Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2011, Article ID 392741, 24 pages
doi:10.1155/2011/392741
Research Article
Strong Convergence of a New Iterative Method
for Infinite Family of Generalized Equilibrium and
Fixed-Point Problems of Nonexpansive Mappings
in Hilbert Spaces
Shenghua Wang
1, 2
and Baohua Guo
1, 2
1
National Engineering Laboratory for Biomass Power Generation Equipment,
North China Electric Power University, Baoding 071003, China
2
Department of Mathematics and Physics, North China Electric Power University, Baoding 071003, China
Correspondence should be addressed to Shenghua Wang, [email protected]
Received 15 October 2010; Accepted 18 November 2010
Academic Editor: Qamrul Hasan Ansari
Copyright q 2011 S. Wang and B. Guo. 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.
We introduce an iterative algorithm for finding a common element of the set of solutions of an
infinite family of equilibrium problems and the set of fixed points of a finite family of nonexpansive
mappings in a Hilbert space. We prove some strong convergence theorems for the proposed
iterative scheme to a fixed point of the family of nonexpansive mappings, which is the unique
solution of a variational inequality. As an application, we use the result of this paper to solve a
multiobjective optimization problem. Our result extends and improves the ones of Colao et al.
2008 and some others.
, ∀x, y ∈ H, 1.2
then f is called a contraction with the constant κ. Recall that an operator A : H → H is called
to be strongly positive with coefficient
γ>0if
Ax, x
≥
γx
2
, ∀x ∈ H.
1.3
Let u ∈ H be a fixed point, A be a strongly positive linear bounded operator on
H and {T}
N
n1
be a finite family of nonexpansive mappings of H into itself such that F
N
n1
FT
n
/
∅.
In 2003, Xu 2 introduced the following iterative scheme:
x
n1
1.5
under suitable hypotheses on
n
and the additional hypothesis:
F F
T
1
T
2
···T
N
F
T
N
T
1
···T
N−1
··· F
T
2
T
3
···T
N
A − γf
x
∗
,x− x
∗
≥ 0, ∀x ∈ F
T
, 1.8
which is the optimality condition for the minimization problem:
min
x∈F
1
2
Ax, x
− h
x
,
1.9
where h is a potential function for γf i.e., h
xγfx for all x ∈ H.
1 − λ
n,2
I,
.
.
.
U
n,N−1
λ
n,N−1
T
N−1
U
n,N−2
1 − λ
n,N−1
I,
W
n
≡ U
n,N
λ
n,N
T
N
U
x
n
, ∀n ≥ 1. 1.11
Then he proved that the iterative scheme 1.10 strongly converges to the unique solution x
∗
of the variational inequality:
A − γf
x
∗
,x− x
∗
≥ 0, ∀x ∈ F, 1.12
where F
N
n1
FT
n
, which is the optimality condition for the minimization problem:
min
x∈F
1
2
Ax, x−h
x
x, y
Az, y − z
≥ 0, ∀y ∈ C
. 1.15
In the case of A ≡ 0, EPG, A is deduced to EP.InthecaseofG ≡ 0, EPG, A is also denoted
by VIC, A.
4 Fixed Point Theory and Applications
In 2007, S. Takahashi and W. Takahashi 6 introduced a viscosity approximation
method for finding a common element of EPG and FT from an arbitrary initial element
x
1
∈ H
G
u
n
,y
1
r
n
y − u
n
} and
{u
n
} both converge strongly to z P
FT∩EPG
fz.
By combing the schemes 1.7 and 1.16, Plubtieg and Punpaeng 7 proposed the
following algorithm:
G
u
n
,y
1
r
n
y − u
n
,u
n
− x
n
≥ 0, ∀y ∈ C,
x
n1
∩ EP
G
, 1.18
which is the optimality condition for the minimization problem:
min
x∈FT∩EP
G
1
2
Ax, x
− h
x
,
1.19
where h is a potential function for γf.
Very recently, for finding a common element of the set of a finite family of
nonexpansive mappings and the set of solutions of an equilibrium problem, by combining
the schemes 1.11 and 1.17, Colao et al. 5 proposed the following explicit scheme:
G
u
I −
n
A
W
n
u
n
, ∀n ≥ 1,
1.20
and proved under some certain hypotheses that both sequences {x
n
} and {u
n
} converge
strongly to a point x
∗
∈ F which is an equilibrium point for G and is the unique solution
of the variational inequality:
A − γf
x
∗
,x− x
∗
≥ 0, ∀x ∈ F ∩ EP
-inverse-strongly monotone
mappings. Let {r
n
} be a sequence such that r
n
⊂ r, 2k
n
with r>0 for each n ≥ 1. Define the
mapping T
r
i
: H → C by
T
r
i
x
z ∈ C : G
i
z, y
1
r
i
n
x
n
n
i1
α
i−1
− α
i
T
r
i
I − r
i
A
i
x
n
,
x
n1
n
γf
} are three sequences in 0,1, A and B are both strongly
positive linear bounded operators on H, W
n
is defined by 1.10, and prove that, under
some certain appropriate hypotheses on the control sequences, the sequence {x
n
} strongly
converges to a point x
∗
∈ Ω, which is the unique solution of the variational inequality:
A − γf
x
∗
,x− x
∗
≥ 0, ∀x ∈ Ω. 1.24
If A
i
≡ A
0
, G
i
≡ G and r
i
≡ r, then 1.23 is reduced to the iterative scheme:
z
n
n
I − δ
n
B −
n
A
W
n
z
n
, ∀n ≥ 1.
1.25
The proof method of the main result of this paper is different with ones of others in the
literatures and our result extends and improves the ones of Colao et al. 5 and some others.
6 Fixed Point Theory and Applications
2. Preliminaries
Let C be a closed convex subset of a Hilbert space H. For any point x ∈ H, there exists a
unique nearest point in C, denoted by P
C
x, such that
x − P
C
x
≤
2.2
Let A be a mapping from C into H, then A is called monotone if
x − y, Ax − Ay
≥ 0 2.3
for all x, y ∈ C. However, A is called an α-inverse-strongly monotone mapping if there exists
a positive real number α such that
x − y, Ax − Ay
≥ α
Ax − Ay
2
2.4
for all x, y ∈ C.LetI denote the identity mapping of H, then for all x, y ∈ C and λ>0, one
has 20
I − λA
x −
I − λA
of the variational inequality is denoted by VIC, A.
In this paper, we need the following lemmas.
Fixed Point Theory and Applications 7
Lemma 2.1 see 21. Given x ∈ H and y ∈ C.ThenP
C
x y if and only if there holds the
inequality
x − y, y − z
≥ 0, ∀z ∈ C. 2.7
Lemma 2.2 see 22. Let {s
n
} be a sequence of nonnegative real numbers satisfying
s
n1
≤
1 − η
n
s
n
η
n
τ
n
ξ
n
, ∀n ≥ 0, 2.8
∞
n0
ξ
n
< ∞.
Then lim
n →∞
s
n
0.
Let H be a Hilbert space. For all x, y ∈ H, the following equality holds:
x y
2
x
2
2
y, x y
−
Lemma 2.5 see 2. Assume that {a
n
} is a sequence of nonnegative numbers such that
a
n1
≤
1 − γ
n
a
n
δ
n
, ∀n ≥ 0, 2.11
where {γ
n
} is a sequence in 0, 1 and δ
n
is a sequence in R such that
1
∞
n1
γ
n
∞;
2 lim sup
n →∞
δ
≤ G
x, y
;
2.12
A4 For each x ∈ C, y → Gx, y is convex and lower semicontinuous.
For x ∈ H and r>0, define a mapping T
r
: H → C by
T
r
x
z ∈ C : G
z, y
1
r
y − z, z − x
≥ 0, ∀y ∈ C
3 FT
r
EPG;
4 EPG is closed and convex.
It is easy to see that if there exists some point v ∈ C such that v T
r
I − rAv, where
A : C → H is an α-inverse strongly monotone mapping, then v ∈ EPG, A. In fact, since
v T
r
I − rAv, one has
G
v, y
1
r
y − v, v −
I − rA
v
≥ 0, ∀y ∈ C,
2.15
that is,
G
1
I,
U
2
λ
2
T
2
U
1
1 − λ
2
I,
.
.
.
U
N−1
λ
N−1
T
N−1
U
N−2
N
see 5, 24, 25.
Lemma 2.7 see 26. Let C be a nonempty closed convex subset of a Banach space. Let
T
1
,T
2
, ,T
N
be nonexpansive mappings of C into itself such that
N
i1
FT
i
/
∅ and let
λ
1
,λ
2
, ,λ
N
be real numbers such that 0 <λ
i
< 1 for each i 1, 2, ,N − 1 and 0 <λ
N
≤ 1.Let
W be the W-mapping of C generated by T
that λ
n,i
→ λ
i
for each i 1, 2, ,N. Moreover, for each n ∈ N,letW and W
n
be the W-
mappings generated by T
1
,T
2
, ,T
N
and λ
1
,λ
2
, ,λ
N
and T
1
,T
2
, ,T
N
and λ
n,1
,λ
n,2
, ,λ
} : C → H be an infinite family of inverse-strongly
monotone mappings with constants {k
n
} such that Ω
N
i1
FT
i
∩
∞
i1
EPG
i
,A
i
/
∅.Let
{
n
} and {δ
n
} be two sequences in 0, 1, {λ
n,i
}
N
i1
be asequence in a, b with 0 <a≤ b<1, {r
n
n
≤ 1 for all n ≥ 1;
E4 {δ
n
}⊂0, min{c, 1/β, 2BB
2
−β
β −B
2
− 2B
2
8βB/4βB} with
c<1;
E5
∞
n1
|
n1
−
n
| < ∞,
∞
n1
|α
n1
Proof. Since
n
→ 0asn →∞by the condition E1, we may assume, without loss of
generality, that
n
< 1 − δ
n
BA
−1
for all n ≥ 1. Noting that A and B are both the linear
bounded self-adjoint operators, one has
A sup
{|
Ax, x
|
: x ∈ H, x 1
}
,
B sup
{|
Bx,x
|
: x ∈ H, x 1
}
.
3.2
Observing that
I − δ
n
A
sup
{
I − δ
n
B −
n
A
x, x
: x ∈ H, x 1
}
sup
{
1 −
δ
n
B
n
A
x, x
: x ∈ H, x 1
Note that
f
0
f
2
B
−
β
B
2
2β
B
0, 3.6
f
⎛
⎜
⎜
⎝
2
⎟
⎟
⎠
1.
3.7
Hence, for each δ
n
satisfying the condition E4, one has
0 < 2
β
B
δ
2
n
β −
B
2
− 2
B
δ
n
i
I −r
i
A
i
is nonexpansive for each i ≥ 1. Noting that {α
n
} is strictly decreasing, α
0
1, we have
z
n
− p
α
n
x
n
− p
n
i1
≤ α
n
x
n
− p
n
i1
α
i−1
− α
i
T
r
i
I − r
i1
α
i−1
− α
i
x
n
− p
x
n
− p
3.10
and hence
W
n
z
n
. 3.11
Then, from 3.4 and 3.11, it follows that noting that B is linear and
β>2B B
2
⇒ β>
B
x
n1
− p
n
γf
x
n
− Ap
δ
n
− Ap
δ
n
B
x
n
− p
I − δ
n
B −
n
A
W
n
z
n
− p
x
n
− p
1 − δ
n
β −
n
γ
W
n
z
n
− p
≤
n
γκ
x
n
− p
n
γ
x
n
− p
≤
1 −
n
γ − γκ
x
n
− p
n
γf
,
γf
p
− Ap
γ − γκ
, ∀n ≥ 1, 3.13
which shows that {x
n
} is bounded, so is {z
n
}.
Step 2. x
n1
− x
n
→0asn →∞.
First, we prove
lim
n →∞
W
n1
z
T
i
U
n,i−1
z
n
< ∞.
3.15
It follows from the definition of W
n
that
U
n1,N−i
z
n
− U
n,N−i
z
n
λ
n1,N−i
T
N−i
U
N−i
U
n1,N−i−1
z
n
− T
N−i
U
n,N−i−1
z
n
|
λ
n1,N−i
− λ
n,N−i
|
T
N−i
U
n,N−i−1
z
n
|
λ
n,N−i−1
z
n
|
λ
n1,N−i
− λ
n,N−i
|
≤
U
n1,N−i−1
z
n
− U
n,N−i−1
z
n
M
1
|
λ
n1,N−i
− λ
n,N−i
|
n1,i
− λ
n,i
|
|
λ
n1,1
− λ
n,1
|
z
n
T
1
z
n
≤ M
1
N
i1
|
α
i−1
− α
i
T
r
i
I − r
i
A
i
x
n
− α
n−1
x
n−1
−
n
i1
α
i−1
− α
i
≤ α
n
x
n
− x
n−1
|
α
n
− α
n−1
|
x
n−1
n
i1
α
i−1
x
n
− x
n−1
|
α
n
− α
n−1
|
x
n−1
|
α
n−1
− α
n
|
T
r
n
I − r
A
n
x
n−1
}.
Next, we prove lim
n →∞
x
n1
− x
n
0. Observe noting that B is linear that
x
n1
− x
n
n
γ
f
x
n
− f
x
n−1
n
z
n
− W
n
z
n−1
I − δ
n
B −
n
A
W
n
z
n−1
−
n−1
γf
x
n−1
− δ
n−1
Bx
B
x
n
− x
n−1
I − δ
n
B −
n
A
W
n
z
n
− W
n
z
n−1
n
−
n−1
BW
n−1
z
n−1
− δ
n
BW
n
z
n−1
n−1
AW
n−1
z
n−1
−
n
AW
n
z
n−1
n
γ
z
n
− W
n
z
n−1
n
−
n−1
γf
x
n−1
δ
n
− δ
n−1
Bx
n−1
z
n−1
n−1
−
n
AW
n−1
z
n−1
n
A
W
n−1
z
n−1
− W
n
z
n−1
.
3.19
14 Fixed Point Theory and Applications
β −
n
γ
z
n
− z
n−1
|
n
−
n−1
|
γ
f
x
n−1
|
δ
n−1
δ
n
B
W
n−1
z
n−1
− W
n
z
n−1
|
n−1
−
n
|
AW
n−1
z
n−1
n−1
1 − δ
n
β −
n
γ
x
n
− x
n−1
|
α
n
− α
n−1
|
L
|
n
− W
n−1
z
n−1
|
δ
n−1
− δ
n
|
BW
n−1
z
n−1
δ
n
B
W
n−1
z
n−1
− W
n
z
1 −
δ
n
β −
B
n
γ − γκ
x
n
− x
n−1
L
|
α
n
− α
n−1
|
W
n−1
z
n−1
− W
n
z
n−1
,
3.20
where M
2
sup
n
{γfx
n−1
Bx
n−1
BW
n−1
z
n−1
AW
n−1
z
n−1
}.
n−1
δ
n
n
M
3
×
1
δ
n
n
M
3
δ
n
B
δ
n−1
L
|
α
n
− α
n−1
|
2
|
n−1
−
n
|
M
2
2
|
δ
n−1
− δ
n
|
M
2
.
3.21
Set
δ
n
n
M
3
n
A
δ
n
n
M
3
W
n−1
z
n−1
− W
M
2
.
3.22
Fixed Point Theory and Applications 15
Then it follows from 3.21 that
x
n1
− x
n
≤
1 − η
n
x
n
− x
n−1
η
n
τ
n
ξ
n
. 3.23
Step 3. x
n
− W
n
z
n
→ 0asn →∞.
For all n ≥ 1, we have
x
n
− W
n
z
n
≤
x
n
− x
n1
x
n1
− W
n
z
W
n
z
n
− W
n
z
n
≤
x
n
− x
n1
n
γf
x
n
− AW
n
z
n
z
n
δ
n
B
x
n
− W
n
z
n
3.25
and hence noting 3.9
x
n
− W
n
z
n
≤
1
1 − δ
z
n
.
3.26
It follows from the assumption conditions E1, E2,andStep 2 that
x
n
− W
n
z
n
−→ 0
n −→ ∞
. 3.27
Step 4. x
n
− z
n
→ 0asn →∞.
16 Fixed Point Theory and Applications
Notice that, for any x ∈ Ω,
z
n
− x
A
i
x
n
− T
r
i
I − r
i
A
i
x
2
≤ α
n
x
n
− x
2
n
x
n
− x
2
n
i1
α
i−1
− α
i
x
n
− x
2
r
i
r
i
− 2k
r
i
r
i
− 2k
i
A
i
x
n
− A
i
x
2
.
3.28
Let y
n
γfx
n
−AW
n
z
n
and λ sup{γfx
n
Bx
n
− Bx
n
γf
x
n
− AW
n
z
n
2
≤
I − δ
n
B
W
n
n
z
n
− W
n
x
δ
n
B
x
n
− x
2
2
n
y
n
,x
n1
− x
≤
1 − δ
n
x
n
− x
2
2λ
2
n
≤
1 − δ
n
β
x
n
− x
2
n
i1
α
i−1
− α
− x
2
2δ
n
1 − δ
n
β
B
x
n
− x
2
2λ
2
n
1 −
δ
n
β − δ
n
α
i−1
− α
i
r
i
r
i
− 2k
i
A
i
x
n
− A
i
x
2
2λ
2
n
≤
i
x
n
− A
i
x
2
2λ
2
n
.
3.29
This shows that
1 − δ
n
β
n
i1
α
i−1
− α
i
r
2
2λ
2
n
≤
x
n
− x
x
n1
− x
x
n
− x
n1
2λ
2
n
3.30
2
≤
x
n
− x
2
−
x
n1
− x
2
2λ
2
n
≤
x
n
− x
x
n1
i
x
n
− A
i
x
0,i≥ 1.
3.32
Now, for x ∈ Ω, we have, f rom Lemma 2.2,
T
r
i
I − r
i
A
i
x
n
− x
2
T
r
i
x
n
− T
r
i
I − r
i
A
i
x,
I − r
i
A
i
x
n
−
I − r
i
A
i
x
n
− x, A
i
x − A
i
x
n
1
2
T
r
i
I − r
i
A
i
x
n
− x
2
x
i
A
i
x
n
− x, A
i
x − A
i
x
n
3.33
and hence
T
r
i
I − r
i
A
i
x
n
− x
2
I − r
i
A
i
x
n
− x, A
i
x − A
i
x
n
.
3.34
Therefore,
z
n
− x
2
≤ α
n
x
n
− x
x
n
− x
2
n
i1
α
i−1
− α
i
x
n
− x
2
−
x
n
− T
r
i
x
n
− x
2
−
n
i1
α
i−1
− α
i
x
n
− T
r
i
I − r
i
A
i
x − A
i
x
n
.
3.35
18 Fixed Point Theory and Applications
By using 3.8, 3.9, 3.35, Lemmas 2.3 and 2.4, we have noting that δ
n
< 1/β
x
n1
− x
2
I − δ
n
B
W
n
z
n
− x
W
n
z
n
− x
δ
n
Bx
n
− Bx
2
2
n
y
n
,x
n1
− x
I − δ
n
1 − δ
n
β
z
n
− x
2
δ
n
B
2
x
n
− x
2
2δ
n
1 − δ
n
β
α
i−1
− α
i
x
n
− T
r
i
I − r
i
A
i
x
n
2
2
n
i1
α
i−1
− α
i
x
n
− x
2
2δ
n
1 − δ
n
β
B
x
n
− x
2
2λ
2
n
1 −
δ
n
i1
α
i−1
− α
i
x
n
− T
r
i
I − r
i
A
i
x
n
2
2
1 − δ
n
β
2λ
2
n
≤
x
n
− x
2
−
1 − δ
n
β
n
i1
α
i−1
− α
i
x
n
− T
T
r
i
I − r
i
A
i
x
n
− x, A
i
x − A
i
x
n
2λ
2
n
3.36
and hence
1 − δ
n
β
n
2
−
x
n1
− p
2
2
1 − δ
n
β
×
n
i1
α
i−1
− α
i
r
i
T
x
n1
− x
x
n
− x
n1
2
1 − δ
n
β
n
i1
α
i−1
− α
i
r
i
T
α
i−1
− α
i
x
n
− T
r
i
I − r
i
A
i
x
n
2
≤
x
n
− x
T
r
i
I − r
i
A
i
x
n
− x, A
i
x − A
i
x
n
2λ
2
n
.
3.38
Since {α
n
} is strictly decreasing, δ
n
→ 0,
n
− x
n
n
i1
α
i−1
− α
i
T
r
i
x
n
− x
n
we get
z
n
− x
n
≤
n
i1
α
− x
n
−→ 0, as n −→ ∞ . 3.41
Step 5. lim sup
n →∞
γf − Ax
∗
,x
n
− x
∗
≤0.
To prove this, we pick a subsequence {x
n
j
} of {x
n
} such that
lim sup
n →∞
γf − A
x
∗
,x
n
− x
∗
i
x
n
→ 0, x
n
j
→ x and T
r
i
I − r
i
A
i
is
nonexpansive, by demiclosed principle of nonexpansive mapping we have
x ∈ F
T
r
i
I − r
i
A
i
EP
G
i
i
FW. Assume that x
/
∈ FW.
Then x
/
W x. Since x ∈ FT
r
i
I − r
i
A
i
for each i ≥ 1, by Step 3, 3.44 and Opial’s property
of the Hilbert space H, we have
lim inf
n →∞
x
n
m
− x
< lim inf
n →∞
x
n
m
− W x
W
n
m
x − W x
≤ lim inf
n →∞
x
n
m
− W
n
m
z
n
m
z
n
m
− x
W
x
n
m
− T
r
i
I − r
i
A
i
x
n
m
x − T
r
i
I − r
i
A
i
x
n
m
m
x
n
m
− T
r
i
I − r
i
A
i
x
n
m
T
r
i
I − r
i
A
i
,
3.45
which is a contradiction. Therefore, x ∈ FW. Hence, x ∈ Ω
N
i1
FT
i
∩
∞
i1
EPG
i
,A
i
.
Step 6. The sequence {x
n
} strongly converges to some point x
∗
∈ H.
By using Lemmas 2.3 and 2.4, we have
x
n1
− x
∗
2
x
n
− Ax
∗
2
≤
I − δ
n
B −
n
A
W
n
z
n
− x
∗
δ
n
Bx
n
z
n
− x
∗
δ
n
B
x
n
− x
∗
2
2
n
γf
x
n
− Ax
∗
,x
2
2
n
γf
x
n
− Ax
∗
,x
n1
− x
∗
1 −
n
γ
x
n
− x
∗
2
2
n
γκ
x
n
− x
∗
x
n1
− x
∗
2
n
γf
x
∗
− Ax
∗
,x
n1
− x
∗
∗
2
2
n
γf
x
∗
− Ax
∗
,x
n1
− x
∗
,
3.46
Fixed Point Theory and Applications 21
which implies that
x
n1
− x
∗
2
∗
− Ax
∗
,x
n1
− x
∗
1 − 2
n
γ
n
γκ
1 −
n
γκ
x
n
− x
∗
2
2
∗
≤
1 −
2
n
γ − κγ
1 −
n
γκ
x
n
− x
∗
2
2
n
γ − κγ
1 −
n
γκ
where M
is an appropriate constant such that M
sup
n≥1
{x
n
− x
∗
}.Put
s
n
2
n
γ − κγ
1 −
n
κγ
,
t
n
1
γ − κγ
γf
2
≤
1 − s
n
x
n
− x
∗
s
n
t
n
.
3.49
It follows from the assumption condition E1 and 3.42 that
lim
n →∞
s
n
0,
∞
n1
s
n
i
∩ EPG, A
/
∅.Let{ε
n
} and
{δ
n
} be two sequences in 0, 1, {λ
n,i
}
N
i1
be a sequence in a, b with 0 <a≤ b<1, r be a number in
0, 2α, and {α
n
} be a sequence 0, 1. Take a fixed number γ>0 with 0 < γ − γκ < 1. Assume that
22 Fixed Point Theory and Applications
E1 lim
n →∞
n
0 and
∞
n1
ε
n
∞;
E2 lim
n
| < ∞,
∞
n1
|α
n1
− α
n
| < ∞,
∞
n1
|δ
n1
− δ
n
| < ∞.
Then the sequence {x
n
} defined by 1.25 converges strongly to x
∗
∈ Ω, which is the unique solution
of the variational inequality:
x
∗
P
Ω
I −
,
min h
2
x
,
x ∈ C, 4.1
where h
1
x and h
2
x are both the convex and lower semicontinuous functions defined on a
closed convex subset of C of a Hilbert space H.
We denote by A the set of solutions of the problem 4.1 and assume that A
/
∅.Also,
we denote the sets of solutions of the following two optimization problems by A
1
and A
2
,
respectively,
min h
1
x
,x∈ C, 4.2
and
x
,G
2
x, y
h
2
y
− h
2
x
, ∀
x, y
∈ C × C, 4.4
respectively. It is easy to see that EPG
1
A
1
and EPG
2
A
2
n
T
r
1
x
n
1 − α
n
T
r
2
x
n
,
x
n1
n
γf
x
n
1 −
n
4 Y. Yao, “A general iterative method for a finite family of nonexpansive mappings,” Nonlinear Analysis:
Theory, Methods & Applications, vol. 66, no. 12, pp. 2676–2687, 2007.
5 V. Colao, G. Marino, and H K. Xu, “An iterative method for finding common solutions of equilibrium
and fixed point problems,” Journal of Mathematical Analysis and Applications, vol. 344, no. 1, pp. 340–
352, 2008.
6 S. Takahashi and W. Takahashi, “Viscosity approximation methods for equilibrium problems and
fixed point problems in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 331, no.
1, pp. 506–515, 2007.
7 S. Plubtieng and R. Punpaeng, “A general iterative method for equilibrium problems and fixed point
problems in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 336, no. 1, pp. 455–
469, 2007.
8 L C. Ceng and J C. Yao, “Hybrid viscosity approximation schemes for equilibrium problems
and fixed point problems of infinitely many nonexpansive mappings,” Applied Mathematics and
Computation, vol. 198, no. 2, pp. 729–741, 2008.
9 L C. Ceng, S. Al-Homidan, Q. H. Ansari, and J C. Yao, “An iterative scheme for equilibrium
problems and fixed point problems of strict pseudo-contraction mappings,” Journal of Computational
and Applied Mathematics, vol. 223, no. 2, pp. 967–974, 2009.
10 L. C. Ceng, A. Petrus¸el, and J. C. Yao, “Iterative approaches to solving equilibrium problems and
fixed point problems of infinitely many nonexpansive mappings,” Journal of Optimization Theory and
Applications, vol. 143, no. 1, pp. 37–58, 2009.
11 S S. Chang, Y. J. Cho, and J. K. Kim, “Approximation methods of solutions for equilibrium problem
in Hilbert spaces,” Dynamic Systems and Applications, vol. 17, no. 3-4, pp. 503–513, 2008.
24 Fixed Point Theory and Applications
12 S. Chang, H. W. Joseph Lee, and C. K. Chan, “A new method for solving equilibrium problem
fixed point problem and variational inequality problem with application to optimization,” Nonlinear
Analysis: Theory, Methods & Applications, vol. 70, no. 9, pp. 3307–3319, 2009.
13 P. Kumam and P. Katchang, “A viscosity of extragradient approximation method for finding equi-
librium problems, variational inequalities and fixed point problems for nonexpansive mappings,”
Nonlinear Analysis: Hybrid Systems, vol. 3, no. 4, pp. 475–486, 2009.
14 A. Moudafi, “Weak convergence theorems for nonexpansive mappings and equilibrium problems,”
26 S. Atsushiba and W. Takahashi, “Strong convergence theorems for a finite family of nonexpansive
mappings and applications,” Indian Journal of Mathematics, vol. 41, no. 3, pp. 435–453, 1999.
27 T. Suzuki, “Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter non-
expansive semigroups without Bochner integrals,” Journal of Mathematical Analysis and Applications,
vol. 305, no. 1, pp. 227–239, 2005.
28 Y. J. Cho and X. Qin, “Convergence of a general iterative method for nonexpansive mappings in
Hilbert spaces,” Journal of Computational and Applied Mathematics, vol. 228, no. 1, pp. 458–465, 2009.
Copyright: Tài liệu đại học ©