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 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**

Let*H*be a real Hilbert space with inner product ·*,*· and induced norm · ,and let*C*be
a nonempty-closed convex subset of*H. Letϕ* :*H* → *R*∪ {∞}be a function and let*F*be
a bifunction from*C*×*C*to *R*such that*C*∩dom*ϕ /*∅, where *R*is the set of real numbers
and dom*ϕ* {x∈ *H* : *ϕx<* ∞}. Flores-Baz´an 1introduced the following generalized
equilibrium problem:

Find*x*∈*C*such that*Fx, 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
problem1.1with*CHR** ^{n}*.

Let*ϕx δ**C*x,∀x∈*H. Hereδ**C*denotes the indicator function of the set*C; that is,*
*δ** _{C}*x 0 if

*x*∈

*C*and

*δ*

*x ∞otherwise. Then the problem1.1becomes the following equilibrium problem:*

_{C}Finding*x*∈*C*such that*Fx, 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.

If*Fx, y gy−gx*for all*x, y*∈*C, whereg*:*C* → *R*is a function, then the problem
1.1becomes a problem of finding*x*∈*C*which 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.

If*ϕ* :*H* → *R*∪ {∞}is replaced by a real-valued function*φ* :*C* → *R, the problem*
1.1reduces to the following mixed equilibrium problem introduced by Ceng and Yao6:

Find *x*∈*C*such that*Fx, y φy*−*φx*≥0, ∀y∈*C.* 1.4
Recall that a mapping*T* :*C* → *C*is said to be a*κ-strict pseudocontraction*7if there
exists 0≤*κ <*1,such that

Tx−*Ty*^{2}≤ x−*y*^{2}*κ*I−*Tx*−I−*Ty*^{2}*,* ∀x, y∈*C,* 1.5
where*I* denotes the identity operator on*C. Whenκ*0,*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 of*S*by 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 an*α-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 an*α-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**

Let*H*be a real Hilbert space with inner product ·*,*· and norm · . Let*C*be a nonempty-
closed convex subset of*H. Let symbols* → anddenote strong and weak convergences,
respectively. In a real Hilbert space*H, it is well known that*

*λx* 1−*λy*^{2}*λx*^{2} 1−*λy*^{2}−*λ1*−*λx*−*y*^{2} 2.1
for all*x, y*∈*H*and*λ*∈0,1.

For any*x*∈*H, there exists a unique nearest point inC, denoted byP**C*x, such that
x−*P**C*x ≤ x−*y*for all*y*∈*C. The mappingP**C*is called the metric projection of*H*onto
*C. We know thatP** _{C}*is a nonexpansive mapping from

*H*onto

*C. It is also known thatP*

_{C}*x*∈

*C*and

*x*−*P**C*x, P*C*x−*y*

≥0 2.2

for all*x*∈*H*and*y*∈*C.*

For each*B*⊆*H, we denote by convB*the convex hull of*B. A multivalued mapping*
*G* : *B* → 2* ^{H}* is said to be a KKM map if, for every finite subset {x1

*, x*2

*, . . . , x*

*n*} ⊆

*B,*conv{x1

*, x*

_{2}

*, . . . , x*

*}⊆*

_{n}_{∞}

*n1**Gx**i*.

We will use the following results in the sequel.

**Lemma 2.1**see22. Let*Bbe a nonempty subset of a Hausdorﬀtopological vector spaceXand let*
*G*:*B* → 2^{X}*be a KKM map. IfGxis closed for allx*∈*Band is compact for at least onex*∈*B, then*

*x∈B**Gx/*∅.

For solving the generalized equilibrium problem, let us give the following assump-
tions for the bifunction*F,ϕ,*and the set*C:*

A1*Fx, x *0 for all*x*∈*C;*

A2*F*is monotone, that is,*Fx, y Fy, x*≤0 for any*x, y*∈*C;*

A3for each*y*∈*C,* *x*→*Fx, y*is weakly upper semicontinuous;

A4for each*x*∈*C, y*→*Fx, y*is convex;

A5for each*x*∈*C, y*→*Fx, y*is lower semicontinuous;

B1For each*x*∈*H*and*r >*0, there exist a bounded subset*D** _{x}*⊆

*C*and

*y*

*∈*

_{x}*C*∩dom

*ϕ*such that for any

*z*∈

*C*\

*D*

*x*,

*F* *z, y**x*

*ϕ* *y**x*

1
*r*

*y**x*−*z, z*−*x*

*< ϕz;* 2.3

B2*C*is a bounded set.

**Lemma 2.2. Let**Cbe 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*∩dom*ϕ /*∅. For*r >0 andx*∈*H,define a mappingS**r* :*H* → *Cas follows:*

*S**r*x

*z*∈*C*:*Fz, y ϕy * 1

*r*y−*z, z*−*x ≥ϕz,*∀y∈*C*

2.4

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

1*for eachx*∈*H,* *S**r*x*/*∅;

2*S**r**is single-valued;*

3*S**r**is firmly nonexpansive, that is, for anyx, y*∈*H,*
*S**r*x−*S**r*y^{2}≤

*S**r*x−*S**r*y, x−*y*

; 2.5

4FixS*r* GEPF, ϕ;

5GEPF, ϕ*is closed and convex.*

*Proof. Letx*0be any given point in*E. For eachy*∈*C, we define*
*Gy *

*z*∈*C*:*Fz, y ϕy * 1
*r*

*y*−*z, z*−*x*_{0}

≥*ϕz*

*.* 2.6

Note that for each*y*∈*C∩domϕ,Gy*is nonempty since*y*∈*Gy*and for each*y*∈*C\domϕ,*
*Gy C. We will prove thatG*is a KKM map on*C*∩dom*ϕ. Suppose that there exists a finite*
subset{y1*, y*_{2}*, . . . , y** _{n}*}of

*C*∩dom

*ϕ*and

*μ*

*≥0 for all*

_{i}*i*1,2, . . . , nwith

_{n}*i1**μ** _{i}* 1 such that

*z*_{n}

*i1**μ**i**y**i**/*∈Gy*i*for each*i*1,2, . . . , n. Then we have

*F* *z, y* *i*

*ϕ* *y**i*

−*ϕz *1
*r*

*y**i*−*z,* *z*−*x*0

*<*0 2.7

for each*i*1,2, . . . , n. ByA4and the convexity of*ϕ, we have*

0*Fz,z * *ϕz* −*ϕz * 1
*r*

*z*−*z,* *z*−*x*0

≤^{n}

*i1*

*μ*_{i}

*F* *z, y* _{i}

*ϕ* *y*_{i}

−*ϕz*
1

*r*
_{n}

*i1*

*μ*_{i}

*y** _{i}*−

*z,*

*z*−

*x*

_{0}

*<*0,

2.8

which is a contradiction. Hence,*G*is a KKM map on*C*∩dom*ϕ. Note thatGy** ^{w}*the weak
closure of

*Gy*is a weakly closed subset of

*C*for each

*y*∈

*C. Moreover, if*B2holds, then

*Gy*

*is also weakly compact for each*

^{w}*y*∈

*C. If*B1holds, then for

*x*

_{0}∈

*E, there exists a*bounded subset

*D*

*x*0 ⊆

*C*and

*y*

*x*0∈

*C*∩dom

*ϕ*such that for any

*z*∈

*C*\

*D*

*x*0,

*F* *z, y**x*0

*ϕ* *y**x*0

1
*r*

*y**x*0−*z, z*−*x*0

*< ϕz.* 2.9

This shows that

*G* *y*_{x}_{0}

*z*∈*C*:*F* *z, y*_{x}_{0}

*ϕ* *y*_{x}_{0}
1

*r*

*y*_{x}_{0}−*z, z*−*x*_{0}

≥*ϕz*

⊆*D*_{x}_{0}*.* 2.10

Hence, *Gy**x*0* ^{w}* is weakly compact. Thus, in both cases, we can use Lemma 2.1and have

*y∈C∩dom**ϕ**Gy*^{w}*/*∅.

Next, we will prove that*Gy*^{w}*Gy*for each*y*∈*C; that is,Gy*is weakly closed.

Let*z*∈*Gy** ^{w}*and let

*z*

*m*be a sequence in

*Gy*such that

*z*

*m*

*z. Then,*

*F* *z**m**, y*

*ϕy *1
*r*

*y*−*z**m**, z**m*−*x*0

≥*ϕ* *z**m*

*.* 2.11

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

lim sup

*m→ ∞*

*y*−*z**m**, z**m*−*x*0

≤

*z*−*y, x*0−*z*

*.* 2.12

It follows fromA3and the weak lower semicontinuity of*ϕ*that
*ϕz*≤lim inf

*m→ ∞* *ϕ* *z**m*

≤lim sup

*m*→ ∞

*F* *z**m**, y*

*ϕy * 1
*r*

*y*−*z**m**, z**m*−*x*0

≤lim sup

*m→ ∞*

*F* *z*_{m}*, y*

*ϕy*
1

*r*lim sup

*m*→ ∞

*y*−*z*_{m}*, z** _{m}*−

*x*

_{0}

≤*Fz, y ϕy * 1
*r*

*z*−*y, x*0−*z*
*.*

2.13

This implies that *z* ∈ *Gy. Hence,* *Gy* is weakly closed. Hence, *S** _{r}*x0

*y∈C**Gy *

*y∈C∩dom**ϕ**Gy *

*y∈C∩dom**ϕ**Gy*^{w}*/*∅. Hence, from the arbitrariness of*x*0, we conclude that
*S** _{r}*x

*/*∅, ∀x∈

*H.*

We observe that *S** _{r}*x ⊆ dom

*ϕ. So by similar argument with that in the proof of*Lemma 2.3 in9, we can easily show that

*S*

*r*is single-valued and

*S*

*r*is a firmly nonexpansive- type map. Next, we claim that FixS

*r*GEPF, ϕ. Indeed, we have the following:

*u*∈Fix *S**r*

⇐⇒*uS**r*u

⇐⇒*Fu, y ϕy *1

*r*y−*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, Since*S**r* is firmly nonexpansive,
*S** _{r}* is also nonexpansive. By23, Proposition 5.3, we know that GEPF, ϕ FixS

*r*is closed and convex.

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

**Lemma 2.4**see24,25. Assume that{α*n*}*is a sequence of nonnegative real numbers such that*
*α** _{n1}*≤ 1−

*γ*

_{n}*α*_{n}*δ*_{n}*,* 2.15

*whereγ**n**is a sequence in*0,1*and*{δ*n*}*is a sequence such that*
i ^{∞}

*n1*

*γ**n*∞;

ii lim sup

*n*→ ∞

*δ*_{n}

*γ** _{n}* ≤

*0 or*∞

*n1*

*δ*_{n}*<*∞.

2.16

*Then, lim*_{n}_{→ ∞}*α** _{n}*0.

**Lemma 2.5. In a real Hilbert space**H, there holds the following inequality:

x*y*^{2}≤ x^{2}2y, x*y* 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 {γ*n*},{r*n*},{α*n*},{ζ_{1}^{n}},
{ζ^{n}_{2} }, . . . ,{ζ^{n}* _{N}*},and{β

*n*}:

C1lim_{n}_{→ ∞}*α** _{n}*0 and

_{∞}

*n1**α** _{n}*∞;

C21*>*lim sup_{n}_{→ ∞}*β** _{n}*≥lim inf

_{n}_{→ ∞}

*β*

_{n}*>*0;

C3{γ*n*} ⊂c, dfor some*c, d*∈ε,1and lim*n*→ ∞|γ*n1*−*γ**n*|0;

C4lim inf*n*→ ∞*r**n**>*0 and lim*n*→ ∞|r*n1*−*r**n*|0;

C5lim_{n}_{→ ∞}|ζ^{n1}* _{j}* −

*ζ*

^{n}

*|0 for all*

_{j}*j*1,2, . . . , 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*∩dom*ϕ /*∅. 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 that*Ω

_{N}*j1*FixT*j*∩GEPF, ϕ*/*∅. Assume for each*n,* {ζ^{n}* _{j}* }

^{N}

_{j1}*is a finite sequence of*

*positive numbers such that*

_{N}*j1**ζ*^{n}_{j}*1 for all* *n* *and inf*_{n≥1}*ζ*^{n}_{j}*>* *0 for all 0* ≤ *j* ≤ *N. Let*
*ε* max{ε*j* : 1 ≤ *j* ≤ *N}. Assume that either (B1) or (B2) holds. Let* *f* *be a contraction ofC*
*into itself and let*{x*n*},{u*n*},*and*{y*n*}*be sequences generated by*

*x*1*x*∈*C,*
*F* *u*_{n}*, y*

*ϕy * 1
*r**n*

*y*−*u*_{n}*, u** _{n}*−

*x*

_{n}≥*ϕ* *u*_{n}

*,* ∀y∈*C,*
*y*_{n}*γ*_{n}*u** _{n}* 1−

*γ*

_{n}

^{N}*j1*

*ζ*^{n}_{j}*T*_{j}*u*_{n}*,*
*x*_{n1}*α*_{n}*f* *x*_{n}

*β*_{n}*x** _{n}* 1−

*α*

*−*

_{n}*β*

_{n}*y*

_{n}3.1

*for everyn* 1,2, . . ., where {γ*n*},{r*n*},{α*n*},{ζ_{1}^{n}},{ζ^{n}_{2} }, . . . ,{ζ^{n}* _{N}*},

*and*{β

*n*}

*are sequences of*

*numbers satisfying the conditions (C1)–(C5). Then,*{x

*n*},{u

*n*},

*and*{y

*n*}

*converge strongly tow*

*P*

_{Ω}

*fw.*

*Proof. We show thatP*_{Ω}*f* is a contraction of*C*into itself. In fact, there exists*a* ∈0,1such
thatfx−*fy ≤ax*−*y*for all*x, y*∈*C. So, we have*

*P*_{Ω}*fx*−*P*_{Ω}*fy*≤*f*x−*fy*≤*ax*−*y* 3.2
for all *x, y* ∈ *C.*Since*H* is complete, there exists a unique element*u*0 ∈ *C*such that*u*0
*P*_{Ω}*fu*0.

Let*u* ∈ Ωand let{S*r**n*}be a sequence of mappings defined as inLemma 2.2. From
*u*_{n}*S*_{r}* _{n}*x

*n*∈

*C, we have*

*u**n*−*uS**r**n* *x**n*

−*S**r**n*u≤*x**n*−*u.* 3.3

We define a mapping*W** _{n}*by

*W**n**x*^{N}

*j1*

*ζ*^{n}_{j}*T**j**x,* ∀x∈*C.* 3.4

By 21, Proposition 2.6, we know that *W**n* is an *ε-strict pseudocontraction and* *FW**n*

_{N}

*j1*FixT*j*. It follows from3.3,*y*_{n}*γ*_{n}*u** _{n}* 1−

*γ*

*W*

_{n}*n*

*u*

*and*

_{n}*uW*

_{n}*u*such that

*y*

*−*

_{n}*u*

^{2}

*γ*

_{n}*u*

*−*

_{n}*u*

^{2}1−

*γ*

_{n}*W*

_{n}*u*

*−*

_{n}*u*

^{2}−

*γ*

*1−*

_{n}*γ*

_{n}*u*

*−*

_{n}*W*

_{n}*u*

_{n}^{2}

≤*γ**n**u**n*−*u*^{2} 1−*γ**n**u**n*−*u*^{2}*εu**n*−*W**n**u**n*^{2}

−*γ**n* 1−*γ**n**u**n*−*W**n**u**n*^{2}
*u** _{n}*−

*u*

^{2}1−

*γ*

_{n}*ε*−

*γ*

_{n}*u*

*−*

_{n}*W*

_{n}*u*

_{n}^{2}

≤*u**n*−*u*^{2}*.*

3.5
Put*M*0max{x1−u,1/1−afu−u}.It is obvious thatx1−u ≤*M*0*.*Suppose
x*n*−*u ≤M*0*.*From3.3,3.5, and*x*_{n1}*α**n**f*x*n* *β**n**x**n* 1−*α**n*−*β**n*y*n*, we have

*x** _{n1}*−

*uα*

*n*

*f*

*x*

*n*

*β**n**x**n* 1−*α**n*−*β**n*

*y**n*−*u*

≤*α*_{n}*f* *x*_{n}

−*f*u*α*_{n}*fu*−*uβ*_{n}*x** _{n}*−

*u*1−

*α*

*−*

_{n}*β*

_{n}*y*

*−*

_{n}*u*

≤*α*_{n}*ax** _{n}*−

*uα*

_{n}*fu*−

*uβ*

_{n}*x*

*−*

_{n}*u*1−

*α*

*−*

_{n}*β*

_{n}*u*

*−*

_{n}*u*

≤*α**n**ax**n*−*uα**n**fu*−*u* 1−*α**n**x**n*−*u*
1−*aα**n*

*fu*−*u*

1−*a*

1−1−*aα**n**x** _{n}*−

*u,*

≤1−*aα**n**M*_{0}

1−1−*aα**n*

*M*_{0}*M*_{0}

3.6

for every*n* 1,2, . . . .Therefore,{x*n*}is bounded. From3.3and3.5, we also obtain that
{y*n*}and{u*n*}are bounded.

Following26, define*B**n*:*C* → *C*by

*B**n**γ**n**I* 1−*γ**n*

*W**n**.* 3.7

As shown in26, each*B**n*is a nonexpansive mapping on*C. SetM*1 sup* _{n≥1}*{u

*n*−

*W*

*n*

*u*

*n*}, we have

*y** _{n1}*−

*y*

*n*

*B*

_{n1}*u*

_{n1}−*B**n* *u**n*

≤*B*_{n1}*u*_{n1}

−*B*_{n1}*u*_{n}*B*_{n1}*u*_{n}

−*B*_{n}*u*_{n}

≤*u** _{n1}*−

*u*

*n*

*M*1

*γ*

*−*

_{n1}*γ*

*n*1−

*γ*

_{n1}*W*

_{n1}*u*

*n*−

*W*

*n*

*u*

*n*

≤*u** _{n1}*−

*u*

_{n}*M*

_{1}

*γ*

*−*

_{n1}*γ*

*1−*

_{n}*γ*

_{n1}

^{N}*j1*

*ζ*^{n1}* _{j}* −

*ζ*

^{n}

_{j}*T*

_{j}*u*

_{n}*.*

3.8

On the other hand, from*u**n**T**r**n*x*n*and*u*_{n1}*T**r** _{n1}*x

*n1*, 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

Putting*yu** _{n1}*in3.9and

*yu*

*in3.10, we have*

_{n}*F* *u*_{n}*, u*_{n1}

*ϕ* *u** _{n1}*
1

*r**n*

*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 of*F, we get*

*u** _{n1}*−

*u*

*n*

*,u*

*−*

_{n}*x*

_{n}*r**n* −*u** _{n1}*−

*x*

_{n1}*r*

_{n1}

≥0, 3.12

hence

*u** _{n1}*−

*u*

*n*

*, u*

*n*−

*u*

_{n1}*u*

*−*

_{n1}*x*

*n*−

*r*

*n*

*r*_{n1}*u** _{n1}*−

*x*

_{n1}≥0. 3.13

Without loss of generality, let us assume that there exists a real number*b*such that*r*_{n}*> b >*0
for all*n*∈*N.*Then,

*u** _{n1}*−

*u*

*n*

^{2}≤

*u** _{n1}*−

*u*

*n*

*, x*

*−*

_{n1}*x*

*n*

1− *r*_{n}

*r*_{n1}*u** _{n1}*−

*x*

_{n1}≤*u** _{n1}*−

*u*

_{n}*x*

*−*

_{n1}*x*

*1−*

_{n}*r*

_{n}*r*_{n1}

*u** _{n1}*−

*x*

_{n1}*,*

3.14

hence

*u** _{n1}*−

*u*

*n*≤

*x*

*−*

_{n1}*x*

*n*1

*r*_{n1}*r** _{n1}*−

*r*

*n*

*u*

*−*

_{n1}*x*

_{n1}≤*x** _{n1}*−

*x*

*1*

_{n}*br** _{n1}*−

*r*

_{n}*M*

_{2}

*,*

3.15

where*M*_{2}sup{u*n*−*x** _{n}*:

*n*≥1}.

It follows from3.8and3.15that
*y** _{n1}*−

*y*

*≤*

_{n}*x*

*−*

_{n1}*x*

*1*

_{n}*br** _{n1}*−

*r*

_{n}*M*

_{2}

*M*

_{1}

*γ*

*−*

_{n1}*γ*

*1−*

_{n}*γ*

_{n1}

^{N}*j1*

*ζ*_{j}^{n1}−*ζ*^{n}_{j}*T**j**u**n**.*

3.16

Define a sequence{v*n*}such that

*x*_{n1}*β**n**x**n* 1−*β**n*

*v**n**,* ∀n≥1. 3.17

Then, we have

*v** _{n1}*−

*v*

*n*

*x*

*−*

_{n2}*β*

_{n1}*x*

_{n1}1−*β** _{n1}* −

*x*

*−*

_{n1}*β*

*n*

*x*

*n*

1−*β*_{n}

*α*_{n1}*f*x* _{n1}* 1−

*α*

*−*

_{n1}*β*

*y*

_{n1}

_{n1}1−*β** _{n1}* −

*α*

*n*

*fx*

*n*1−

*α*

*n*−

*β*

*n*y

*n*

1−*β**n*

*α*_{n1}

1−*β*_{n1}*f* *x*_{n1}

− *α** _{n}*
1−

*β*

*n*

*f*

*x*

*n*

*y** _{n1}*−

*y*

*n*

*α*

_{n}1−*β**n**y**n*− *α** _{n1}*
1−

*β*

_{n1}*y*

_{n1}*.*

3.18

From3.18and3.16, we have
*v** _{n1}*−

*v*

*−*

_{n}*x*

*−*

_{n1}*x*

_{n}≤ *α*_{n1}

1−*β*_{n1}*f* *x*_{n1}*y*_{n1}*α** _{n}*
1−

*β*

*n*

*f* *x*_{n}*y*_{n}*y** _{n1}*−

*y*

*−*

_{n}*x*

*−*

_{n1}*x*

_{n}≤ *α*_{n1}

1−*β*_{n1}*f* *x*_{n1}*y*_{n1}*α**n*

1−*β**n* *f* *x**n**y**n*
1

*br** _{n1}*−

*r*

_{n}*M*

_{2}

*M*

_{1}

*γ*

*−*

_{n1}*γ*

*1−*

_{n}*γ*

_{n1}

^{N}*j1*

*ζ*^{n1}* _{j}* −

*ζ*

^{n}

_{j}*T*

_{j}*u*

_{n}*.*

3.19

It follows fromC1–C5that lim sup

*n*→ ∞ *v** _{n1}*−

*v*

*−*

_{n}*x*

*−*

_{n1}*x*

*≤0. 3.20*

_{n}Hence, by27, Lemma 2.2, we have lim*n*→ ∞v*n*−*x** _{n}*0. Consequently,

*n*lim→ ∞*x** _{n1}*−

*x*

*n*lim

*n*→ ∞ 1−*β**n**v**n*−*x**n*0. 3.21

Since*x*_{n1}*α**n**f*x*n* *β**n**x**n* 1−*α**n*−*β**n*y*n*, we have
*x** _{n}*−

*y*

*≤*

_{n}*x*

*−*

_{n1}*x*

_{n}*x*

*−*

_{n1}*y*

_{n}≤*x** _{n1}*−

*x*

*n*

*α*

*n*

*f*

*x*

*n*

−*y**n**β**n**x**n*−*y**n**,* 3.22

thus

*x**n*−*y**n*≤ 1

1−*β*_{n}*x** _{n1}*−

*x*

*n*

*α*

*n*

*fx*

*n*−

*y*

*n*

*.*3.23 It follows fromC1andC2that lim

*n*→ ∞x

*n*−

*y*

*n*0.

Since*x*_{n1}*α**n**f*x*n* *β**n**x**n* 1−*α**n*−*β**n*y*n**,*for*u*∈Ω, it follows from3.5and3.3
that

*x** _{n1}*−

*u*

^{2}

*α*

*n*

*f*

*x*

*n*

*β**n**x**n* 1−*α**n*−*β**n*

*y**n*−*u*^{2}

≤*α*_{n}*f* *x*_{n}

−*u*^{2}*β*_{n}*x** _{n}*−

*u*

^{2}1−

*α*

*−*

_{n}*β*

_{n}*y*

*−*

_{n}*u*

^{2}

≤*α**n**f* *x**n*

−*u*^{2}*β**n**x**n*−*u*^{2}

1−*α** _{n}*−

*β*

_{n}*u*

*−*

_{n}*u*

^{2}1−

*γ*

_{n}*ε*−

*γ*

_{n}*u*

*−*

_{n}*W*

_{n}*u*

_{n}^{2}

≤*α**n**f* *x**n*

−*u*^{2} 1−*α**n**x**n*−*u*^{2}

1−*α**n*−*β**n* 1−*γ**n* *ε*−*γ**n**u**n*−*W**n**u**n*^{2}*,*

3.24

from which it follows that
*u**n*−*W**n**u**n*^{2}≤ *α*_{n}

1−*α**n*−*β**n*1−*γ**n*γ*n*−*ε* *f* *x**n*

−*u*^{2}−*x**n*−*u*^{2}
1

1−*α**n*−*β**n*1−*γ**n*γ*n*−*ε* *x** _{n}*−

*u*

^{2}−

*x*

*−*

_{n1}*u*

^{2}

*.*

≤ *α**n*

1−*α** _{n}*−

*β*

*1−*

_{n}*γ*

*γ*

_{n}*n*−

*ε*

*f*

*x*

_{n}−*u*^{2}−*x** _{n}*−

*u*

^{2}1

1−*α** _{n}*−

*β*

*1−*

_{n}*γ*

*γ*

_{n}*n*−

*ε*

*x*

*−*

_{n}*ux*

*−*

_{n1}*ux*

*−*

_{n1}*x*

_{n}*.*

3.25

It follows fromC1–C3andx*n1*−*x**n* → 0 that

*u**n*−*W**n**u**n*−→0. 3.26

For*u*∈Ω, we have fromLemma 2.2,
*u**n*−*u*^{2}*S**r**n**x**n*−*S**r**n**u*^{2} ≤

*S**r**n**x**n*−*S**r**n**u, x**n*−*u*

*u**n*−*u, x**n*−*u*
1

2*u**n*−*u*^{2}*x**n*−*u*^{2}−*x**n*−*u**n*^{2}
*.*

3.27

Hence,

*u**n*−*u*^{2}≤*x**n*−*u*^{2}−*x**n*−*u**n*^{2}*.* 3.28

By3.24and3.28, we have
*x** _{n1}*−

*u*

^{2}≤

*α*

_{n}*f*

*x*

_{n}−*u*^{2}*β*_{n}*x** _{n}*−

*u*

^{2}1−

*α*

*−*

_{n}*β*

_{n}*u*

*−*

_{n}*u*

^{2}

≤*α**n**f* *x**n*

−*u*^{2}*β**n**x**n*−*u*^{2} 1−*α**n*−*β**n**x**n*−*u*^{2}−*x**n*−*u**n*^{2}
*.*
3.29

Hence,

1−*α**n*−*β**n**x**n*−*u**n*^{2}≤*α**n**f* *x**n*

−*u*^{2}−*α**n**x**n*−*u*^{2}*x**n*−*u*^{2}−*x** _{n1}*−

*u*

^{2}

≤*α**n**f* *x**n*

−*u*^{2}−*α**n**x**n*−*u*^{2}

*x**n*−*ux** _{n1}*−

*ux*

*n*−

*x*

_{n1}*.*

3.30

It follows fromC1,C2, andx*n*−*x** _{n1}* → 0 that lim

_{n}_{→ ∞}x

*n*−

*u*

*0.*

_{n}Next, we show that

lim sup

*n→ ∞*

*fu*0−*u*_{0}*, x** _{n}*−

*u*

_{0}

≤0, 3.31

where*u*_{0}*P*_{Ω}*fu*0. To show this inequality, we can choose a subsequence{x*n**i*}of{x*n*}such
that

*i*lim→ ∞

*fu*0−*u*0*, x**n**i* −*u*0

lim sup

*n*→ ∞

*fu*0−*u*0*, x**n*−*u*0

*.* 3.32

Since{x*n**i*} is bounded, there exists a subsequence {x*n** _{ij}*} of {x

*n*

*i*} which converges weakly to

*w. Without loss of generality, we can assume that*{x

*n*

*i*}

*w.*Fromx

*n*−

*u*

*→ 0,*

_{n}we obtain that*u**n**i* * w. From*x*n*−*y**n* → 0, we also obtain that*y**n**i* * w. Since*{u*n**i*} ⊂*C*
and*C*is closed and convex, we obtain*w*∈*C.*

We first show that*w* ∈_{N}

*k1*FixT*k*.To see this, we observe that we may assumeby
passing to a further subsequence if necessary*ζ*^{n}_{k}^{i}^{} → *ζ**k*as*i* → ∞for*k*1,2, . . . , N.It is
easy to see that*ζ**k**>*0 and_{N}

*k1**ζ**k*1. We also have

*W*_{n}_{i}*x*−→*Wx* as*i*−→ ∞∀x∈*C,* 3.33

where*W* _{N}

*k1**ζ*_{k}*T** _{k}*. Note that by21, Proposition 2.6,

*W*is an

*ε-strict pseudocontraction*

and FixW _{N}

*i1*FixT*i*. Since

*u*_{n}* _{i}*−

*Wu*

_{n}*≤*

_{i}*u*

_{n}*−*

_{i}*W*

_{n}

_{i}*u*

_{n}

_{i}*W*

_{n}

_{i}*u*

_{n}*−*

_{i}*Wu*

_{n}

_{i}≤*u**n**i*−*W**n**i**u**n**i*^{N}

*k1*

*ζ*^{n}_{k}^{i}^{}−*ζ**k**T**k**u**n**i**,* 3.34

it follows from3.26and*ζ*^{n}_{k}^{i}^{} → *ζ** _{k}*that

*u*_{n}* _{i}*−

*Wu*

_{n}*−→0. 3.35 So by the demiclosedness principle21, Proposition 2.6ii, it follows that*

_{i}*w*∈ FixW

_{N}*i1*FixT*i*.

We now show*w*∈GEPF, ϕ.By*u*_{n}*T*_{r}_{n}*x*_{n}*,*we know that

*F* *u*_{n}*, y*

*ϕy * 1
*r**n*

*y*−*u*_{n}*, u** _{n}*−

*x*

_{n}≥*ϕ* *u*_{n}

*,* ∀y∈*C.* 3.36

It follows fromA2that

*ϕy * 1
*r**n*

*y*−*u*_{n}*, u** _{n}*−

*x*

_{n}≥*F* *y, u*_{n}

*ϕ* *u*_{n}

*,* ∀y∈*C.* 3.37

Hence,

*ϕy *

*y*−*u*_{n}_{i}*,u**n**i*−*x**n**i*

*r**n**i*

≥*F* *y, u*_{n}_{i}

*ϕ* *u*_{n}

*,* ∀y∈*C.* 3.38

It follows fromA4,A5and the weakly lower semicontinuity of*ϕ,*u*n**i*−*x**n**i*/r*n**i* →
0,and*u*_{n}_{i}* w*that

*Fy, w ϕw*≤*ϕy,* ∀y∈*C.* 3.39

For*t*with 0*< t*≤1 and*y*∈*C*∩dom*ϕ, lety**t**ty* 1−*tw.*Since*y*∈*C*∩dom*ϕ*and
*w*∈*C*∩dom*ϕ, we obtainy**t*∈*C*∩dom*ϕ,*and hence*Fy**t**, w ϕw*≤*ϕy**t*. So byA4and
the convexity of*ϕ, we have*

0*F* *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* *y**t**, y*

*ϕy*−*ϕ* *y**t*

*.*

3.40

Dividing by*t, we get*

*F* *y*_{t}*, y*

*ϕy*−*ϕ* *y*_{t}

≥0. 3.41

Letting*t* → 0, it follows fromA3and the weakly lower semicontinuity of*ϕ*that

*Fw, y ϕy*≥*ϕw* 3.42

for all *y* ∈ *C*∩dom*ϕ. Observe that ify* ∈ *C*\dom*ϕ, thenF*w, y *ϕy* ≥ *ϕw*holds.

Moreover, hence*w*∈GEPF, ϕ. This implies*w*∈Ω.Therefore, we have
lim sup

*n*→ ∞

*fu*0−*u*0*, x**n*−*u*0

lim

*i*→ ∞

*fu*0−*u*0*, x**n**i*−*u*0

*fu*0−*u*0*, w*−*u*0

≤0. 3.43

Finally, we show that*x**n* → *u*0, where*u*0*P*_{Ω}*fu*0.
FromLemma 2.5, we have

*x** _{n1}*−

*u*0

^{2}

*α*

*n*

*f*

*x*

*n*

−*u*0

*β**n* *x**n*−*u*0

1−*α**n*−*β**n* *y**n*−*u*0^{2}

≤*β**n* *x**n*−*u*0

1−*α**n*−*β**n* *y**n*−*u*0^{2}2α*n*

*f* *x**n*

−*u*0*, x** _{n1}*−

*u*0

≤ 1−*α**n*−*β**n**y**n*−*u*0*β**n**x**n*−*u*0^{2}2α*n*

*f* *x**n*

−*u*0*, x** _{n1}*−

*u*0

≤ 1−*α**n*

_{2}*x**n*−*u*0^{2}2α*n*

*f* *x**n*

−*f* *u*0

*, x** _{n1}*−

*u*0

2α*n*

*f* *u*0

−*u*0*, x** _{n1}*−

*u*0

≤ 1−*α**n*

_{2}*x**n*−*u*0^{2}2α*n**ax**n*−*u*0*x** _{n1}*−

*u*02α

*n*

*f* *u*0

−*u*0*, x** _{n1}*−

*u*0

≤ 1−*α**n*

_{2}*x**n*−*u*0^{2}*α**n**a* *x**n*−*u*0^{2}*x** _{n1}*−

*u*0

^{2}2α

*n*

*f* *u*0

−*u*0*, x** _{n1}*−

*u*0

*,*

3.44

thus

*x** _{n1}*−

*u*

_{0}

^{2}≤

1−21−*aα**n*

1−*aα*_{n}

*x** _{n}*−

*u*

_{0}

^{2}21−

*aα*

*n*

1−*aα**n*

*α*_{n}

21−*ax** _{n}*−

*u*

_{0}

^{2}1 1−

*a*

2f *u*_{0}

−2u_{0}*, x** _{n1}*−

*u*

_{0}

*.*

3.45

It follows fromC1,3.43,3.45, andLemma 2.4that lim_{n}_{→ ∞}x*n*−*u*_{0} 0. From
x*n*−*u** _{n}* → 0 andy

*n*−

*x*

*→ 0, we have*

_{n}*u*

*→*

_{n}*u*

_{0}and

*y*

*→*

_{n}*u*

_{0}. 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*∩dom*ϕ /*∅. 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 that*Ω

_{N}*j1*FixT*j*∩GEPF, ϕ*/*∅. Assume for each*n,* {ζ^{n}* _{j}* }

^{N}

_{j1}*is a finite sequence of*

*positive numbers such that*

_{N}*j1**ζ*^{n}_{j}*1 for all* *n* *and inf*_{n≥1}*ζ*^{n}_{j}*>* *0 for all 0* ≤ *j* ≤ *N. Let*
*ε* max{ε*j* : 1 ≤ *j* ≤ *N}. Assume that either (B1) or (B2) holds. Let* *vbe an arbitrary point in*
*Cand let*{x*n*}, {u*n*},*and*{y*n*}*be sequences generated by*

*x*1*x*∈*C,*
*F* *u**n**, y*

*ϕy * 1
*r*_{n}

*y*−*u**n**, u**n*−*x**n*

≥*ϕ* *u**n*

*,* ∀y∈*C,*

*y**n**γ**n**u**n* 1−*γ**n*

^{N}

*j1*

*ζ*^{n}_{j}*T**j**u**n**,*
*x*_{n1}*α**n**vβ**n**x**n* 1−*α**n*−*β**n*

*y**n*

3.46

*for every* *n* 1,2, . . . ,*where* {γ*n*},{r*n*},{α*n*},{ζ_{1}^{n}},{ζ^{n}_{2} }, . . . ,{ζ^{n}* _{N}*},

*and*{β

*n*}

*are sequences of*

*numbers satisfying the conditions (C1)–(C5). Then,*{x

*n*}, {u

*n*},

*and*{y

*n*}

*converge strongly tow*

*P*

_{Ω}

*v.*

*Proof. Letfx v*for all*x*∈*C, by*Theorem 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: for*j* 1,2, . . . , N, let*T*_{1} *T*_{2} · · ·*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*∩ dom*ϕ /*∅. 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*{x*n*}, {u*n*},*and*{y*n*}*be sequences generated*
*by*

*x*1*x*∈*C,*
*F* *u**n**, y*

*ϕy * 1
*r*_{n}

*y*−*u**n**, u**n*−*x**n*

≥*ϕ* *u**n*

*,* ∀y∈*C,*
*y*_{n}*γ*_{n}*u** _{n}* 1−

*γ*

_{n}*Tu*_{n}*,*
*x*_{n1}*α*_{n}*f* *x*_{n}

*β*_{n}*x** _{n}* 1−

*α*

*−*

_{n}*β*

_{n}*y*

_{n}4.1

*for everyn* 1,2, . . . ,*where* {γ*n*}, {r*n*}, {α*n*},*and* {β*n*}*are sequences of numbers satisfying the*
*conditions (C1)–(C4). Then,*{x*n*}, {u*n*},*and*{y*n*}*converge strongly towP*FixT∩GEPF,ϕ*fw.*

**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*∩ dom*ϕ /*∅. 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*{x*n*}, {u*n*},*and*{y*n*}*be sequences generated*
*by*

*x*1*x*∈*C,*
*F* *u**n**, y*

*ϕy * 1
*r*_{n}

*y*−*u**n**, u**n*−*x**n*

≥*ϕ* *u**n*

*,* ∀y∈*C,*
*y**n* *γ**n**u**n* 1−*γ**n*

*Tu**n**,*
*x*_{n1}*α*_{n}*vβ*_{n}*x** _{n}* 1−

*α*

*−*

_{n}*β*

_{n}*y*_{n}

4.2

*for everyn* 1,2, . . . ,*where* {γ*n*}, {r*n*}, {α*n*},*and* {β*n*}*are sequences of numbers satisfying the*
*conditions (C1)–(C4). Then,*{x*n*}, {u*n*},*and*{y*n*}*converge strongly towP*FixT∩GEPF,ϕ*v.*

We need the following two assumptions.

B3For each*x*∈ *H*and*r >*0, there exist a bounded subset*D** _{x}* ⊆

*C*and

*y*

*∈*

_{x}*C*such that for any

*z*∈

*C*\

*D*

*x*,

*F* *z, y**x*

1
*r*

*y**x*−*z, z*−*x*

*<*0. 4.3

B4For each*x*∈*H*and*r >*0, there exist a bounded subset*D**x*⊆*C*and*y**x*∈*C*∩dom*ϕ*
such that for any*z*∈*C*\*D** _{x}*,

*g* *y**x*

*ϕ* *y**x*

1
*r*

*y**x*−*z, z*−*x*

*< ϕz gz.* 4.4

Let *ϕx * *δ**C*x,∀x ∈ *H, by Theorems* 3.1 and 3.2, respectively, we obtain the
following results.