RESEA R C H Open Access
Some extragradient methods for common
solutions of generalized equilibrium problems
and fixed points of nonexpansive mappings
Jian-Wen Peng
Correspondence: jwpeng6@yahoo.
com.cn
School of Mathematics, Chongqing
Normal University, Chongqing
400047, PR China
Abstract
In this article, we introduce some new iterative schemes based on the extragradient
method (and the hybrid method) for find ing a common element of the set of
solutions of a generalized equilibrium problem, and the set of fixed points of a
family of infinitely nonexpansive mappings and the set of solutions of the variational
inequality for a monotone, Lipschitz-continuous mapping in Hilbert spaces. We
obtain some strong convergence theorems and weak convergence theorems. The
results in this article generalize, improve, and unify some well-known convergence
theorems in the literature.
Keywords: Generalized equilibrium problem, Extragra dient method, Hybrid method,
Nonex-pansive mapping, Strong convergence, Weak convergence
1. Introduction
Let H be a real Hilbert space with inner product 〈.,.〉 and induced norm ||·||. Let C be
a nonempty closed convex subset of H.LetF beabifunctionfromC×Cto R and let
B : C ® H be a nonlinear mapping, where R is the set of real numbers. Moudafi [1],
Moudafi and Thera [2], Peng and Yao [3,4], Takahashi and Takahashi [5] considered
the following generalized equilibrium problem:
Find x ∈ C Such that F
(
x, y
)

provide d the original work is properly cited.
Let the mapping A : C ® H be monotone and k-Lipschitz-continuous. The varia-
tional inequality problem is to find x Î C such that
Ax,
y
− x≥
0
for all y Î C. The set of solutions of the variational inequality problem is denoted by
VI(C, A).
Several algorithms have been proposed for finding the solution of problem (1.1). Mou-
dafi [1] introduced an iterative scheme for finding a common element of the set of solu-
tions of problem (1.1) and the set of fixed points of a nonexpansive mapping in a Hilbert
space, and proved a weak convergence theorem. Moudafi and Thera [2] introduced an
auxiliary scheme for finding a solution of problem (1.1) in a Hilbert space and obtained a
weak convergence theorem. Peng and Yao [3,4] introduced some iterative schemes for
finding a common element of the set of solutions of problem (1.1), the set of fixed points
of a nonexpansive mapping and the set of solutions of the variational inequality for a
monotone, Lipschitz-continuous mapping and obtain both strong convergence theorems,
and weak convergence theorems for the sequences generated by the corresponding pro-
cesses in Hilbert spaces. Takahashi and Takahashi [5] introduced an iterative scheme for
finding a common element of the set of solutions of problem (1.1) and the set of fixed
points of a nonexpansive mapping in a Hilbert space, and proved a strong convergence
theorem.
Some methods also have been proposed to solve the problem (1.2); see, for instance,
[8-19] and the references therein. Takahashi and Takahashi [9] introduced an iterative
scheme by the viscosity approximation m ethod for fin ding a common element of the
set of solutions of problem (1.2) and the set of fixed points of a non-expansive map-
ping, and proved a strong convergence theorem in a Hilbert space. Su et al. [10] intro-
duced and researched an iterative scheme by t he viscosity approximation method for
finding a co mmon element of the set of solutions of problem (1.2) and the set of fixed

mappings and the set of solutions of a variational inequality without the W-mapping
gene rated by a family of infinitely (finitely) nonexpansive mapping s? In this article, we
will give a positive answer to this question.
Recently, OHaraa et al. [22] introduced and researched an iterative approach for
finding a nearest point of infinitely many nonexpansive mappings in a Hilbert spaces
without using the W-mapping generated by a family of infinitely (finitely) nonexpansive
mappings. Inspired by the ideas in [1-6,8-16,22] and the references therein, we intro-
duce some new iterative schemes based on the extragradient method (and the hybrid
method) for finding a common element of the set of solutions of a generalized equili-
brium problem, the set of fixed points of a family of infinitely nonexpansive mappings,
and the set of solutions of the variational inequality for a monotone, Lipschitz–contin-
uous mapping without using the W-mapping generated by a family of infinitely
(finitely) nonexpansive mappings. We obtain both strong convergence theorems and
weak convergence theorems for the sequences generated by the corresponding pro-
cesses. The results in this article generalize, improve, and unify some well-known con-
vergence theorems in the literature.
2. Preliminaries
Let H be a real Hilbert space with inner product 〈·,·〉 and norm ||·||. Let C be a none-
mpty closed convex subset of H. Let symbols ® and ⇀ denote strong and weak con-
vergences, respectively. In a real Hilbert space H, it is well known that


λx +(1− λ)y


2
= λ

x


x − P
C
(
x
)
, P
C
(
x
)
− y≥0
(2:1)
for all x Î H and y Î C.
It is easy to see that (2.1) is equivalent to


x − y


2
≥


x − P
C
(x)


2
+



Ax − Ay


≤ k


x − y


for all x, y Î C. It is easy to see that if A is a-inverse-strongly monotone, then A is
monotone and Lipschitz-con tinuous. The conver se is not true in general. The class of
a-inverse-strongly monotone mappings does not contain some important classes o f
mappings even in a finite-dimensional case. For example, if the matrix in the corre-
sponding linear compl ementarity problem is positively semidefinite, but not positi vely
definite, then the mapping A will be monotone and Lipschitz-continuous, but not a-
inverse-strongly monotone (see [23]).
Let A be a monotone mapping of C into H. In the context of the variational inequal-
ity problem, the characterization of projection (2.1) implies the following:
u
∈ VI
(
C, A
)
⇒ u = P
C
(
u − λAu
)



x
n
− y


holds for every y Î H with x ≠ y.
A set-valued mapping T : H ® 2
H
is called monotone if for all x, y Î H, f Î Tx and g Î
Ty imply 〈x-y, f-g〉 ≥ 0. A monotone mapping T : H ® 2
H
is maximal if its graph G(T)
of T is not properly contained in the graph of any other monotone mapping. It is known
that a monotone mapping T is maximal if and only if for (x, f) Î H × H, 〈x-y, f-g〉 ≥ 0
for every (y, g) Î G(T) implies f Î Tx. Let A be a monotone, k-Li pschitz-continuous
mapping of C into H and N
C
v be normal cone to C at v Î C, i.e., N
C
v ={w Î H : 〈v-u,
w〉 ≥ 0, ∀u Î C}. Define
Tv =

Av + N
C
v if v ∈ C
,
∅ if v /∈ C

1
r
y − z, z − x≥0, ∀y ∈ C
}
for all x Î H. Then, the following statements hold:
(1) T
r
is single-valued;
(2) T
r
is firmly nonexpansive, i.e., for any x, y Î H,


T
r
(x) − T
r
(y)


2
≤T
r
(x) − T
r
(y), x − y
;
(3) F(T
r
)=EP (F);

x
∈
K
||S
n
x − S
i
(S
n
x)|| =0. (
)
Let {x
n
}, {u
n
}, {y
n
} and {z
n
} be sequences generated by
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪

− x
n
≥0, ∀y ∈ C,
y
n
=(1− γ
n
)u
n
+ γ
n
P
C
(u
n
− λ
n
Au
n
),
z
n
=(1− α
n
− β
n
)x
n
+ α
n

2
||Au
n
||
2
}
,
Q
n
= {z ∈ C : x
n
− z, x − x
n
≥0},
x
n+1
= P
C
n

Q
n
x
(3:1)
Peng Fixed Point Theory and Applications 2011, 2011:12
/>Page 5 of 19
for every n = 1, 2, where {l
n
} ⊂ [a, b] for some
a, b ∈ (0,

n
→∞
β
n
>
0
;
(iv)
lim
n
→
∞
γ
n
=
1
and
γ
n
>
3
4
for all n Î N;
Then, {x
n
}, {u
n
}, {y
n
} and {z

+ α
n
)
b
2

Au
n

2
}
,
we also have that C
n
is convex for every n = 1, 2, It is easy to see that 〈x
n
-z, x-
x
n
〉 ≥ 0forallz Î Q
n
and by (2.1),
x
n
= P
Q
n
x
.Lett
n

n
= T
r
n
(x
n
− r
n
Bx
n
) ∈
C
and the a-
inverse strongly monotonicity of B, we have

u
n
− u

2
=


T
r
n
(x
n
− r
n

− u

2
− 2r
n
x
n
− u, Bx
n
− Bu + r
2
n

Bx
n
− Bu

2
≤

x
n
− u

2
− 2r
n
α

Bx

2
≤

x
n
− u

2
.
(3:2)
From (2.2), the monotonicity of A, and u Î VI(C, A), we have

t
n
− u

2
≤


u
n
− λ
n
Ay
n
− u


2

n
Ay
n
, u − t
n

=

u
n
− u

2
−

u
n
− t
n

2
+2λ
n
(Ay
n
− Au, u − y
n
 + Au, u − y
n
 + Ay


≤

u
n
− u

2
−


u
n
− y
n


2
− 2u
n
− y
n
, y
n
− t
n
−


y


2
−


y
n
− t
n


2
+2u
n
− λ
n
Ay
n
− y
n
, t
n
− y
n
.
Further, Since y
n
=(1- g
n
)u

n
Au
n
− y
n
, t
n
− y
n
 + λ
n
Au
n
− λ
n
Ay
n
, t
n
− y
n

≤u
n
− λ
n
Au
n
− (1 − γ
n

− y
n


≤ γ
n
u
n
− λ
n
Au
n
− P
C
(u
n
− λ
n
Au
n
), t
n
− y
n
−(1 − γ
n
)λ
n
Au
n

n
− λ
n
Au
n
− P
C
(u
n
− λ
n
Au
n
), t
n
− y
n

= u
n
− λ
n
Au
n
− P
C
(u
n
− λ
n

(u
n
− λ
n
Au
n
), t
n
− u
n

+γ
n
u
n
− λ
n
Au
n
− P
C
(u
n
− λ
n
Au
n
), t
n
− P


t
n
− u
n

≤ (1 − γ
n
)λ
n

u
n
− Au
n
− u
n

(


t
n
− y
n


+



n


).
It follows from
b <
1
4
k
,
γ
n
>
3
4
and (3.2) that

t
n
− u

2
≤

u
n
− u

2
−


t
n
− y
n


+


y
n
− u
n


)
+2(1 − γ
n
)b

Au
n



t
n
− y
n

n
− y
n


2
−


y
n
− t
n


2
+(1− γ
n
)(2b
2

Au
n

2
+


t
n

− y
n


2
)+bk(


u
n
− y
n


2
+


t
n
− y
n


2
)
=

u
n

2

Au
n

2
≤

u
n
− u

2
+3(1− γ
n
)b
2

Au
n

2
≤

x
n
− u

2
+3

n
(P
C
(u
n
− λ
n
Au
n
) − u)


2
≤ (1 − γ
n
)

u
n
− u

2
+ γ
n


P
C
(u
n

2
≤ (1 − γ
n
)

u
n
− u

2
+ γ
n
[

u
n
− u

2
− 2λ
n
Au
n
, u
n
− u + λ
2
n

Au


2
.
(3:4)
Therefore, from (3.2) to (3.4) and z
n
=(1- a
n
- b
n
)x
n
+ a
n
y
n
+ b
n
S
n
t
n
and u = S
n
u,
we have

z
n
− u


x
n
− u

2
+ α
n


y
n
− u


2
+ β
n

S
n
t
n
− u

2
≤ (1 − α
n
− β
n


x
n
− u

2
+ α
n
[

u
n
− u

2
+ b
2

Au
n

2
]
+β
n
[

u
n
− u

n

2
],
(3:5)
for every n = 1, 2, and hence u Î C
n
.So,Ω ⊂ C
n
for every n = 1, 2, Next, let us
show by mathematical induction that x
n
is well defined and Ω ⊂ C
n
∩ Q
n
for every n
= 1, 2, For n =1wehavex
1
= x Î C and Q
1
= C. Hence, we obtain Ω ⊂C
1
∩ Q
1
.
Suppose that x
k
is given and Ω ⊂ C
k

∩ Q
k
.SinceΩ ⊂ C
k
∩ Q
k
,wehave〈x
k+1
- z, x - x
k+1
〉 ≥ 0for
every z Î Ω and hence Ω ⊂ Q
k+1
. Therefore, we obtain Ω ⊂ C
k+1
∩ Q
k+1
.
Peng Fixed Point Theory and Applications 2011, 2011:12
/>Page 7 of 19
Let l
0
= P
Ω
x. From
x
n+1
= P
C
n

}, {Au
n
}, {t
n
}and{z
n
} are bounded. Since
x
n+1
Î C
n
∩ Q
n
⊂ C
n
and
x
n
= P
Q
n
x
, we have

x
n
− x

≤


x
n+1
− x

2
−

x
n
− x

2
for every n = 1, 2, This implies that
lim
n
→∞

x
n+1
− x
n

=0
.
Since x
n+1
Î C
n
,wehave||z
n

n®∞
||z
n
- x
n+1
|| = 0. Since
||
x
n
− z
n
||
≤
||
x
n
− x
n+1
||
+
||
x
n+1
− z
n
||
for every n = 1, 2, , we have ||x
n
-z
n

− u||
2
≤
(
3 − 3γ
n
+ α
n
)
b
2
||Au
n
||
2
.
Since lim
n®∞
g
n
= 1 and lim
n®∞
a
n
=0,{x
n
}, {y
n
}, {Au
n

||S
n
t
n
− u||
2
−||x
n
− u||
2
=0
.
From (3.3) and u = S
n
u, we have
lim
n→∞
||S
n
t
n
− u||
2
−||x
n
− u||
2
≤ lim
n→∞
||t

=0.
From (3.3) and (3.2), we have
(γ
n
− bk)||u
n
− y
n
||
2
+(2γ
n
− 1 − bk)||t
n
− y
n
||
2
≤||x
n
− u||
2
−||t
n
− u||
2
+3
(
1 − γ
n

Peng Fixed Point Theory and Applications 2011, 2011:12
/>Page 8 of 19
The assumptions on g
n
and l
n
imply that
γ
n
− bk >
1
2
and
2γ
n
− 1 − bk >
1
4
. Conse-
quently, lim
n®∞
||u
n
-y
n
|| = lim
n®∞
||t
n
-y

as
z
n
− x
n
= α
n
(
y
n
− x
n
)
+ β
n
(
S
n
t
n
− x
n
).
From lim
n®∞
||z
n
-x
n
|| = 0, lim

)||x
n
− u||
2
+ α
n
[||u
n
− u||
2
+ b
2
||Au
n
||
2
]+β
n
[||u
n
− u||
2
+3(1− γ
n
)b
2
||Au
n
||
2

+ β
n
[||x
n
− u||
2
+ r
n
(r
n
− 2α)||Bx
n
− Bu||
2
+3(1− γ
n
)b
2
||Au
n
||
2
]
≤||x
n
− u||
2
+
(
α

||
2
].
(3:7)
Hence, we have
(α
n
+ β
n
)d(2α − e)||Bx
n
− Bu||
2
≤ (α
n
+ β
n
)r
n
(2α − r
n
)||Bx
n
− Bu||
2
≤||x
n
− u||
2
−||z

(
3β
n
− 3β
n
γ
n
+ α
n
)
b
2
||Au
n
||
2
.
Since lim
n®∞
a
n
= 1, lim inf
n®∞
b
n
> 0, lim
n®∞
g
n
=1,||x

(u − r
n
Bu)||
2
≤T
r
n
(x
n
− r
n
Bx
n
) − T
r
n
(u − r
n
Bu), x
n
− r
n
Bx
n
− (u − r
n
Bu)
=
1
2

{||u
n
− u||
2
+ ||x
n
− u||
2
−||x
n
− r
n
Bx
n
− (u − r
n
Bu) − (u
n
− u)||
2
}
=
1
2
{||u
n
− u||
2
+ ||x
n

n
− u||
2
−||x
n
− u
n
||
2
+2r
n
Bx
n
− Bu, x
n
− u
n
−r
2
n
||Bx
n
− Bu||
2
≤
||
x
n
− u
||

n
− u||
2
+ α
n
[||u
n
− u||
2
+ b
2
||Au
n
||
2
]+β
n
[||u
n
− u||
2
+3(1− γ
n
)b
2
||Au
n
||
2
]

||Au
n
||
2
]
+ β
n
[(||x
n
− u||
2
−||x
n
− u
n
||
2
+2r
n
Bx
n
− Bu, x
n
− u
n
)+3(1− γ
n
)b
2
||Au

n
− Bu|| ||x
n
− u
n
|| +
(
3β
n
− 3β
n
γ
n
+ α
n
)
b
2
||Au
n
||
2
Hence,
(α
n
+ β
n
)||x
n
− u

n
)b
2
||Au
n
||
2
≤
(
||x
n
− u|| + ||z
n
− u||
)
||x
n
− z
n
|| +2r
n
(
α
n
+ β
n
)
||Bx
n
− Bu|| ||x

n®∞
g
n
=1,||x
n
- z
n
|| ® 0, ||Bx
n
- Bu|| ® 0
and the sequences {x
n
}, {u
n
} and {z
n
} are bounded, we obtain ||x
n
- u
n
|| ® 0. From ||z
n
-
t
n
|| ≤ ||z
n
-x
n
||+||x

|| ® 0.
Since z
n
=(1- a
n
- b
n
)x
n
+ a
n
y
n
+ b
n
S
n
t
n
,wehaveb
n
(S
n
t
n
-t
n
)=(1- a
n
- b

n
)
||t
n
− x
n
|| + α
n
||t
n
− y
n
|| + ||z
n
− t
n
|
|
and hence ||S
n
t
n
-t
n
|| ® 0. At the same time, observe that for all i Î {1, 2, },
|
|S
i
t
n

≤ 2||S
n
t
n
− t
n
|| +sup
x∈
K
||S
i
(S
n
x) − S
n
x||.
It follows from (3.8) and the condition (*) that for all i Î {1, 2, },
lim
n
→∞
||S
i
t
n
− t
n
|| =0
.
(3:9)
As {x

n
= T
r
n
(x
n
− r
n
Bx
n
) ∈
C
, we know that
F(u
n
, y)+Bx
n
, y − u
n
 +
1
r
n
y − u
n
, u
n
− x
n
≥0, ∀y ∈ C

,
u
n
i
− x
n
i
r
n
i
≥F(y, u
n
i
), ∀y ∈ C
.
(3:10)
For t with 0 <t≤ 1andy Î C,lety
t
= t
y
+(1-t)w.Sincey Î C and w Î C,we
obtain y
t
Î C. So, from (3.10) we have

y
t
− u
n
i

u
n
i
− x
n
i
r
n
i
 + F(y
t
, u
n
i
)
= y
t
− u
n
i
, By
t
− Bu
n
i
 + y
t
− u
n
i

u
n
i
− x
n
i
||
→
0
, we have
||
Bu
n
i
− Bx
n
i
||
→
0
. Further, from the inverse-strongly
monotonicity of B,wehave
y
t
− u
n
i
, By
t
− Bu

t
, w
),
(3:11)
as i ® ∞. From (A1), (A4) and (3.11), we also have
0=F(y
t
, y
t
) ≤ tF(y
t
, y)+(1− t)F(y
t
, w
)
≤ tF(y
t
, y)+(1− t)y
t
− w, By
t

= tF
(
y
t
, y
)
+
(

We next show that
w ∈∩
∞
i
=1
Fix(S
i
)
. Assume
w /∈∩
∞
i
=1
Fix(S
i
)
.Since
t
n
i

w
and
w = S
i
0
w
for some i
0
Î {1, 2, } from the Opial condition, we have

i
− S
i
0
w||
}
≤ lim inf
i
→∞
||t
n
i
− w||.
This is a contradiction. Hence, we get
w ∈∩
∞
i
=1
Fix(S
i
)
.
Finally we show w Î VI(C, A). Let
Tv =

Av + N
C
v if v ∈ C
,
∅ if v ∈ C

n
, t
n
− v≥
0
and hence
v − t
n
,
t
n
− u
n
λ
n
+ Ay
n
≥0
.
Therefore, we have
v − t
n
i
, g≥v − t
n
i
,
A
v
≥v − t

n
i
λ
n
i

= v − t
n
i
, Av − At
n
i
+ At
n
i
− Ay
n
i
−
t
n
i
− u
n
i
λ
n
i

= v − t

i
, At
n
i
− Ay
n
i
−v − t
n
i
,
t
n
i
− u
n
i
λ
n
i

Hence, we obtain 〈v-w, g〉 ≥ 0asi ® ∞.SinceT is maximal monotone, we have w
Î T
-1
0 and hence w Î VI(C, A). This implies that w Î Ω.
Peng Fixed Point Theory and Applications 2011, 2011:12
/>Page 11 of 19
From l
0
= P

From
x
n
i
− x  w −
x
,wehave
x
n
i
− x → w −
x
, and hence
x
n
i
→
w
. Since
x
n
= P
Q
n
x
and l
0
Î Ω ⊂ C
n
∩ Q

i
, x − l
0

.
As i ® ∞, we obtain - ||l
0
-w||
2
≥ 〈l
0
-w, x-l
0
〉 ≥ 0byl
0
= P
Ω
x and w Î Ω. Hence,
we have w = l
0
. This implies that x
n
® l
0
.Itiseasytoseeu
n
® l
0
, y
n

i
) ∩ VI(C, A) ∩ GEP(F, B) =
∅
. Assume that for all i Î
{1, 2, } and for any bounded subset K of C, thenthere holds
lim
n→∞
sup
x
∈
K
||S
n
x − S
i
(S
n
x)|| =0. (
)
Let {x
n
}, {u
n
} and {y
n
} be the sequences generated by
⎧
⎪
⎪
⎪

= P
C
(u
n
− λ
n
Au
n
),
x
n+1
= β
n
x
n
+(1− β
n
)S
n
P
C
(u
n
− λ
n
Ay
n
)
(3:12)
for every n =1,2, If{l

Let F be a bifunction from C×C to R satisfying (A1)-(A4). Let B be an a-inverse-
strongly monotone mapping of C into H.LetS
1
, S
2
, be a family of infinitely
Peng Fixed Point Theory and Applications 2011, 2011:12
/>Page 12 of 19
nonexpansive mappings of C into itself such that

= ∩
∞
i=1
Fix(S
i
) ∩ GEP(F, B) =
∅
.
Assume that for all i Î {1, 2, } and for any bounded subset K of C, thenthere holds
lim
n→∞
sup
x
∈
K
||S
n
x − S
i
(S

⎩
x
1
= x ∈ C,
F(u
n
, y)+Bx
n
, y − u
n
 +
1
r
n
y − u
n
, u
n
− x
n
≥0, ∀y ∈ C
,
z
n
=(1− α
n
− β
n
)x
n

n+1
= P
C
n
∩Q
n
x
for every n = 1, 2, where {r
n
} ⊂ [d, e]forsomed, e Î (0, 2a), and {a
n
}, {b
n
}are
sequences in [0, 1] satisfying the conditions:
(i) a
n
+ b
n
≤ 1 for all n Î N;
(ii)
lim
n
→
∞
α
n
=0
;
(iii)

i
) ∩ GEP(F, B) =
∅
.Assume
that for all i Î {1, 2, } and for any bounded subset K of C, thenthere holds
lim
n→∞
sup
x
∈
K
||S
n
x − S
i
(S
n
x)|| =0. (
)
Let {x
n
} and {u
n
} be sequences generated by
⎧
⎪
⎪
⎨
⎪
⎪

)S
n
u
n
for every n = 1, 2, If {b
n
} ⊂ [δ, ε]forsomeδ, ε Î (0, 1) and {r
n
} ⊂ [d, e]forsome
d, e Î (0, 2a). Then, {x
n
} and {u
n
} converge weakly to w Î ∑, where w = lim
n®∞
P
∑
x
n
.
Theorem 4.3.LetC be a nonempty closed convex subset of a real Hilbert space H.
Let F be a bifunction from C×C to R satisfying (A1)-(A4). Let A be a monotone and k-
Lipschitz-continuous mapping of C into H and B be an a-inverse-strongly monotone
mapping of C into H. Let S
1
, S
2
, be a family of infinitely nonexpansive mappings of C
into itself such that


n
}, and {z
n
} be sequences generated by
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
x
1
= x ∈ C,
F(u
n
, y)+Bx
n

n
+ β
n
S
n
P
C
(u
n
− λ
n
Ay
n
),
C
n
= {z ∈ C : ||z
n
− z||
2
≤||x
n
− z||
2
,
Q
n
= {z ∈ C : x
n
− z, x − x

>
0
.Then,{x
n
},
{u
n
}, {y
n
}, and {z
n
} converge strongly to w = P
Ω
(x).
Proof. Putting g
n
= 1 and a
n
= 0, by Theorem 3.1, we obtain the desired result.
Let B = 0, by Theorems 3.1, 3.2, and 4.3, we obtain the following results.
Theorem 4.4.LetC be a nonempty closed convex subset of a real Hilbert space H.
Let F be a bifunction from C×C to R satisfying (A1)-(A4). Let A be a monotone and k-
Lipschitz-continuous mapping of C into H.LetS
1
, S
2
, be a family of infinitely nonex-
pansive mappings of C into itself such that
 = ∩
∞

} be the sequences generated by
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
x
1
= x ∈ C,
F(u
n
, y)+
1
r
n
y − u

n
+ α
n
y
n
+ β
n
S
n
P
C
(u
n
− λ
n
Ay
n
),
C
n
= {z ∈ C : ||z
n
− z||
2
≤||x
n
− z||
2
+(3− 3γ
n

1
4
k
)
,{r
n
} ⊂ [d, +∞)for
some d>0, and {a
n
}, {b
n
}, {g
n
} are three sequences in [0, 1] satisfying the following
conditions:
(i) a
n
+ b
n
≤ 1 for all n Î N;
(ii)
lim
n
→∞
α
n
=0
;
(iii)
lim inf

} converge strongly to w = P
Λ
(x).
Theorem 4.5.LetC be a nonempty closed convex subset of a real Hilbert space H.
Let F be a bifunction from C×C to R satisfying (A1)-(A4). Let A be a monotone and k-
Lipschitz-continuous mapping of C into H.LetS
1
, S
2
, be a family of infinitely nonex-
pansive mappings of C into itself such that
 = ∩
∞
i
=1
Fix(S
i
) ∩ VI(C, A) ∩ EP(F) =
∅
.
Assume that for all i Î {1, 2, } and for any bounded subset K of C, thenthere holds
Peng Fixed Point Theory and Applications 2011, 2011:12
/>Page 14 of 19
lim
n→∞
sup
x
∈
K
||S

, y)+
1
r
n
y − u
n
, u
n
− x
n
≥0, ∀y ∈ C
,
y
n
= P
C
(u
n
− λ
n
Au
n
),
x
n+1
= β
n
x
n
+(1− β

} converge weakly to
w Î Λ, where w = lim
n®∞
P
Λ
x
n
.
Theorem 4.6.LetC be a nonempty closed convex subset of a real Hilbert space H.
Let F be a bifunction from C×Cto R satisfying (A1)-(A4). Let A be a monotone and
k-Lipschitz-continuous mapping of C into H. Let S
1
, S
2
, be a family of infinitely non-
expansive mappings of C into itself such that
 = ∩
∞
i=1
Fix(S
i
) ∩ VI(C, A) ∩ EP(F) =
∅
.
Assume that for all i Î {1, 2, } and for any bounded subset K of C, thenthere holds
lim
n→∞
sup
x
∈

⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
x
1
= x ∈ C,
F(u
n
, y)+
1
r
n
y − u
n
, u
n
− x
n
≥0, ∀y ∈ C
,
y
n

= {z ∈ C : ||z
n
− z||
2
≤||x
n
− z||
2
,
Q
n
= {z ∈ C : x
n
− z, x − x
n
≥0},
x
n+1
= P
C
n
∩Q
n
x
for every n = 1, 2, where {l
n
} ⊂ [a, b]forsome
a, b ∈ (0,
1
4

Let A be a monotone and k-Lipschitz-continuous mapping of C into H.LetS
1
, S
2
, be
a family of infinitely nonexpansive mappings of C into itself such that
 = ∩
∞
i
=1
Fix(S
i
) ∩ VI(C, A) =
∅
. Assume that for all i Î {1, 2, } and for any bounded
subset K of C, thenthere holds
lim
n→∞
sup
x
∈
K
||S
n
x − S
i
(S
n
x)|| =0. (
)

n
+ γ
n
P
C
(x
n
− λ
n
Ax
n
),
z
n
=(1− α
n
− β
n
)x
n
+ α
n
y
n
+ β
n
S
n
P
C

n
= {z ∈ C : x
n
− z, x − x
n
≥0},
x
n+1
= P
C
n
∩
Q
n
x
Peng Fixed Point Theory and Applications 2011, 2011:12
/>Page 15 of 19
for every n = 1, 2, where {l
n
} ⊂ [a, b] for some
a, b ∈ (0,
1
4
k
)
, and {a
n
}, {b
n
}, {g

∞
γ
n
=
1
and
γ
n
>
3
4
for all n Î N;
Then, {x
n
}, {y
n
}, and {z
n
} converge strongly to w = P
Γ
(x).
Theorem 4.8.LetC be a nonempty closed convex subset of a real Hilbert space H.
Let A be a monotone and k-Lipschitz-continuous mapping of C into H.LetS
1
, S
2
, be
a family of infinitely nonexpansive mappings of C into itself such that
 = ∩
∞

⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
x
1
= x ∈ C,
y
n
= P
C
(x
n
− λ
n
Ax
n
),
z
n
=(1− β

= {z ∈ C : x
n
− z, x − x
n
≥0},
x
n+1
= P
C
n
∩
Q
n
x
for every n = 1, 2, where {l
n
} ⊂ [a, b]forsome
a, b ∈ (0,
1
4
k
)
,and{b
n
}isa
sequence in [0, 1] satisfying
lim inf
n
→∞
β

. Assume that for all i Î {1, 2, } and for any
bounded subset K of C, thenthere holds
l
im
n→∞
sup
x
∈
K
||S
n
x − S
i
(
S
n
x
)
|| =0.
(

)
Let {x
n
}, {u
n
}, and {y
n
} be the sequences generated by
⎧

),
x
n+1
= α
n
x
n
+(1− α
n
)S
n
P
C
(u
n
− λ
n
Ay
n
)
Peng Fixed Point Theory and Applications 2011, 2011:12
/>Page 16 of 19
for every n = 1, 2, if {l
n
} ⊂ [a, b]forsome
a, b ∈ (0,
1
k
)
,{b

Lipschitz continuity of A.
The following result illustrates that there are the nonexpansive mappings S
1
, S
2
,
satisfying the condition (*).
Lemma 4.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let
T beanonexpansivemappingofC into itself such that Fix(T) ≠ ∅.Ifwedefine
S
n
(x)=
1
n

n−1
j
=0
T
j
x
for n Î {1, 2, }, and x Î C, then the following results hold:
(a) For any bounded subset K of C, there holds
lim
n→∞
sup
x
∈
K
||S

(b) It follows from (a) that
∩
∞
i
=1
Fix(S
i
) ⊆ Fix(T
)
.
Moreover, it is obvious that
∩
∞
i
=1
Fix(S
i
) ⊇ Fix(T
)
. Hence,
∩
∞
i
=1
Fix(S
i
)=Fix(T
)
.
(c) It can be proved by mathematical induction. In fact, it is clear that this conclu-

x∈K
||S
n
x − S
m
(S
n
x)|| + lim
n→∞
sup
x∈K
||S
m
(S
n
x) − S
m+1
(S
n
x)|
|
≤ lim
n→∞
sup
x∈K
||S
n
x − S
m
(S

It is easy to verify that S
1
, S
2
, are nonexpansive mappings. It follows from (4.1) and
(4.2) that for any bounded subset K of C, there holds
lim
n→∞
sup
x∈
K
||S
n
x − S
m+1
(S
n
x)|| =0
.
From Lemma 4.1, we know that by Theorems 3.1 and 3.2, respectively, we can obtain
the following results.
Theorem 4.10. Let C be a nonempty closed convex subset of a real Hilbert space H.
Let F be a bifunction from C×Cto R satisfying (A1)-(A4). Let A be a monotone and
k-Lipschitz-continuous mapping of C into H and B be an a-inverse-strongly monotone
mapping of C into H.LetT be a nonexpansive mapping of C into itself such that Θ =
Fix(T)∩VI(C, A)∩GE P(F, B) ≠ ∅.Let{l
n
} ⊂ [a, b]forsome
a, b ∈ (0,
1

n
→∞
β
n
>
0
;
(iv)
lim
n
→
∞
γ
n
=
1
and
γ
n
>
3
4
for all n Î N; If we define
S
n
(x)=
1
n

n−1

n
}
⊂ [δ, ε]forsomeδ, ε Î (0, 1), and {r
n
} ⊂ [d, e]somed, e Î (0, 2a). If we define
S
n
(x)=
1
n

n−1
j
=0
T
j
x
for n Î {1, 2, } and x Î C, then the sequences {x
n
}, {u
n
}, and {y
n
}
generated by algorithm (3.12) converge weakly to w Î Θ, where w = lim
n®∞
P
Θ
x
n

11. Tada, A, Takahashi, W: Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium
problem. J Optim Theory Appl. 133, 359–370 (2007). doi:10.1007/s10957-007-9187-z
12. Plubtieng, S, Punpaeng, R: A new iterative method for equilibrium problems and fixed point problems of nonexpansive
mappings and monotone mappings. Appl Math Comput. 197, 548–558 (2008). doi:10.1016/j.amc.2007.07.075
13. Chang, SS, Joseph Lee, HW, Chan, CK: A new method for solving equilibrium problem, fixed point problem and
variational inequality problem with application to optimization. Nonlinear Anal. 70, 3307–3319 (2009). doi:10.1016/j.
na.2008.04.035
14. Yao, YH, Liou, YC, Yao, JC: Convergence theorem for equilibrium problems and fixed point problems of infinite family
of nonexpansive mappings. Fixed Point Theory Appl 12 (2007). Article ID 64363
15. Ceng, LC, Yao, JC: Hybrid viscosity approximation schemes for equilibrium problems and fixed point problems of
infinitely many nonexpansive mappings. Appl Math Comput. 198, 729–741 (2007)
16. Colao, V, Marino, G, Xu, H-K: An iterative method for finding common solutions of equilibrium and fixed point
problems. J Math Anal Appl. 344, 340–352 (2008). doi:10.1016/j.jmaa.2008.02.041
17. Iusem, AN, Sosa, W: Iterative algorithms for equilibrium problems. Optimization. 52, 301–316 (2003). doi:10.1080/
0233193031000120039
18. Nguyen, TTV, Strodiot, JJ, Nguyen, VH: A bundle method for solving equilibrium problems. Math Program Ser B. 116,
529–552 (2009). doi:10.1007/s10107-007-0112-x
19. Ceng, LC, AI-Homidan, S, Ansari, QH, Yao, JC: An iterative scheme for equilibrium problems and fixed point problems of
strict pseudo-contraction mappings. J Comput Appl Math. 223, 967–974 (2008)
20. Takahashi, W, Shimoji, K: Convergence theorems for nonexpansive mappings and feasibility problems. Math Comput
Model. 32, 1463–1471 (2000). doi:10.1016/S0895-7177(00)00218-1
21. Shimoji, K, Takahashi, W: Strong convergence to common fixed points of infinite nonexpansive mappings and
applications. Taiwanese J Math. 5(2):387
–404 (2001)
22. OHaraa, JG, Pillay, P, Xu, HK: Iterative approaches to finding nearest common fixed points of nonexpansive mappings in
Hilbert spaces. Nonlinear Anal. 54, 1417–1426 (2003). doi:10.1016/S0362-546X(03)00193-7
23. Nadezhkina, N, Takahashi, W: Strong convergence theorem by a hybrid method for non-expansive mappings and
Lipschitz-continuous monotone mappings. SIAM J Optim. 16(4):1230–1241 (2006). doi:10.1137/050624315
24. Opial, Z: Weak convergence of the sequence of successive approximation for nonexpansive mappings. Bull Am Math
Soc. 73, 561–597 (1967)