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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,
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
In this paper, we always assume that H is a real Hilbert space with inner product ·, ·
and induced norm ·and C is a nonempty closed convex subset of H, S : C → C is a
nonexpansive mapping; that is, Sx − Sy≤x − y for all x,y ∈ C, P
C
denotes the metric
projection of H onto C,andFixS denotes the fixed points set of S.
Let {F
k
}
k∈Γ
be a countable family of bifunctions from C × C to R, where R is the set of
real numbers. Combettes and Hirstoaga 1 introduced the following system of equilibrium
problems:
finding x ∈ C, such that ∀k ∈ Γ, ∀y ∈ C, F
k
x, y
≥ 0, 1.1
where Γ is an arbitrary index set. If Γ is a singleton, the problem 1.1 becomes the following
equilibrium problem:
finding x ∈ C, such that F
x, y
≥ 0, ∀y ∈ C. 1.2
2 Fixed Point Theory and Applications
The set of solutions of 1.2 is denoted by EPF. And it is easy to see that the set of solutions
of 1.1 can be written as
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
. 1.4
Mann iteration algorithm has been extensively investigated for nonexpansive mappings, for
example, please see 6, 7. Takahashi et al. 8 modified the Mann iteration method 1.4 and
introduced the following hybrid projection algorithm:
x
0
∈ H, C
1
C, x
1
P
C
1
x
0
,
y
n
α
n
x
n
1 − α
n
Sx
n
,
C
n1
z ∈ C
n
:
y
n
− z
≤
x
n
− z
,
x
n1
P
C
n1
x
0
, ∀n ∈ N,
1.5
where 0 ≤ α
n
<a<1. They proved that the sequence {x
n
} generated by 1.5 converges
strongly to P
FixS
x
0
.
In 1976, Korpelevi
ˇ
c 9 introduced the following so-called extragradient algorithm:
x
0
x ∈ C,
y
n
P
C
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
space: starting with an arbitrary initial x
1
∈ C, define sequences {x
n
} and {u
n
} recursively by
F
u
n
,y
1
r
n
y − u
n
,u
n
− x
n
≥ 0, ∀y ∈ C,
x
n1
α
n
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
/
∅. Takahashi
and Zembayashi 11 introduced the following hybrid method in Banach space: let {x
n
} be a
sequence generated by x
0
x ∈ C, C
0
C,and
y
n
J
−1
α
n
Jx
n
1 − α
n
JSx
n
,
u
n
∈ C, such that f
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
z, x
n
,
x
n1
C
n1
x
1.8
for every n ∈ N ∪{0}, where J is the duality napping on E, φx, yy
2
− 2y, Jx
x
2
for all x, y ∈ E,and
C
x arg min
y∈C
φy, x for all x ∈ E. They proved that the
sequence {x
n
} generated by 1.8 converges strongly to
FixS∩EPf
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
variational inequality for monotone and k-Lipschitz continuous mappings in a Hilbert space
4 Fixed Point Theory and Applications
and obtain some strong convergence theorems. Our results unify, extend, and improve those
corresponding results in 8, 11 and the references therein.
2. Preliminaries
Let symbols → and denote strong and weak convergence, respectively. It is well known
that
λx
1 − λ
y
2
λ
x
2
1 − λ
y
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
C
x
,P
C
x
− y
≥ 0 2.2
for all x ∈ H and y ∈ C.
It is easy to see that 2.2 is equivalent to
x − y
2
≥
x − P
C
x
2
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
u − λAu
, for some λ>0 ⇒ u ∈ VI
C, A
.
2.4
For solving the equilibrium problem, let us assume that the bifunction F satisfies the
following conditions which were imposed in 2:
A1 Fx, x0 for all x ∈ C;
A2 F is monotone; that is, Fx, yFy, x ≤ 0 for any x, y ∈ C;
A3 for each x, y, z ∈ C,
lim
t↓0
F
tz
1 − t
x, y
≤ F
x, y
;
2.5
A4 for each x ∈ C, y → Fx, y is convex and lower semicontinuous.
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
z ∈ C : F
z, y
1
r
y − z, z − x
≥ 0, ∀y ∈ C
2.7
for all x ∈ H. Then, the following statements hold:
1 T
F
r
is single-valued;
2 T
F
r
is firmly nonexpansive; that is, for any x, y ∈ H,
T
F
r
x
− T
F
r
y
2
≤
T
F
r
x
− T
F
r
y
,x− y
;
2.8
3 FixT
F
r
EPF;
4 EP F is closed and convex.
3. Main Results
In this section, we will introduce a new algorithm based on hybrid and 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 variational
inequality for monotone and k-Lipschitz continuous mappings in a Hilbert space and show
that the sequences generated by the processes converge strongly to a same point.
Theorem 3.1. 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), let A 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 ∩
M
k1
EPF
k
/
∅. Pick any 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
T
F
M
r
M·n
T
F
M−1
r
M−1,n
···T
F
2
r
2,n
T
F
1
r
1,n
x
n
,
y
n
P
C
u
n
− λ
n
Au
n
,
w
n
α
n
x
n
1 − α
n
SP
C
u
n
− λ
n
Ay
n
,
C
n1
{
z ∈ C
n
:
w
n
− z
≤
x
n
− z
}
,
x
n1
P
C
n1
x
0
3.1
6 Fixed Point Theory and Applications
for each n ∈ N.If{λ
n
}⊂a, b for some a, b ∈ 0, 1/k, {α
n
}⊂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
}, {y
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
− x
n
,x
n
− z
≤ 0
, 3.2
we also have that C
n
is convex for each n ∈ N.Thus,{x
n
}, {u
n
}, {y
n
},and{w
n
} are
welldefined. By taking Θ
k
n
T
F
k
r
k·n
T
F
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
.Suppose
that Ω ⊂ C
n
for some n ∈ N.Letv ∈ Ω. Then, by Lemma 2.2 and v P
C
v − λ
n
AvΘ
M
n
v,
we have
u
n
− v
Θ
M
n
x
n
− Θ
M
n
v
≤
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
≤
u
n
− λ
n
Ay
n
− v
2
−
u
n
− λ
n
Ay
n
− v
n
2
u
n
− v
2
−
u
n
− v
n
2
2λ
n
Ay
n
,v− v
n
u
n
− v
2
−
u
n
− v
n
2
2λ
n
Ay
n
− Av, v − y
n
Av, v − y
n
Ay
n
,y
n
− v
n
≤
u
n
− v
2
−
u
n
− v
n
2
2λ
n
Ay
n
,y
n
− v
n
u
n
− v
2
−
u
n
− y
n
2
− 2
u
n
− y
n
,y
n
− v
n
−
y
n
− v
n
2
2λ
n
Ay
n
,y
n
− v
n
u
n
− v
2
−
u
n
− y
n
2
−
y
n
− v
n
2
2
u
n
− λ
n
Ay
n
− y
n
,v
n
− y
n
.
3.4
Moreover, from y
n
P
C
u
n
− λ
n
Au
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
u
n
− λ
n
Au
n
− y
n
,v
n
− y
n
λ
n
Au
n
− λ
n
Ay
n
,v
n
− y
n
≤
λ
n
Au
n
− λ
n
Ay
n
,v
n
− y
n
≤ λ
n
k
u
n
− y
n
v
n
− y
n
.
3.6
So, we have
v
n
− v
2
≤
u
n
− v
2
−
u
n
− y
n
2
−
y
n
− v
n
2
2λ
n
k
u
n
− y
n
v
n
− y
n
≤
u
n
− v
2
−
u
n
− y
n
2
−
y
n
− v
n
2
λ
2
n
k
2
u
n
− y
n
2
v
n
− y
n
2
u
n
− v
2
λ
2
n
k
2
− 1
u
n
− y
n
2
≤
u
n
− v
2
.
3.7
From 3.7 and the definition of w
n
, we have
w
n
− v
2
≤ α
n
x
n
− v
2
1 − α
n
Sv
n
− v
2
≤ α
n
x
n
− v
2
1 − α
n
v
n
− v
2
≤ α
n
x
n
− v
2
1 − α
n
u
n
− v
2
λ
2
n
k
2
− 1
u
n
− y
n
2
3.8
≤ α
n
x
n
− v
2
1 − α
n
x
n
− v
2
1 − α
n
λ
2
n
k
2
− 1
u
n
− y
n
2
x
n
− v
2
1 − α
n
λ
2
n
k
2
− 1
u
n
− y
n
2
≤
x
n
− v
2
,
3.9
and hence v ∈ C
n1
. This implies that Ω ⊂ C
n
for all n ∈ N.
Step 2. We show that lim
n →∞
x
n
− w
n
→0 and lim
n →∞
u
n
− y
n
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
, ∀n ∈ N. 3.10
Therefore, {x
n
} is bounded. From 3.3–3.9, we also obtain that {w
n
}, {v
n
},and{u
n
} are
bounded. Since x
n1
∈ C
n1
⊆ C
n
and x
n
P
C
n
x
0
, we have
x
n
− 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
x
0
∈ C
n1
⊂ C
n
, we have
x
0
− x
n
,x
n
− x
n1
≥ 0, ∀n ∈ N. 3.12
So
x
n
− x
n1
2
x
n
− x
0
x
0
− x
n1
2
x
n
− x
0
2
2
x
n
− x
0
,x
0
− x
n1
x
0
− x
n1
2
x
n
− x
0
2
2
x
n
− x
0
,x
0
− x
n
x
n
− 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
n
,x
n
− x
n1
x
0
− x
n1
2
≤x
n
− x
0
2
− 2x
n
− x
0
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
n
0.
3.14
Since x
n1
∈ C
n1
, we have w
n
− x
n1
≤x
n
− x
n1
, and hence
x
n
− w
n
≤
x
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
2
≤
1
1 − α
n
1 − λ
2
n
k
2
x
n
− v
2
−
w
n
− v
2
1
1 − α
n
1 − λ
2
n
k
2
x
n
− v
−
w
n
− v
x
n
− v
w
n
− v
≤
1
1 − α
n
1 − λ
2
n
k
2
x
n
− w
n
x
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
− Θ
k−1
n
x
n
0,k 1, 2, ,M.
3.17
Fixed Point Theory and Applications 9
Indeed, let v ∈ Ω. It follows form the firmly nonexpansiveness of T
F
k
r
k,n
that we have, for each
k ∈{1, 2, ,M},
Θ
k
n
x
n
− v
2
T
F
k
r
k,n
Θ
k−1
n
x
n
− T
F
k
r
k,n
v
2
≤
Θ
k
n
x
n
− v, Θ
k−1
n
x
n
− v
1
2
Θ
k
n
x
n
− v
2
Θ
k−1
n
x
n
− v
2
−
Θ
k
n
x
n
− Θ
k−1
n
x
n
2
.
3.18
Thus, we get
Θ
k
n
x
n
− v
2
≤
Θ
k−1
n
x
n
− v
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
2
≤
Θ
0
n
x
n
− v
2
−
Θ
k
n
x
n
− Θ
k−1
n
x
n
2
−
Θ
k−1
n
x
n
− Θ
k−2
n
x
n
2
−···−
Θ
2
n
x
n
− Θ
1
n
x
n
2
−
Θ
1
n
x
n
− Θ
0
n
x
n
2
≤
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
,and3.20, we have, for each k ∈{1, 2, ,M},
w
n
− v
2
≤ α
n
x
n
− v
2
1 − α
n
u
n
− v
2
≤ α
n
x
n
− v
2
1 − α
n
Θ
k
n
x
n
− v
2
, ∀k ∈
{
1, 2, ,M
}
≤ α
n
x
n
− v
2
1 − α
n
x
n
− v
2
−
Θ
k
n
x
n
− Θ
k−1
n
x
n
2
≤
x
n
− v
2
−
1 − α
n
Θ
k
n
x
n
− Θ
k−1
n
x
n
2
,
3.21
which implies that
1 − α
n
Θ
k
n
x
n
− Θ
k−1
x
n
≤
x
n
− v
2
−
w
n
− v
2
x
n
− v
w
n
− v
x
n
− v
−
w
n
− v
≤
x
n
− v
w
n
− v
x
n
− w
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
− y
n
,we
get
lim
n →∞
x
n
− y
n
0.
3.23
We observe that
v
n
− y
n
P
C
u
n
− λ
n
Ay
n
− P
C
u
n
− λ
n
Au
n
≤
λ
n
Au
n
− λ
n
Ay
n
≤ λ
n
k
u
n
− y
n
,
3.24
which implies that
lim
n →∞
v
n
− y
n
0.
3.25
Since x
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
Since Sv
n
− v
n
≤Sv
n
− x
n
x
n
− y
n
y
n
− v
n
,weget
lim
n →∞
Sv
n
− v
n
0.
3.27
Step 5. We show that x
n
→ w, where w P
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
− w
n
→0, v
n
− y
n
→0, and u
n
− y
n
→0thatw
n
i
w,
y
n
i
w,andv
n
i
w.
In order to show that w ∈ Ω, we first show that w ∈
M
k1
EPF
k
. Indeed, by definition
of T
F
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
k−1
n
x
n
≥ 0, ∀y ∈ C.
3.28
From A2, we also have
1
r
k,n
y − Θ
k
n
x
n
, Θ
k
n
x
n
− Θ
k−1
n
x
n
≥ F
k
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
i
x
n
i
r
k,n
i
≥ F
k
y, Θ
k
n
i
x
n
i
, ∀y ∈ C.
3.30
From A4, Θ
k
n
i
x
n
i
−Θ
k−1
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
t
ty 1 − tw. Since y ∈ C and
w ∈ C, we have y
t
∈ C, and hence F
k
y
t
,w ≤ 0. So, from A1 and A4, we have, for each
k ∈{1, 2, ,M},
0 F
k
y
t
,y
t
≤ tF
k
y
t
,y
1 − t
F
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
i
wand w
/
Sw,
from Opial’s condition 13, we have
lim inf
i →∞
v
n
i
− w
< lim inf
i →∞
v
n
i
− Sw
≤ lim sup
i →∞
v
n
i
− Sv
n
i
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
⎧
⎨
⎩
Av N
C
v, v ∈ C,
∅,v/∈ C.
3.34
It is worth to note that in this case the mapping T is maximal monotone and 0 ∈ Tv if and
only if v ∈ VIC, Asee 14.Letv, u ∈ GT. Since u − Av ∈ N
C
v and v
n
∈ C, we have
12 Fixed Point Theory and Applications
v − v
n
,u − Av≥0. On the other hand, from v
n
P
C
u
n
− λ
n
Ay
n
and v ∈ C, we have
v − v
n
,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
,Av
≥
v − v
n
i
,Av
−
v − v
n
i
,
v
n
i
− u
n
i
λ
n
i
Ay
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
i
,Av
n
i
− Ay
n
i
−
v − v
n
i
,
v
n
i
− u
n
i
λ
n
i
≥
v − v
n
i
,Av
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
0andA is k-Lipschitz continuous, we obtain that lim
n →∞
Av
n
−
Ay
n
0. From v
n
i
w, lim inf
n →∞
λ
n
> 0, and lim
n →∞
v
n
− u
n
0, we obtain
v − w, u
≥ 0. 3.36
Since T is maximal monotone, we have w ∈ T
−1
0, and hence w ∈ VIC, A, which implies that
w ∈ Ω. Finally, we show that x
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
≤
w − x
0
≤ lim inf
i →∞
x
n
i
− x
0
≤ lim sup
i →∞
x
n
i
− x
0
≤
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
and the Kadec-Klee property of H, we have x
n
i
− x
0
→ w − x
0
,
and hence x
n
i
→ w. This implies that x
n
→ w.Itiseasytoseethatu
n
→ w, y
n
→ w,and
w
n
→ w. The proof is now complete.
By Theorem 3.1, we can easily obtain some new results as follows.
Corollary 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H.LetF 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 let S be a nonexpansive mapping from C into itself such that Ω
Fixed Point Theory and Applications 13
FixS ∩ VIC, A ∩ EPF
/
∅. Pick any 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
r
n
y − u
n
,u
n
− x
n
≥ 0, ∀y ∈ C,
y
n
P
C
u
n
− λ
n
Au
n
,
w
n
α
n
x
n
1 − α
n
SP
C
u
n
− λ
n
Ay
n
,
C
n1
{
z ∈ C
n
:
w
n
− z
≤
x
n
− z
}
,
x
n1
P
C
n1
x
0
3.40
for each n ∈ N.If{λ
n
}⊂a, b for some a, b ∈ 0, 1/k, {α
n
}⊂c, d for some c,d ∈ 0, 1, and
{r
n
}⊂0, ∞ satisfies lim inf
n →∞
r
n
> 0,then{x
n
}, {u
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
/
∅. Pick any
x
0
∈ H, and set C
1
C.Let{x
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
r
2,n
T
F
1
r
1,n
x
n
,
w
n
α
n
x
n
1 − α
n
Su
n
,
C
n1
{
z ∈ C
n
:
w
n
− z
≤
x
n
− z
}
,
x
n1
P
C
n1
x
0
3.41
for each n ∈ N.If{α
n
}⊂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
}, {y
n
}, and {w
n
} be sequences generated by x
1
P
C
1
x
0
and
y
n
P
C
x
n
− λ
n
Ax
n
,
w
n
α
n
x
n
1 − α
n
SP
C
u
n
− λ
n
Ay
n
,
C
n1
{
z ∈ C
n
:
w
n
− z
≤
x
n
− z
}
,
x
n1
P
C
n1
x
0
3.42
14 Fixed Point Theory and Applications
for each n ∈ N.If{λ
n
}⊂a, b for some a, b ∈ 0, 1/k, {α
n
}⊂c, d for some c,d ∈ 0, 1,then
{x
n
}, {y
n
}, and {w
n
} converge strongly to P
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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