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, [email protected]
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
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.
− 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
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
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
} 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
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
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
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
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
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
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
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, ∀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
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
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
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
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
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
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
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
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
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
− 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
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
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
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
,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
y
n1
− y
n
≤x
n1
− x
n
1 −
r
n
r
n1
y
n1
− x
n1
≤x
n1
− x
n
M
n1
B
y
n1
− P
C
I − s
n
B
y
n
≤
I − s
n1
B
y
n1
−
I − s
n
B
y
− 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
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
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
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
ASρ
n
ρ
n1
− ρ
n
−x
n1
− x
n
≤
α
n1
1 − β
n1
γf
x
n1
ASρ
n1
.
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
1 − β
n
I − α
n
A
Sρ
n
− x
n
≤ α
n
γf
x
n
− Ax
n
1 − β
n
− α
n
γ
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
− 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
γf
x
n
− Ap
2
1 − β
n
Sρ
n
− p
− α
n
A
Sρ
n
− p
β
n
− p
α
n
γf
x
n
− ASρ
n
2
≤
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
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
− 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
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
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
.
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
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
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
y
n
−
I − S
n
B
p −
ρ
n
− p
2
≤
1
2
y
n
− p
2
ρ
n
− p
2
−
− 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
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 − 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
− 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
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
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.
,
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
0 F
η
t
,η
t
≤ tF
η
t
,η
1 − t
F
η
t
,w
≤ tF
η
t
,η
. 3.52
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
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
w
1
− ρ
n
i
,w
2
≥
w
1
− ρ
n
i
,Bw
1
≥
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
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
− 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
n
I − α
n
A
Sρ
n
− q
β
n
x
n
− q
2
α
2
n
γf
x
n
− Aq
2
2β
− 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
γα
n
Sρ
n
− q, f
x
n
− f
q
2
1 − β
n
α
n
Sρ
n
− q, γf
q
− Aq
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
,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
− Aq
− 2α
2
n
A
Sρ
n
− q
·γf
q
− Aq
1 − 2
γ − αγ
α
n
x
n
− q
2
α
n
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
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
− 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
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
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
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
| < ∞ 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
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
|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
7 F. Deutsch and I. Yamada, “Minimizing certain convex functions over the intersection of the fixed
point sets of nonexpansive mappings,” Numerical Functional Analysis and Optimization, vol. 19, no.
1-2, pp. 33–56, 1998.
8 G. Marino and H K. Xu, “A general iterative method for nonexpansive mappings in Hilbert spaces,”
Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 43–52, 2006.
9 H K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society,
vol. 66, no. 1, pp. 240–256, 2002.
10 H. K. Xu, “An iterative approach to quadratic optimization,” Journal of O ptimization Theory and
Applications, vol. 116, no. 3, pp. 659–678, 2003.
11 I. Yamada, “The hybrid steepest descent method for the variational inequality problem over the
intersection of fixed point sets of nonexpansive mappings,” in Inherently Parallel Algorithms in
Feasibility and Optimization and Their Applications (Haifa, 2000), D. Butnariu, Y. Censor, and S. Reich,
Eds., vol. 8 of Studies in Computational Mathematics, pp. 473–504, North-Holland, Amsterdam, The
Netherlands, 2001.
12 A. Moudafi, “Viscosity approximation methods for fixed-points problems,” Journal of Mathematical
Analysis and Applications, vol. 241, no. 1, pp. 46–55, 2000.
13 W. Takahashi and M. Toyoda, “Weak convergence theorems for nonexpansive mappings and
monotone mappings,” Journal of Optimization Theory and Applications, vol. 118, no. 2, pp. 417–428,
2003.
14
H. Iiduka and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and inverse-
strongly monotone mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 61, no. 3, pp.
341–350, 2005.
15 J. Chen, L. Zhang, and T. Fan, “Viscosity approximation methods for nonexpansive mappings and
monotone mappings,” Journal of Mathematical Analysis and Applications, vol. 334, no. 2, pp. 1450–1461,
2007.
Journal of Inequalities and Applications 25
16 Y. Yao and J C. Yao, “On modified iterative method for nonexpansive mappings and monotone
mappings,” Applied Mathematics and Computation, vol. 186, no. 2, pp. 1551–1558, 2007.
17 Y. Su, M. Shang, and X. Qin, “An iterative method of solution for equilibrium and optimization
Copyright: Tài liệu đại học ©