx∗

of the variational inequality:

A−γfx∗, x−x∗≥0, ∀x∈F, 1.12

whereFN_n₁FTn, which is the optimality condition for the minimization problem:

min x∈F

1

2Ax, x −hx, 1.13

wherehis a potential function forγfHowever, Colao et al. pointed out in5that there is a gap in Yao’s proof.

LetCbe a nonempty closed convex subset ofHandG :C×C → Rbe a bifunction. The equilibrium problem for the functionG is to determine the equilibrium points, that is, the set

EPG x∈C:Gx, y≥0, ∀y∈C . 1.14

LetA:C → Hbe a nonlinear mapping. LetEPG, Adenote the set of all solutions to the following equilibrium problem:

EPcG, A x∈C:Gx, yAz, y−z≥0, ∀y∈C . 1.15

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In 2007, S. Takahashi and W. Takahashi 6 introduced a viscosity approximation method for finding a common element ofEPGandFTfrom an arbitrary initial element

x1∈H

Gun, y

1

rn

y−un, un−xn

≥0, ∀y∈C,

xn1nfxn 1−nTun, ∀n≥1,

1.16

and proved that, under certain appropriate conditions overnandrn, the sequences{xn}and

{un}both converge strongly tozPFT∩EPGfz.

By combing the schemes1.7 and 1.16, Plubtieg and Punpaeng7proposed the following algorithm:

Gun, y

1

rn

y−un, un−xn

≥0, ∀y∈C,

xn1nγfxn I−nATun, ∀n≥1,

1.17

and proved that the iterative schemes{xn}and{un}converge strongly to the unique solution

zof the variational inequality:

A−γfz, x−z≥0, ∀x∈FT∩EPG, 1.18

which is the optimality condition for the minimization problem:

min x∈FT∩EPG

1

2Ax, x −hx, 1.19

wherehis a potential function forγf.

Very recently, for finding a common element of the set of a finite family of nonexpansive mappings and the set of solutions of an equilibrium problem, by combining the schemes1.11and1.17, Colao et al.5proposed the following explicit scheme:

Gun, y

1

rn

y−un, un−xn

≥0, ∀y∈C,

xn1nγfxn βxn

1−βI−nA

Wnun, ∀n≥1,

1.20

and proved under some certain hypotheses that both sequences {xn} and {un} converge strongly to a pointx∗ ∈ F which is an equilibrium point forG and is the unique solution of the variational inequality:

A−γfx∗, x−x∗≥0, ∀x∈F∩EPG, 1.21

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The equilibrium problems have been considered by many authors; see, for example,

6,8–19and the reference therein. But, in these references, the authors only considered at most finite family of equilibrium problems and few of authors investigate the infinite family of equilibrium problems in a Hilbert space or Banach space. In this paper, we consider a new iterative scheme for obtaining a common element in the solution set of an infinite family of generalized equilibrium problems and in the common fixed-point set of a finite family of nonexpansive mappings in a Hilbert space. Let {Tn}Nn1 N ≥ 1 be a finite family of

nonexpansive mappings of H into itself, be{Gn} : C×C → R be an infinite family of bifunctions, and be{An} : C → H be an infinite family of kn-inverse-strongly monotone mappings. Let{rn}be a sequence such thatrn⊂r,2knwithr >0 for eachn≥1. Define the mappingTri :H → Cby

Trix

z∈C:Gi

z, y 1 ri

y−z, z−x≥0, ∀y∈C

, x∈H, i≥1. 1.22

Assume that Ω N_i₁FTi∞i1EPGi, Ai/∅. For an arbitrary initial pointx1 ∈ H, we

define the iterative scheme{xn}by

znαnxn n

i1

αi−1−αiTriI−riAixn,

xn1nγfxn δnBxn I−δnB−nAWnzn, ∀n≥1,

1.23

whereα0 1,{αn},{n}and {δn} are three sequences in0,1,AandBare both strongly positive linear bounded operators on H, Wn is defined by 1.10, and prove that, under some certain appropriate hypotheses on the control sequences, the sequence{xn}strongly converges to a pointx∗∈Ω, which is the unique solution of the variational inequality:

A−γfx∗, x−x∗≥0, ∀x∈Ω. 1.24

IfAi≡A0,Gi≡Gandri ≡r, then1.23is reduced to the iterative scheme:

znαnxn 1−αnTrI−rA0xn,

xn1nγfxn δnBxn I−δnB−nAWnzn, ∀n≥1.

1.25

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2. Preliminaries

LetCbe a closed convex subset of a Hilbert spaceH. For any pointx ∈ H, there exists a unique nearest point inC, denoted byPCx, such that

x−PCx ≤x−y, ∀y∈C. 2.1

ThenPCis called the metric projection ofHontoC. It is well known thatPCis a nonexpansive mapping ofHontoCand satisfies the following:

x−y, PCx−PCy

≥PCx−PCy2, ∀x, y∈H. 2.2

LetAbe a mapping fromCintoH, thenAis called monotone if

x−y, Ax−Ay≥0 2.3

for allx, y∈C. However,Ais called anα-inverse-strongly monotone mapping if there exists a positive real numberαsuch that

x−y, Ax−Ay≥αAx−Ay2 2.4

for allx, y∈C. LetI denote the identity mapping ofH, then for allx, y∈Candλ >0, one has20

_I₋_λAx₋_I₋_λAy2 _≤

x−y2λλ−2αAx−Ay2. 2.5

Hence, ifλ∈0,2α, thenI−λAis a nonexpansive mapping ofCintoH. If there existsu∈Csuch that

v−u, Au ≥0 2.6

for allv∈C, thenuis called the solution of this variational inequality. The set of all solutions of the variational inequality is denoted by VIC, A.

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Lemma 2.1see21. Givenx ∈ H and y ∈ C. ThenPCx y if and only if there holds the inequality

x−y, y−z≥0, ∀z∈C. 2.7

Lemma 2.2see22. Let{sn}be a sequence of nonnegative real numbers satisfying

sn1≤

1−ηn

snηnτnξn, ∀n≥0, 2.8

where{ηn},{τn}, and{ξn}satisfy the conditions:

1{ηn} ⊂0,1,∞n1ηn∞or, equivalently,∞n01−ηn 0;

2lim sup_n_{→ ∞}τn ≤0;

3ξn≥0 n≥0,∞n0ξn<∞.

Thenlimn→ ∞sn0.

LetHbe a Hilbert space. For allx, y∈H, the following equality holds:

xy2 x 22y, xy−y2. 2.9

Therefore, the following lemma naturally holds.

Lemma 2.3. LetHbe a real Hilbert space. The following identity holds:

xy2≤ x 22y, xy, ∀x, y∈H. 2.10

Lemma 2.4see3. Assume thatAis a strongly positive linear bounded operator on a Hilbert spaceHwith coeﬃcientγ >0and0< ρ≤ A −1_{. Then} _I₋_ρA_≤₁₋_ργ_.

Lemma 2.5see2. Assume that{an}is a sequence of nonnegative numbers such that

an1≤

1−γn

anδn, ∀n≥0, 2.11

where{γn}is a sequence in0,1andδnis a sequence inRsuch that

1∞_n₁γn∞;

2lim sup_n_{→ ∞}δn/γn≤0or

_∞

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Lemma 2.6see23. Let C be a nonempty closed convex subset of a Hilbert space H and letG :

C×C → Rbe a bifunction which satisfies the following: A1Gx, x 0for allx∈C;

A2Gis monotone, that is,Gx, y Gy, x≤0for allx, y∈C; A3For eachx, y, z∈C,

lim t↓0 G

tz 1−tx, y≤Gx, y; _2.12

A4For eachx∈C,y→Gx, yis convex and lower semicontinuous. Forx∈Handr >0, define a mappingTr :H → Cby

Trx

z∈C:Gz, y1 r

y−z, z−x≥0, ∀y∈C

. 2.13

ThenTris well defined and the following hold:

1Tris single-valued;

2Tris firmly nonexpansive, that is, for anyx, y∈H,

_T_r_x₋_T_r_y2_≤

Trx−Try, x−y

; 2.14

3FTr EPG;

4EPGis closed and convex.

It is easy to see that if there exists some pointv∈Csuch thatvTrI−rAv, where

A : C → His anα-inverse strongly monotone mapping, thenv ∈ EPG, A. In fact, since

vTrI−rAv, one has

Gv, y1 r

y−v, v−I−rAv≥0, ∀y∈C, 2.15

that is,

Gv, yy−v, Av ≥0, ∀y∈C. 2.16

Hence,v∈EPG, A.

LetCbe a nonempty convex subset of a Banach space. Let{Ti}Ni1be a finite family of

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for eachi1,2, . . . , N. Define a mappingWofCinto itself as follows:

U1λ1T1 1−λ1I,

U2λ2T2U1 1−λ2I,

.. .

UN−1λN−1TN−1UN−2 1−λN−1I,

WUNλNTNUN−1 1−λNI.

2.17

Such a mappingW is called theW-mappinggenerated byT1, T2, . . . , TN andλ1, λ2, . . . , λN

see5,24,25.

Lemma 2.7 see 26. Let C be a nonempty closed convex subset of a Banach space. Let T1, T2, . . . , TN be nonexpansive mappings of C into itself such that Ni1FTi/∅ and let

λ1, λ2, . . . , λN be real numbers such that0< λi <1for eachi1,2, . . . , N−1and0< λN ≤1. Let

Wbe theW-mapping ofCgenerated byT1, T2, . . . , TNandλ1, λ2, . . . , λN. ThenFW

_N

i1FTi. Lemma 2.8 see 5. Let C be a nonempty convex subset of a Banach space. Let {Ti}Ni1 be a

finite family of nonexpansive mappings ofCinto itself and let{λn,i}Ni1 be sequences in0,1such

that λn,i → λi for each i 1,2, . . . , N. Moreover, for each n ∈ N, let W and Wn be the W -mappings generated byT1, T2, . . . , TNandλ1, λ2, . . . , λN andT1, T2, . . . , TNandλn,1, λn,2, . . . , λn,N, respectively. Then, for allx∈C, it follows that

lim

n→ ∞ Wnx−Wx 0. 2.18

3. Main Results

Now, we give our main results in this paper.

Theorem 3.1. LetHbe a Hilbert space andCbe a nonempty closed convex subset ofH. Letf :H → Hbe a contraction with coeﬃcient0 < κ <1,A,B :H → H be strongly positive linear bounded self-adjoint operators with coeﬃcientsγ > 0 andβ > 2 B B 2, respectively,{Tn}Nn1 : H →

H N≥1be a finite family of nonexpansive mappings,{Gn}: C×C → R be an infinite family of bifunctions satisfying A1–A4, and{An}:C → Hbe an infinite family of inverse-strongly monotone mappings with constants{kn}such thatΩ Ni1FTi∩∞i1EPGi, Ai/∅. Let

{n}and{δn}be two sequences in0,1,{λn,i}Ni1be asequence in a, bwith0< a≤ b <1,{rn} be a sequence inr,2knwithr > 0and{αn}be a strictly decreasing sequence0,1. Setα0 1.

Take a fixed numberγ >0with0< γ−γκ <1. Assume that E1limn→ ∞n0 and ∞n1n∞;

E2limn→ ∞|λn1,i−λn,i|0 for eachi1,2, . . . , N;

E30≤δnn≤1 for alln≥1;

E4{δn} ⊂0,min{c,1/β, 2 B B 2−β

β− B 2−2 B 28β B /4β B }with c <1;

E5∞_n₁|n1−n|<∞,

_∞

n1|αn1−αn|<∞,

_∞

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Then the sequence{xn}defined by1.23converges strongly tox∗∈Ω, which is the unique solution of the variational inequality:1.24, that is,

x∗PΩI−A−γfx∗. 3.1

Proof. Since n → 0 as n → ∞ by the condition E1, we may assume, without loss of generality, thatn < 1−δn B A −1for alln≥1. Noting thatAandBare both the linear bounded self-adjoint operators, one has

A sup{|Ax, x|:x∈H, x 1},

B sup{|Bx, x|:x∈H, x 1}.

3.2

Observing that

I−δnB−nAx, x1−δnBx, x −nAx, x

≥1−δn B −n A

≥0,

3.3

we obtain thatI−δnB−nAis positive for alln≥1. It follows that

I−δnB−nA sup{I−δnB−nAx, x:x∈H, x 1}

sup{1− δnBnAx, x:x∈H, x 1}

≤1−δnβ−nγ.

3.4

For eachn≥1, define a quadratic functionfδninδnas follows:

fδn 2β B δn2

β− B 2−2 B δn. 3.5

Note that

f0 f

2 B −β B 2

2β B

0, 3.6

f ⎛ ⎜ ⎜ ⎝

2 B B 2−ββ− B 2−2 B 28β B

4β B

⎞ ⎟ ⎟

⎠1. 3.7

Hence, for eachδnsatisfying the conditionE4, one has

0<2β B δ2

n

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Moreover, it follows from3.7,f1/ B >1 andE4that

δn< 1

B , ∀n≥1. 3.9

Next, we proceed the proof with following steps.

Step 1. {xn}is bounded.

Let p ∈ Ω. Lemma 2.6 shows that every Tri is firmly nonexpansive and hence nonexpansive. Sincer < ri<2ki,I−riAiis nonexpansive for eachi≥1. Therefore,TriI−riAi is nonexpansive for eachi≥1. Noting that{αn}is strictly decreasing,α01, we have

zn−p

αn

xn−p

n

i1

αi−1−αi

TriI−riAixn−TriI−riAip

≤αnxn−p

n

i1

αi−1−αiTriI−riAixn−TriI−riAip

≤αnxn−p

n

i1

αi−1−αixn−p

xn−p

3.10

and hence

Wnzn−pWnzn−Wnp≤zn−p≤xn−p. 3.11

Then, from3.4and3.11, it follows thatnoting thatBis linear andβ >2 B B 2_⇒_{β >}

B

xn1−p

n

γfxn−Ap

δn

Bxn−Bp

I−δnB−nA

Wnzn−p

≤nγfxn−ApδnB

xn−p I−δnB−nA Wnzn−p

≤nγκxn−pnγf

p−Apδn B xn−p

1−δnβ−nγ

Wnzn−p

≤nγκxn−pnγf

p−Apδn B xn−p

1−δnβ−nγ

xn−p

≤1−n

γ−γκxn−pnγf

p−Ap.

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It follows fromn ∈0,1and 0< γ−γκ <1 that 0< nγ−γκ<1. Therefore, by the simple induction, we have

xn−p≤max

x1−p,

_γf

p−Ap γ−γκ

, ∀n≥1, 3.13

which shows that{xn}is bounded, so is{zn}. Step 2. xn1−xn → 0 asn → ∞.

First, we prove

lim

n→ ∞ Wn1zn−Wnzn 0. 3.14

Leti∈ {0,1, . . . , N−2}and set

M1sup

n

zn T1zn N

i2

TiUn,i−1zn

<∞. 3.15

It follows from the definition ofWnthat

Un1,N−izn−Un,N−izn

λn1,N−iTN−iUn1,N−i−1zn 1−λn1,N−izn−λn,N−iTN−iUn,N−i−1zn−1−λn,N−izn

≤λn1,N−i TN−iUn1,N−i−1zn−TN−iUn,N−i−1zn

|λn1,N−i−λn,N−i| TN−iUn,N−i−1zn |λn1,N−i−λn,N−i| zn

≤ Un1,N−i−1zn−Un,N−i−1zn zn TN−iUn,N−i−1zn |λn1,N−i−λn,N−i|

≤ Un1,N−i−1zn−Un,N−i−1zn M1|λn1,N−i−λn,N−i|

3.16

for eachi∈ {0,1, . . . , N−2}. Thus, using the above recursive inequalities repeatedly, we have

Wn1zn−Wnzn Un1,Nzn−Un,Nzn

≤M1

N

i2

|λn1,i−λn,i||λn1,1−λn,1| zn T1zn

≤M1

N

i1

|λn1,i−λn,i|.

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Also, we have

zn−zn−1

αnxn

n

i1

αi−1−αiTriI−riAixn−αn−1xn−1

−n

i1

αi−1−αiTriI−riAixn−1 αn−1−αnTrnI−rnAnxn−1

≤αn xn−xn−1 |αn−αn−1| xn−1

n

i1

αi−1−αi xn−xn−1 |αn−1−αn| TrnI−rnAnxn−1

3.18

whereLsup{ xn−1 TrnI−rnAnxn−1 }.

Next, we prove limn→ ∞ xn1−xn 0. Observenoting thatBis linearthat

xn1−xnnγ

fxn−fxn−1

nγfxn−1 δnBxn−xn−1 δnBxn−1

I−δnB−nAWnzn−Wnzn−1 I−δnB−nAWnzn−1

−n−1γfxn−1−δn−1Bxn−1−I−δn−1B−n−1AWn−1zn−1

nγ

fxn−fxn−1

δnBxn−xn−1 I−δnB−nAWnzn−Wnzn−1

n−n−1γfxn−1 δn−δn−1Bxn−1 Wnzn−1−Wn−1zn−1

δn−1BWn−1zn−1−δnBWnzn−1 n−1AWn−1zn−1−nAWnzn−1

nγ

fxn−fxn−1

δnBxn−xn−1 I−δnB−nAWnzn−Wnzn−1

n−n−1γfxn−1 δn−δn−1Bxn−1 Wnzn−1−Wn−1zn−1

δn−1−δnBWn−1zn−1δnBWn−1zn−1−Wnzn−1

n−1−nAWn−1zn−1nAWn−1zn−1−Wnzn−1.

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Hence, by3.4and3.18, we get

xn1−xn ≤nγκ xn−xn−1 δn B xn−xn−1

1−δnβ−nγ

zn−zn−1

|n−n−1|γfxn−1|δn−δn−1| Bxn−1 Wnzn−1−Wn−1zn−1

|δn−1−δn| BWn−1zn−1 δn B Wn−1zn−1−Wnzn−1

|n−1−n| AWn−1zn−1 n A Wn−1zn−1−Wnzn−1

≤nγκ xn−xn−1 δn B xn−xn−1

1−δnβ−nγ

xn−xn−1 |αn−αn−1|L

|n−n−1|γfxn−1|δn−δn−1| Bxn−1 Wnzn−1−Wn−1zn−1

|δn−1−δn| BWn−1zn−1 δn B Wn−1zn−1−Wnzn−1

|n−1−n| AWn−1zn−1 n A Wn−1zn−1−Wnzn−1

≤1−δn

β− B n

γ−γκ xn−xn−1 L|αn−αn−1|2|n−1−n|M2

2|δn−1−δn|M2 1δn B n A Wn−1zn−1−Wnzn−1 ,

3.20

whereM2sup_n{γ fxn−1 Bxn−1 BWn−1zn−1 AWn−1zn−1 }.

SetM3 min{β− B , γ −γκ}. It follows from 0 ≤ γ −γκ < 1 andβ > B due to

β >2 B B 2_{that 0}_≤_M

2 <1. Thus we have

xn1−xn ≤1−δnnM3 xn−xn−1 δnnM3

×

1

δnnM3

δn B

δnnM3

n A

δnnM3

× Wn−1zn−1−Wnzn−1 L|αn−αn−1|2|n−1−n|M22|δn−1−δn|M2.

3.21

Set

ηn δnnM3,

τn

1

δnnM3

δn B

δnnM3

n A

δnnM3

Wn−1zn−1−Wnzn−1 ,

ξnL|αn−αn−1|2|n−1−n|M22|δn−1−δn|M2.

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Then it follows from3.21that

xn1−xn ≤

1−ηn

xn−xn−1 ηnτnξn. 3.23

It follows from the assumption conditionE1,E3,E5, and3.14that

ηn⊂0,1,

∞

n1

ηn∞, lim n→ ∞τn0,

∞

n1

ξn<∞. 3.24

By applyingLemma 2.2to3.23, we obtain xn1−xn → 0 asn → ∞. Step 3. xn−Wnzn → 0 asn → ∞.

For alln≥1, we have

xn−Wnzn ≤ xn−xn1 xn1−Wnzn

xn−xn1 nγfxn δnBxn I−δnB−nAWnzn−Wnzn

≤ xn−xn1 nγfxn−AWnznδn Bxn−BWnzn

≤ xn−xn1 nγfxn−AWnznδn B xn−Wnzn

3.25

and hencenoting3.9

xn−Wnzn ≤ 1 1−δn B

xn−xn1 n

1−δn

!

γfxn AWnzn

"

. 3.26

It follows from the assumption conditionsE1,E2, andStep 2that

xn−Wnzn−→0 n−→ ∞. 3.27

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Notice that, for anyx∈Ω,

zn−x 2

αnxn−x n

i1

αi−1−αiTriI−riAixn−TriI−riAix

2

≤αn xn−x 2

n

i1

αi−1−αi I−riAixn−I−riAix 2

≤αn xn−x 2

n

i1

αi−1−αi

xn−x 2riri−2ki Aixn−Aix 2

xn−x 2 n

i1

αi−1−αiriri−2ki Aixn−Aix 2.

3.28

Letynγfxn−AWnznandλsup{ γfxn−AWnzn :n≥1}. By using3.8,3.9,3.28, Lemmas 2.3, and 2.4, we havenoting thatδn<1/β

xn1−x 2I−δnBWnzn−x δnBxn−Bx n

γfxn−AWnzn2

≤ I−δnBWnzn−x δnBxn−Bx 22n

yn, xn1−x

I−δnBWnzn−Wnx δnBxn−x 22n

yn, xn1−x

≤1−δnβ

zn−x 2δn B 2 xn−x 22δn

1−δnβ

B xn−x 22λ2n

≤1−δnβ

#

xn−x 2 n

i1

αi−1−αiriri−2ki Aixn−Aix 2

$

δn B 2 xn−x 22δn

1−δnβ

B xn−x 22λ2n

1−δnβ−δn B 2−2δn

1−δnβ

B xn−x 2

1−δnβ

n

i1

αi−1−αiriri−2ki Aixn−Aix 22λ2n

≤ xn−x 2

1−δnβ

n

i1

αi−1−αiriri−2ki Aixn−Aix 22λ2n.

3.29

This shows that

1−δnβ

n

i1

αi−1−αiri2ki−ri Aixn−Aix 2

≤ xn−x 2− xn1−x 22λ2n

≤ xn−x xn1−x xn−xn1 2λ2n

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and hence, for eachi≥1,

1−δnβ

αi−1−αiri2ki−ri Aixn−Aix 2

≤ xn−x 2− xn1−x 22λ2n

≤ xn−x xn1−x xn−xn1 2λ2n.

3.31

Sinceδn → 0, xn−xn1 → 0 andαi−1−αi>0, we have

lim

n→ ∞ Aixn−Aix 0, i≥1. 3.32

Now, forx∈Ω, we have, fromLemma 2.2,

TriI−riAixn−x

2

TriI−riAixn−TriI−riAix

2

≤ TriI−riAixn−TriI−riAix,I−riAixn−I−riAix

TriI−riAixn−x, xn−xriTriI−riAixn−x, Aix−Aixn

1

2

%

2 _x

n−x 2− xn−TriI−riAixn

2&

riTriI−riAixn−x, Aix−Aixn

3.33

and hence

2_≤_x

n−x 2− xn−TriI−riAixn

2

2riTriI−riAixn−x, Aix−Aixn.

3.34

Therefore,

zn−x 2≤αn xn−x 2 n

i1

αi−1−αi TriI−riAixn−x

2

≤αn xn−x 2

n

i1

αi−1−αi

xn−x 2− xn−TriI−riAixn

2

2riTriI−riAixn−x, Aix−Aixn

xn−x 2− n

i1

αi−1−αi xn−TriI−riAixn

2

2 n

i1

αi−1−αiriTriI−riAixn−x, Aix−Aixn.

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By using3.8,3.9,3.35, Lemmas2.3and2.4, we havenoting thatδn<1/β

xn1−x 2

I−δnBWnzn−x δnBxn−Bx n

γfxn−AWnzn2

≤ I−δnBWnzn−x δnBxn−Bx 22n

yn, xn1−x

I−δnBWnzn−Wnx δnBxn−x 22n

yn, xn1−x

≤1−δnβ

zn−x 2δn B 2 xn−x 22δn

1−δnβ

B xn−x 22λ2n

≤1−δnβ

#

xn−x 2− n

i1

2

2 n

i1

αi−1−αiriTriI−riAixn−x, Aix−Aixn

$

δn B 2 xn−x 22δn

1−δnβ

B xn−x 22λ2n

1−δnβ−δn B 2−2δn

1−δnβ

B xn−x 2

−1−δnβ

n

i1

2

21−δnβ

n

i1

αi−1−αiriTriI−riAixn−x, Aix−Aixn2λ

2

n

≤ xn−x 2−

1−δnβ

n

i1

2

21−δnβ

n

i1

2

n

3.36

and hence

1−δnβ

n

i1

2

≤ xn−x 2−xn1−p22

1−δnβ

×n

i1

2

n

≤ xn−x xn1−x xn−xn1

21−δnβ

n

i1

2

n.

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This shows that for, eachi≥1,

1−δnβ

2

≤ xn−x xn1−x xn−xn1

21−δnβ

n

i1

2

n.

3.38

Since{αn}is strictly decreasing,δn → 0,n → 0,Aixn−Aix → 0 and xn−xn1 → 0, we

have, for eachi≥1,

xn−TriI−riAixn −→0, n−→ ∞. 3.39

Now, fromzn−xn

_n

i1αi−1−αiTrixn−xnwe get

zn−xn ≤ n

i1

αi−1−αi Trixn−xn . 3.40

Since xn−Trixn → 0 and 0< αi−1−αifor eachi≥1, one has

zn−xn −→0, asn−→ ∞. 3.41

Step 5. lim sup_n_{→ ∞}γf−Ax∗, xn−x∗ ≤0.

To prove this, we pick a subsequence{xnj}of{xn}such that

lim sup n→ ∞

γf−Ax∗, xn−x∗

lim

j→ ∞

'

γf−Ax∗, xnj−x∗

(

. 3.42

Without loss of generality, we may further assume thatxnj x). Obviously, to proveStep 5, we only need to prove thatx)∈Ω.

Indeed, for eachi ≥ 1, since xn−TriI−riAixn → 0,xnj → x)and TriI−riAiis nonexpansive, by demiclosed principle of nonexpansive mapping we have

)

x∈FTriI−riAi EPGi, Ai, i≥1. 3.43

Assume thatλnm,k → λk ∈ 0,1for eachk 1,2, . . . , N. LetW be theW-mapping generated byT1, . . . , TNandλ1, . . . , λN. Then, byLemma 2.8, we have

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Moreover, it follows fromLemma 2.7thatN_n₁FTi FW. Assume thatx /)∈FW. Thenx /)Wx). Sincex)∈FTriI−riAifor eachi≥1, byStep 3,3.44and Opial’s property of the Hilbert spaceH, we have

lim inf

n→ ∞ xnm−x)

<lim inf

n→ ∞ xnm−Wx)

≤lim inf

n→ ∞ xnm−Wnmznm Wnmznm−Wnmx) Wnmx)−Wx) ≤lim inf

n→ ∞ xnm−Wnmznm znm−x) Wnmx)−Wx) ≤lim inf

n→ ∞ xnm−Wnmznm znm−xnm xnm −TriI−riAixnm

x)−TriI−riAixnm Wnmx)−Wx) ≤lim inf

n→ ∞ xnm−Wnmznm znm−xnm xnm −TriI−riAixnm

TriI−riAi)x−TriI−riAixnm Wnmx)−Wx) ≤lim inf

n→ ∞ xnm−x) ,

3.45

which is a contradiction. Therefore,x)∈FW. Hence,x)∈Ω N_i₁FTi∩∞i1EPGi, Ai. Step 6. The sequence{xn}strongly converges to some pointx∗∈H.

By using Lemmas2.3and2.4, we have

xn1−x∗ 2 I−δnB−nAWnzn−x∗ δnBxn−Bx∗ n

γfxn−Ax∗2

≤ I−δnB−nAWnzn−x∗ δnBxn−Bx∗ 2

2n

γfxn−Ax∗, xn1−x∗

≤!1−δn B −nγ

zn−x∗ δn B xn−x∗

"2

2n

≤!1−δn B −nγ

xn−x∗ δn B xn−x∗

"2

2n

!1−nγ

xn−x∗

"2

2n

γfxn−Ax∗, xn1−x∗

≤1−nγ

2

xn−x∗ 22nγκ xn−x∗ xn1−x∗

2n

γfx∗−Ax∗, xn1−x∗

≤1−nγ

2

xn−x∗ 2nγκ

xn−x∗ 2 xn1−x∗ 2

2n

γfx∗−Ax∗, xn1−x∗

,

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which implies that

xn1−x∗ 2

≤

1−nγ

2

nγκ 1−nγκ xn−x

∗ 2 2n 1−nγκ

γfx∗₋_Ax∗_{, x}

n1−x∗

1−2nγnγκ

1−nγκ

xn−x∗ 2

2

nγ2 1−nγκ

xn−x∗ 2 2n 1−nγκ

γfx∗₋_Ax∗_{, x}

n1−x∗

≤

#

1−2n

γ−κγ

1−nγκ

$

xn−x∗ 2

2n

γ−κγ

1−nγκ

#

1

γ−κγ

γfx∗₋_Ax∗_{, x}

n1−x∗

nγ2 2γ−κγM

$

,

3.47

whereMis an appropriate constant such thatMsup_n_≥₁{ xn−x∗ }. Put

sn 2n

γ−κγ

1−nκγ

,

tn 1

γ−κγ

γfx∗₋_Ax∗_{, x}

n1−x∗

nγ2 2γ−κγM

_.

3.48

Then we have

xn1−x∗ 2≤1−sn xn−x∗ sntn. 3.49

It follows from the assumption conditionE1and3.42that

lim n→ ∞sn0,

∞

n1

sn∞, lim sup

n→ ∞ tn ≤0. 3.50

Thus, applyingLemma 2.5to3.49, it follows thatxn → x∗asn → ∞. This completes the proof.

ByTheorem 3.1, we have the following direct corollaries.

Corollary 3.2. LetH be a Hilbert space andCbe a nonempty closed convex subset ofH. Letf :

H → H be a contraction with coeﬃcient0 < κ < 1,A : H → H be strongly positive linear bounded self-adjoint operator with coeﬃcientγ >0,{Tn}Nn1:H → HN ≥1be a finite family of

nonexpansive mappings,G :C×C → Rbe a bifunction satisfying (A1)–(A4), andA0 :C → H

be anα-inverse strongly monotone mapping such thatΩ N_i₁FTi∩EPG, A/∅. Let{εn}and

{δn}be two sequences in0,1,{λn,i}Ni1be a sequence ina, bwith0< a≤b <1,rbe a number in

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E1limn→ ∞n0and∞n1εn∞;

E2limn→ ∞|λn1,i−λn,i|0for eachi1,2, . . . , N;

E30≤δnn≤1for alln≥1;

E4{δn} ⊂0,min{c,1/β, 2 B B 2−β

β− B 2−₂ _B 2₈_β _B /₄_β _B }_with

c <1;

E5∞_n₁|n1−n|<∞,∞n1|αn1−αn|<∞,∞n1|δn1−δn|<∞.

Then the sequence{xn}defined by1.25converges strongly tox∗∈Ω, which is the unique solution of the variational inequality:

x∗PΩI−A−γfx∗. 3.51

Remark 3.3. In the proof process ofTheorem 3.1, we do not use Suzuki’s Lemmasee27, which was used by many others for obtaining xn1−xn → 0 asn → ∞see4,5,28. The proof method ofx)∈∞_i₁EPGi, Aiis simple and diﬀerent with ones of others.

4. Applications for Multiobjective Optimization Problem

In this section, we study a kind of multiobjective optimization problem by using the result of this paper. That is, we will give an iterative algorithm of solution for the following multiobjective optimization problem with the nonempty set of solutions:

min h1x,

min h2x,

x∈C, 4.1

whereh1xandh2xare both the convex and lower semicontinuous functions defined on a

closed convex subset ofCof a Hilbert spaceH.

We denote byAthe set of solutions of the problem4.1and assume thatA /∅. Also, we denote the sets of solutions of the following two optimization problems byA1 andA2,

respectively,

min h1x, x∈C, 4.2

and

minh2x, x∈C. 4.3

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Now, letG1andG2be two bifunctions fromC×CtoRdefined by

G1

x, yh1

y−h1x, G2

x, yh2

y−h2x, ∀

x, y∈C×C, 4.4

respectively. It is easy to see thatEPG1 A1 andEPG2 A2, whereEPGidenotes the set of solutions of the equilibrium problem:

Gi

x, y≥0, ∀y∈C, i1,2, 4.5

respectively. In addition, it is easy to see thatG1andG2satisfy the conditionsA1–A4. Let

{αn}be a sequence in0,1andr1, r2∈0,1. Define a sequence{xn}by

x1∈H,

znαnTr1xn 1−αnTr2xn,

xn1nγfxn 1−nzn, ∀n≥1.

4.6

By Theorem 3.1with A I,N 1,T1 I and δn 0 for all n ≥ 1, the sequence {xn} converges strongly to a solutionx∗ PA1∩A2γfx∗, which is a solution of the multiobjective optimization problem4.1.

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