Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2011, Article ID 232163, 15 pages
doi:10.1155/2011/232163
Research Article
A Hybrid-Extragradient Scheme for System of
Equilibrium Problems, Nonexpansive Mappings,
and Monotone Mappings
Jian-Wen Peng,
1
Soon-Yi Wu,
2
and Gang-Lun Fan
2
1
School of Mathematics, Chongqing Normal University, Chongqing 400047, China
2
Department of Mathematics, National Cheng Kung University, Tainan 701, Taiwan
Correspondence should be addressed to Jian-Wen Peng, [email protected]
Received 21 October 2010; Accepted 24 November 2010
Academic Editor: Jen Chih Yao
Copyright q 2011 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.
We introduce a new iterative scheme based on both hybrid method and extragradient method
for finding a common element of the solutions set of a system of equilibrium problems, the fixed
points set of a nonexpansive mapping, and the solutions set of a variational inequality problems
for a monotone and k-Lipschitz continuous mapping in a Hilbert space. Some convergence results
for the iterative sequences generated by these processes are obtained. The results in this paper
extend and improve some known results in the literature.
1. Introduction
k∈Γ
EPF
k
.
Given a mapping T : C → H,letFx, yTx,y − x for all x, y ∈ C. Then, the
problem 1.2 becomes the following variational inequality:
finding x ∈ C, such that
Tx,y − x
≥ 0, ∀y ∈ C. 1.3
The set of solutions of 1.3 is denoted by VIC, A.
The problem 1.1 is very general in the sense that it includes, as special cases,
optimization problems, variational inequalities, minimax problems, Nash equilibrium
problem in noncooperative games, and others; see, for instance, 1–4.
In 1953, Mann 5 introduced t he following iteration algorithm: let x
0
∈ C be an
arbitrary point, let {α
n
} be a real sequence in 0, 1, and let the sequence {x
n
} be defined
by
x
n1
α
n
x
n
1 − α
n
Sx
n
,
C
n1
z ∈ C
n
:
y
n
− z
≤
x
n
− z
,
x
n
− λAx
n
,
x
n1
P
C
x
n
− λAy
n
1.6
for all n ≥ 0, where λ ∈ 0, 1/k, A is monotone and k-Lipschitz continuous mapping of C
into R
n
. She proved that, if VIC, A is nonempty, the sequences {x
n
} and {y
n
}, generated by
1.6, converge to the same point z ∈ VIC, A.
Some methods have been proposed to solve the problem 1.2; see, for instance, 10,
11 and the references therein. S. Takahashi and W. Takahashi 10 introduced the following
iterative scheme by the viscosity approximation method for finding a common element of the
Fixed Point Theory and Applications 3
set of the solution 1.2 and the set of fixed points of a nonexpansive mapping in a real Hilbert
f
x
n
1 − α
n
Su
n
,n≥ 1.
1.7
They proved that under certain appropriate conditions imposed on {α
n
} and {r
n
},the
sequences {x
n
} and {u
n
} converge strongly to z ∈ FixS ∩ EPF, where z P
FixS∩EPF
fz.
Let E be a uniformly smooth and uniformly convex Banach space, and let C be a
nonempty closed convex subset of E.Letf be a bifunction from C × C to R,andletS be
a relatively nonexpansive mapping from C into itself such that FixS ∩ EPf
/
u
n
,y
1
r
n
y − u
n
,Ju
n
− Jy
n
≥ 0, ∀y ∈ C,
C
n1
z ∈ C
n
: φ
z, u
n
≤ φ
x if {α
n
}⊂0, 1 satisfies
lim inf
n →∞
α
n
1 − α
n
> 0and{r
n
}⊂a, ∞ for some a>0.
On the other hand, Combettes and Hirstoaga 1 introduced an iterative scheme
for finding a common element of the set of solutions of problem 1.1 in a Hilbert space
and obtained a weak convergence theorem. Peng and Yao 4 introduced a new viscosity
approximation scheme based on the extragradient method for finding a common element
of the set of solutions of problem 1.1, the set of fixed points of an infinite family of
nonexpansive mappings, and the set of solutions to the variational inequality for a monotone,
Lipschitz continuous mapping in a Hilbert space and obtained two strong convergence
theorems. Colao et al. 3 introduced an implicit method for finding common solutions of
variational inequalities and systems of equilibrium problems and fixed points of infinite
family of nonexpansive mappings in a Hilbert space and obtained a strong convergence
theorem. Peng et al. 12 introduced a new iterative scheme based on extragradient method
and viscosity approximation method for finding a common element of the solutions set of
a system of equilibrium problems, fixed points set of a family of infinitely nonexpansive
mappings, and the solution set of a variational inequality for a relaxed coercive mapping
in a Hilbert space and obtained a strong convergence theorem.
In this paper, motivated by the above results, we introduce a new hybrid extragradient
method to find a common element of the set of solutions to a system of equilibrium
problems, the set of fixed points of a nonexpansive mapping, and the set of solutions of the
2
− λ
1 − λ
x − y
2
2.1
for all x, y ∈ H and λ ∈ R.
For any x ∈ H, there exists a unique nearest point in C denoted by P
C
x such that
x − P
C
x≤x − y for all y ∈ C. The mapping P
C
is called the metric projection of H
onto C. We know that P
C
is a nonexpansive mapping from H onto C. It is also known that
P
C
x ∈ C and
x − P
y − P
C
x
2
2.3
for all x ∈ H and y ∈ C.
A mapping A of C into H is called monotone if Ax − Ay, x − y≥0 for all x, y ∈ C.A
mapping A : C → H is called L-Lipschitz continuous if there exists a positive real number L
such that Ax − Ay≤Lx − y for all x, y ∈ C.
Let A be a monotone mapping of C into H. In the context of the variational inequality
problem, the characterization of projection 2.2 implies the following:
u ∈ VI
C, A
⇒ u P
C
u − λAu
, ∀λ>0,
u P
C
We recall some lemmas which will be needed in the rest of this paper.
Fixed Point Theory and Applications 5
Lemma 2.1 See 2. Let C be a nonempty closed convex subset of H, and let F be a bifunction from
C × C to R satisfying (A1)–(A4). Let r>0 and x ∈ H. Then, there exists z ∈ C such that
F
z, y
1
r
y − z, z − x
≥ 0, ∀y ∈ C.
2.6
Lemma 2.2 See 1. Let C be a nonempty closed convex subset of H, and let F be a bifunction from
C × C to R satisfying (A1)–(A4). For r>0 and x ∈ H, define a mapping T
F
r
: H → 2
C
as follows:
T
F
r
x
x
− T
F
r
y
2
≤
T
F
r
x
− T
F
r
y
,x− y
;
2.8
n
}, {y
n
}, {w
n
}, and {u
n
} be sequences generated by x
1
P
C
1
x
0
and
u
n
T
F
M
r
M·n
T
F
M−1
r
M−1,n
···T
F
2
1 − α
n
SP
C
u
n
− λ
n
Ay
n
,
C
n1
{
z ∈ C
n
:
w
n
− z
≤
x
n
}, and
{w
n
} generated by 3.1 converge strongly to P
Ω
x
0
.
Proof. It is obvious that C
n
is closed for each n ∈ N. Since
C
n1
z ∈ C
n
:
w
n
− x
n
2
2
w
n
k−1
r
k−1,n
···T
F
2
r
2,n
T
F
1
r
1,n
for k ∈{1, 2, ,M} and n ∈ N, Θ
0
n
I
for each n ∈ N, where I is the identity mapping on H. Then, it is easy to see that u
n
Θ
M
n
x
n
.
We divide the proof into several steps.
Step 1. We show by induction that Ω ⊂ C
n
for n ∈ N. It is obvious that Ω ⊂ C C
1
≤
x
n
− v
, ∀n ∈ N. 3.3
Putting v
n
P
C
u
n
− λ
n
Ay
n
for each n ∈ N,from2.3 and the monotonicity of A, we have
v
n
− v
2
≤
2
−
u
n
− v
n
2
2λ
n
Ay
n
,v− v
n
u
n
− v
2
−
u
n
− v
n
2
−
u
n
− v
n
2
2λ
n
Ay
n
,y
n
− v
n
u
n
− v
2
−
u
Ay
n
,y
n
− v
n
u
n
− v
2
−
u
n
− y
n
2
−
y
n
− v
n
and 2.2, we have
u
n
− λ
n
Au
n
− y
n
,v
n
− y
n
≤ 0. 3.5
Fixed Point Theory and Applications 7
Since A is k-Lipschitz continuous, it follows that
u
n
− λ
n
Ay
n
− y
n
,v
n
− y
n
≤
λ
n
Au
n
− λ
n
Ay
n
,v
n
− y
n
≤ λ
n
k
u
n
− y
n
2
−
y
n
− v
n
2
2λ
n
k
u
n
− y
n
v
n
− y
n
2
n
k
2
u
n
− y
n
2
v
n
− y
n
2
u
n
− v
2
w
n
− v
2
≤ α
n
x
n
− v
2
1 − α
n
Sv
n
− v
2
≤ α
n
x
n
n
− v
2
λ
2
n
k
2
− 1
u
n
− y
n
2
3.8
≤ α
n
x
n
− v
− y
n
2
x
n
− v
2
1 − α
n
λ
2
n
k
2
− 1
u
n
− y
0.
Let l
0
P
Ω
x
0
.Fromx
n
P
C
n
x
0
and l
0
∈ Ω ⊂ C
n
, we have
x
n
− x
0
≤
l
0
− x
0
≤
x
n1
− x
0
, ∀n ∈ N. 3.11
Therefore, lim
n →∞
x
n
− x
0
exists.
8 Fixed Point Theory and Applications
From x
n
P
C
n
x
0
and x
n1
P
C
n1
n
− x
0
x
0
− x
n1
2
x
n
− x
0
2
2
x
n
− x
0
,x
0
− x
− x
n1
x
0
− x
n1
2
x
n
− x
0
2
− 2
x
0
− x
n
,x
0
− x
n
2
x
0
− x
n1
2
−
x
n
− x
0
2
x
0
− x
n1
2
,
3.13
which implies that
lim
n →∞
x
n1
x
n1
− w
n
≤ 2
x
n
− x
n1
, ∀n ∈ N. 3.15
It follows from 3.14 that x
n
− w
n
→0.
For v ∈ Ω, it follows from 3.9 that
u
n
− y
n
1
1 − α
n
1 − λ
2
n
k
2
x
n
− v
−
w
n
− v
x
n
− v
n
− v
w
n
− v
,
3.16
which implies that lim
n →∞
u
n
− y
n
0.
Step 3. We now show that
lim
n →∞
Θ
k
n
x
n
2
T
F
k
r
k,n
Θ
k−1
n
x
n
− T
F
k
r
k,n
v
2
≤
Θ
k
n
Θ
k−1
n
x
n
− v
2
−
Θ
k
n
x
n
− Θ
k−1
n
x
n
2
.
Θ
k
n
x
n
− Θ
k−1
n
x
n
2
,k 1, 2, ,M,
3.19
which implies that, for each k ∈{1, 2, ,M},
Θ
k
n
x
n
− v
n
2
−
Θ
k−1
n
x
n
− Θ
k−2
n
x
n
2
−···−
Θ
2
n
x
n
− v
2
−
Θ
k
n
x
n
− Θ
k−1
n
x
n
2
.
3.20
By 3.8, u
n
Θ
M
n
x
n
− v
2
1 − α
n
Θ
k
n
x
n
− v
2
, ∀k ∈
{
1, 2, ,M
}
≤ α
n
x
n
2
≤
x
n
− v
2
−
1 − α
n
Θ
k
n
x
n
− Θ
k−1
x
n
− v
2
−
w
n
− v
2
x
n
− v
w
n
− v
x
n
3.22
It follows from x
n
− w
n
→0and0<c≤ α
n
≤ d<1that3.17 holds.
Step 4. We now show that lim
n →∞
Sv
n
− v
n
0.
10 Fixed Point Theory and Applications
It follows from 3.17 that x
n
− u
n
→0. Since x
n
− y
n
≤x
n
− u
n
u
n
u
n
− λ
n
Ay
n
− P
C
u
n
− λ
n
Au
n
≤
λ
n
Au
n
− λ
n
Ay
n
− w
n
x
n
− α
n
x
n
− 1 − α
n
Sv
n
1 − α
n
x
n
− Sv
n
,weobtain
lim
n →∞
x
n
− Sv
n
0.
3.26
Ω
x
0
.
As {x
n
} is bounded, there exists a subsequence {x
n
i
} which converges weakly to w.
From Θ
k
n
x
n
− Θ
k−1
n
x
n
→0 for each k 1, 2, ,M,weobtainthatΘ
k
n
i
x
n
i
wfor k
1, 2, ,M. It follows from x
n
k
r
k,n
, we have that, for each k ∈{1, 2, ,M},
F
k
Θ
k
n
x
n
,y
1
r
k,n
y − Θ
k
n
x
n
, Θ
k
n
x
n
− Θ
y, Θ
k
n
x
n
, ∀y ∈ C.
3.29
Fixed Point Theory and Applications 11
And hence
y − Θ
k
n
i
x
n
i
,
Θ
k
n
i
x
n
i
− Θ
k−1
n
n
i
x
n
i
/r
k,n
i
→ 0andΘ
k
n
i
x
n
i
wimply that, for each k ∈{1, 2, ,M},
F
k
y, w
≤ 0, ∀y ∈ C. 3.31
Since x
n
i
⊂ C, x
n
i
wand C is closed and convex, C is weakly closed, and hence
w ∈ C.Thus,fort with 0 <t≤ 1andy ∈ C,lety
k
y
t
,w
≤ tF
k
y
t
,y
, 3.32
and hence, for each k ∈{1, 2, ,M},0 ≤ F
k
y
t
,y.FromA3, we have, f or each k ∈
{1, 2, ,M},0≤ F
k
w, y, for all y ∈ C.Thus,w ∈
M
k1
EPF
k
.
We now show that w ∈ FixS. Assume that w/∈ FixS. Since v
n
lim inf
i →∞
Sv
n
i
− Sw
lim inf
i →∞
Sv
n
i
− Sw
≤ lim inf
i →∞
v
n
i
− w
,
3.33
which is a contradiction. Thus, we obtain w ∈ FixS.
We next show that w ∈ VIC, A.Let
Tv
,v
n
− u
n
− λ
n
Ay
n
≥0, and hence v − v
n
, v
n
− u
n
/λ
n
Ay
n
≥0. Therefore, we have
v − v
n
i
,u
≥
v − v
n
i
v − v
n
i
,Av− Ay
n
i
−
v
n
i
− u
n
i
λ
n
i
v − v
n
i
,Av− Av
n
i
v − v
n
n
i
− Ay
n
i
−
v − v
n
i
,
v
n
i
− u
n
i
λ
n
i
.
3.35
Since lim
n →∞
v
n
− y
n
→ w, where
w P
Ω
x
0
. 3.37
Since x
n
P
C
n
x
0
and w ∈ Ω ⊂ C
n
, we have x
n
− x
0
≤w − x
0
. It follows from
l
0
P
Ω
x
0
and the lower semicontinuousness of the norm that
l
0
− x
0
.
3.38
Thus, we obtain w l
0
and
lim
i →∞
x
n
i
− x
0
w − x
0
.
3.39
From x
n
i
− x
0
w− x
0
∈ H, and set C
1
C.Let{x
n
}, {y
n
}, {w
n
}, and {u
n
}
be sequences generated by x
1
P
C
1
x
0
and
u
n
∈ C, such that F
u
n
,y
1
1 − α
n
SP
C
u
n
− λ
n
Ay
n
,
C
n1
{
z ∈ C
n
:
w
n
− z
≤
n
}, {y
n
}, and {w
n
} converge strongly to
P
Ω
x
0
.
Proof. Putting F
M
F
M−1
··· F
1
F in Theorem 3.1,weobtainCorollary 3.2.
Corollary 3.3. Let C be a nonempty closed convex subset of a real Hilbert space H.LetF
k
,
k ∈{1, 2, ,M} be a family of bifunctions from C × C to R satisfying (A1)–(A4), and let S be
a nonexpansive mapping from C into itself such that ΩFixS ∩
M
k1
EPF
k
/
M−1,n
···T
F
2
r
2,n
T
F
1
r
1,n
x
n
,
w
n
α
n
x
n
1 − α
n
Su
n
,
C
n1
}⊂c, d for some c, d ∈ 0, 1 and {r
k,n
}⊂0, ∞ satisfies lim inf
n →∞
r
k,n
> 0
for each k ∈{1, 2, ,M},then{x
n
}, {u
n
}, and {w
n
} converge strongly to P
Ω
x
0
.
Proof. Let A 0inTheorem 3.1, then complete the proof.
Corollary 3.4. Let C be a nonempty closed convex subset of a real Hilbert space H.LetA be a
monotone and k-Lipschitz continuous mapping of C into H, and let S be a nonexpansive mapping
from C into itself such that ΩFixS ∩ VIC, A
/
∅. Pick any x
0
∈ H, and set C
1
C.Let
{x
n
n
1 − α
n
SP
C
u
n
− λ
n
Ay
n
,
C
n1
{
z ∈ C
n
:
w
n
− z
≤
Ω
x
0
.
Proof. Putting F
M
F
M−1
··· F
1
0inTheorem 3.1,weobtainCorollary 3.4.
Remark 3.5. Letting F
M
F
M−1
··· F
1
F in Corollary 3.3, we obtain the Hilbert space
version of Theorem 3.1 in 11. Letting A 0inCorollary 3.4, we recover Theorem 4.1 in 8.
Hence, Theorem 3.1 unifies, generalizes, and extends t he corresponding results in 8, 11 and
the references therein.
Acknowledgments
This research was supported by the National Natural Science Foundation of China Grants
10771228 and 10831009, the Natural Science Foundation of Chongqing Grant no. CSTC,
2009BB8240, and the Research Project of Chongqing Normal University Grant 08XLZ05.
The authors are grateful to the referees for the detailed comments and helpful suggestions,
which have improved the presentation of this paper.
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Fixed Point Theory and Applications 15
13 Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive
mappings,” Bulletin of the American Mathematical Society, vol. 73, pp. 591–597, 1967.
14 R. T. Rockafellar, “On the maximality of sums of nonlinear monotone operators,” Transactions of the
American Mathematical Society, vol. 149, pp. 75–88, 1970.
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