Hybrid algorithm for generalized mixed equilibrium problems and variational inequality problems and fixed point problems

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Computers and Mathematics with Applications

journal homepage:www.elsevier.com/locate/camwa

Hybrid algorithm for generalized mixed equilibrium problems and

variational inequality problems and fixed point problems

✩

Gang Cai

∗

, Shangquan Bu

Department of Mathematical Sciences, Tsinghua University, 100084 Beijing, China

a r t i c l e i n f o

Article history:

Received 18 August 2011

Received in revised form 25 October 2011 Accepted 25 October 2011

Keywords:

Strong convergence

General mixed equilibrium problem Variational inequality

Fixed point

a b s t r a c t

In this paper, we introduce a new iterative algorithm by hybrid method for finding a common element of the set of solutions of finite general mixed equilibrium problems and the set of solutions of a general variational inequality problem for finite inverse strongly monotone mappings and the set of common fixed points of infinite family of strictly pseudocontractive mappings in a real Hilbert space. Then we prove strong convergence of the scheme to a common element of the three above described sets. Our result improves and extends the corresponding results announced by many others.

©2011 Elsevier Ltd. All rights reserved.

1. Introduction

In this paper, we always assume thatHis a real Hilbert space with inner product

⟨

., .

⟩

and induced norm

∥

.

∥

andCis a nonempty closed convex subset ofH.PCdenotes the metric projection ofHontoCandF

(

T

)

denotes the fixed points set of

a mappingT.

A mappingT

:

C

→

Cis said to be nonexpansive if

∥

Tx

−

Ty

∥ ≤ ∥

x

−

y

∥

,

∀

x

,

y

∈

C

.

A mappingT

:

C

→

His said to beL-Lipschitzian if there existsL

>

0 such that

∥

Tx

−

Ty

∥ ≤

L

∥

x

−

y

∥

,

∀

x

,

y

∈

C

.

A mappingT

:

C

→

Cis said to bek-strictly pseudocontractive if there exists a constantk

∈ [

0

,

1

)

such that

∥

Tx

−

Ty

∥

2

≤ ∥

_x

−

_y

∥

2

+

_k

∥

(

_I

−

_T

)

_x

−

(

_I

−

_T

)

_y

∥

2

,

∀

_x

,

_y

∈

_C

.

Ifk

=

0, then the mappingT is nonexpansive. Iterative approximation of fixed point ofk-strictly pseudocontractive mappings has been studied extensively by many authors (see, e.g., [1–4] and the references contained therein).

A mappingA

:

C

→

His said to be monotone if

⟨

Ax

−

Ay

,

x

−

y

⟩ ≥

0

,

∀

x

,

y

∈

C

.

A mappingA

:

C

→

His said to be

α

-inverse-strongly monotone if there exists a positive real number

α

such that

⟨

Ax

−

Ay

,

x

−

y

⟩ ≥

α

∥

Ax

−

Ay

∥

2

,

∀

x

,

y

∈

C

.

✩_{This work was supported by the NSF of China (No. 10731020, No. 11171172) and the Specialized Research Fund for the Doctoral Program of Higher}

Education (No. 200800030059).

∗_{Corresponding author.}

E-mail addresses:[email protected](G. Cai),[email protected](S. Bu). 0898-1221/$ – see front matter©2011 Elsevier Ltd. All rights reserved.

LetA

:

C

→

Hbe a nonlinear mapping. The variational inequality problem is to find ax∗

∈

Csuch that



Ax∗

,

y

−

x∗



≥

0

,

∀

y

∈

C

.

(1.1)

(see, e.g., [5,6]). The set of solution of the variational inequality problem(1.1)is denoted byVI

(

C

,

A

)

.

Let

ϕ

:

C

→

_Ris a real-valued function andA

:

C

→

Ha nonlinear mapping. SupposeF

:

C

×

C

→

_Rbe a bifunction. The generalized mixed equilibrium problem is to findx

∈

C(see, e.g., [7,8]) such that

F

(

x

,

y

)

+

ϕ(

y

)

−

ϕ(

x

)

+ ⟨

Ax

,

y

−

x

⟩ ≥

0

,

∀

y

∈

C

.

(1.2) We denote the set of(1.2)byGMEP

(

F

, ϕ,

A

)

.

If

ϕ

=

0, then problem(1.2)reduces to generalized equilibrium problem studied by many authors (see, e.g., [9–11]), which is to findx

∈

Csuch that

F

(

x

,

y

)

+ ⟨

Ax

,

y

−

x

⟩ ≥

0

,

∀

y

∈

C

.

(1.3) The set of solutions of(1.3)is denoted byEP

(

F

,

A

)

.

IfA

=

0, then problem(1.2)reduces to mixed equilibrium problem considered by many authors (see, e.g., [12–14]), which is to findx

∈

Csuch that

F

(

x

,

y

)

+

ϕ(

y

)

−

ϕ(

x

)

≥

0

,

∀

y

∈

C

.

(1.4) The set of solutions of(1.4)is denoted byMEP

(

F

, ϕ)

.

If

ϕ

=

0

,

A

=

0, then problem(1.2)reduces to equilibrium problem studied by many authors (see, e.g., [15–17]), which is to findx

∈

Csuch that

F

(

x

,

y

)

≥

0

,

∀

y

∈

C

.

(1.5)

The set of solutions of(1.5)is denoted byEP

(

F

)

.

The generalized mixed equilibrium problems include fixed point problems, optimization problems, variational inequality problems, Nash equilibrium problems, and equilibrium problems as special cases. Numerous problems in Physics, optimization, and economics reduce to find a solution of problem(1.2). Several methods have been proposed to solve the fixed point problems, variational inequality problems and equilibrium problems in the literature (see, e.g., [18–30]).

LetA

,

B

:

C

→

Hbe two mappings. Ceng et al. [27] considered the following problem of finding

(

x∗

,

_y∗

)

∈

_C

×

_Csuch

that



λ

Ay∗

+

x∗

−

y∗

,

x

−

x∗



≥

0

,

∀

x

∈

C

,



µ

Bx∗

+

y∗

−

x∗

,

x

−

y∗

≥

0

,

∀

x

∈

C

,

(1.6)

which is called a general system of variational inequalities where

λ >

0 and

µ >

0 are two constants. In particular, ifA

=

B, then problem(1.6)reduces to finding

(

x∗

,

_y∗

)

∈

_C

×

_{Csuch that}



λ

Ay∗

+

x∗

−

y∗

,

x

−

x∗



≥

0

,

∀

x

∈

C

,



µ

Ax∗

+

y∗

−

x∗

,

x

−

y∗



≥

0

,

∀

x

∈

C

.

(1.7)

Further, if we add up the requirement thatx∗

=

_y∗_{, the problem(1.6)reduces to the classical variational inequality problem}

(1.1).

In order to find the common element of the solutions of problem(1.6)and the set of fixed points of one nonexpansive mappingT, Ceng et al. [27] studied the following algorithm:x1

=

u

∈

Cand



yn

=

PC

(

xn

−

µ

Bxn

),

xn+1

=

α

nu

+

β

nxn

+

γ

nSPC

(

yn

−

λ

Ayn

).

(1.8)

Under appropriate conditions they obtained one strong convergence theorem.

Very recently, Cho et al. [2] introduced a hybrid projection method for finding a common element of the set of solutions of a generalized equilibrium problem, the set of solutions of a variational inequality and the set of fixed points of a strict pseudocontraction in a real Hilbert space.

In this paper, motivated and inspired by the above facts, we introduce the following system of variational inequalities in a Hilbert spaceH: LetCbe a nonempty closed convex subset of a real Hilbert spaceH. Let

{

Ai

}

Ni=1

:

C

→

Hbe a family of

mappings. First we consider the following problem of finding

(

x∗ 1

,

x ∗ 2

, . . . ,

x ∗ N

)

∈

C

×

C

· · · ×

Csuch that





























λ

NANx∗N

+

x ∗ 1

−

x ∗ N

,

x

−

x ∗ 1



≥

0

,

∀

x

∈

C

,



λ

N−1AN−1x∗N−1

+

x ∗ N

−

x ∗ N−1

,

x

−

x ∗ N



≥

0

,

∀

x

∈

C

,

...



λ

2A2x∗2

+

x ∗ 3

−

x ∗ 2

,

x

−

x ∗ 3



≥

0

,

∀

x

∈

C

,



λ

1A1x∗1

+

x ∗ 2

−

x ∗ 1

,

x

−

x ∗ 2



≥

0

,

∀

x

∈

C

,

(1.9)

which is called a more general system of variational inequalities in Hilbert spaces, where

λ

i

>

0 for alli

∈ {

1

,

2

, . . . ,

N

}

.

The set of solutions to(1.9)is denoted byΩ. In particular, ifN

=

2

,

A1

=

B

,

A2

=

A

, λ

1

=

µ, λ

2

=

λ,

x∗1

=

x ∗

,

_x∗

2

=

y ∗_{, then}

problem(1.9)reduces to problem(1.6). Then we study a new iterative algorithm by hybrid method for finding a common element of the set of solutions of finite general mixed equilibrium problems and the set of solutions of a general variational inequality problem for finite inverse strongly monotone mappings and the set of common fixed points of infinite family of strictly pseudocontractive mappings in a real Hilbert space. Furthermore we prove a strong convergence of the scheme to a common element of the three above described sets. Our result extends the corresponding results announced by many others.

2. Preliminaries

LetHbe a real Hilbert space. It is well known that

∥

x

−

y

∥

2

= ∥

x

∥

2

− ∥

y

∥

2

−

2

⟨

x

−

y

,

y

⟩

(2.1) and

∥

λ

x

+

(

1

−

λ)

y

∥

2

=

λ

∥

x

∥

2

+

(

1

−

λ)

∥

y

∥

2

−

λ(

1

−

λ)

∥

x

−

y

∥

2 (2.2) for allx

,

y

∈

Hand

λ

∈ [

0

,

1

]

.

LetCbe a nonempty closed convex subset ofH. For every pointx

∈

H, there exists a unique nearest point inC, denoted byPCx, such that

∥

x

−

PCx

∥ ≤ ∥

x

−

y

∥

for ally

∈

C

.

(2.3)

Pis called a metric projection ofHontoC. It is well known thatPCis a nonexpansive mapping ofHontoCand satisfies

⟨

x

−

y

,

PCx

−

PCy

⟩ ≥ ∥

PCx

−

PCy

∥

2 (2.4)

for everyx

,

y

∈

H. Moreover,PCxis characterized by the following properties:PCx

∈

Cand

⟨

x

−

PCx

,

y

−

PCx

⟩ ≤

0 (2.5)

∥

x

−

y

∥

2

≥ ∥

_x

−

_P

Cx

∥

2

+ ∥

y

−

PCx

∥

2 (2.6)

for allx

∈

H

,

y

∈

C.

In the context of the variational inequality problem, this implies that

u

∈

VI

(

A

,

C

)

⇔

u

=

PC

(

u

−

λ

Au

),

for all

λ >

0

.

(2.7)

IfAis

α

-inverse-strongly monotone mapping ofCintoH, then it is obvious thatAis 1_α-Lipschitz continuous. We also have that, for allu

, v

∈

Cand

λ >

0,

∥

(

I

−

λ

A

)

u

−

(

I

−

λ

A

)v

∥

2

= ∥

(

u

−

v)

−

λ(

Au

−

A

v)

∥

2

= ∥

u

−

v

∥

2

−

2

λ

⟨

Ax

−

Ay

,

x

−

y

⟩ +

λ

2

∥

Au

−

A

v

∥

2

≤ ∥

u

−

v

∥

2

+

λ(λ

−

2

α)

∥

Au

−

A

v

∥

2

.

(2.8) So, if

λ

≤

2

α

, thenI

−

λ

Ais a nonexpansive mapping fromCtoH.

For solving the equilibrium problem, let us assume that the bifunctionFsatisfies the following conditions: (A1) F

(

x

,

x

)

=

0 for allx

∈

C;

(A2) Fis monotone, i.e.,F

(

x

,

y

)

+

F

(

y

,

x

)

≤

0 for anyx

,

y

∈

C; (A3) Fis upper-semicontinuous, i.e., for eachx

,

y

,

z

∈

C,

lim sup

t→0+

F

(

tz

+

(

1

−

t

)

x

,

y

)

≤

F

(

x

,

y

)

;

(A4) F

(

x

, .)

is convex and weakly lower semicontinuous for eachx

∈

C;

(B1) for eachx

∈

Handr

>

0, there exists a bounded subsetDx

⊆

Candyx

∈

Csuch that for anyz

∈

C

\

Dx, F

(

z

,

yx

)

+

ϕ(

yx

)

−

ϕ(

z

)

+

1

r

⟨

yx

−

z

,

z

−

x

⟩

<

0

;

(B2) Cis a bounded set.

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

Lemma 2.1 ([21]).Assume that F

:

C

×

C

→

_Rsatisfies

(

A1

)

–

(

A4

)

and let

ϕ

:

C

→

_Rbe a proper lower semicontinuous and convex function. Assume that either

(

B1

)

or

(

B2

)

holds. For r

>

0and x

∈

H, define a mapping Tr(F,ϕ)

:

H

→

C as follows:

T_r(F,ϕ)

(

x

)

=



z

∈

C

:

F

(

z

,

y

)

+

ϕ(

y

)

−

ϕ(

z

)

+

1

r

⟨

y

−

z

,

z

−

x

⟩ ≥

0

,

∀

y

∈

C



for all z

∈

H. Then the following hold:

(1) for each x

∈

H

,

Tr(F,ϕ)

(

x

)

̸= ∅

;

(3) Tr(F,ϕ)is firmly nonexpansive, that is, for any x

,

y

∈

H,



T

_r(F,ϕ)x

−

T_r(F,ϕ)y





2

≤



T_r(F,ϕ)x

−

T_r(F,ϕ)y

,

x

−

y



;

(4) F

(

Tr(F,ϕ)

)

=

MEP

(

F

, ϕ)

;

(5) MEP

(

F

, ϕ)

is closed and convex.

Lemma 2.2 ([31]).Let C be a nonempty closed convex subset of a real Hilbert space H. Let T

:

C

→

C be a k-strict pseudo-contractive mapping such that F

(

T

)

̸= ∅

.

(1) (Demi-closed principle) T is demi-closed on C , that is, if xn

⇀

x

∈

C and xn

−

Txn

→

0, then x

=

Tx.

(2) T satisfies the Lipschitz condition

∥

Tx

−

Ty

∥ ≤

L

∥

x

−

y

∥ =

1

+

k

1

−

k

∥

x

−

y

∥

,

∀

x

,

y

∈

C

.

(3) The fixed point set F

(

T

)

of T is closed and convex so that the projection PF(T)is well defined.

Lemma 2.3 ([32]).Let C be a nonempty closed convex subset of a Banach space and let

{

Tn

}

be a sequence of mappings of C into itself. Suppose

∞

_n=1sup_x∈C

∥

Tn+1x

−

Tnx

∥

<

∞

. Then for each y

∈

C

,

{

Tny

}

converges strongly to some point of C . Moreover, let T be a mapping of C into itself defined by Ty

=

limn→∞Tny

,

∀

y

∈

C . Thenlimn→∞supx∈C

∥

Tnx

−

Tx

∥ =

0.

Lemma 2.4 ([33]).Let H be a real Hilbert space, let C be a closed convex subset of H, and let T

:

C

→

C be a nonexpansive mapping with F

(

T

)

̸= ∅

. If

{

xn

}

is a sequence in C weakly converging to x and if

(

I

−

T

)

xnconverges strongly to y, then

(

I

−

T

)

x

=

y. Lemma 2.5. Let C be a nonempty closed convex subset of a real Hilbert space H. Let Ai

:

C

→

H be an

α

i-inverse-strongly accretive mapping, where i

∈ {

1

,

2

, . . . ,

N

}

. Let G

:

C

→

C be a mapping defined by

G

(

x

)

=

PC

(

I

−

λ

NAN

)

PC

(

I

−

λ

N−1AN−1

)

· · ·

PC

(

I

−

λ

2A2

)

PC

(

I

−

λ

1A1

)

x

,

∀

x

∈

C

.

If 0

< λ

i

≤

2

α

i

,

i

=

1

,

2

, . . . ,

N, then G

:

C

→

C is nonexpansive.

Proof. PutΩi

=

_P

C

(

I

−

λ

iAi

)

PC

(

I

−

λ

i−1Ai−1

)

· · ·

PC

(

I

−

λ

2A2

)

PC

(

I

−

λ

1A1

),

i

=

1

,

2

, . . . ,

NandΩ0

=

I, whereIis identity

mapping. ThenG

=

ΩN_{. For allx}

,

_y

∈

_{C, it follows from(2.8)that}

∥

Gx

−

Gy

∥ =





ΩNx

−

ΩNy





=



P

C

(

I

−

λ

NAN

)

ΩN−1x

−

PC

(

I

−

λ

NAN

)

ΩN−1y





≤





(

I

−

λ

_NA_N

)

ΩN−1x

−

(

I

−

λ

_NA_N

)

ΩN−1y





≤





ΩN−1x

−

ΩN−1y





...

≤





Ω0x

−

Ω0y





,

= ∥

x

−

y

∥

which impliesGis nonexpansive. This completes the proof.

Lemma 2.6. Let C be a nonempty closed convex subset of a real Hilbert space H. Let Ai

:

C

→

H be nonlinear mapping, where i

=

1

,

2

, . . . ,

N. For given x∗ i

∈

C

,

i

=

1

,

2

, . . . ,

N

, (

x ∗ 1

,

x ∗ 2

, . . . ,

x ∗

N

)

is a solution of problem(1.9)if and only if x∗_i

=

PC

(

I

−

λ

i−1Ai−1

)

x∗i−1

,

x

∗

1

=

QC

(

I

−

λ

NAN

)

x∗N

,

i

=

2

, . . . ,

N

.

That is

x∗₁

=

PC

(

I

−

λ

NAN

)

QC

(

I

−

λ

N−1AN−1

)

· · ·

PC

(

I

−

λ

2A2

)

PC

(

I

−

λ

1A1

)

x∗1

.

Proof. We can rewrite(1.9)as

































x∗₁

−

(

x∗_N

−

λ

_NANx∗N

),

x

−

x ∗ 1



≥

0

,

∀

x

∈

C

,



x∗_N

−

(

x∗_N−1

−

λ

N−1AN−1x∗N−1

),

x

−

x ∗ N



≥

0

,

∀

x

∈

C

,

...



x∗₃

−

(

x∗₂

−

λ

₂A2x∗2

),

x

−

x ∗ 3



≥

0

,

∀

x

∈

C

.



x∗₂

−

(

x∗₁

−

λ

₁A1x∗1

),

x

−

x ∗ 2



≥

0

,

∀

x

∈

C

.

(2.9)

From(2.5), we deduce that(2.9)is equivalent to x∗_i

=

PC

(

I

−

λ

i−1Ai−1

)

x∗i−1

,

x ∗ 1

=

PC

(

I

−

λ

NAN

)

x∗N

,

i

=

2

, . . . ,

N

.

Therefore we have x∗₁

=

PC

(

I

−

λ

NAN

)

PC

(

I

−

λ

N−1AN−1

)

· · ·

PC

(

I

−

λ

2A2

)

PC

(

I

−

λ

1A1

)

x∗1

.

This completes the proof.

3. Main results

Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let

{

Fk

}

Mk=1be a family of bifunctions from C

×

C

→

_Rsatisfying

(

A1

)

–

(

A4

)

,

{

ϕ

_k

}

M_k=1be a family of proper lower semicontinuous functions from C into_R

∪ {+∞}

and

{

Bk

}

Mk=1be a family of

µ

k-inverse-strongly monotone mappings of C into H. Let Aibe

η

i-inverse-strongly monotone, respectively, where i

∈ {

1

,

2

, . . . ,

N

}

. Let

{

Sn

}

i∞=1be a family of

δ

i-strict pseudocontractive mappings of C into itself such that F

=

(

∩

Mi=1 GMEP

(

Fi

, ϕ

i

,

Bi

))

∩

F

(

G

)

∩

(

∩

∞i=1F

(

Si

))

̸= ∅

, where G is defined byLemma2.5. Assume that either

(

B1

)

or

(

B2

)

holds. Pick any x0

∈

H and set C1

=

C . Let

{

xn

}

be a sequence generated by x1

=

PC1x0and



























un

=

Tr(FMM,n,ϕM)

(

I

−

rM,nBM

)

T (FM−1,ϕM−1) rM−1,n

(

I

−

rM−1,nBM−1

)

· · ·

T (F1,ϕ1) r1,n

(

I

−

r1,nB1

)

xn

,

yn

=

PC

(

I

−

λ

NAN

)

PC

(

I

−

λ

N−1AN−1

)

· · ·

PC

(

I

−

λ

2A2

)

PC

(

I

−

λ

1A1

)

un

,

zn

=

α

nyn

+

(

1

−

α

n

)

Snyn

,

Cn+1

= {

z

∈

Cn

: ∥

zn

−

z

∥ ≤ ∥

xn

−

z

∥}

,

xn+1

=

PCn+1x0

,

∀

n

≥

1

.

(3.1) Assume that

{

λ

_i

} ⊂

(

0

,

2

η

i

),

i

∈ {

1

,

2

, . . . ,

N

}

,



rk,n



⊂ [

ek

,

fk

] ⊂

(

0

,

2

µ

k

)

, where k

∈ {

1

,

2

, . . . ,

M

}

. Let

δ

=

supi≥1

δ

iand assume

δ

≤

α

_n

≤

g

<

1. Suppose

∞

_n=1supx∈B

∥

Sn+1x

−

Snx

∥

<

∞

for any bounded subset B of C and let S be a mapping of C into itself defined by Sx

=

limn→∞Snx for all x

∈

C . Suppose that F

(

S

)

= ∩

∞n=1F

(

Sn

)

. Then

{

xn

}

converges strongly to PFx0. Proof. It is obvious thatCnis closed for eachn

∈

N. Since

∥

zn

−

z

∥ ≤ ∥

xn

−

z

∥

is equivalent to

∥

zn

−

xn

∥

2

+

2

⟨

zn

−

xn

,

xn

−

z

⟩ ≤

0

,

we have thatCnis convex for eachn

∈

N. Thus

{

xn

}

is well defined. Putting

Θk n

=

Tr(Fk,kn,ϕk)

(

I

−

rk,nBM

)

T (Fk−1,ϕk−1) rk−1,n

(

I

−

rk−1,nBk−1

)

· · ·

T (F1,ϕ1) r1,n

(

I

−

r1,nB1

),

∀

k

∈ {

1

,

2

, . . . ,

M

}

,

n

∈

N

,

and Ωi

=

_P C

(

I

−

λ

iAi

)

PC

(

I

−

λ

i−1Ai−1

)

· · ·

PC

(

I

−

λ

2A2

)

PC

(

I

−

λ

1A1

),

∀

i

∈ {

1

,

2

, . . . ,

N

}

,

Θ0 n

=

Ω

0

=

_{I, whereIis the identity mapping onH. Then we have thatu}

n

=

ΘnMxnandyn

=

ΩNun. FromLemma 2.1

and(2.8), it can be seen easily thatΘk

nandΩiare nonexpansive, wherek

∈ {

1

,

2

, . . . ,

M

}

,

i

∈ {

1

,

2

, . . . ,

N

}

. We divide the

proof into four steps.

Step1 We show by induction thatF

⊂

Cnfor eachn

∈

N. It is obvious thatF

⊂

C1

=

C. Suppose thatF

⊂

Cnfor some n

∈

_N. Letp

∈

F, byLemma 2.1, we have

∥

un

−

p

∥ =





Θ_nMx_n

−

Θ_nMp





≤ ∥

x_n

−

p

∥

.

(3.2)

It follows from(3.2)that

∥

yn

−

p

∥ =





ΩNu_n

−

ΩNp





≤ ∥

un

−

p

∥

≤ ∥

xn

−

p

∥

.

(3.3)

From(2.2)and(3.3), we have

∥

zn

−

p

∥

2

= ∥

α

n

(

yn

−

p

)

+

(

1

−

α

n

)(

Snyn

−

p

)

∥

2

=

α

_n

∥

yn

−

p

∥

2

+

(

1

−

α

n

)

∥

Snyn

−

p

∥

2

−

α

n

(

1

−

α

n

)

∥

yn

−

Snyn

∥

2

≤

α

_n

∥

yn

−

p

∥

2

+

(

1

−

α

n

)(

∥

yn

−

p

∥

2

+

δ

∥

yn

−

Snyn

∥

2

)

−

α

n

(

1

−

α

n

)

∥

yn

−

Snyn

∥

2

= ∥

yn

−

p

∥

2

+

(

1

−

α

n

)(δ

−

α

n

)

∥

yn

−

Snyn

∥

2

≤ ∥

yn

−

p

∥

2

≤ ∥

xn

−

p

∥

2

.

(3.4)

Step2 We prove that limn→∞

∥

xn

−

zn

∥ =

0.

Let

v

=

PFx0. Fromxn

=

PCnx0and

v

∈

F

⊂

Cn, we obtain

∥

xn

−

x0

∥ ≤ ∥

v

−

x0

∥

.

(3.5)

This implies that

{

xn

}

is bounded. Sincexn+1

∈

Cn+1

⊂

Cnandxn

=

PCnx0, we have

∥

xn

−

x0

∥ ≤ ∥

xn+1

−

x0

∥

,

∀

n

∈

N

.

Therefore limn→∞

∥

xn

−

x0

∥

exists. Fromxn

=

PCnx0

,

xn+1

∈

Cn+1

⊂

Cnand(2.6), we obtain

∥

xn+1

−

xn

∥

2

≤ ∥

x0

−

xn+1

∥

2

− ∥

x0

−

xn

∥

2

,

which implies lim

n→∞

∥

xn+1

−

xn

∥ =

0

.

(3.6)

It follows fromxn+1

∈

Cn+1that

∥

zn

−

xn+1

∥ ≤ ∥

xn

−

xn+1

∥

, and hence

∥

xn

−

zn

∥ ≤ ∥

xn

−

xn+1

∥ + ∥

xn+1

−

zn

∥ ≤

2

∥

xn

−

xn+1

∥

.

From(3.6), we have lim

n→∞

∥

xn

−

zn

∥ =

0

.

(3.7)

Step3 We show that limn→∞

∥

xn

−

un

∥ =

0

,

limn→∞

∥

un

−

yn

∥ =

0.

First we prove that lim n→∞





Θ_nkxn

−

Θnk−1xn





=

0

,

k

=

1

,

2

, . . . ,

M

.

(3.8)

Forp

∈

F, it follows from(2.8)that





Θ_nkx_n

−

p





2

=







T (Fk,ϕk) rk,n

(

I

−

rk,nBk

)

Θ k−1 n xn

−

Tr(kF,kn,ϕk)

(

I

−

rk,nBk

)

p







2

≤





(

I

−

rk,nB_k

)

Θ_nk−1x_n

−

(

I

−

rk,nB_k

)

p





2

≤





Θ_nk−1x_n

−

p





2

+

r_k,n

(

r_k,n

−

2

µ

k

)



B

kΘnk−1xn

−

Bkp





2

≤ ∥

x_n

−

p

∥

2

+

r_k,n

(

r_k,n

−

2

µ

_k

)



B

_kΘ_nk−1x_n

−

B_kp





2

.

(3.9) By(3.3)and(3.4), we have

∥

zn

−

p

∥

2

≤ ∥

un

−

p

∥

2

=





Θ_nMx_n

−

p





2

≤





Θ_nkx_n

−

p





2

.

(3.10)

Substituting(3.9)into(3.10), we have

∥

zn

−

p

∥

2

≤ ∥

xn

−

p

∥

2

+

rk,n

(

rk,n

−

2

µ

k

)



B

kΘnk−1xn

−

Bkp





2

,

which implies that

r_k,n

(

2

µ

k

−

rk,n

)



B

kΘnk−1xn

−

Bkp





2

≤ ∥

xn

−

p

∥

2

− ∥

zn

−

p

∥

2

≤ ∥

xn

−

zn

∥

(

∥

xn

−

p

∥ + ∥

zn

−

p

∥

).

Since



r_k,n



⊂ [

ek

,

fk

] ⊂

(

0

,

2

µ

k

)

, wherek

∈ {

1

,

2

, . . . ,

M

}

, and(3.7), we obtain

lim n→∞



B

kΘnk−1xn

−

Bkp





=

0

,

k

=

1

,

2

, . . . ,

M

.

(3.11)

ByLemma 2.1and(2.1), we have





Θ_nkx_n

−

p





2

=







T (Fk,ϕk) rk,n

(

I

−

rk,nBk

)

Θ k−1 n xn

−

Tr(Fk,kn,ϕk)

(

I

−

rk,nBk

)

p







2

≤



(

I

−

r_k,nBk

)

Θnk−1xn

−

(

I

−

rk,nBk

)

p

,

Θnkxn

−

p



=

1 2







(

I

−

rk,nB_k

)

Θ_nk−1x_n

−

(

I

−

rk,nB_k

)

p





2

+





Θ_nkx_n

−

p





2

−





(

I

−

rk,nB_k

)

Θ_nk−1x_n

−

(

I

−

rk,nB_k

)

p

−

(

Θ_nkx_n

−

p

)





2



≤

1 2







Θk −1 n xn

−

p





2

+





Θ_nkxn

−

p





2

−





Θk −1 n xn

−

Θnkxn

−

rk,n

(

BkΘnk−1xn

−

Bkp

)





2



,

which implies





Θ_nkx_n

−

p





2

≤





Θ_nk−1x_n

−

p





2

−





Θ_nk−1x_n

−

Θ_nkx_n

−

rk,n

(

B_kΘ_nk−1x_n

−

B_kp

)





2

=





Θ_nk−1x_n

−

p





2

−





Θ_nk−1x_n

−

Θ_nkx_n





2

−

r_k,n2



B

kΘnk−1xn

−

Bkp





2

+

2r_k,n



Θ_nk−1xn

−

Θnkxn

,

BkΘnk−1xn

−

Bkp



≤





Θk −1 n xn

−

p





2

−





Θk −1 n xn

−

Θnkxn





2

+

2r_k,n





Θk −1 n xn

−

Θnkxn







B

kΘnk−1xn

−

Bkp





≤ ∥

xn

−

p

∥

2

−





Θ_nk−1x_n

−

Θ_nkx_n





2

+

2r_k,n





Θ_nk−1x_n

−

Θ_nkx_n







B

kΘnk−1xn

−

Bkp





.

(3.12)

Substituting(3.12)into(3.10), we obtain

∥

z_n

−

p

∥

2

≤ ∥

x_n

−

p

∥

2

−





Θk −1 n xn

−

Θnkxn





2

+

2r_k,n





Θk −1 n xn

−

Θnkxn







B

_kΘ_nk−1x_n

−

B_kp





,

which implies





Θk −1 n xn

−

Θnkxn





2

≤ ∥

xn

−

p

∥

2

− ∥

zn

−

p

∥

2

+

2rk,n





Θk −1 n xn

−

Θnkxn







B

kΘnk−1xn

−

Bkp





≤ ∥

xn

−

zn

∥

(

∥

xn

−

p

∥ + ∥

zn

−

p

∥

)

+

2rk,n





Θ_nk−1x_n

−

Θ_nkx_n







B

kΘnk−1xn

−

Bkp





.

From(3.7)and(3.11), we have that(3.8)holds. Therefore we get

∥

xn

−

un

∥ =





Θ_n0x_n

−

Θ_nMx_n





≤





Θ_n0xn

−

Θn1xn





+





Θ_n1xn

−

Θn2xn





+ · · · +





ΘM −1 n xn

−

ΘnMxn





→

0 asn

→ ∞

.

(3.13)

Next we show that lim n→∞



A

iΩi−1un

−

AiΩi−1p





=

0

,

i

=

1

,

2

, . . . ,

N

.

(3.14) By(2.8), we have





ΩNu_n

−

ΩNp





2

=



P

C

(

I

−

λ

NAN

)

ΩN−1un

−

PC

(

I

−

λ

NAN

)

ΩN−1p





2

≤





(

I

−

λ

_NA_N

)

ΩN−1u_n

−

(

I

−

λ

_NA_N

)

ΩN−1p





2

≤





ΩN −1 un

−

ΩN −1 p





2

+

λ

_N

(λ

_N

−

2

η

N

)



A

NΩN −1 un

−

ANΩN −1 p





2

.

(3.15) By induction, we get





ΩNu_n

−

ΩNp





2

≤ ∥

un

−

p

∥

2

+

N



i=1

λ

i

(λ

i

−

2

η

i

)



A

iΩi−1un

−

AiΩi−1p





2

≤ ∥

xn

−

p

∥

2

+

N



i=1

λ

i

(λ

i

−

2

η

i

)



A

iΩi−1un

−

AiΩi−1p





2

.

(3.16) By(3.4), we have

∥

zn

−

p

∥

2

≤ ∥

yn

−

p

∥

2

=





ΩNu_n

−

ΩNp





2

.

(3.17)

Substituting(3.16)into(3.17), we have

∥

zn

−

p

∥

2

≤ ∥

xn

−

p

∥

2

+

N



i=1

λ

i

(λ

i

−

2

η

i

)



A

iΩi−1un

−

AiΩi−1p





2

,

which implies N



i=1

λ

i

(

2

η

i

−

λ

i

)



A

iΩi−1un

−

AiΩi−1p





2

≤ ∥

xn

−

p

∥

2

− ∥

zn

−

p

∥

2

≤ ∥

xn

−

zn

∥

(

∥

xn

−

p

∥ + ∥

zn

−

p

∥

).

Since

{

λ

_i

} ⊂

(

0

,

2

η

i

)

,wherei

∈ {

1

,

2

, . . . ,

N

}

and(3.7), we have(3.14)holds. From(2.4)and(2.1), we obtain





ΩNu_n

−

ΩNp





2

=



P

C

(

I

−

λ

NAN

)

ΩN−1un

−

PC

(

I

−

λ

NAN

)

ΩN−1p





2

≤



(

I

−

λ

_NAN

)

ΩN−1un

−

(

I

−

λ

NAN

)

ΩN−1p

,

ΩNun

−

ΩNp



=

1 2







(

I

−

λ

_NA_N

)

ΩN−1u_n

−

(

I

−

λ

_NA_N

)

ΩN−1p





2

+





ΩNu_n

−

ΩNp





2

−





(

I

−

λ

NAN

)

ΩN−1un

−

(

I

−

λ

NAN

)

ΩN−1p

−

(

ΩNun

−

ΩNp

)





2



≤

1 2







ΩN−1u_n

−

ΩN−1p





2

+





ΩNu_n

−

ΩNp





2

−





ΩN −1 un

−

ΩNun

+

ΩNp

−

ΩN −1 p

−

λ

_N

(

ANΩN −1 un

−

ANΩN −1 p

)





2



,

which implies





ΩNu_n

−

ΩNp





2

≤





ΩN−1u_n

−

ΩN−1p





2

−





ΩN−1u_n

−

ΩNu_n

+

ΩNp

−

ΩN−1p

−

λ

_N

(

A_NΩN−1u_n

−

A_NΩN−1p

)





2

=





ΩN −1 un

−

ΩN −1 p





2

−





ΩN −1 un

−

ΩNun

+

ΩNp

−

ΩN −1 p





2

−

λ

2_N



A

NΩN−1un

−

ANΩN−1p





2

+

2

λ

N



ΩN−1 un

−

ΩNun

+

ΩNp

−

ΩN −1 p

,

ANΩN −1 un

−

ANΩN −1 p



≤





ΩN −1_u n

−

ΩN−1p





2

−





ΩN −1_u n

−

ΩNun

+

ΩNp

−

ΩN−1p





2

+

2

λ

N





ΩN −1 un

−

ΩNun

+

ΩNp

−

ΩN −1 p







A

NΩN −1 un

−

ANΩN −1 p





.

(3.18) By induction, we have





ΩNu_n

−

ΩNp





2

≤ ∥

un

−

p

∥

2

−

N



i=1





Ωi−1u_n

−

Ωiu_n

+

Ωip

−

Ωi−1p





2

+

N



i=1 2

λ

i





Ωi−1u_n

−

Ωiu_n

+

Ωip

−

Ωi−1p







A

iΩi−1un

−

AiΩi−1p





≤ ∥

xn

−

p

∥

2

−

N



i=1





Ωi −1_u n

−

Ωiun

+

Ωip

−

Ωi−1p





2

+

N



i=1 2

λ

i





Ωi−1u_n

−

Ωiu_n

+

Ωip

−

Ωi−1p







A

iΩi−1un

−

AiΩi−1p





.

(3.19)

Substituting(3.19)into(3.17), we have

∥

zn

−

p

∥

2

≤ ∥

xn

−

p

∥

2

−

N



i=1





Ωi−1u_n

−

Ωiu_n

+

Ωip

−

Ωi−1p





2

+

N



i=1 2

λ

i





Ωi−1u_n

−

Ωiu_n

+

Ωip

−

Ωi−1p







A

iΩi−1un

−

AiΩi−1p





,

which implies N



i=1





Ωi−1u_n

−

Ωiu_n

+

Ωip

−

Ωi−1p





2

≤ ∥

xn

−

p

∥

2

− ∥

zn

−

p

∥

2

+

N



i=1 2

λ

i





Ωi −1_u n

−

Ωiun

+

Ωip

−

Ωi−1p







A

iΩi−1un

−

AiΩi−1p





≤ ∥

xn

−

zn

∥

(

∥

xn

−

p

∥ + ∥

zn

−

p

∥

)

+

N



i=1 2

λ

i





Ωi−1u_n

−

Ωiu_n

+

Ωip

−

Ωi−1p







A

iΩi−1un

−

AiΩi−1p





.

It follows from(3.7)and(3.14), we have lim n→∞





Ωi−1u_n

−

Ωiu_n

+

Ωip

−

Ωi−1p





=

0

,

i

=

1

,

2

, . . . ,

N

.

(3.20) From(3.20), we obtain

∥

un

−

yn

∥ =





Ω0u_n

−

ΩNu_n





≤

N



i=1





Ωi −1_u n

−

Ωiun

+

Ωip

−

Ωi−1p





→

0 asn

→ ∞

.

(3.21)

From(3.13)and(3.21), we have

∥

xn

−

yn

∥ ≤ ∥

xn

−

un

∥ + ∥

un

−

yn

∥

→

0 asn

→ ∞

.

(3.22) We observe

∥

zn

−

xn

∥ = ∥

α

n

(

yn

−

xn

)

+

(

1

−

α

n

)(

Snyn

−

xn

)

∥

≥

(

1

−

α

_n

)

∥

Snyn

−

xn

∥ −

α

n

∥

yn

−

xn

∥

,

which implies

∥

Snyn

−

xn

∥ ≤

1 1

−

α

_n

(α

n

∥

yn

−

xn

∥ + ∥

zn

−

xn

∥

) .

From(3.7),(3.22)and

α

n

≤

f

<

1, we obtain

lim

n→∞

∥

Snyn

−

xn

∥ =

0

.

(3.23)

Combining(3.22)and(3.23), we have

∥

Snyn

−

yn

∥ ≤ ∥

Snyn

−

xn

∥ + ∥

xn

−

yn

∥

→

0 asn

→ ∞

.

(3.24)

It follows fromLemma 2.3and(3.24)that

∥

Syn

−

yn

∥ ≤ ∥

Syn

−

Snyn

∥ + ∥

Snyn

−

yn

∥

→

0 asn

→ ∞

.

(3.25)

Step4 We show thatxn

→

v

, where

v

=

PFx0.

Since

{

xn

}

is bounded, there exists a subsequence



xni



which converges weakly to

ω

. From(3.8),(3.13)and(3.21), we have thatΘk

nixni

⇀ ω,

uni

⇀ ω,

yni

⇀ ω

, wherek

∈ {

1

,

2

, . . . ,

M

}

. From(3.25)andLemma 2.2, we have that

w

∈

F

(

S

)

=

∩

∞

n=1F

(

Sn

)

. From(3.21),Lemmas 2.4and2.5, we obtain

w

∈

F

(

G

)

. Next we prove that

ω

∈ ∩

Mk=1GMEP

(

Fk

, ϕ

k

,

Bk

)

. Since

Θk nxn

=

Tr(Fk,kn,ϕk)

(

I

−

rk,nBk

)

Θ k−1 n xn

,

n

≥

1

,

k

∈ {

1

,

2

, . . . ,

M

}

, we have Fk

(

Θnkxn

,

y

)

+

ϕ

k

(

y

)

−

ϕ

k

(

Θnkxn

)

+



BkΘnk−1xn

,

y

−

Θnkxn



+

1 r_k,n



y

−

Θ_nkxn

,

Θnkxn

−

Θnk−1xn



≥

0

.

By (A2), we have

ϕ

k

(

y

)

−

ϕ

k

(

Θnkxn

)

+



BkΘk −1 n xn

,

y

−

Θnkxn



+

1 r_k,n



y

−

Θ_nkxn

,

Θnkxn

−

Θk −1 n xn



≥

Fk

(

y

,

Θnkxn

)

Letzt

=

ty

+

(

1

−

t

)ω

for allt

∈

(

0

,

1

]

andy

∈

C. This implies thatzt

∈

C. Then, we have



zt

−

Θnkxn

,

Bkzt



≥

ϕ

_k

(

Θ_nkxn

)

−

ϕ

k

(

zt

)

+



zt

−

Θnkxn

,

Bkzt



−



zt

−

Θnkxn

,

BkΘnk−1xn



−



zt

−

Θnkxn

,

Θ k nxn

−

Θnk−1xn rk,n



+

Fk

(

zt

,

Θnkxn

)

=

ϕ

_k

(

Θ_nkxn

)

−

ϕ

k

(

zt

)

+



zt

−

Θnkxn

,

Bkzt

−

BkΘnkxn



+



zt

−

Θnkxn

,

BkΘnkxn

−

BkΘk −1 n xn



−



zt

−

Θnkxn

,

Θ k nxn

−

Θnk−1xn r_k,n



+

Fk

(

zt

,

Θnkxn

).

By(3.8), we have



B

kΘnkxn

−

BkΘnk−1xn





→

0 asn

→ ∞

. Furthermore, by the monotonicity ofB_k, we obtain

⟨

z_t

−

Θ_nkx_n

,

Bkzt

−

BkΘnkxn

⟩ ≥

0. Then, by (A4) we obtain,

⟨

zt

−

ω,

Bkzt

⟩ ≥

ϕ

k

(ω)

−

ϕ

k

(

zt

)

+

Fk

(

zt

, ω).

(3.26)

Using (A1), (A4) and(3.26), we obtain 0

=

Fk

(

zt

,

zt

)

+

ϕ

k

(

zt

)

−

ϕ

k

(

zt

)

≤

tFk

(

zt

,

y

)

+

(

1

−

t

)

Fk

(

zt

, ω)

+

t

ϕ

k

(

y

)

+

(

1

−

t

)ϕ

k

(ω)

−

ϕ

k

(

zt

)

≤

t

[

Fk

(

zt

,

y

)

+

ϕ

k

(

y

)

−

ϕ

k

(

zt

)

] +

(

1

−

t

)

⟨

zt

−

ω,

Bkzt

⟩

=

t

[

Fk

(

zt

,

y

)

+

ϕ

k

(

y

)

−

ϕ

k

(

zt

)

] +

(

1

−

t

)

t

⟨

y

−

ω,

Bkzt

⟩

,

and hence 0

≤

Fk

(

zt

,

y

)

+

ϕ

k

(

y

)

−

ϕ

k

(

zt

)

+

(

1

−

t

)

⟨

y

−

ω,

Bkzt

⟩

.

Lettingt

→

0, we have, for eachy

∈

C,

0

≤

Fk

(ω,

y

)

+

ϕ

k

(

y

)

−

ϕ

k

(ω)

+ ⟨

y

−

ω,

Bk

ω

⟩

.

This implies that

ω

∈

GMEP

(

Fk

, ϕ

k

,

Bk

)

. Hence

ω

∈ ∩

Mk=1GMEP

(

Fk

, ϕ

k

,

Bk

)

. Therefore, z

∈

F

=

(

∩

M_i=1GMEP

(

Fi

, ϕ

i

,

Bi

))

∩

(

F

(

G

))

∩

(

∩

∞ i=1F

(

Si

)).

Since

ω

∈

F

, v

=

PFx0,(3.5)and the lower semicontinuousness of the norm, we obtain

∥

v

−

x0

∥ ≤ ∥

ω

−

x0

∥ ≤

lim inf i→∞



x

ni

−

x0





≤

lim sup i→∞



x

ni

−

x0





≤ ∥

v

−

x₀

∥

.

Thus, we obtain

ω

=

v

and limi→∞



x

ni

−

x0





= ∥

ω

−

x₀

∥

. Fromx_ni

−

x₀

⇀ ω

−

x₀and the Kadec–Klee property ofH, we

havexni

−

x0

→

ω

−

x0, and hencexni

→

ω

. This implies thatxn

→

ω

=

v

. This completes the proof.

References

[1] G.L. Acedo, H.K. Xu, Iterative methods for strict pseudo-contractions in Hilbert spaces, Nonlinear Anal. 67 (2007) 2258–2271.

[2] Y.J. Cho, X. Qin, J.I. Kang, Convergence theorems based on hybrid methods for generalized equilibrium problems and fixed point problems, Nonlinear Anal. 71 (2009) 4203–4214.

[3] Y. Liu, A general iterative method for equilibrium problems and strict pseudo-contractions in Hilbert spaces, Nonlinear Anal. 71 (2009) 4852–4861. [4] H. Zhou, Convergence theorems of fixed points fork-strict pseudo-contractions in Hilbert spaces, Nonlinear Anal. 69 (2008) 456–462.

[5] L.-C. Ceng, et al., Viscosity approximation methods for strongly positive and monotone operators, Fixed Point Theory 10 (2009) 35–72. [6] R.T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Optim. 14 (1976) 877–898.

[7] M. Liu, S. Chang, P. Zuo, On a hybrid method for generalized mixed equilibrium problem and fixed point problem of a family of quasi-φ-asymptotically nonexpansive mappings in Banach spaces, Fixed Point Theory and Applications, 2010, Article ID 157278, 18 pages.doi:10.1155/2010/157278. [8] S. Zhang, Generalized mixed equilibrium problem in Banach spaces, Appl. Math. Mech. (English Ed.) 30 (2009) 1105–1112.

[9] X. Qin, Y.J. Cho, S.M. Kang, Viscosity approximation methods for generalized equilibrium problems and fixed point problems with applications, Nonlinear Anal. 72 (2010) 99–112.

[10] Y. Shehu, Fixed point solutions of generalized equilibrium problems for nonexpansive mappings, J. Comput. Appl. Math. 234 (2010) 892–898. [11] S. Takahashi, W. Takahashi, Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space,

Nonlinear Anal. 69 (2008) 1025–1033.

[12] J.-W. Peng, J.-C. Yao, Strong convergence theorems of iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems, Math. Comput. Modelling 49 (2009) 1816–1828.

[13] S. Plubtieng, K. Sombut, Weak convergence theorems for a system of mixed equilibrium problems and nonspreading mappings in a Hilbert space, Journal of Inequalities and Applications, 2010, Article ID 246237, 12 pages.doi:10.1155/2010/246237.

[14] Y. Yao, Y.-C. Liou, J.-C. Yao, A new hybrid iterative algorithm for fixed-point problems, variational inequality problems, and mixed equilibrium problems, Fixed Point Theory and Applications, 2008, Article ID 417089, 15 pages.doi:10.1155/2008/417089.

[15] S. Plubtieng, R. Punpaeng, A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings, Appl. Math. Comput. 197 (2008) 548–558.

[16] X. Qin, M. Shang, Y. Su, Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems, Math. Comput. Modelling 48 (2008) 1033–1046.

[17] Y. Su, M. Shang, X. Qin, An iterative method of solution for equilibrium and optimization problems, Nonlinear Anal. 69 (2008) 2709–2719. [18] L.-C. Ceng, J.-C. Yao, A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, J. Comput. Appl. Math. 214 (2008) 186–201.

[19] P. Kumam, A new hybrid iterative method for solution of equilibrium problems and fixed point problems for an inverse strongly monotone operator and a nonexpansive mapping, J. Appl. Math. Comput. 29 (2009) 263–280.

[20] X. Qin, Y.J. Cho, S.M. Kang, Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces, J. Comput. Appl. Math. 225 (2009) 20–30.

[21] J.-W. Peng, J.-C. Yao, A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems, Taiwanese J. Math. 12 (2008) 1401–1432.

[22] Y. Shehu, Strong convergence theorems for an infinite family of quasi-nonexpansive mappings and generalized equilibrium problems and variational inequality problems, Comput. Math. Appl. 61 (2011) 357–366.

[23] Y. Yao, et al., Mixed equilibrium problems and optimization problems, J. Math. Anal. Appl. 354 (2009) 319–329.

[24] L.-C. Ceng, J.-C. Yao, Convergence analysis of a hybrid Mann iterative scheme with perturbed mapping for variational inequalities and fixed point problems, Optimization 59 (2010) 929–944.

[25] L.-C. Ceng, et al., Hybrid shrinking projection method for a generalized equilibrium problem, a maximal monotone operator and a countable family of relatively nonexpansive mappings, Comput. Math. Appl. 61 (2011) 2468–2479.

[26] Y. Yao, et al., Some new algorithms for solving mixed equilibrium problems, Comput. Math. Appl. 60 (2010) 1351–1359.

[27] L.C. Ceng, C. Wang, J.C. Yao, Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities, Math. Methods Oper. Res. 67 (2008) 375–390.

[28] H. Iiduka, A new iterative algorithm for the variational inequality problem over the fixed point set of a firmly nonexpansive mapping, Optimization 59 (2010) 873–885.

[29] Y.J. Cho, S.M. Kang, X. Qin, Strong convergence of an implicit iterative process for an infinite family of strict pseudocontractions, Bull. Korean Math. Soc. 47 (2010) 1259–1268.

[30] L.-C. Ceng, S.M. Guu, J.-C. Yao, Hybrid viscosity-like approximation methods for nonexpansive mappings in Hilbert spaces, Comput. Math. Appl. 58 (2009) 605–617.

[31] G. Marino, H.-K. Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 329 (2007) 336–346. [32] K. Aoyama, et al., Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Anal. 67

(2007) 2350–2360.