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Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 943275, 25 pages
doi:10.1155/2010/943275
Research Article
Some Iterative Methods for Solving Equilibrium
Problems and Optimization Problems
Huimin He,
1
Sanyang Liu,
1
and Qinwei Fan
2
1
Department of Mathematics, Xidian University, Xi’An 710071, China
2
Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China
Correspondence should be addressed to Huimin He,
Received 3 September 2010; Accepted 29 October 2010
Academic Editor: Vijay Gupta
Copyright q 2010 Huimin He 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 for finding a common element of the set of solutions of
the equilibrium problems, the set of solutions of variational inequality for a relaxed cocoercive
mapping, and the set of fixed points of a nonexpansive mapping. The results presented in this
paper extend and improve some recent results of Ceng and Yao 2008,Yao2007, S. Takahashi
and W. Takahashi 2007, Marino and Xu 2006, Iiduka and Takahashi 2005,Suetal.2008,and
many others.
1. Introduction
Throughout this paper, we always assume that H is a real Hilbert space with inner product
·, · and norm ·, respectively, C is a nonempty closed and convex subset of H,andP
C
is
the metric projection of H onto C. In the following, we denote by “ → ” strong convergence,
by “” weak convergence, and by “R” the real number set. Recall that a mapping S : C → C
is called nonexpansive i f
Sx − Sy≤x − y, ∀x, y ∈ C. 1.1
We denote by FS the set of fixed points of the mapping S.
For a given nonlinear operator A, consider the problem of finding u ∈ C such that
Au, v − u≥0, ∀v ∈ C, 1.2
2 Journal of Inequalities and Applications
which is called the variational inequality. For the recent applications, sensitivity analysis,
dynamical systems, numerical methods, and physical formulations of the variational
inequalities, see 1–24 and the references therein.
For a given z ∈ H, u ∈ C satisfies the inequality
u − z, v − u≥0, ∀v ∈ C, 1.3
if and only if u P
C
z, where P
C
is the projection of the Hilbert space onto the closed convex
set C.
It is known that projection operator P
C
is nonexpansive. It is also known that P
C
satisfies
x − y, P
C
x − P
C
y≥P
C
x − P
C
y
2
, ∀x, y ∈ H.
1.4
Moreover, P
C
x is characterized by the properties P
C
x ∈ C and x − P
C
x, P
C
x − y≥0 for all
y ∈ C.
Using characterization of the projection operator, one can easily show that the
variational inequality 1.2 is equivalent to finding the fixed point problem of finding u ∈ C
which satisfies the relation
u P
C
u − λAu
, 1.5
where λ>0 is a constant.
This fixed-point formulation has been used to suggest the following iterative scheme.
For a given u
0
∈ C,
u
n1
P
C
u
n
− λAu
n
,n 1, 2, , 1.6
which is known as the projection iterative method for solving the variational inequality
1.2. The convergence of this iterative method requires that the operator A must be strongly
monotone and Lipschitz continuous. These strict conditions rule out their applications in
many important problems arising in the physical and engineering sciences. To overcome
these drawbacks, Noor 2, 3 used the technique of updating the solution to suggest the two-
step or predictor-corrector method for solving the variational inequality 1.2. For a given
u
0
∈ C,
w
n
P
C
u
n
− λAu
n
,
u
n1
P
C
w
n
− λAw
n
,n 0, 1, 2, ,
1.7
which is also known as the modified double-projection method. For the convergence analysis
and applications of this method, see the works of Noor 3 and Y. Yao and J C. Yao 16.
Numerous problems in physics, optimization, and economics reduce to find a
solution of 2.12. Some methods have been proposed to solve the equilibrium problem;
see 4, 5. Combettes and Hirstoaga 4 introduced an iterative scheme for finding the best
approximation to the initial data when EPF is nonempty and proved a strong convergence
Journal of Inequalities and Applications 3
theorem. Very recently, S. Takahashi and W. Takahashi 6 also introduced a new iterative
scheme,
F
y
n
,u
1
r
n
u − y
n
,y
n
− x
n
≥0, ∀u ∈ C,
x
n1
α
n
f
x
n
1 − α
n
Ty
n
,
1.8
for approximating a common element of the set of fixed points of a nonexpansive nonself
mapping and the set of solutions of the equilibrium problem and obtained a strong
convergence theorem in a real Hilbert space.
Iterative methods for nonexpansive mappings have recently been applied to solve
convex minimization problems; see 7–11 and the references therein. A typical problem is
to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping
on a real Hilbert space H:
min
x∈C
1
2
Ax, x−x,b,
1.9
where A is a linear bounded operator, C is the fixed point set of a nonexpansive mapping S,
and b is a given point in H.In10, 11, it is proved that the sequence {x
n
} defined by the
iterative method below, with the initial guess x
0
∈ H chosen arbitrarily,
x
n1
I − α
n
A
Sx
n
α
n
b, n ≥ 0, 1.10
converges strongly to the unique solution of the minimization problem 1.9 provided the
sequence {α
n
} satisfies certain conditions. Recently, Marino and Xu 8 introduced a new
iterative scheme by the viscosity approximation method 12:
x
n1
I − α
n
A
Sx
n
α
n
γf
x
n
,n≥ 0. 1.11
They proved that the sequence {x
n
} generated by the above iterative scheme converges
strongly to the unique solution of the variational inequality
A − γf
x
∗
,x− x
∗
≥0,x∈ C, 1.12
which is the optimality condition for the minimization problem
min
x∈C
1
2
Ax, x−h
x
,
1.13
where C is the fixed point set of a nonexpansive mapping S and h a potential function for γf
i.e., h
xγfx for x ∈ H.
4 Journal of Inequalities and Applications
For finding a common element of the set of fixed points of nonexpansive mappings
and t he set of solution of variational inequalities f or α-cocoercive map, Takahashi and Toyoda
13 introduced the following iterative process:
x
n1
α
n
x
n
1 − α
n
SP
C
x
n
− λ
n
Ax
n
, 1.14
for every n 0, 1, 2, , where A is α-cocoercive, x
0
x ∈ C, {α
n
} is a sequence in 0,1,
and {λ
n
} is a sequence in 0, 2α. They showed that, if FS ∩ VIC, A is nonempty, then the
sequence {x
n
} generated by 1.14 converges weakly to some z ∈ FS ∩ VIC, A. Recently,
Iiduka and Takahashi 14 proposed another iterative scheme as follows:
x
n1
α
n
x
1 − α
n
SP
C
x
n
− λ
n
Ax
n
, 1.15
for every n 0, 1, 2, , where A is α-cocoercive, x
0
x ∈ C, {α
n
} is a sequence in 0,1,
and {λ
n
} is a sequence in 0, 2α. They proved that the sequence {x
n
} converges strongly to
z ∈ FS ∩ VIC, A.
Recently, Chen et al. 15 studied the following iterative process:
x
n1
α
n
f
x
n
1 − α
n
SP
C
x
n
− λ
n
Ax
n
1.16
and also obtained a strong convergence theorem by viscosity approximation method.
Inspired and motivated by the ideas and techniques of Noor 2, 3 and Y. Yao and J C.
Yao 16 introduce the following iterative scheme.
Let C be a closed convex subset of real Hilbert space H.LetA be an α-inverse strongly
monotone mapping of C into H, and let S be a nonexpansive mapping of C into itself such
that z ∈ FS ∩ VIC, A
/
∅. Suppose that x
1
u ∈ C and {x
n
}, {y
n
} are given by
y
n
P
C
x
n
− λ
n
Ax
n
,
x
n1
α
n
u β
n
x
n
γ
n
SP
C
y
n
− λ
n
Ay
n
,
1.17
where {α
n
}, {β
n
},and{γ
n
} are the sequences in 0, 1 and {λ
n
} is a sequence in 0, 2α. They
proved that the sequence {x
n
} defined by 1.17 converges strongly to common element of
the set of fixed points of a nonexpansive mapping and the set of solutions of the variational
inequality for α-inverse-strongly monotone mappings under some parameters controlling
conditions.
In this paper motivated by the iterative schemes considered in 6, 15, 16, we introduce
a general iterative process as follows:
F
y
n
,u
1
r
n
u − y
n
,y
n
− x
n
≥0, ∀u ∈ C,
x
n1
α
n
γf
x
n
β
n
x
n
1 − β
n
I − α
n
A
SP
C
I − s
n
B
y
n
,
1.18
where A is a linear bounded operator and B is relaxed cocoercive. We prove that the sequence
{x
n
} generated by the above iterative scheme converges strongly to a common element of
Journal of Inequalities and Applications 5
the set of fixed points of a nonexpansive mapping, the set of solutions of the variational
inequalities for a relaxed cocoercive mapping, and the set of solutions of the equilibrium
problems 2.12, which solves another variational inequality
γf
q
− Aq, q − P ≤0, ∀p ∈ F, 1.19
where F FS ∩ VIC, B ∩ EPF and is also the optimality condition for the minimization
problem min
x∈F
1/2Ax, x−hx, where h is a potential function for γf i.e., h
xγfx
for x ∈ H. The results obtained in this paper improve and extend the recent ones announced
by S. Takahashi and W. Takahashi 6, Iiduka and Takahashi 14,MarinoandXu8, Chen
et al. 15,Y.YaoandJ C.Yao16, Ceng and Yao 22,Suetal.17, and many others.
2. Preliminaries
For solving the equilibrium problem for a bifunction F : C × C → R, let us assume that F
satisfies the following conditions:
A1 Fx, x0 for all x ∈ C;
A2 F is monotone, that is, Fx, yFy,x ≤ 0 for all x, y ∈ C;
A3 for each x, y, z ∈ C, lim
t → 0
Ftz 1 − tx, y ≤ Fx, y;
A4 for each x ∈ C, y → Fx, y is convex and lower semicontinuous.
Recall the following.
1 B is called ν-strong monotone if for all x, y ∈ C, we have
Bx − By, x − y≥νx − y
2
,
2.1
for a constant ν>0. This implies that
Bx − By≥νx − y, 2.2
that is, B is ν-expansive, and when ν 1, it is expansive.
2 B is said to be μ-cocoercive 2, 3 if for all x, y ∈ C, we have
Bx − By, x − y
≥ μBx − By
2
, for a constant μ>0.
2.3
Clearly, every μ-cocoercive map B is 1/μ-Lipschitz continuous.
3 B is called −μ-cocoercive if there exists a constant μ>0 such that
Bx − By, x − y≥−μBx − By
2
, ∀x, y ∈ C.
2.4
4 B is said to be relaxed μ, ν-cocoercive if there exists two constants μ, ν > 0 such that
Bx − By, x − y≥−μBx − By
2
νx − y
2
, ∀x, y ∈ C,
2.5
6 Journal of Inequalities and Applications
for μ 0, B is ν-strongly monotone. This class of maps are more general than the class of
strongly monotone maps. It is easy to see that we have the following implication: ν-strongly
monotonicity ⇒ relaxed μ, ν-cocoercivity.
We will give the practical example of the relaxed μ, ν-cocoercivity and Lipschitz
continuous operator.
Example 2.1. Let Tx κx, for all x ∈ C, for a constant κ>1; then, T is relaxed μ, ν-cocoercive
and Lipschitz continuous. Especially, T is ν-strong monotone.
Proof. 1. Since Tx κx, for all x ∈ C, we have T : C → C. For for all x, y ∈ C, for all μ ≥ 0, we
also have the below
Tx − Ty,x − y κx − y
2
≥−μTx − Ty
2
κ − 1
x − y
2
.
2.6
Taking ν κ − 1, it is clear that T is relaxed μ, ν-cocoercive.
2. Obviously, for for all x, y ∈ C
Tx − Ty≤
κ 1
x − y. 2.7
Then, T is κ 1 Lipschitz continuous.
Especially, Taking μ 0, we observe that
Tx − Ty,x − y≥
κ − 1
x − y
2
. 2.8
Obviously, T is ν-strong monotone.
The proof is completed.
5 A mapping f : H → H is said to be a contraction if there exists a coefficient
α 0 ≤ α<1 such that
f
x
− f
y
≤αx − y, ∀x, y ∈ H. 2.9
6 An operator A is strong positive if there exists a constant
γ>0 with the property
Ax, x≥
γx
2
, ∀x ∈ H.
2.10
7 A set-valued mapping T : H → 2
H
is called monotone if for all x, y ∈ H, f ∈ Tx,
and g ∈ Ty imply x − y, f − g≥0. A monotone mapping T : H → 2
H
is maximal if the
graph of GT of T is not properly contained in the graph of any other monotone mapping.
It is well known that a monotone mapping T is maximal if and only if for x, f ∈ H × H,
x − y, f − g≥0 for every y,g ∈ GT implies f ∈ Tx.
Journal of Inequalities and Applications 7
Let B be a monotone map of C into H and let N
C
v be the normal cone to C at v ∈ C,
that is, N
C
v {w ∈ H : v − u, w≥0, ∀u ∈ C} and define
Tv
⎧
⎨
⎩
Bv N
C
v, v ∈ C,
∅,v
∈ C.
2.11
Then T is the maximal monotone and 0 ∈ Tv if and only if v ∈ VIC, B;see1.
Related to the variational inequality problem 1.2, we consider the equilibrium
problem, which was introduced by Blum and Oettli 19 and Noor and Oettli 20 in 1994. To
be more precise, let F be a bifunction of C × C into R, where R is the set of real numbers.
For given bifunction F·, · : C × C → R, we consider the problem of finding x ∈ C
such that
F
x, y
≥ 0, ∀y ∈ C 2.12
which is known as the equilibrium problem. The set of solutions of 2.12 is denoted by EPF.
Given a mapping T : C → H,letFx, yTx,y
− x for all x, y ∈ C. Then x ∈ EPF if
and only if Tx,y − x≥0 for all y ∈ C,thatis,x is a solution of the variational inequality.
That is to say, the variational inequality problem is included by equilibrium problem, and the
variational inequality problem is the special case of equilibrium problem.
Assume that
T is a potential function for T i.e., ∇TxTx for all x ∈ C, it is well
known that x ∈ C satisfies the optimality condition Tx,y − x≥0 for all y ∈ C if and only if
find a point x ∈ C such that
Tx min
y∈C
T
y
.
2.13
We can rewrite the variational inequality Tx,y − x≥0 for all y ∈ C as, for any γ>0,
x −
x − γTx
,y− x
≥ 0 ∀y ∈ C. 2.14
If we introduce the nearest point projection P
C
from H onto C,
P
C
x arg min
u∈C
1
2
x − u
2
,x∈ H,
2.15
which is characterized b y the inequality
C x P
C
x ⇐⇒ x − x, y − x≤0, ∀y ∈ C, 2.16
then we see from the above 2.14 that the minimization 2.13 is equivalent to the fixed point
problem
P
C
x − γTx
x. 2.17
8 Journal of Inequalities and Applications
Therefore, they have a relation as follows:
finding x ∈ C, x ∈ EP
F
Finding x ∈ C, F
x, y
≥ 0, ∀y ∈ C, let F
x, y
γTx,y − x≥0, ∀γ>0, ∀y ∈ C.
min
y∈C
T
y
, where ∇T
x
T
x
, ∀x ∈ C.
x ∈ Fix
P
C
I − γT
.
2.18
In addition to this, based on the result 3 of Lemma 2.7,FixT
r
EPF,weknowif
the element x ∈ F : FixS ∩ EPF ∩ VIC, B, we have x is the solution of the nonlinear
equation
x − SP
C
I − γB
T
r
x 0, ∀γ>0, 2.19
where T
r
is defined as in Lemma 2.7. Once we have the solutions of the equation 2.19,
then it simultaneously solves the fixed points problems, equilibrium points problems, and
variational inequalities problems. Therefore, the constrained set F : FixS∩EPF∩VIC, B
is very important and applicable.
We now recall some well-known concepts and results. It is well-known that for all
x, y ∈ H and λ ∈ 0, 1 there holds
λx
1 − λ
y
2
λx
2
1 − λ
y
2
− λ
1 − λ
x − y
2
.
2.20
A space X is said to satisfy Opial’s condition 18 if for each sequence {x
n
} in X which
converges weakly to point x ∈ X, we have
lim inf
n →∞
x
n
− x < lim inf
n →∞
x
n
− y, ∀y ∈ X, y
/
x.
2.21
Lemma 2.2 see 9, 10. Assume that {α
n
} is a sequence of nonnegative real numbers such that
α
n1
≤
1 − γ
n
α
n
δ
n
, 2.22
where γ
n
is a sequence in (0,1) and {δ
n
} is a sequence such that
i
∞
n1
γ
n
∞;
ii lim sup
n →∞
δ
n
/γ
n
≤ 0 or
∞
n1
|δ
n
| < ∞.
Then lim
n →∞
α
n
0.
Journal of Inequalities and Applications 9
Lemma 2.3. In a real Hilbert space H, the following inequality holds:
x y
2
≤x
2
2
y, x y
, ∀x, y ∈ H.
2.23
Lemma 2.4 Marino and Xu 8. Assume that B is a strong positive linear bounded operator on a
Hilbert space H with coefficient
γ>0 and 0 <ρ≤B
−1
.ThenI − ρB≤1 − ργ.
Lemma 2.5 see 21. Let {x
n
} and {y
n
} be bounded sequences in a Banach space X and let {β
n
} be
a sequence in 0, 1 with 0 < lim inf
n →∞
β
n
≤ lim sup
n →∞
β
n
< 1. Suppose x
n1
1 −β
n
z
n
β
n
x
n
for all integers n ≥ 0 and lim sup
n →∞
z
n1
− z
n
−x
n1
− x
n
≤ 0. Then, lim
n →∞
z
n
− x
n
0.
Lemma 2.6 Blum and Oettli 19. Let C be a nonempty closed convex subset of H and let F be a
bifunction of C × C into 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.
2.24
Lemma 2.7 Combettes and Hirstoaga 4. Assume that F : C ×C → R satisfies (A1)–(A4). For
r>0 and x ∈ H, define a mapping T
r
: H → C as follows:
T
r
x
z ∈ C : F
z, y
1
r
y − z, z − x≥0, ∀y ∈ C
2.25
for all z ∈ H. Then, the following hold:
1 T
r
is single-valued;
2 T
r
is firmly nonexpansive, that is, for any x, y ∈ H, T
r
x − T
r
y
2
≤T
r
x − T
r
y, x − y;
3 FT
r
EPF;
4 EPF is closed and convex.
3. Main Results
Theorem 3.1. Let C be a nonempty closed convex subset of a Hilbert space H.LetF be a bifunction
of C × C into R which satisfies (A1)–(A4), let S be a nonexpansive mapping of C into H, and let B be a
λ-Lipschitzian, relaxed μ, ν-cocoercive map of C into H such that F FS ∩EPF∩VIC, B
/
∅.
Let A be a strongly positive linear bounded operator with coefficient
γ>0. Assume that 0 <γ<γ/α.
Let f be a contraction of H into itself with a coefficient α 0 <α<1 and let {x
n
} and {y
n
} be
sequences generated by x
1
∈ H and
F
y
n
,η
1
r
n
η − y
n
,y
n
− x
n
≥0, ∀η ∈ C,
x
n1
α
n
γf
x
n
β
n
x
n
1 − β
n
I − α
n
A
SP
C
I − s
n
B
y
n
3.1
10 Journal of Inequalities and Applications
for all n,where{α
n
}, {β
n
}⊂0, 1 and {r
n
}, {s
n
}⊂0, ∞ satisfy
C1 lim
n →∞
α
n
0;
C2
∞
n1
α
n
∞;
C3 0 < lim inf
n →∞
β
n
≤ lim sup
n →∞
β
n
< 1;
C4
∞
n1
|α
n1
− α
n
| < ∞,
∞
n1
|r
n1
− r
n
| < ∞ and
∞
n1
|s
n1
− s
n
| < ∞;
C5 lim inf
n →∞
r
n
> 0;
C6 {s
n
}∈a, b for some a, b with 0 ≤ a ≤ b ≤ 2ν − μλ
2
/λ
2
.
Then, both {x
n
} and {y
n
} converge strongly to q ∈ F,whereq P
F
γf I − Aq,which
solves the following variational inequality:
γf
q
− Aq, p − q≤0, ∀p ∈ F. 3.2
Proof. Note that from the condition C1, we may assume, without loss of generality, that
α
n
≤ 1 − β
n
A
−1
. Since A is a strongly positive bounded linear operator on H, then
A sup
{|
Ax, x
|
: x ∈ H, x 1
}
. 3.3
observe that
1 − β
n
I − α
n
A
x, x 1 − β
n
− α
n
Ax, x
≥ 1 − β
n
− α
n
A
≥ 0,
3.4
that is to say 1 − β
n
I − α
n
A is positive. It follows that
1 − β
n
I − α
n
A sup
1 − β
n
I − α
n
A
x, x : x ∈ H, x 1
sup
1 − β
n
− α
n
Ax, x : x ∈ H, x 1
≤ 1 − β
n
− α
n
γ.
3.5
Journal of Inequalities and Applications 11
First, we show that I − s
n
B is nonexpansive. Indeed, from the relaxed μ, ν-cocoercive and
λ-Lipschitzian definition on B and condition C6, we have
I − s
n
B
x −
I − s
n
B
y
2
x − y
− s
n
Bx − By
2
x − y
2
− 2s
n
x − y, Bx − By s
2
n
Bx − By
2
≤x − y
2
− 2s
n
−μBx − By
2
νx − y
2
s
2
n
Bx − By
2
≤x − y
2
2s
n
λ
2
μx − y
2
− 2s
n
νx − y
2
λ
2
s
2
n
x − y
2
1 2s
n
λ
2
μ − 2s
n
ν λ
2
s
2
n
x − y
2
≤x − y
2
,
3.6
which implies that the mapping I − s
n
B is nonexpansive.
Now, we observe that {x
n
} is bounded. Indeed, take p ∈ F,sincey
n
T
r
n
x
n
, we have
y
n
− p T
r
n
x
n
− T
r
n
p≤x
n
− p. 3.7
Put ρ
n
P
C
I − s
n
By
n
,sincep ∈ VIC, B, we have p P
C
I − s
n
Bp. Therefore, we have
ρ
n
− p P
C
I − s
n
B
y
n
− P
C
I − s
n
B
p
≤
I − s
n
B
y
n
−
I − s
n
B
p
≤y
n
− p≤x
n
− p.
3.8
Due to 3.5, it follows that
x
n1
− p α
n
γf
x
n
− Ap
β
n
x
n
− p
1 − β
n
I − α
n
A
Sρ
n
− p
≤
1 − β
n
− α
n
γ
x
n
− p β
n
x
n
− p α
n
γf
x
n
− Ap
≤
1 − α
n
γ
x
n
− p α
n
γf
x
n
− f
p
α
n
γf
p
− Ap
≤
1 − α
n
γ
x
n
− p α
n
γαx
n
− p α
n
γf
p
− Ap
1 −
γ − γα
α
n
x
n
− p α
n
γf
p
− Ap.
3.9
It follows from 3.9 that
x
n
− p≤max
x
0
− p,
γf
p
− Ap
γ − γα
,n≥ 0. 3.10
Hence, {x
n
} is bounded, so are {fx
n
}, y
n
,andρ
n
.
12 Journal of Inequalities and Applications
Next, we show that
lim
n →∞
x
n1
− x
n
0.
3.11
Observing that y
n
T
r
n
x
n
and y
n1
T
r
n1
x
n1
, we have
F
y
n
,η
1
r
n
η − y
n
,y
n
− x
n
≥0, ∀η ∈ C,
3.12
F
y
n1
,η
1
r
n1
η − y
n1
,y
n1
− x
n1
≥ 0, ∀η ∈ C.
3.13
Putting η y
n1
in 3.12 and η y
n
in 3.13, we have
F
y
n
,y
n1
1
r
n
y
n1
− y
n
,y
n
− x
n
≥0, ∀η ∈ C,
F
y
n1
,y
n
1
r
n1
y
n
− y
n1
,y
n1
− x
n1
≥ 0, ∀η ∈ C.
3.14
It follows from A2 that
y
n1
− y
n
,
y
n
− x
n
r
n
−
y
n1
− x
n1
r
n1
≥ 0. 3.15
That is,
y
n1
− y
n
,y
n
− y
n1
y
n1
− x
n
−
r
n
r
n1
y
n1
− x
n1
≥ 0. 3.16
Without loss of generality, let us assume that there exists a real number m such that r
n
>m>0
for all n. It follows that
y
n1
− y
n
2
≤y
n1
− y
n
x
n1
− x
n
1 −
r
n
r
n1
y
n1
− x
n1
. 3.17
It follows that
y
n1
− y
n
≤x
n1
− x
n
1 −
r
n
r
n1
y
n1
− x
n1
≤x
n1
− x
n
M
1
m
|
r
n1
− r
n1
|
,
3.18
Journal of Inequalities and Applications 13
where M
1
is an appropriate constant such that sup
n≥1
y
n
− x
n
≤M
1
.Notethat
ρ
n1
− ρ
n
P
C
I − s
n1
B
y
n1
− P
C
I − s
n
B
y
n
≤
I − s
n1
B
y
n1
−
I − s
n
B
y
n
I − s
n1
B
y
n1
−
I − s
n1
B
y
n
s
n
− s
n1
By
n
≤y
n1
− y
n
|
s
n
− s
n1
|
By
n
.
3.19
Substituting 3.18 into 3.19 yields that
ρ
n1
− ρ
n
≤x
n1
− x
n
M
2
|
r
n1
− r
n
|
|
s
n1
− s
n
|
, 3.20
where M
2
is an appropriate constant such that M
2
max{sup
n≥1
By
n
,M
1
/m}.
Define
x
n1
1 − β
n
z
n
β
n
x
n
,n≥ 0. 3.21
Observe that from the definition of y
n
,weobtain
z
n1
− z
n
x
n2
− β
n1
x
n1
1 − β
n1
−
x
n1
− β
n
x
n
1 − β
n
α
n1
γf
x
n1
1 − β
n1
I − α
n1
A
Sρ
n1
1 − β
n1
−
α
n
γf
x
n
1 − β
n
I − α
n
A
Sρ
n
1 − β
n
α
n1
1 − β
n1
γf
x
n1
−
α
n
1 − β
n
γf
x
n
α
n
1 − β
n
ASρ
n
−
α
n1
1 − β
n1
ASρ
n1
Sρ
n1
− Sρ
n
α
n1
1 − β
n1
γf
x
n1
− ASρ
n1
α
n
1 − β
n
ASρ
n
− γf
x
n
Sρ
n1
− Sρ
n
.
3.22
14 Journal of Inequalities and Applications
It follows that with
z
n1
− z
n
−x
n1
− x
n
≤
α
n1
1 − β
n1
γf
x
n1
ASρ
n1
α
n
1 − β
n
γf
x
n
ASρ
n
ρ
n1
− ρ
n
−x
n1
− x
n
≤
α
n1
1 − β
n1
γf
x
n1
ASρ
n1
α
n
1 − β
n
γf
x
n
ASρ
n
M
2
|
r
n1
− r
n
|
|
s
n1
− s
n
|
.
3.23
This together with C1, C3,andC4 implies that
lim sup
n →∞
z
n1
− z
n
−x
n1
− x
n
≤ 0.
3.24
Hence, by Lemma 2.5,weobtainz
n
− x
n
→0asn →∞.
Consequently,
lim
n →∞
x
n1
− x
n
lim
n →∞
1 − β
n
z
n
− x
n
0.
3.25
Note that
x
n1
− x
n
α
n
γf
x
n
− Ax
n
1 − β
n
I − α
n
A
Sρ
n
− x
n
≤ α
n
γf
x
n
− Ax
n
1 − β
n
− α
n
γ
Sρ
n
− x
n
.
3.26
This together with 3.25 implies that
Sρ
n
− x
n
−→0. 3.27
For p ∈ F, we have
y
n
− p
2
T
r
n
x
n
− T
r
n
p
2
≤T
r
n
x
n
− T
r
n
p, x
n
− p
y
n
− p, x
n
− p
1
2
y
n
− p
2
x
n
− p
2
−x
n
− y
n
2
3.28
and hence
y
n
− p
2
≤x
n
− p
2
−x
n
− y
n
2
.
3.29
Journal of Inequalities and Applications 15
Set λ>0 as a constant such that
λ>sup
k
γf
x
k
− ASρ
k
, x
k
− p
.
3.30
By 3.29 and 3.30, we have
x
n1
− p
2
α
n
γf
x
n
β
n
x
n
1 − β
n
I − α
n
A
Sρ
n
− p
2
1 − β
n
I − α
n
A
Sρ
n
− p
β
n
x
n
− p
α
n
γf
x
n
− Ap
2
1 − β
n
Sρ
n
− p
− α
n
A
Sρ
n
− p
β
n
x
n
− p
α
n
γf
x
n
− Ap
2
1 − β
n
Sρ
n
− p
β
n
x
n
− p
α
n
γf
x
n
− ASρ
n
2
≤
1 − β
n
Sρ
n
− p
β
n
x
n
− p
2
2α
n
γf
x
n
− ASρ
n
,x
n1
− p
≤
1 − β
n
Sρ
n
− p
β
n
x
n
− p
2
2α
n
λ
2
≤
1 − β
n
Sρ
n
− p
2
β
n
x
n
− p
2
2α
n
λ
2
≤
1 − β
n
ρ
n
− p
2
β
n
x
n
− p
2
2α
n
λ
2
≤
1 − β
n
y
n
− p
2
β
n
x
n
− p
2
2α
n
λ
2
≤x
n
− p
2
−
1 − β
n
y
n
− x
n
2
2α
n
λ
2
.
3.31
It follows that
y
n
− x
n
2
≤
1
1 − β
n
x
n
− p
2
−x
n1
− p
2
2α
n
λ
2
1
1 − β
n
x
n
− p−x
n1
− p
x
n
− p x
n1
− p
2α
n
λ
2
≤
1
1 − β
n
x
n
− x
n1
x
n
− p x
n1
− p
2α
n
λ
2
.
3.32
By x
n
− x
n1
→0andα
n
→ 0, as n →∞,and{x
n
} is bounded, we obtain that
lim
n →∞
y
n
− x
n
0.
3.33
16 Journal of Inequalities and Applications
For p ∈ F, we have
ρ
n
− p
2
P
C
I − s
n
B
y
n
− P
C
I − S
n
B
p
2
≤
y
n
− p
− s
n
By
n
− Bp
2
y
n
− p
2
− 2s
n
y
n
− p, By
n
− Bp s
2
n
By
n
− Bp
2
≤x
n
− p
2
− 2s
n
−μBy
n
− Bp
2
νy
n
− p
2
s
2
n
By
n
− Bp
2
≤x
n
− p
2
2s
n
μBy
n
− Bp
2
− 2s
n
νy
n
− p
2
s
2
n
By
n
− Bp
2
≤x
n
− p
2
2s
n
μ s
2
n
−
2s
n
ν
λ
2
By
n
− Bp
2
.
3.34
Observe 3.31 that
x
n1
− p
2
≤
1 − β
n
ρ
n
− p
2
β
n
x
n
− p
2
2α
n
λ
2
.
3.35
Substituting 3.34 into 3.35, we have
x
n1
− p
2
≤x
n
− p
2
2s
n
μ s
2
n
−
2s
n
ν
λ
2
By
n
− Bp
2
2α
n
λ
2
. 3.36
It follows from condition C6 that
2aν
λ
2
− 2bμ − b
2
By
n
− Bp
2
≤x
n
− p
2
−x
n1
− p
2
2α
n
λ
2
≤x
n
− x
n1
x
n
− p x
n1
− p
2α
n
λ
2
.
3.37
From condition C1 and 3.25, we h ave that
lim
n →∞
By
n
− Bp 0.
3.38
Journal of Inequalities and Applications 17
On the other hand, we have
ρ
n
− p
2
P
C
I − s
n
B
y
n
− P
C
I − S
n
B
p
2
≤
I − s
n
B
y
n
−
I − S
n
B
p, ρ
n
− p
1
2
I − s
n
B
y
n
−
I − S
n
B
p
2
ρ
n
− p
2
−
I − s
n
B
y
n
−
I − S
n
B
p −
ρ
n
− p
2
≤
1
2
y
n
− p
2
ρ
n
− p
2
−
y
n
− ρ
n
− s
n
By
n
− Bp
2
≤
1
2
x
n
− p
2
ρ
n
− p
2
−y
n
− ρ
n
2
− s
2
n
By
n
− Bp
2
2s
n
y
n
− ρ
n
,Ay
n
− Ap
,
3.39
which yields that
ρ
n
− p
2
≤x
n
− p
2
−y
n
− ρ
n
2
2s
n
y
n
− ρ
n
By
n
− Bp.
3.40
Substituting 3.40 into 3.35 yields that
x
n1
− p
2
≤x
n
− p
2
−
1 − β
n
y
n
− ρ
n
2
2s
n
y
n
− ρ
n
By
n
− Bp 2α
n
λ
2
.
3.41
It follows that
y
n
− ρ
n
2
≤
1
1 − β
n
x
n
− p
2
−x
n1
− p
2
2s
n
1 − β
n
y
n
− ρ
n
By
n
− Bp
2α
n
λ
2
1 − β
n
≤
1
1 − β
n
x
n1
− x
n
x
n
− p x
n1
− p
2s
n
1 − β
n
y
n
− ρ
n
By
n
− Bp
2α
n
λ
2
1 − β
n
.
3.42
From condition C1, 3.25,and3.38, we have that
lim
n →∞
y
n
− ρ
n
0.
3.43
18 Journal of Inequalities and Applications
Observe that
y
n
− Sy
n
≤y
n
− x
n
x
n
− Sρ
n
Sρ
n
− Sy
n
≤y
n
− x
n
x
n
− Sρ
n
ρ
n
− y
n
.
3.44
From 3.27, 3.33,and3.43, we have
lim
n →∞
y
n
− Sy
n
0.
3.45
Observe that P
F
γf I − A is a contraction. Indeed, for all x, y ∈ H, we have
P
F
γf
I − A
x − P
F
γf
I − A
y≤
γf
I − A
x −
γf
I − A
y
≤ γf
x
− f
y
I − Ax − y
≤ γαx − y
1 −
γ
x − y
1 −
γ − γα
x − y.
3.46
Banach’s Contraction Mapping Principle guarantees that P
F
γf I − A has a unique fixed
point, say q ∈ H,thatis,q P
F
γf I − Aq.
Next, we show that
lim sup
n →∞
γf
q
− Aq, x
n
− q≤0.
3.47
To see this, we choose a subsequence {x
n
i
} of {x
n
} such that
lim sup
n →∞
γf
q
− Aq, x
n
− q lim sup
i →∞
γf
q
− Aq, x
n
i
− q.
3.48
Correspondingly, there exists a subsequence {y
n
i
} of {y
n
}. Since {y
n
i
} is bounded, there exists
a subsequence {y
n
i
j
} of {y
n
i
} which converges weakly to w. Without loss of generality, we can
assume that y
n
i
w.
Next, we show that w ∈ F.First,weprovew ∈ EPF. Since y
n
T
r
n
x
n
, we have
F
y
n
,η
1
r
n
η − y
n
,y
n
− x
n
≥0, ∀η ∈ C.
3.49
It follows from A2 that,
η − y
n
,
y
n
− x
n
r
n
≥ F
η, y
n
. 3.50
Journal of Inequalities and Applications 19
It follows that
η − y
n
i
,
y
n
i
− x
n
i
r
n
i
≥ F
η, y
n
i
.
3.51
Since y
n
i
− x
n
i
/r
n
i
→ 0, y
n
i
w,andA4, we have Fη, w ≤ 0 for all η ∈ C. For t with
0 <t≤ 1andη ∈ C,letη
t
tη 1 − tw. Since η ∈ C and w ∈ C, we have η
t
∈ C and hence
Fη
t
,w ≤ 0. So, from A1 and A4, we have
0 F
η
t
,η
t
≤ tF
η
t
,η
1 − t
F
η
t
,w
≤ tF
η
t
,η
. 3.52
That is, Fη
t
,η ≤ 0. It follows from A3 that Fw, η ≥ 0 for all η ∈ C and hence w ∈ EPF.
Since Hilbert spaces satisfy Opial’s condition, from 3.43, suppose w
/
Sw; we have
lim inf
i →∞
y
n
i
− w < lim inf
i →∞
y
n
i
− Sw
lim inf
i →∞
y
n
i
− Sy
n
i
Sy
n
i
− Sw
≤ lim inf
i →∞
Sy
n
i
− Sw
< lim inf
i →∞
y
n
i
− w
3.53
which is a contradiction. Thus, we have w ∈ FS.
Next, let us show that w ∈ VIC, B.Put
Tw
1
⎧
⎨
⎩
Bw
1
N
C
w
1
,w
1
∈ C,
∅,w
1
∈ C.
3.54
Since B is relaxed μ, ν-cocoercive and from condition C6, we have
Bx − By, x − y≥
−μ
Bx − By
2
νx − y
2
≥
ν − μλ
2
x − y
2
≥ 0, 3.55
which yields that B is monotone. Thus T is maximal monotone. Let w
1
,w
2
∈ GT. Since
w
2
− Bw
1
∈ N
C
w
1
and ρ
n
∈ C, we have
w
1
− ρ
n
,w
2
− Bw
1
≥0. 3.56
20 Journal of Inequalities and Applications
On the other hand, from ρ
n
P
C
I − s
n
By
n
, we have
w
1
− ρ
n
,ρ
n
−
I − s
n
B
y
n
≥ 0. 3.57
and hence
w
1
− ρ
n
,
ρ
n
− y
n
s
n
By
n
≥ 0. 3.58
It follows that
w
1
− ρ
n
i
,w
2
≥
w
1
− ρ
n
i
,Bw
1
≥
w
1
− ρ
n
i
,Bw
1
−
w
1
− ρ
n
i
,
ρ
n
i
− y
n
i
s
n
i
By
n
i
w
1
− ρ
n
i
,Bw
1
−
ρ
n
i
− y
n
i
s
n
i
− By
n
i
w
1
− ρ
n
i
,Bw
1
− Bρ
n
i
w
1
− ρ
n
i
,Bρ
n
i
− By
n
i
−
w
1
− ρ
n
i
,
ρ
n
i
− y
n
i
s
n
i
≥
w
1
− ρ
n
i
,Bρ
n
i
− By
n
i
−
w
1
− ρ
n
i
,
ρ
n
i
− y
n
i
s
n
i
,
3.59
which implies that w
1
− w, w
2
≥0, We have w ∈ T
−1
0 and hence w ∈ VIC, B.Thatis,
w ∈ F.
Since q P
F
γf I − Aq, we have
lim sup
n →∞
γf
q
− Aq, x
n
− q lim sup
i →∞
γf
q
− Aq, x
n
i
− q
γf
q
− Aq, w − q≤0.
3.60
That is, 3.47 holds.
Finally, we show that x
n
→ q, where q P
F
γf I − Aq, which solves the following
variational inequality:
γf
q
− Aq, p − q≤0, ∀p ∈ F. 3.61
Journal of Inequalities and Applications 21
We consider
x
n1
− q
2
1 − β
n
I − α
n
A
Sρ
n
− q
β
n
x
n
− q
α
n
γf
x
n
− Aq
2
1 − β
n
I − α
n
A
Sρ
n
− q
β
n
x
n
− q
2
α
2
n
γf
x
n
− Aq
2
2β
n
α
n
x
n
− q, γf
x
n
− Aq
2α
n
1 − β
n
I − α
n
A
Sρ
n
− q
,γf
x
n
− Aq
≤
1 − β
n
I − α
n
γ
Sρ
n
− q β
n
x
n
− q
2
α
2
n
γf
x
n
− Aq
2
2β
n
γα
n
x
n
− q, f
x
n
− f
q
2β
n
α
n
x
n
− q, γf
q
− Aq
2
1 − β
n
γα
n
Sρ
n
− q, f
x
n
− f
q
2
1 − β
n
α
n
Sρ
n
− q, γf
q
− Aq
− 2α
2
n
A
Sρ
n
− q
,γf
q
− Aq
,
3.62
which implies that
x
n1
− q
2
≤
1 − α
n
γ
2
2αβ
n
γα
n
2α
1 − β
n
γα
n
x
n
− q
2
α
2
n
γf
x
n
− Aq
2
2β
n
α
n
x
n
− q, γf
q
− Aq
2
1 − β
n
α
n
Sρ
n
− q, γf
q
− Aq−2α
2
n
A
Sρ
n
− q
,γf
q
− Aq
≤
1 − 2
γ − αγ
α
n
x
n
− q
2
γ
2
α
2
n
x
n
− q
2
α
2
n
γf
x
n
− Aq
2
2β
n
α
n
x
n
− q, γf
q
− Aq 2
1 − β
n
α
n
Sρ
n
− q, γf
q
− Aq
− 2α
2
n
A
Sρ
n
− q
·γf
q
− Aq
1 − 2
γ − αγ
α
n
x
n
− q
2
α
n
α
n
γ
2
x
n
− q
2
γf
x
n
− Aq
2
2A
Sρ
n
− q
·γf
q
− Aq
2β
n
x
n
− q, γf
q
− Aq 2
1 − β
n
Sρ
n
− q, γf
q
− Aq
.
3.63
Since {x
n
}, {fx
n
},and{Sρ
n
} are bounded, we can take a constant M
2
> 0 such that
M
2
≥ γ
2
x
n
− q
2
γf
x
n
− Aq
2
2A
Sρ
n
− q
·γf
q
− Aq
3.64
22 Journal of Inequalities and Applications
for all n ≥ 0. It then follows that
x
n1
− q
2
≤
1 − 2
γ − αγ
α
n
x
n
− q
2
α
n
ξ
n
,
3.65
where
ξ
n
2β
n
x
n
− q, γf
q
− Aq 2
1 − β
n
Sρ
n
− q, γf
q
− Aq α
n
M
2
. 3.66
From 3.27 and 3.47, we also have
lim sup
n →∞
γf
q
− Aq, Sρ
n
− q lim sup
n →∞
γf
q
− Aq, Sρ
n
− x
n
lim sup
n →∞
γf
q
− Aq, x
n
− q
≤ lim sup
n →∞
γf
q
− Aq, x
n
− q
≤ 0.
3.67
By C1, 3.47,and3.67, we get lim sup
n →∞
ξ
n
≤ 0. Now applying Lemma 2.2 to 3.65
concludes that x
n
→ q n →∞.
This completes the proof.
Remark 3.2. Some iterative algorithms were presented in Yamada 11, Combettes 24,and
Iiduka-Yamada 25, f or example, the steepest descent method, the hybrid steepest descent
method, and the conjugate gradient methods; these methods have common form
x
n1
x
n
ω
n
d
n
, 3.68
where x
n
is the nth approximation to the solution, ω
n
> 0 is a step size, and d
n
is a search
direction. In this paper, We define T : SP
C
I − sBT
r
; the method 3.1 will be changed as
x
n1
α
n
γf
x
n
β
n
x
n
1 − β
n
I − α
n
A
SP
C
I − s
n
B
Tx
n
x
n
1 − β
n
−x
n
Tx
n
α
n
γf
x
n
− ATx
n
.
3.69
Take ω
n
d
n
1 − β
n
−x
n
Tx
n
α
n
γfx
n
− ATx
n
, the method 3.1 will be changed as
3.68.
Remark 3.3. The computational possibility of the resolvent, T
r
,ofF in Lemma 2.7 and
Theorem 3.1 is well defined mathematically, but, in general, the computation of T
r
is very
difficult in large-scale finite spaces and infinite spaces.
Journal of Inequalities and Applications 23
4. Applications
Theorem 4.1. Let C be a nonempty closed convex subset of a Hilbert space H.LetF be a bifunction
of C × C into R which satisfies (A1)–(A4); let S be a nonexpansive mapping of C into H such that
F FS∩EPF∩ VIC, B
/
∅.LetA be a strongly positive linear bounded operator with coefficient
γ>0. Assume that 0 <γ<γ/α.Letf be a contraction of H into itself with a coefficient α 0 <α<1
and let {x
n
} and {Y
n
} be sequences generated by x
1
∈ H and
F
y
n
,η
1
r
n
η − y
n
,y
n
− x
n
≥0, ∀η ∈ C,
x
n1
α
n
γf
x
n
β
n
x
n
1 − β
n
I − α
n
A
Sy
n
,
4.1
for all n,where{α
n
}, {β
n
}⊂0, 1 and {r
n
}, {s
n
}⊂0, ∞ satisfy
C1 lim
n →∞
α
n
0;
C2
∞
n1
α
n
∞;
C3 0 < lim inf
n →∞
β
n
≤ lim sup
n →∞
β
n
< 1;
C4
∞
n1
|α
n1
− α
n
| < ∞ and
∞
n1
|γ
n1
− γ
n
| < ∞;
C5 lim inf
n →∞
γ
n
> 0.
Then, both {x
n
} and {y
n
} converge strongly to q ∈ F,whereq P
F
γf I − Aq,which
solves the following variational inequality:
γf
q
− Aq, p − q≤0, ∀p ∈ F. 4.2
Proof. Taking {s
n
} 0inTheorem 3.1, we can get the desired conclusion easily.
Theorem 4.2. Let C be a nonempty closed convex subset of a Hilbert space H,letS be a nonexpansive
mapping of C into H, and let B be a λ-Lipschitzian, relaxed μ, ν-cocoercive map of C into H such
that F FS ∩ VIC, B
/
∅.LetA be a strongly positive linear bounded operator with coefficient
γ>0. Assume that 0 <γ<γ/α.Letf be a contraction of H into itself with a coefficient α 0 <α<1
and let {x
n
} and {y
n
} be sequences generated by x
1
∈ H and
x
n1
α
n
γf
x
n
β
n
x
n
1 − β
n
I − α
n
A
SP
C
I − s
n
B
P
C
x
n
, 4.3
for all n,where{α
n
}, {β
n
}⊂0, 1 and {r
n
}, {s
n
}⊂0, ∞ satisfy
C1 lim
n →∞
α
n
0;
C2
∞
n1
α
n
∞;
C3 0 < lim inf
n →∞
β
n
≤ lim sup
n →∞
β
n
< 1;
C4
∞
n1
|α
n1
− α
n
| < ∞ and
∞
n1
|s
n1
− s
n
| < ∞;
C5 lim inf
n →∞
γ
n
> 0;
C6 {s
n
}∈a, b for some a, b with 0 ≤ a ≤ b ≤ 2ν − μλ
2
/λ
2
.
24 Journal of Inequalities and Applications
Then, both {x
n
} and {y
n
} converge strongly to q ∈ F,whereq P
F
γf I − Aq,which
solves the following variational inequality:
γf
q
− Aq, p − q≤0, ∀p ∈ F. 4.4
Proof. Put Fx, y0 for all x, y ∈ C and γ
n
1 for all n in Theorem 3.1. Then we have
y
n
P
C
x
n
. we can obtain the desired conclusion easily.
Acknowledgments
This work is supported by the Fundamental Research Funds for the Central Universities, no.
JY10000970006 and National Science Foundation of China, no. 60974082.
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