An Iterative Algorithm Combining Viscosity Method with Parallel Method for a Generalized Equilibrium Problem and Strict Pseudocontractions

(1)

Volume 2009, Article ID 794178,21pages doi:10.1155/2009/794178

Research Article

An Iterative Algorithm Combining

Viscosity Method with Parallel Method for a

Generalized Equilibrium Problem and Strict

Pseudocontractions

Jian-Wen Peng,

1

_{Yeong-Cheng Liou,}

2

_{and Jen-Chih Yao}

3

1_{College of Mathematics and Computer Science, Chongqing Normal University, Chongqing 400047, China} 2_{Department of Information Management, Cheng Shiu University, Kaohsiung, Taiwan 833, Taiwan} 3_{Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan 804, Taiwan}

Correspondence should be addressed to Yeong-Cheng Liou,simplex [email protected]

Received 5 August 2008; Accepted 4 January 2009

Recommended by Hichem Ben-El-Mechaiekh

We introduce a new approximation scheme combining the viscosity method with parallel method for finding a common element of the set of solutions of a generalized equilibrium problem and the set of fixed points of a family of finitely strict pseudocontractions. We obtain a strong convergence theorem for the sequences generated by these processes in Hilbert spaces. Based on this result, we also get some new and interesting results. The results in this paper extend and improve some well-known results in the literature.

Copyrightq2009 Jian-Wen Peng et al. 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

LetHbe a real Hilbert space with inner product ·,· and induced norm · ,and letCbe a nonempty-closed convex subset ofH. Letϕ :H → R∪ {∞}be a function and letFbe a bifunction fromC×Cto Rsuch thatC∩domϕ /∅, where Ris the set of real numbers and domϕ {x∈ H : ϕx< ∞}. Flores-Baz´an 1introduced the following generalized equilibrium problem:

Findx∈Csuch thatFx, y ϕy≥ϕx, ∀y∈C. 1.1

(2)

Letϕx δCx,∀x∈H. HereδCdenotes the indicator function of the setC; that is,

δCx 0 ifx∈CandδCx ∞otherwise. Then the problem1.1becomes the following

equilibrium problem:

Findingx∈Csuch thatFx, y≥0, ∀y∈C. 1.2

The set of solutions of1.2is denoted by EPF. The problem1.2includes, as special cases, the optimization problem, the variational inequality problem, the fixed point problem, the nonlinear complementarity problem, the Nash equilibrium problem in noncooperative games, and the vector optimization problem. For more detail, please see 3–5 and the references therein.

IfFx, y gy−gxfor allx, y∈C, whereg:C → Ris a function, then the problem

1.1becomes a problem of findingx∈Cwhich is a solution of the following minimization problem:

min

y∈C

ϕy gy. 1.3

The set of solutions of1.3is denoted by Argming, ϕ.

Ifϕ :H → R∪ {∞}is replaced by a real-valued functionφ :C → R, the problem

1.1reduces to the following mixed equilibrium problem introduced by Ceng and Yao6:

Find x∈Csuch thatFx, y φy−φx≥0, ∀y∈C. 1.4

Recall that a mappingT :C → Cis said to be aκ-strict pseudocontraction7if there exists 0≤κ <1,such that

Tx−Ty2≤ x−y2κI−Tx−I−Ty2, ∀x, y∈C, 1.5

whereI denotes the identity operator onC. Whenκ0,T is said to be nonexpansive. Note that the class of strict pseudocontraction mappings strictly includes the class of nonexpansive mappings. We denote the set of fixed points ofSby FixS.

(3)

a common element of the set of solutions of problem 1.2 and the set of fixed points of a nonexpansive mapping in a Hilbert space and obtained both strong convergence theorem and weak convergence theorem. Ceng et al.15introduced an iterative algorithm for finding a common element of the set of solutions of problem1.2and the set of fixed points of a strict pseudocontraction mapping. Chang et al.16introduced some iterative processes based on the extragradient method for finding the common element of the set of fixed points of a family of infinitely nonexpansive mappings, the set of problem1.2, and the set of solutions of a variational inequality problem for anα-inverse strongly monotone mapping. Colao et al.

17introduced an iterative method for finding a common element of the set of solutions of problem 1.2 and the set of fixed points of a finite family of nonexpansive mappings in a Hilbert space and proved the strong convergence of the proposed iterative algorithm to the unique solution of a variational inequality, which is the optimality condition for a minimization problem. To the best of our knowledge, there is not any algorithms for solving problem1.1.

On the other hand, Marino and Xu19 and Zhou 20introduced and researched some iterative scheme for finding a fixed point of a strict pseudocontraction mapping. Acedo and Xu21introduced some parallel and cyclic algorithms for finding a common fixed point of a family of finite strict pseudocontraction mappings and obtained both weak and strong convergence theorems for the sequences generated by the iterative schemes.

In the present paper, we introduce a new approximation scheme combining the viscosity method with parallel method for finding a common element of the set of solutions of the generalized equilibrium problem and the set of fixed points of a family of finitely strict pseudocontractions. We obtain a strong convergence theorem for the sequences generated by these processes. Based on this result, we also get some new and interesting results. The results in this paper extend and improve some well-known results in the literature.

2. Preliminaries

LetHbe a real Hilbert space with inner product ·,· and norm · . LetCbe a nonempty-closed convex subset ofH. Let symbols → anddenote strong and weak convergences, respectively. In a real Hilbert spaceH, it is well known that

_λx ₁₋_λ_y2

λx2 1−λy2−λ1−λx−y2 2.1

for allx, y∈Handλ∈0,1.

For anyx∈H, there exists a unique nearest point inC, denoted byPCx, such that x−PCx ≤ x−yfor ally∈C. The mappingPCis called the metric projection ofHonto

C. We know thatPCis a nonexpansive mapping fromHontoC. It is also known thatPCx∈C

and

x−PCx, PCx−y

≥0 2.2

for allx∈Handy∈C.

For eachB⊆H, we denote by convBthe convex hull ofB. A multivalued mapping

G : B → 2H _{is said to be a KKM map if, for every finite subset} _{_x

1, x2, . . . , xn} ⊆ B,

(4)

We will use the following results in the sequel.

Lemma 2.1see22. LetBbe a nonempty subset of a Hausdorﬀtopological vector spaceXand let

G:B → 2X_{be a KKM map. If}_G_x_{is closed for all}_x_∈_B_{and is compact for at least one}_x_∈_B_{, then}

x∈BGx/∅.

For solving the generalized equilibrium problem, let us give the following assump-tions for the bifunctionF,ϕ,and the setC:

A1Fx, x 0 for allx∈C;

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

A3for eachy∈C, x→Fx, yis weakly upper semicontinuous;

A4for eachx∈C, y→Fx, yis convex;

A5for eachx∈C, y→Fx, yis lower semicontinuous;

B1For eachx∈Handr >0, there exist a bounded subsetDx⊆Candyx∈C∩domϕ

such that for anyz∈C\Dx,

F z, yx

ϕ yx

1

r

yx−z, z−x

< ϕz; 2.3

B2Cis a bounded set.

Lemma 2.2. LetCbe a nonempty-closed convex subset ofH. LetF be a bifunction fromC×Cto

Rsatisfying (A1)–(A4) and letϕ : H → R∪ {∞}be a proper lower semicontinuous and convex function such thatC∩domϕ /∅. Forr >0andx∈H,define a mappingSr :H → Cas follows:

Srx

z∈C:Fz, y ϕy 1

ry−z, z−x ≥ϕz,∀y∈C

2.4

for allx∈H. Assume that either (B1) or (B2) holds. Then, the following conclusions hold:

1for eachx∈H, Srx/∅;

2Sris single-valued;

3Sris firmly nonexpansive, that is, for anyx, y∈H,

_S_r_x₋_S_r_y2_≤

Srx−Sry, x−y

; 2.5

4FixSr GEPF, ϕ;

5GEPF, ϕis closed and convex.

Proof. Letx0be any given point inE. For eachy∈C, we define

Gy

z∈C:Fz, y ϕy 1 r

y−z, z−x0

≥ϕz

(5)

Note that for eachy∈C∩domϕ,Gyis nonempty sincey∈Gyand for eachy∈C\domϕ,

Gy C. We will prove thatGis a KKM map onC∩domϕ. Suppose that there exists a finite subset{y1, y2, . . . , yn}ofC∩domϕandμi ≥0 for alli1,2, . . . , nwith

_n

i1μi 1 such that

zn_i₁μiyi/∈Gyifor eachi1,2, . . . , n. Then we have

F z, y i

ϕ yi

−ϕz 1 r

yi−z, z−x0

<0 2.7

for eachi1,2, . . . , n. ByA4and the convexity ofϕ, we have

0Fz, z ϕz−ϕz 1 r

z−z, z−x0

≤n i1

μi

F z, y i

ϕ yi

−ϕz 1 r

_n

i1

μi

yi−z, z−x0

<0,

2.8

which is a contradiction. Hence,Gis a KKM map onC∩domϕ. Note thatGywthe weak closure ofGyis a weakly closed subset ofCfor eachy∈C. Moreover, ifB2holds, then

Gywis also weakly compact for eachy ∈ C. IfB1holds, then forx0 ∈ E, there exists a

bounded subsetDx0 ⊆Candyx0∈C∩domϕsuch that for anyz∈C\Dx0,

F z, yx0

ϕ yx0

1

r

yx0−z, z−x0

< ϕz. 2.9

This shows that

G yx0

z∈C:F z, yx0

ϕ yx0

1

r

yx0−z, z−x0

≥ϕz

⊆Dx0. 2.10

Hence, Gyx0 w

is weakly compact. Thus, in both cases, we can use Lemma 2.1and have

y∈C∩domϕGy w

/

∅.

Next, we will prove thatGyw Gyfor eachy∈C; that is,Gyis weakly closed. Letz∈Gywand letzmbe a sequence inGysuch thatzm z. Then,

F zm, y

ϕy 1 r

y−zm, zm−x0

≥ϕ zm

. 2.11

Since · 2_{is weakly lower semicontinuous, we can show that}

lim sup

m→ ∞

y−zm, zm−x0

≤z−y, x0−z

(6)

It follows fromA3and the weak lower semicontinuity ofϕthat

ϕz≤lim inf

m→ ∞ ϕ zm

≤lim sup

m→ ∞

F zm, y

ϕy 1 r

y−zm, zm−x0

≤lim sup

m→ ∞

F zm, y

ϕy1

rlim sup_m_{→ ∞}

y−zm, zm−x0

≤Fz, y ϕy 1 r

z−y, x0−z

.

2.13

This implies that z ∈ Gy. Hence, Gy is weakly closed. Hence, Srx0

y∈CGy

y∈C∩domϕGy

y∈C∩domϕGy w

/

∅. Hence, from the arbitrariness ofx0, we conclude that

Srx/∅, ∀x∈H.

We observe that Srx ⊆ domϕ. So by similar argument with that in the proof of

Lemma 2.3 in9, we can easily show thatSris single-valued andSris a firmly

nonexpansive-type map. Next, we claim that FixSr GEPF, ϕ. Indeed, we have the following:

u∈Fix Sr

⇐⇒uSru

⇐⇒Fu, y ϕy 1

ry−u, u−u ≥ϕu, ∀y∈C

⇐⇒Fu, y ϕy≥ϕu, ∀y∈C

⇐⇒u∈GEPF, ϕ.

2.14

At last, we claim that GEPF, ϕis a closed convex. Indeed, SinceSr is firmly nonexpansive,

Sr is also nonexpansive. By23, Proposition 5.3, we know that GEPF, ϕ FixSris closed

and convex.

Remark 2.3. It is easy to see thatLemma 2.2is a generalization of9, Lemma 2.3.

Lemma 2.4see24,25. Assume that{αn}is a sequence of nonnegative real numbers such that

αn1≤ 1−γn

αnδn, 2.15

whereγnis a sequence in0,1and{δn}is a sequence such that

i

∞

n1

γn∞;

ii lim sup

n→ ∞

δn

γn ≤

0 or

∞

n1

δn<∞.

2.16

Then,limn→ ∞αn0.

Lemma 2.5. In a real Hilbert spaceH, there holds the following inequality:

xy2≤ x22y, xy 2.17

(7)

3. Strong Convergence Theorems

In this section, we show a strong convergence of an iterative algorithm based on both viscosity approximation method and parallel method which solves the problem of finding a common element of the set of solutions of a generalized equilibrium problem and the set of fixed points of a family of finitely strict pseudocontractions in a Hilbert space.

We need the following assumptions for the parameters {γn},{rn},{αn},{ζ₁n}, {ζ₂n}, . . . ,{ζ_Nn},and{βn}:

C1limn→ ∞αn0 and

_∞

n1αn∞;

C21>lim sup_n_{→ ∞}βn≥lim infn→ ∞βn>0;

C3{γn} ⊂c, dfor somec, d∈ε,1and limn→ ∞|γn1−γn|0;

C4lim infn→ ∞rn>0 and limn→ ∞|rn1−rn|0;

C5limn→ ∞|ζjn1−ζ

n

j |0 for allj1,2, . . . , N.

Theorem 3.1. Let C be a nonempty-closed convex subset of a real Hilbert space H. Let F be a bifunction from C× C to R satisfying (A1)–(A5), and let ϕ : C → R ∪ {∞} be a proper lower semicontinuous and convex function such that C∩domϕ /∅. Let N ≥ 1 be an integer. For each1 ≤ j ≤ N, let Tj : C → C be anεj-strict pseudocontraction for some 0 ≤ εj < 1 such thatΩ N_j₁FixTj∩GEPF, ϕ/∅. Assume for eachn, {ζ_jn}

N

j1 is a finite sequence of

positive numbers such that N_j₁ζ_jn 1 for all n and infn≥1ζjn > 0 for all 0 ≤ j ≤ N. Let

ε max{εj : 1 ≤ j ≤ N}. Assume that either (B1) or (B2) holds. Let f be a contraction ofC into itself and let{xn},{un},and{yn}be sequences generated by

x1x∈C,

F un, y

ϕy 1 rn

y−un, un−xn

≥ϕ un

, ∀y∈C,

ynγnun 1−γn

N

j1

ζ_jnTjun,

xn1 αnf xn

βnxn 1−αn−βn

yn

3.1

for everyn 1,2, . . ., where {γn},{rn},{αn},{ζ₁n},{ζ₂n}, . . . ,{ζ_Nn},and {βn} are sequences of numbers satisfying the conditions (C1)–(C5). Then,{xn},{un},and{yn}converge strongly tow

P_Ωfw.

Proof. We show thatPΩf is a contraction ofCinto itself. In fact, there existsa ∈0,1such

thatfx−fy ≤ax−yfor allx, y∈C. So, we have

_P_Ω_f_x₋_P_Ω_f_y_≤_f_x₋_f_y_≤_a_x₋_y _3.2

for all x, y ∈ C.SinceH is complete, there exists a unique elementu0 ∈ Csuch thatu0

(8)

Letu ∈ Ωand let{Srn}be a sequence of mappings defined as inLemma 2.2. From

unSrnxn∈C, we have

_u_n₋_u_S_r

n xn

−Srnu≤xn−u. 3.3

We define a mappingWnby

Wnx N

j1

ζ_jnTjx, ∀x∈C. 3.4

By 21, Proposition 2.6, we know that Wn is an ε-strict pseudocontraction and FWn

_N

j1FixTj. It follows from3.3,ynγnun 1−γnWnunanduWnusuch that

yn−u2γnun−u2 1−γnWnun−u2−γn 1−γnun−Wnun2

≤γnun−u2 1−γnun−u2εun−Wnun2

−γn 1−γnun−Wnun2

un−u2 1−γn ε−γnun−Wnun2

≤un−u2.

3.5

PutM0max{x1−u,1/1−afu−u}.It is obvious thatx1−u ≤M0.Suppose

xn−u ≤M0.From3.3,3.5, andxn1αnfxn βnxn 1−αn−βnyn, we have

_x_n₁₋_u_α_n_f _x_n

yn−u

≤αnf xn

−fuαnfu−uβnxn−u 1−αn−βnyn−u

≤αnaxn−uαnfu−uβnxn−u 1−αn−βnun−u

≤αnaxn−uαnfu−u 1−αnxn−u

1−aαn

_f_u₋_u

1−a

1−1−aαnxn−u,

≤1−aαnM0

1−1−aαn

M0M0

3.6

for everyn 1,2, . . . .Therefore,{xn}is bounded. From3.3and3.5, we also obtain that {yn}and{un}are bounded.

Following26, defineBn:C → Cby

BnγnI 1−γn

(9)

As shown in26, eachBnis a nonexpansive mapping onC. SetM1 sup_n_≥₁{un−Wnun},

we have

_y_n₁₋_y_n_B_n₁ _u_n₁

−Bn un ≤Bn1 un1

−Bn1 unBn1 un

−Bn un

≤un1−unM1γn1−γn 1−γn1Wn1 un−Wn un

≤un1−unM1γn1−γn 1−γn1

N

j1

ζ_jn1−ζ_jnTjun.

3.8

On the other hand, fromunTrnxnandun1Trn1xn1, we have

F un, y

ϕy 1 rn

y−un, un−xn

≥ϕ un

, ∀y∈C, 3.9

F un1, y

ϕy 1

rn1

y−un1, un1−xn1

≥ϕ un1

, ∀y∈C. 3.10

Puttingyun1in3.9andyunin3.10, we have

F un, un1

ϕ un1

1

rn

un1−un, un−xn

≥ϕ un

,

F un1, un

ϕ un

1

rn1

un−un1, un1−xn1

≥ϕ un1

.

3.11

So, from the monotonicity ofF, we get

un1−un,

un−xn

rn −

un1−xn1

rn1

≥0, 3.12

hence

un1−un, un−un1un1−xn− rn

rn1

un1−xn1

≥0. 3.13

Without loss of generality, let us assume that there exists a real numberbsuch thatrn> b >0

for alln∈N.Then,

_u_n₁₋_u_n2_≤

un1−un, xn1−xn

1− rn

rn1 un1−xn1

≤un1−un

xn1−xn

1− rn

rn1

un1−xn1

,

(10)

hence

_u_n₁₋_u_n_≤_x_n₁₋_x_n 1

rn1

_r_n₁₋_r_n_u_n₁₋_x_n₁

≤xn1−xn

1

brn1−rnM2,

3.15

whereM2sup{un−xn:n≥1}.

It follows from3.8and3.15that

yn1−yn≤xn1−xn

1

brn1−rnM2M1γn1−γn

1−γn1

N

j1

_ζn1

j −ζ

n

j Tjun.

3.16

Define a sequence{vn}such that

xn1βnxn 1−βn

vn, ∀n≥1. 3.17

Then, we have

vn1−vn

xn2−βn1xn1

1−βn1 −

xn1−βnxn

1−βn

αn1fxn1 1−αn1−βn1yn1

1−βn1 −

αnfxn 1−αn−βnyn

1−βn

αn1

1−βn1f xn1

− αn

1−βnf xn

yn1−yn

αn

1−βnyn−

αn1

1−βn1yn1.

3.18

From3.18and3.16, we have

vn1−vn−xn1−xn

≤ αn1

1−βn1

f xn1yn1

αn

1−βn

f xnyn

yn1−yn−xn1−xn

≤ αn1

1−βn1 f xn1

_y_n₁ αn

1−βn f xn

_y_n

1

brn1−rnM2M1γn1−γn 1−γn1 N

j1

ζ_jn1−ζ_jnTjun.

(11)

It follows fromC1–C5that

lim sup

n→ ∞

vn1−vn−xn1−xn≤0. 3.20

Hence, by27, Lemma 2.2, we have limn→ ∞vn−xn0. Consequently,

lim

n→ ∞xn1−xnnlim→ ∞ 1−βnvn−xn0. 3.21

Sincexn1 αnfxn βnxn 1−αn−βnyn, we have

xn−yn≤xn1−xnxn1−yn ≤xn1−xnαnf xn

−ynβnxn−yn,

3.22

thus

_x_n₋_y_n_≤ 1 1−βn

xn1−xnαnfxn−yn. 3.23

It follows fromC1andC2that limn→ ∞xn−yn0.

Sincexn1 αnfxn βnxn 1−αn−βnyn,foru∈Ω, it follows from3.5and3.3

that

_x_n₁₋_u2

αnf xn

yn−u2

≤αnf xn

−u2βnxn−u2 1−αn−βnyn−u2

≤αnf xn

−u2βnxn−u2

1−αn−βnun−u2 1−γn ε−γnun−Wnun2

≤αnf xn

−u2 1−αnxn−u2

1−αn−βn 1−γn ε−γnun−Wnun2,

3.24

from which it follows that

_u_n₋_W_n_u_n2_≤ αn

1−αn−βn1−γnγn−ε f xn

−u2−xn−u2

1

1−αn−βn1−γnγn−ε

xn−u2−xn1−u2

.

≤ αn

f xn

−u2−xn−u2

1

xn−uxn1−uxn1−xn.

(12)

It follows fromC1–C3andxn1−xn → 0 that

_u_n₋_W_n_u_n_−→₀_. _3.26

Foru∈Ω, we have fromLemma 2.2,

_u_n₋_u2

Srnxn−Srnu

2 _≤

Srnxn−Srnu, xn−u

un−u, xn−u

1

2un−u

2

xn−u2−xn−un2

.

3.27

Hence,

_u_n₋_u2_≤

xn−u2−xn−un2. 3.28

By3.24and3.28, we have

xn1−u2≤αnf xn

−u2βnxn−u2 1−αn−βnun−u2

≤αnf xn

−u2βnxn−u2 1−αn−βnxn−u2−xn−un2

.

3.29

Hence,

1−αn−βnxn−un2≤αnf xn

−u2−αnxn−u2xn−u2−xn1−u2

≤αnf xn

−u2−αnxn−u2

xn−uxn1−uxn−xn1.

3.30

It follows fromC1,C2, andxn−xn1 → 0 that limn→ ∞xn−un0.

Next, we show that

lim sup

n→ ∞

fu0−u0, xn−u0

≤0, 3.31

whereu0PΩfu0. To show this inequality, we can choose a subsequence{xni}of{xn}such

that

lim

i→ ∞

fu0−u0, xni −u0

lim sup

n→ ∞

fu0−u0, xn−u0

. 3.32

Since{xni} is bounded, there exists a subsequence {xnij} of {xni} which converges

(13)

we obtain thatuni w. Fromxn−yn → 0, we also obtain thatyni w. Since{uni} ⊂C

andCis closed and convex, we obtainw∈C.

We first show thatw ∈N_k₁FixTk.To see this, we observe that we may assumeby

passing to a further subsequence if necessaryζni

k → ζkasi → ∞fork1,2, . . . , N.It is

easy to see thatζk>0 andNk1ζk1. We also have

Wnix−→Wx asi−→ ∞∀x∈C, 3.33

whereW N_k₁ζkTk. Note that by21, Proposition 2.6,W is anε-strict pseudocontraction

and FixW N_i₁FixTi. Since

uni−Wuni≤uni−WniuniWniuni−Wuni

≤uni−Wniuni N

k1

_ζni

k −ζkTkuni,

3.34

it follows from3.26andζni

k → ζkthat

uni−Wuni−→0. 3.35

So by the demiclosedness principle21, Proposition 2.6ii, it follows thatw ∈ FixW N

i1FixTi.

We now showw∈GEPF, ϕ.ByunTrnxn,we know that

F un, y

ϕy 1 rn

y−un, un−xn

≥ϕ un

, ∀y∈C. 3.36

It follows fromA2that

ϕy 1 rn

y−un, un−xn

≥F y, un

ϕ un

, ∀y∈C. 3.37

Hence,

ϕy

y−uni,

uni−xni

rni

≥F y, uni

ϕ un

(14)

It follows fromA4,A5and the weakly lower semicontinuity ofϕ,uni−xni/rni →

0,anduni wthat

Fy, w ϕw≤ϕy, ∀y∈C. 3.39

Fortwith 0< t≤1 andy∈C∩domϕ, letytty 1−tw.Sincey∈C∩domϕand

w∈C∩domϕ, we obtainyt∈C∩domϕ,and henceFyt, w ϕw≤ϕyt. So byA4and

the convexity ofϕ, we have

0F yt, yt

ϕ yt

−ϕ yt

≤tF yt, y

1−tF yt, w

tϕy 1−tϕw−ϕ yt

≤tF yt, y

ϕy−ϕ yt

.

3.40

Dividing byt, we get

F yt, y

ϕy−ϕ yt

≥0. 3.41

Lettingt → 0, it follows fromA3and the weakly lower semicontinuity ofϕthat

Fw, y ϕy≥ϕw 3.42

for all y ∈ C∩domϕ. Observe that ify ∈ C\domϕ, thenFw, y ϕy ≥ ϕwholds. Moreover, hencew∈GEPF, ϕ. This impliesw∈Ω.Therefore, we have

lim sup

n→ ∞

fu0−u0, xn−u0

lim

i→ ∞

fu0−u0, xni−u0

fu0−u0, w−u0

≤0. 3.43

Finally, we show thatxn → u0, whereu0PΩfu0.

FromLemma 2.5, we have

_x_n₁₋_u₀2

αn f xn

−u0

βn xn−u0

1−αn−βn yn−u02

≤βn xn−u0

1−αn−βn yn−u022αn

f xn

−u0, xn1−u0

≤ 1−αn−βnyn−u0βnxn−u022αn

f xn

−u0, xn1−u0

≤ 1−αn

2

xn−u022αn

f xn

−f u0

, xn1−u0

2αn

f u0

−u0, xn1−u0

≤ 1−αn

2

xn−u022αnaxn−u0xn1−u02αn

f u0

−u0, xn1−u0

≤ 1−αn

2

xn−u02αna xn−u02xn1−u02

2αn

f u0

−u0, xn1−u0

,

(15)

thus

xn1−u02 ≤

1−21−aαn 1−aαn

xn−u02

21−aαn

1−aαn

αn

21−axn−u0

2 1

1−a

2f u0

−2u0, xn1−u0

.

3.45

It follows fromC1,3.43,3.45, andLemma 2.4that limn→ ∞xn−u0 0. From

xn−un → 0 andyn −xn → 0, we haveun → u0 and yn → u0. The proof is now

complete.

Theorem 3.2. Let C be a nonempty-closed convex subset of a real Hilbert space H. Let F be a bifunction from C ×C to R satisfying (A1)–(A5), and let ϕ : H → R ∪ {∞} be a proper lower semicontinuous and convex function such that C∩domϕ /∅. Let N ≥ 1 be an integer. For each1 ≤ j ≤ N, let Tj : C → C be anεj-strict pseudocontraction for some 0 ≤ εj < 1 such thatΩ N_j₁FixTj∩GEPF, ϕ/∅. Assume for eachn, {ζ_jn}

N

j1 is a finite sequence of

positive numbers such that N_j₁ζ_jn 1 for all n and infn≥1ζ_jn > 0 for all 0 ≤ j ≤ N. Let

ε max{εj : 1 ≤ j ≤ N}. Assume that either (B1) or (B2) holds. Let vbe an arbitrary point in

Cand let{xn}, {un},and{yn}be sequences generated by

x1x∈C,

F un, y

ϕy 1 rn

y−un, un−xn

≥ϕ un

, ∀y∈C,

ynγnun 1−γn

N

j1

ζ_jnTjun,

xn1αnvβnxn 1−αn−βn

yn

3.46

for every n 1,2, . . . ,where {γn},{rn},{αn},{ζ₁n},{ζ₂n}, . . . ,{ζ_Nn},and {βn}are sequences of numbers satisfying the conditions (C1)–(C5). Then,{xn}, {un},and{yn}converge strongly tow

PΩv.

Proof. Letfx vfor allx∈C, byTheorem 3.1, we obtain the desired result.

4. Applications

By Theorems 3.1 and 3.2, we can obtain many new and interesting strong convergence theorems. Now, give some examples as follows: forj 1,2, . . . , N, letT1 T2 · · ·TN T,

by Theorems3.1and3.2, respectively, we have the following results.

(16)

(B2) holds. Letfbe a contraction ofCinto itself and let{xn}, {un},and{yn}be sequences generated by

x1x∈C,

F un, y

ϕy 1 rn

y−un, un−xn

≥ϕ un

, ∀y∈C,

yn γnun 1−γn

Tun,

xn1 αnf xn

yn

4.1

for everyn 1,2, . . . ,where {γn}, {rn}, {αn},and {βn}are sequences of numbers satisfying the conditions (C1)–(C4). Then,{xn}, {un},and{yn}converge strongly towPFixT∩GEPF,ϕfw.

Theorem 4.2. Let C be a nonempty-closed convex subset of a real Hilbert space H. Let F be a bifunction fromC×CtoRsatisfying (A1)–(A5), and letϕ : H → R∪ {∞}be a proper lower semicontinuous and convex function such that C∩ domϕ /∅. Let T : C → C be an ε-strict pseudocontraction for some0 ≤ ε < 1such thatFixT∩GEPF, ϕ/∅. Assume that either (B1) or (B2) holds. Letvbe an arbitrary point inC,and let{xn}, {un},and{yn}be sequences generated by

x1x∈C,

F un, y

ϕy 1 rn

y−un, un−xn

≥ϕ un

, ∀y∈C,

yn γnun 1−γn

Tun,

yn

4.2

for everyn 1,2, . . . ,where {γn}, {rn}, {αn},and {βn}are sequences of numbers satisfying the conditions (C1)–(C4). Then,{xn}, {un},and{yn}converge strongly towPFixT∩GEPF,ϕv.

We need the following two assumptions.

B3For eachx∈ Handr >0, there exist a bounded subsetDx ⊆Candyx ∈ Csuch

that for anyz∈C\Dx,

F z, yx

1

r

yx−z, z−x

<0. 4.3

B4For eachx∈Handr >0, there exist a bounded subsetDx⊆Candyx∈C∩domϕ

such that for anyz∈C\Dx,

g yx

ϕ yx

1

r

yx−z, z−x

< ϕz gz. 4.4

Let ϕx δCx,∀x ∈ H, by Theorems 3.1 and 3.2, respectively, we obtain the

(17)

Theorem 4.3. Let C be a nonempty-closed convex subset of a real Hilbert space H. Let F be a bifunction fromC×CtoRsatisfying (A1)–(A5). LetN≥1be an integer. For each1≤j≤N, letTj:

C → Cbe anεj-strict pseudocontraction for some0≤εj<1such thatΓ

_N

j1FixTj∩EPF/∅. Assume for eachn,{ζ_jn}N

j1is a finite sequence of positive numbers such that

N j1ζ

n

j 1for alln andinfn≥1ζ_jn >0for all0≤j ≤N. Letεmax{εj : 1≤j ≤N}. Assume that either (B3) or (B2) holds. Letfbe a contraction ofHinto itself, and let{xn}, {un},and{yn}be sequences generated by

x1x∈C,

F un, y

1

rn

y−un, un−xn

≥0, ∀y∈C,

ynγnun 1−γn

N

j1

ζ_jnTjun,

xn1 αnf xn

yn

4.5

for every n 1,2, . . . ,where {γn},{rn},{αn},{ζ₁n},{ζ₂n}, . . . ,{ζ_Nn},and {βn}are sequences of numbers satisfying the conditions (C1)–(C5). Then,{xn},{un},and{yn}converge strongly tow

P_Γfw.

Theorem 4.4. Let C be a nonempty-closed convex subset of a real Hilbert space H. Let F be a bifunction fromC×CtoRsatisfying (A1)–(A5). LetN≥1be an integer. For each1≤j≤N, letTj:

C → Cbe anεj-strict pseudocontraction for some0≤εj<1such thatΓ

_N

j1FixTj∩EPF/∅. Assume for eachn,{ζ_jn}N

j1is a finite sequence of positive numbers such that

N j1ζ

n

j 1for alln andinfn≥1ζ_jn >0for all0≤j ≤N. Letεmax{εj : 1≤j ≤N}. Assume that either (B3) or (B2) holds. Letvbe an arbitrary point inC,and let{xn},{un},and{yn}be sequences generated by

x1x∈C,

F un, y

1

rn

y−un, un−xn

≥0, ∀y∈C,

ynγnun 1−γn

N

j1

ζ_jnTjun,

yn

4.6

P_Γv.

LetFx, y gy−gxfor allx, y ∈ C, by Theorems3.1and 3.2, respectively, we obtain the following results.

Theorem 4.5. LetCbe a nonempty-closed convex subset of a real Hilbert spaceH. Letg :C → R

(18)

1 ≤ j ≤ N, letTj : C → Cbe anεj-strict pseudocontraction for some0 ≤ εj < 1 such thatΘ

_N

j1FixTj∩Argming, ϕ/∅. Assume for eachn,{ζ_jn} N

j1is a finite sequence of positive numbers

such thatN_j₁ζ_jn 1for allnandinfn≥1ζ_jn >0for all0≤j ≤N. Letεmax{εj : 1≤j ≤N}. Assume that either (B4) or (B2) holds. Letf be a contraction ofHinto itself, and let{xn}, {un}and {yn}be sequences generated by

x1x∈C,

gy ϕy 1 rn

y−un, un−xn

≥g un

ϕ un

, ∀y∈C,

ynγnun 1−γn

N

j1

ζ_jnTjun,

xn1 αnf xn

yn

4.7

for everyn 1,2, . . ., where {γn},{rn},{αn},{ζ₁n},{ζ2n}, . . . ,{ζ

n

N},and {βn} are sequences of numbers satisfying the conditions (C1)–(C5). Then,{xn},{un},and{yn}converge strongly tow

PΘfw.

Theorem 4.6. LetCbe a nonempty-closed convex subset of a real Hilbert spaceH. Letg :C → R

be a lower semicontinuous and convex function, and letϕ : H → R∪ {∞} be a proper lower semicontinuous and convex function such thatC∩domϕ /∅. LetN ≥ 1be an integer. For each

1 ≤ j ≤ N, letTj : C → Cbe anεj-strict pseudocontraction for some0 ≤ εj < 1 such thatΘ

N

j1FixTj∩Argming, ϕ/∅. Assume for eachn,{ζ_jn} N

j1is a finite sequence of positive numbers

such thatN_j₁ζ_jn 1for allnandinfn≥1ζjn >0for all0≤j ≤N. Letεmax{εj : 1≤j ≤N}. Assume that either (B4) or (B2) holds. Letvbe an arbitrary point inC,and let{xn}, {un},and{yn} be sequences generated by

x1x∈C,

gy ϕy 1 rn

y−un, un−xn

≥ϕ un

g un

, ∀y∈C,

ynγnun 1−γn

N

j1

ζ_jnTjun,

yn

4.8

PΘv.

Letϕx δCx,∀x∈H, and letFx, y 0 for allx, y∈C. ThenunPCxnxn. By

Theorems3.1and3.2, we obtain the following results.

(19)

such thatN_j₁FixTj/∅. Assume for eachn,{ζ_jn} N

j1is a finite sequence of positive numbers such

thatN_j₁ζ_jn 1for allnandinfn≥1ζjn >0for all0≤j≤N. Letεmax{εj : 1≤j≤N}. Letf be a contraction ofHinto itself, and let{xn}and{yn}be sequences generated by

x1x∈C,

ynγnxn 1−γn

N

j1

ζ_jnTjxn,

xn1 αnf xn

yn

4.9

for everyn1,2, . . ., where{γn},{αn},{ζ₁n},{ζ₂n}, . . . ,{ζ_Nn},and{βn}are sequences of numbers satisfying the conditions (C1)–(C3) and (C5). Then, {xn}, and {yn} converge strongly to w

P_∩N

j1FixTjfw.

Theorem 4.8. LetCbe a nonempty-closed convex subset of a real Hilbert spaceH. LetN ≥1be an integer. For each1 ≤j ≤N, letTj :C → Cbe anεj-strict pseudocontraction for some0 ≤ εj <1 such thatN_j₁FixTj/∅. Assume for eachn,{ζjn}

N

j1is a finite sequence of positive numbers such

thatN_j₁ζ_jn 1for allnandinfn≥1ζ_jn >0for all0≤j ≤N. Letεmax{εj : 1≤j ≤N}. Letv be an arbitrary point inC,and let{xn}and{yn}be sequences generated by

x1x∈C,

ynγnxn 1−γn

N

j1

ζ_jnTjxn,

yn

4.10

for everyn1,2, . . ., where{γn},{αn},{ζ1n},{ζ

n

2 }, . . . ,{ζ

n

N },and{βn}are sequences of numbers satisfying the conditions (C1)–(C3) and (C5). Then, {xn} and {yn} converge strongly to w

P_∩N

j1FixTjv.

Remark 4.9. 1 Since the nonexpansive mappings have been replaced by the strict pseudocontractions, Theorems3.1,3.2,4.1and4.2extend and improve6, Theorem 3.1,8, Theorem 3.5,9, Theorems 4.1 and 4.2,18, Theorem 4.1, and the main results in9–11,13–

16.

2Since the weak convergence has been replaced by strong convergence, Theorems

3.1,3.2,4.1−4.4extend and improve12, Theorem 3.1,10, Corollary 4.1.

3Theorems 4.7and4.8are strong convergence theorems for strict pseudocontrac-tions withoutCQconstraints and hence they improve the corresponding results in19,21. Theorems3.1and3.2also improve10, Corollary 3.1.

Acknowledgments

The first author was supported by the National Natural Science Foundation of China

(20)

Chinese Ministry of EducationGrant no. 206123, the Education Committee project Research Foundation of Chongqing Normal University Grant no. KJ070816; the second and third authors were partially supported by the Grants NSC97-2221-E-230-017 and NSC96-2628-E-110-014-MY3, respectively.

References

1 F. Flores-Baz´an, “Existence theorems for generalized noncoercive equilibrium problems: the quasi-convex case,”SIAM Journal on Optimization, vol. 11, no. 3, pp. 675–690, 2000.

2 G. Bigi, M. Castellani, and G. Kassay, “A dual view of equilibrium problems,”Journal of Mathematical Analysis and Applications, vol. 342, no. 1, pp. 17–26, 2008.

3 A. N. Iusem and W. Sosa, “Iterative algorithms for equilibrium problems,”Optimization, vol. 52, no. 3, pp. 301–316, 2003.

4 S. D. Fl˚am and A. S. Antipin, “Equilibrium programming using proximal-like algorithms,”

Mathematical Programming, vol. 78, no. 1, pp. 29–41, 1997.

5 E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,”The Mathematics Student, vol. 63, no. 1–4, pp. 123–145, 1994.

6 L.-C. Ceng and J.-C. Yao, “A hybrid iterative scheme for mixed equilibrium problems and fixed point problems,”Journal of Computational and Applied Mathematics, vol. 214, no. 1, pp. 186–201, 2008. 7 F. E. Browder and W. V. Petryshyn, “Construction of fixed points of nonlinear mappings in Hilbert

space,”Journal of Mathematical Analysis and Applications, vol. 20, pp. 197–228, 1967.

8 Y. Yao, Y.-C. Liou, and J.-C. Yao, “A new hybrid iterative algorithm for fixed-point problems, vari-ational inequality problems, and mixed equilibrium problems,”Fixed Point Theory and Applications, vol. 2008, Article ID 417089, 15 pages, 2008.

9 J.-W. Peng and J.-C. Yao, “A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems,”Taiwanese Journal of Mathemat-ics, vol. 12, no. 6, pp. 1401–1432, 2008.

10 J.-W. Peng and J.-C. Yao, “Some new iterative algorithms for generalized mixed equilibrium problems with strict pseudo-contractions and monotone mappings,” to appear inTaiwanese Journal of Mathematics.

11 P. L. Combettes and S. A. Hirstoaga, “Equilibrium programming in Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol. 6, no. 1, pp. 117–136, 2005.

12 S. Takahashi and W. Takahashi, “Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces,”Journal of Mathematical Analysis and Applications, vol. 331, no. 1, pp. 506–515, 2007.

13 Y. Su, M. Shang, and X. Qin, “An iterative method of solution for equilibrium and optimization problems,”Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 8, pp. 2709–2719, 2008. 14 A. Tada and W. Takahashi, “Weak and strong convergence theorems for a nonexpansive mapping and

an equilibrium problem,”Journal of Optimization Theory and Applications, vol. 133, no. 3, pp. 359–370, 2007.

15 L.-C. Ceng, S. AI-Homidan, Q. H. Ansari, and J.-C. Yao, “An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings,”Journal of Computational and Applied Mathematics, vol. 223, no. 2, pp. 967–974, 2009.

16 S. S. Chang, H. W. Joseph Lee, and C. K. Chan, “A new method for solving equilibrium problem, fixed point problem and variational inequality problem with application to optimization,”Nonlinear Analysis: Theory, Methods & Applications. In press.

17 V. Colao, G. Marino, and H.-K. Xu, “An iterative method for finding common solutions of equilibrium and fixed point problems,”Journal of Mathematical Analysis and Applications, vol. 344, no. 1, pp. 340– 352, 2008.

18 S. Takahashi and W. Takahashi, “Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space,”Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 3, pp. 1025–1033, 2008.

19 G. Marino and H.-K. Xu, “Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces,”Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 336–346, 2007. 20 H. Zhou, “Convergence theorems of fixed points forκ-strict pseudo-contractions in Hilbert spaces,”

(21)

21 G. L. Acedo and H.-K. Xu, “Iterative methods for strict pseudo-contractions in Hilbert spaces,”

Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 7, pp. 2258–2271, 2007.

22 K. Fan, “A generalization of Tychonoﬀ’s fixed point theorem,”Mathematische Annalen, vol. 142, pp. 305–310, 1961.

23 K. Goebel and S. Reich,Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, vol. 83 of

Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1984. 24 H.-K. Xu, “Iterative algorithms for nonlinear operators,”Journal of the London Mathematical Society.

Second Series, vol. 66, no. 1, pp. 240–256, 2002.

25 H. K. Xu, “An iterative approach to quadratic optimization,” Journal of Optimization Theory and Applications, vol. 116, no. 3, pp. 659–678, 2003.

26 G. Marino, V. Colao, X. Qin, and S. M. Kang, “Strong convergence of the modified Mann iterative method for strict pseudo-contractions,”Computers & Mathematics with Applications, vol. 57, no. 3, pp. 455–465, 2009.