Viscosity Method with Parallel Method for a Generalized Equilibrium Problem and Strict Pseudocontractions

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,

Yeong-Cheng Liou,

and Jen-Chih Yao

1College of Mathematics and Computer Science, Chongqing Normal University, Chongqing 400047, China

2Department of Information Management, Cheng Shiu University, Kaohsiung, Taiwan 833, Taiwan

3Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan 804, Taiwan

Correspondence should be addressed to Yeong-Cheng Liou,simplex liou@hotmail.com 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 The set of solutions of 1.1 is denoted by GEPF, ϕ. Flores-Baz´an 1 provided some characterizations of the nonemptiness of the solution set for problem1.1in reflexive Banach spaces in the quasiconvex case. Bigi et al. 2 studied a dual problem associated with the problem1.1withCHRⁿ.

Letϕx δCx,∀x∈H. HereδCdenotes the indicator function of the setC; that is, δ_Cx 0 ifx∈Candδ_Cx ∞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:

miny∈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.

Ceng and Yao6, Yao et al. 8, and Peng and Yao9,10introduced some iterative schemes for finding a common element of the set of solutions of the mixed equilibrium problem 1.4 and the set of common fixed points of a family of finitely infinitely nonexpansive mappingsstrict pseudocontractionsin a Hilbert space and obtained some strong convergence theoremsweak convergence theorems. Some methods have been proposed to solve the problem 1.2; see, for instance, 3–5, 11–18 and the references therein. Recently, S. Takahashi and W. Takahashi 12 introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of problem1.2and the set of fixed points of a nonexpansive mapping in a Hilbert space and proved a strong convergence theorem. Su et al.13introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of problem 1.2 and the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for anα-inverse strongly monotone mapping in a Hilbert space. Tada and Takahashi14 introduced two iterative schemes for finding

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−λy²λx² 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 thatP_Cis a nonexpansive mapping fromHontoC. It is also known thatP_Cx∈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 → 2^H is said to be a KKM map if, for every finite subset {x1, x2, . . . , xn} ⊆ B, conv{x1, x₂, . . . , x_n}⊆_∞

n1Gxi.

We will use the following results in the sequel.

Lemma 2.1see22. LetBbe a nonempty subset of a Hausdorfftopological vector spaceXand let G:B → 2^Xbe a KKM map. IfGxis closed for allx∈Band is compact for at least onex∈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 subsetD_x⊆Candy_x∈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 >0 andx∈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, Srx−Sry²≤

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

. 2.6

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, . . . , y_n}ofC∩domϕandμ_i ≥0 for alli1,2, . . . , nwith_n

i1μ_i 1 such that

z_n

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

μ_i

F z, y _i

ϕ y_i

−ϕz 1

r _n

i1

μ_i

y_i−z, z−x₀

<0,

2.8

which is a contradiction. Hence,Gis a KKM map onC∩domϕ. Note thatGy^wthe weak closure ofGyis a weakly closed subset ofCfor eachy∈C. Moreover, ifB2holds, then Gy^wis 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 y_x0

z∈C:F z, y_x0

ϕ y_x0 1

r

y_x0−z, z−x₀

≥ϕz

⊆D_x0. 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 thatGy^w Gyfor eachy∈C; that is,Gyis weakly closed.

Letz∈Gy^wand letzmbe a sequence inGysuch thatzm z. Then,

F zm, y

ϕy 1 r

y−zm, zm−x0

≥ϕ zm

. 2.11

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

lim sup

m→ ∞

y−zm, zm−x0

≤

z−y, x0−z

. 2.12

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 z_m, y

ϕy 1

rlim sup

m→ ∞

y−z_m, z_m−x₀

≤Fz, y ϕy 1 r

z−y, x0−z .

2.13

This implies that z ∈ Gy. Hence, Gy is weakly closed. Hence, S_rx0

y∈CGy

y∈C∩domϕGy

y∈C∩domϕGy^w/∅. Hence, from the arbitrariness ofx0, we conclude that S_rx/∅, ∀x∈H.

We observe that S_rx ⊆ 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, S_r 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 α_n1≤ 1−γ_n

α_nδ_n, 2.15

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

n1

γn∞;

ii lim sup

n→ ∞

δ_n

γ_n ≤0 or ∞ n1

δ_n<∞.

2.16

Then, lim_{n→ ∞}α_n0.

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

xy²≤ x²2y, xy 2.17

for allx, y∈H.

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}, {ζⁿ₂ }, . . . ,{ζⁿ_N},and{βn}:

C1lim_{n→ ∞}α_n0 and_∞

n1α_n∞;

C21>lim sup_{n→ ∞}β_n≥lim inf_{n→ ∞}β_n>0;

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

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

C5lim_{n→ ∞}|ζⁿ¹_j −ζⁿ_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 each 1 ≤ j ≤ N, let T_j : C → C be anε_j-strict pseudocontraction for some 0 ≤ ε_j < 1 such thatΩ _N

j1FixTj∩GEPF, ϕ/∅. Assume for eachn, {ζⁿ_j }^N_j1 is a finite sequence of positive numbers such that _N

j1ζⁿ_j 1 for all n and inf_n≥1ζⁿ_j > 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 u_n, y

ϕy 1 rn

y−u_n, u_n−x_n

≥ϕ u_n

, ∀y∈C, y_nγ_nu_n 1−γ_n^N

j1

ζⁿ_j T_ju_n, x_n1 α_nf x_n

β_nx_n 1−α_n−β_n y_n

3.1

for everyn 1,2, . . ., where {γn},{rn},{αn},{ζ₁ⁿ},{ζⁿ₂ }, . . . ,{ζⁿ_N},and {βn} are sequences of numbers satisfying the conditions (C1)–(C5). Then,{xn},{un},and{yn}converge strongly tow P_Ωfw.

Proof. We show thatP_Ωf is a contraction ofCinto itself. In fact, there existsa ∈0,1such thatfx−fy ≤ax−yfor allx, y∈C. So, we have

P_Ωfx−P_Ωfy≤fx−fy≤ax−y 3.2 for all x, y ∈ C.SinceH is complete, there exists a unique elementu0 ∈ Csuch thatu0 P_Ωfu0.

Letu ∈ Ωand let{Srn}be a sequence of mappings defined as inLemma 2.2. From u_nS_rnxn∈C, we have

un−uSrn xn

−Srnu≤xn−u. 3.3

We define a mappingW_nby

Wnx^N

j1

ζⁿ_j Tjx, ∀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,y_nγ_nu_n 1−γ_nWnu_nanduW_nusuch that y_n−u²γ_nu_n−u² 1−γ_nW_nu_n−u²−γ_n 1−γ_nu_n−W_nu_n²

≤γnun−u² 1−γnun−u²εun−Wnun²

−γn 1−γnun−Wnun² u_n−u² 1−γ_n ε−γ_nu_n−W_nu_n²

≤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, andx_n1αnfxn βnxn 1−αn−βnyn, we have

x_n1−uαnf xn

βnxn 1−αn−βn

yn−u

≤α_nf x_n

−fuα_nfu−uβ_nx_n−u 1−α_n−β_ny_n−u

≤α_nax_n−uα_nfu−uβ_nx_n−u 1−α_n−β_nu_n−u

≤αnaxn−uαnfu−u 1−αnxn−u 1−aαn

fu−u

1−a

1−1−aαnx_n−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

Wn. 3.7

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

y_n1−ynB_n1 u_n1

−Bn un

≤B_n1 u_n1

−B_n1 u_nB_n1 u_n

−B_n u_n

≤u_n1−unM1γ_n1−γn 1−γ_n1W_n1 un−Wn un

≤u_n1−u_nM₁γ_n1−γ_n 1−γ_n1^N

j1

ζⁿ¹_j −ζⁿ_j T_ju_n.

3.8

On the other hand, fromunTrnxnandu_n1Tr_n1xn1, we have

F u_n, y

ϕy 1 r_n

y−u_n, u_n−x_n

≥ϕ u_n

, ∀y∈C, 3.9

F u_n1, y

ϕy 1 r_n1

y−u_n1, u_n1−x_n1

≥ϕ u_n1

, ∀y∈C. 3.10

Puttingyu_n1in3.9andyu_nin3.10, we have

F u_n, u_n1

ϕ u_n1 1

rn

u_n1−u_n, u_n−x_n

≥ϕ u_n , F u_n1, u_n

ϕ u_n 1

r_n1

u_n−u_n1, u_n1−x_n1

≥ϕ u_n1 .

3.11

So, from the monotonicity ofF, we get

u_n1−un,u_n−x_n

rn −u_n1−x_n1 r_n1

≥0, 3.12

hence

u_n1−un, un−u_n1u_n1−xn− rn

r_n1 u_n1−x_n1

≥0. 3.13

Without loss of generality, let us assume that there exists a real numberbsuch thatr_n> b >0 for alln∈N.Then,

u_n1−un²≤

u_n1−un, x_n1−xn

1− r_n

r_n1 u_n1−x_n1

≤u_n1−u_nx_n1−x_n 1− r_n

r_n1

u_n1−x_n1 ,

3.14

hence

u_n1−un≤x_n1−xn 1

r_n1r_n1−rnu_n1−x_n1

≤x_n1−x_n1

br_n1−r_nM₂,

3.15

whereM₂sup{un−x_n:n≥1}.

It follows from3.8and3.15that y_n1−y_n≤x_n1−x_n1

br_n1−r_nM₂M₁γ_n1−γ_n 1−γ_n1^N

j1

ζ_jⁿ¹−ζⁿ_j Tjun.

3.16

Define a sequence{vn}such that

x_n1βnxn 1−βn

vn, ∀n≥1. 3.17

Then, we have

v_n1−vn x_n2−β_n1x_n1

1−β_n1 −x_n1−βnxn

1−β_n

α_n1fx_n1 1−α_n1−β_n1y_n1

1−β_n1 −αnfxn 1−αn−βnyn

1−βn

α_n1

1−β_n1f x_n1

− α_n 1−βnf xn

y_n1−yn α_n

1−βnyn− α_n1 1−β_n1y_n1.

3.18

From3.18and3.16, we have v_n1−v_n−x_n1−x_n

≤ α_n1

1−β_n1 f x_n1y_n1 α_n 1−βn

f x_ny_n y_n1−y_n−x_n1−x_n

≤ α_n1

1−β_n1 f x_n1y_n1 αn

1−βn f xnyn 1

br_n1−r_nM₂M₁γ_n1−γ_n 1−γ_n1^N

j1

ζⁿ¹_j −ζⁿ_j T_ju_n.

3.19

It follows fromC1–C5that lim sup

n→ ∞ v_n1−v_n−x_n1−x_n≤0. 3.20

Hence, by27, Lemma 2.2, we have limn→ ∞vn−x_n0. Consequently,

nlim→ ∞x_n1−xn lim

n→ ∞ 1−βnvn−xn0. 3.21

Sincex_n1 αnfxn βnxn 1−αn−βnyn, we have x_n−y_n≤x_n1−x_nx_n1−y_n

≤x_n1−xnαnf xn

−ynβnxn−yn, 3.22

thus

xn−yn≤ 1

1−β_n x_n1−xnαnfxn−yn. 3.23 It follows fromC1andC2that limn→ ∞xn−yn0.

Sincex_n1 αnfxn βnxn 1−αn−βnyn,foru∈Ω, it follows from3.5and3.3 that

x_n1−u²αnf xn

βnxn 1−αn−βn

yn−u²

≤α_nf x_n

−u²β_nx_n−u² 1−α_n−β_ny_n−u²

≤αnf xn

−u²βnxn−u²

1−α_n−β_nu_n−u² 1−γ_n ε−γ_nu_n−W_nu_n²

≤αnf xn

−u² 1−αnxn−u²

1−αn−βn 1−γn ε−γnun−Wnun²,

3.24

from which it follows that un−Wnun²≤ α_n

1−αn−βn1−γnγn−ε f xn

−u²−xn−u² 1

1−αn−βn1−γnγn−ε x_n−u²−x_n1−u² .

≤ αn

1−α_n−β_n1−γ_nγn−ε f x_n

−u²−x_n−u² 1

1−α_n−β_n1−γ_nγn−ε x_n−ux_n1−ux_n1−x_n.

3.25

It follows fromC1–C3andxn1−xn → 0 that

un−Wnun−→0. 3.26

Foru∈Ω, we have fromLemma 2.2, un−u²Srnxn−Srnu² ≤

Srnxn−Srnu, xn−u

un−u, xn−u 1

2un−u²xn−u²−xn−un² .

3.27

Hence,

un−u²≤xn−u²−xn−un². 3.28

By3.24and3.28, we have x_n1−u²≤α_nf x_n

−u²β_nx_n−u² 1−α_n−β_nu_n−u²

≤αnf xn

−u²βnxn−u² 1−αn−βnxn−u²−xn−un² . 3.29

Hence,

1−αn−βnxn−un²≤αnf xn

−u²−αnxn−u²xn−u²−x_n1−u²

≤αnf xn

−u²−αnxn−u²

xn−ux_n1−uxn−x_n1.

3.30

It follows fromC1,C2, andxn−x_n1 → 0 that lim_{n→ ∞}xn−u_n0.

Next, we show that

lim sup

n→ ∞

fu0−u₀, x_n−u₀

≤0, 3.31

whereu₀P_Ωfu0. To show this inequality, we can choose a subsequence{xni}of{xn}such that

ilim→ ∞

fu0−u0, xni −u0

lim sup

n→ ∞

fu0−u0, xn−u0

. 3.32

Since{xni} is bounded, there exists a subsequence {xn_ij} of {xni} which converges weakly tow. Without loss of generality, we can assume that{xni} w.Fromxn−u_n → 0,

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 necessaryζⁿ_kⁱ → ζkasi → ∞fork1,2, . . . , N.It is easy to see thatζk>0 and_N

k1ζk1. We also have

W_nix−→Wx asi−→ ∞∀x∈C, 3.33

whereW _N

k1ζ_kT_k. Note that by21, Proposition 2.6,W is anε-strict pseudocontraction

and FixW _N

i1FixTi. Since

u_ni−Wu_ni≤u_ni−W_niu_niW_niu_ni−Wu_ni

≤uni−Wniuni^N

k1

ζⁿ_kⁱ−ζkTkuni, 3.34

it follows from3.26andζⁿ_kⁱ → ζ_kthat

u_ni−Wu_ni−→0. 3.35 So by the demiclosedness principle21, Proposition 2.6ii, it follows thatw ∈ FixW _N

i1FixTi.

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

F u_n, y

ϕy 1 rn

y−u_n, u_n−x_n

≥ϕ u_n

, ∀y∈C. 3.36

It follows fromA2that

ϕy 1 rn

y−u_n, u_n−x_n

≥F y, u_n

ϕ u_n

, ∀y∈C. 3.37

Hence,

ϕy

y−u_ni,uni−xni

rni

≥F y, u_ni

ϕ u_n

, ∀y∈C. 3.38

It follows fromA4,A5and the weakly lower semicontinuity ofϕ,uni−xni/rni → 0,andu_ni 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 y_t, y_t

ϕ y_t

−ϕ y_t

≤tF y_t, y

1−tF y_t, w

tϕy 1−tϕw−ϕ y_t

≤t

F yt, y

ϕy−ϕ yt

.

3.40

Dividing byt, we get

F y_t, y

ϕy−ϕ y_t

≥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, whereu0P_Ωfu0. FromLemma 2.5, we have

x_n1−u0²αn f xn

−u0

βn xn−u0

1−αn−βn yn−u0²

≤βn xn−u0

1−αn−βn yn−u0²2αn

f xn

−u0, x_n1−u0

≤ 1−αn−βnyn−u0βnxn−u0²2αn

f xn

−u0, x_n1−u0

≤ 1−αn

₂xn−u0²2αn

f xn

−f u0

, x_n1−u0

2αn

f u0

−u0, x_n1−u0

≤ 1−αn

₂xn−u0²2αnaxn−u0x_n1−u02αn

f u0

−u0, x_n1−u0

≤ 1−αn

₂xn−u0²αna xn−u0²x_n1−u0² 2αn

f u0

−u0, x_n1−u0

,

3.44

thus

x_n1−u₀² ≤

1−21−aαn

1−aα_n

x_n−u₀² 21−aαn

1−aαn

α_n

21−ax_n−u₀² 1 1−a

2f u₀

−2u₀, x_n1−u₀ .

3.45

It follows fromC1,3.43,3.45, andLemma 2.4that lim_{n→ ∞}xn−u₀ 0. From xn−u_n → 0 andyn −x_n → 0, we haveu_n → u₀ and y_n → 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 each 1 ≤ j ≤ N, let T_j : C → C be anε_j-strict pseudocontraction for some 0 ≤ ε_j < 1 such thatΩ _N

j1FixTj∩GEPF, ϕ/∅. Assume for eachn, {ζⁿ_j }^N_j1 is a finite sequence of positive numbers such that _N

j1ζⁿ_j 1 for all n and inf_n≥1ζⁿ_j > 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 r_n

y−un, un−xn

≥ϕ un

, ∀y∈C,

ynγnun 1−γn

^N

j1

ζⁿ_j Tjun, x_n1αnvβnxn 1−αn−βn

yn

3.46

for every n 1,2, . . . ,where {γn},{rn},{αn},{ζ₁ⁿ},{ζⁿ₂ }, . . . ,{ζⁿ_N},and {βn}are sequences of numbers satisfying the conditions (C1)–(C5). Then,{xn}, {un},and{yn}converge strongly tow P_Ωv.

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

Theorem 4.1. 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 some 0≤ε <1 such that FixT∩GEPF, ϕ/∅. Assume that either (B1) or

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

x1x∈C, F un, y

ϕy 1 r_n

y−un, un−xn

≥ϕ un

, ∀y∈C, y_n γ_nu_n 1−γ_n

Tu_n, x_n1 α_nf x_n

β_nx_n 1−α_n−β_n y_n

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 some 0 ≤ ε < 1 such that FixT∩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 r_n

y−un, un−xn

≥ϕ un

, ∀y∈C, yn γnun 1−γn

Tun, x_n1α_nvβ_nx_n 1−α_n−β_n

y_n

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 subsetD_x ⊆Candy_x ∈ 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\D_x,

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 following results.