Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2011, Article ID 284363, 20 pages
doi:10.1155/2011/284363
Research Article
A General Iterative Approach to Variational
Inequality Problems and Optimization Problems
Jong Soo Jung
Department of Mathematics, Dong-A University, Busan 604-714, Republic of Korea
Correspondence should be addressed to Jong Soo Jung, [email protected]
Received 4 October 2010; Accepted 14 November 2010
Academic Editor: Jen Chih Yao
Copyright q 2011 Jong Soo Jung. 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 general iterative scheme for finding a common element of the set of solutions
of variational inequality problem for an inverse-strongly monotone mapping and the set of fixed
points of a nonexpansive mapping in a Hilbert space and then establish strong convergence of
the sequence generated by the proposed iterative scheme to a common element of the above
two sets under suitable control conditions, which is a solution of a certain optimization problem.
Applications of the main result are also given.
1. Introduction
Let H be a real Hilbert space with inner product ·, · and induced norm ·.LetC be a
nonempty closed convex subset of H and S : C → C be self-mapping on C. We denote by
FS the set of fixed points of S and by P
C
the metric projection of H onto C.
Let A be a nonlinear mapping of C into H. The variational inequality problem is to
find a u ∈ C such that
v − u, Au≥0, ∀v ∈ C. 1.1
We denote the set of solutions of the variational inequality problem 1.1 by VIC, A.The
,n≥ 1, 1.2
where {α
n
}⊂0, 1 and {λ
n
}⊂0, 2α. They proved that the sequence generated by 1.2
strongly converges strongly to P
FS∩VIC,A
x. In 2010, Jung 10 provided the following new
composite iterative scheme for the fixed point problem and the problem 1.1: x
1
x ∈ C and
y
n
α
n
f
x
n
1 − α
n
SP
C
1.3
where f is a contraction with constant k ∈ 0, 1,{α
n
},{β
n
}∈0, 1,and{λ
n
}⊂0, 2α.He
proved that the sequence {x
n
} generated by 1.3 strongly converges strongly to a point in
FS ∩ VIC, A, which is the unique solution of a certain variational inequality.
On the other hand, the following optimization problem has been studied extensively
by many authors:
min
x∈Ω
μ
2
Bx,x
1
2
x − u
2
− h
N
i1
C
i
and
hxx, b for a given point b in H.
In 2007, related to a certain optimization problem, Marino and Xu 14 introduced the
following general iterative scheme for the fixed point problem of a nonexpansive mapping:
x
n1
α
n
γf
x
n
I − α
n
B
Sx
n
,n≥ 0, 1.5
where {α
n
}∈0, 1 and γ>0. They proved that the sequence {x
where h is a potential function for γf. The result improved the corresponding results of
Moudafi 15 and Xu 16.
Fixed Point Theory and Applications 3
In this paper, motivated by the above-mentioned results, we introduce a new general
composite iterative scheme for finding a common point of the set of solutions of the
variational inequality problem 1.1 for an inverse-strongly monotone mapping and the set
of fixed points of a nonexapansive mapping and then prove that the sequence generated by
the proposed iterative scheme converges strongly to a common point of the above two sets,
which is a solution of a certain optimization problem. Applications of the main result are also
discussed. Our results improve and complement the corresponding results of Chen et al. 6,
Iiduka and Takahashi 8,Jung10, and others.
2. Preliminaries and Lemmas
Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. We write
x
n
xto indicate that the sequence {x
n
} converges weakly to x. x
n
→ x implies that {x
n
}
converges strongly to x.
First we recall that a mapping f : C → C is a contraction on C if there exists a constant
k ∈ 0, 1 such that fx − fy≤kx − y, x, y ∈ C. A mapping T : C → C is called
nonexpansive if Tx− Ty≤x − y,x,y∈ C. We denote by FT the set of fixed points of T.
For every point x ∈ H, there exists a unique nearest point in C, denoted by P
C
x, such
that
y
≥
P
C
x
− P
C
y
2
2.2
for every x, y ∈ H. Moreover, P
C
x is characterized by the properties:
x − y
2
≥
2.3
In the context of the variational inequality problem for a nonlinear mapping A, this implies
that
u ∈ VI
C, A
⇐⇒ u P
C
u − λAu
, for any λ>0. 2.4
It is also well known that H satisfies the Opial condition, that is, for any sequence {x
n
} with
x
n
x, the inequality
lim inf
n →∞
x
n
− x
< lim inf
n →∞
≥ η
x − y
2
2.7
for all x, y ∈ C. In such a case, we say A is η-strongly monotone. If A is η-strongly monotone
and κ- Lipschitz continuous,thatis,Ax − Ay≤κx − y for all x, y ∈ C, then A is η/κ
2
-
inverse-strongly monotone. If A is an α-inverse-strongly monotone mapping of C into H,
then it is obvious that A is 1/α-Lipschitz continuous. We also have that for all x, y ∈ C and
λ>0,
I − λA
x −
I − λA
y
2
2
λ
λ − 2α
Ax − Ay
2
.
2.8
So, if λ ≤ 2α, then I − λA is a nonexpansive mapping of C into H. The following result for the
existence of solutions of the variational inequality problem for inverse strongly-monotone
mappings was given in Takahashi and Toyoda 9.
Proposition 2.1. Let C be a bounded closed convex subset of a real Hilbert space and let A be an
α-inverse-strongly monotone mapping of C into H. Then, VIC, A is nonempty.
A set-valued mapping T : H → 2
H
is called monotone if for all x, y ∈ H, f ∈ Tx,and
g ∈ Tyimply x−y, f−g≥0. A monotone mapping T : H → 2
H
is maximal if the graph GT
of T is not properly contained in the graph of any other monotone mapping. It is 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.LetA be an inverse-strongly monotone mapping of C into H
and let N
2
2
y, x y
,
2.10
for all x, y ∈ H.
Lemma 2.3 Xu 12. Let {s
n
} be a sequence of nonnegative real numbers satisfying
s
n1
≤
1 − λ
n
s
n
β
n
γ
n
,n≥ 1, 2.11
where {λ
n
} and {β
n
} satisfy the following conditions:
≥ 0 n ≥ 1,
∞
n1
γ
n
< ∞.
Then lim
n →∞
s
n
0.
Lemma 2.4 Marino and Xu 14. Assume that A is a strongly positive linear bounded operator on
a Hilbert space H with constant
γ>0 and 0 <ρ≤B
−1
.ThenI − ρB≤1 − ργ.
The following lemma can be found in 20, 21see also Lemma 2.2 in 22.
Lemma 2.5. Let C be a nonempty closed convex subset of a real Hilbert space H, and let g : C →
R ∪{∞}be a proper lower semicontinunous differentiable convex function. If x
∗
isasolutiontothe
minimization problem
g
x
∗
inf
x∈C
x − u
2
− h
x
,
2.14
then
u
γf −
I μB
x
∗
,x− x
∗
≤ 0,x∈ C, 2.15
where h is a potential function for γf.
6 Fixed Point Theory and Applications
3. Main Results
In this section, we present a new general composite iterative scheme for inverse-strongly
monotone mappings and a nonexpansive mapping.
Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H such that C ± C ⊂
SP
C
x
n
− λ
n
Ax
n
,
x
n1
1 − β
n
y
n
β
n
SP
C
y
n
− λ
n
⊂ 0,a for all n ≥ 0 and for some a ∈ 0, 1;
iii λ
n
∈ c, d for some c, d with 0 <c<d<2α;
iv
∞
n1
|α
n1
− α
n
| < ∞,
∞
n1
|β
n1
− β
n
| < ∞,
∞
n1
|λ
n1
− λ
n
| < ∞.
Then {x
Proof. We note that from the control condition i, we may assume, without loss of generality,
that α
n
≤ 1 μB
−1
. Recall that if B is bounded linear self-adjoint operator on H, then
B
sup
{|
Bu, u
|
: u ∈ H,
u
1
}
. 3.1
Observe that
I − α
n
I μB
u, u
1 − α
sup
I − α
n
I μB
u, u
: u ∈ H,
u
1
sup
1 − α
n
− α
n
μ
Bu, u
: u ∈ H,
u
1
C
y
n
−
λ
n
Ay
n
for every n ≥ 1. Let p ∈ FS ∩ VIC, A. Since I − λ
n
A is nonexpansive and p
P
C
p − λ
n
Ap from 2.4, we have
z
n
− p
≤
x
n
− λ
≤
y
n
− p
. 3.5
Now, set
B I μB.Letp ∈ FS ∩ VIC, A. Then, from IS and 3.4,weobtain
y
n
− p
α
n
u α
n
γf
x
z
n
− p
α
n
u
α
n
γ
f
x
n
− f
p
α
n
α
n
γk
x
n
− p
α
n
γf
p
−
Bp
1 −
1 μ
u
1 μ
γ − γk
.
3.6
8 Fixed Point Theory and Applications
From 3.5 and 3.6, it follows that
x
n1
− p
1 − β
n
y
n
− p
β
n
1 − β
n
y
n
− p
β
n
y
n
− p
y
n
− p
≤ max
⎧
⎨
⎬
⎭
.
3.7
By induction, it follows from 3.7 that
x
n
− p
≤ max
⎧
⎨
⎩
x
1
− p
,
γf
p
n
}, {Ay
n
},and{BSz
n
} are
bounded. Moreover, since Sz
n
− p≤x
n
− p and Sw
n
− p≤y
n
− p, {Sz
n
} and {Sw
n
}
are also bounded. And by the condition i, we have
y
n
− Sz
n
α
n
BSz
n
−→ 0
as n −→ ∞
.
3.9
Step 2. We show that lim
n →∞
x
n1
− x
n
0 and lim
n →∞
y
n1
− y
n
0. Indeed, since I − λ
n
A
and P
C
are nonexpansive and z
n
n−1
Ax
n−1
≤
x
n
− x
n−1
|
λ
n
− λ
n−1
|
Ax
n−1
.
3.10
Similarly, we get
w
n
− w
α
n
u γf
x
n
I − α
n
B
Sz
n
− α
n−1
u γf
x
n−1
−
I − α
n−1
− f
x
n−1
I − α
n
B
Sz
n
− Sz
n−1
.
3.12
Fixed Point Theory and Applications 9
So, we obtain
y
n
− y
n−1
Sz
n−1
α
n
γk
x
n
− x
n−1
1 −
1 μ
γα
n
z
n
− z
n−1
≤
|
α
n
γk
x
n
− x
n−1
1 −
1 μ
γα
n
x
n
− x
n−1
|
λ
n
Sw
n−1
− y
n−1
β
n
Sw
n
− Sw
n−1
.
3.14
By 3.11, 3.13,and3.14, we have
x
n1
− x
n
≤
1 − β
n
w
n
− w
n−1
≤
1 − β
n
y
n
− y
n−1
β
n
y
n
− y
n−1
β
y
n−1
≤
y
n
− y
n−1
|
λ
n
− λ
n−1
|
Ay
n−1
x
n
− x
n−1
|
α
n
− α
n−1
|
u
γ
f
x
n−1
β
n
− β
n−1
Sw
n−1
y
n−1
≤
1 −
1 μ
γ − γk
n
− β
n−1
,
3.15
where M
1
sup{u γfx
n
BT
n
z
n
: n ≥ 1}, M
2
sup{Ay
n
Ax
n
: n ≥ 1},and
M
3
sup{Sw
n
y
n
: n ≥ 1}. From the conditions i and iv,itiseasytoseethat
lim
n−1
|
M
2
|
λ
n
− λ
n−1
|
M
3
β
n
− β
n−1
< ∞.
3.16
10 Fixed Point Theory and Applications
Applying Lemma 2.3 to 3.15,weobtain
lim
n →∞
x
n1
n
0 and lim
n →∞
x
n
− Sz
n
0. Indeed,
x
n1
− y
n
β
n
Sw
n
− y
n
≤ β
n
≤ a
y
n
− x
n
Sz
n
− y
n
≤ a
y
n
− x
≤
a
1 − a
x
n1
− x
n
Sz
n
− y
n
.
3.20
Obviously, by 3.9 and Step 2, we have x
n1
− y
n
→0asn →∞. This implies that
x
x
n
− y
n
y
n
− Sz
n
−→ 0asn −→ ∞ . 3.22
Fixed Point Theory and Applications 11
Step 4. We show that lim
n →∞
x
n
− z
n
0 and lim
n →∞
y
n
− z
n
u γf
x
n
−
Bp
I − α
n
B
Sz
n
− p
2
≤
α
n
2
≤ α
n
u γf
x
n
−
Bp
2
1 − α
n
1 μ
γ
z
n
n
− p
≤ α
n
u γf
x
n
−
Bp
2
1 − α
n
1 μ
γ
1 μ
γ
γu f
x
n
−
Bp
z
n
− p
≤ α
n
u γf
d − 2α
Ax
n
− Ap
2
2α
n
u γf
x
n
−
Bp
z
n
− p
γu f
x
n
−
Bp
2
x
n
− p
y
n
− p
z
n
− p
≤ α
n
γu f
x
n
−
Bp
2
x
n
−
Bp
z
n
− p
.
3.24
12 Fixed Point Theory and Applications
Since α
n
→ 0 from the condition i and x
n
− y
n
→0fromStep 3, we have Ax
n
− Ap→
0 n →∞. Moreover, from 2.4 we obtain
z
n
− p
− λ
n
Ax
n
−
p − λ
n
Ap
,z
n
− p
1
2
x
n
− λ
n
Ax
n
−
n
Ap
−
z
n
− p
2
≤
1
2
x
n
− p
2
z
n
− Ap
2
,
3.25
and so
z
n
− p
2
≤
x
n
− p
2
−
x
n
− z
y
n
− p
2
≤ α
n
u γf
x
n
−
Bp
2
1 − α
n
1 μ
γ
z
n
− p
≤ α
n
u γf
x
n
−
Bp
2
x
n
n
x
n
− z
n
,Ax
n
− Ap
−
1 − α
n
1 μ
γ
λ
2
n
Ax
n
− Ap
2
1 μ
γ
x
n
− z
n
2
≤ α
n
u γf
x
n
−
Bp
2
2
1 − α
n
1 μ
γ
λ
n
x
n
− z
n
,Ax
n
− Ap
−
1 − α
n
1 μ
γ
− p
≤ α
n
u γf
x
n
−
Bp
2
x
n
− p
n
,Ax
n
− Ap
−
1 − α
n
1 μ
γ
λ
2
n
Ax
n
− Ap
2
2α
n
n
→0. Also by 3.21
y
n
− z
n
≤
y
n
− x
n
x
n
− z
n
−→ 0
n −→ ∞
α
n
u γf
x
n
−
BSz
n
y
n
− z
n
,
3.30
from 3.9 and 3.29 , we have lim
− q
≤ 0,
3.31
where q is a solution of the optimization problem OP1. First we prove that
lim sup
n →∞
u
γf −
B
q, Sz
n
− q
≤ 0.
3.32
14 Fixed Point Theory and Applications
Since {z
n
} is bounded, we can choose a subsequence {z
n
i
} of {z
n
} such that
lim sup
n →∞
z
/
∈ FS. Since z
n
i
zand Sz
/
z, by the Opial condition and Step 5,weobtain
lim inf
i →∞
z
n
i
− z
< lim inf
i →∞
z
n
i
− Sz
≤ lim inf
i →∞
z
n
3.34
which is a contradiction. Thus we have z ∈ FS.
Next, let us show that z ∈ VIC, A.Let
Tv
⎧
⎨
⎩
Av N
C
v, v ∈ C,
∅,v
/
∈ C.
3.35
Then T is maximal monotone. Let v, w ∈ GT. Since w − Av ∈ N
C
v and z
n
∈ C, we have
v − z
n
,w− Av≥0. 3.36
On the other hand, from z
n
P
C
x
n
− λ
n
n
i
,w≥
v − z
n
i
,Av
≥
v − z
n
i
,Av
−
v − z
n
i
,
z
n
i
− x
n
i
λ
n
n
i
v − z
n
i
,Az
n
i
− Ax
n
i
−
v − z
n
i
,
z
n
i
− x
n
i
λ
n
i
3.38
Since z
n
− x
n
→0inStep 4 and A is α-inverse-strongly monotone, we have v − z, w≥0
as i →∞. Since T is maximal monotone, we have z ∈ T
−1
0 and hence z ∈ VIC, A.
Therefore, z ∈ FS ∩ VIC, A. Now from Lemma 2.5 and Step 5,weobtain
lim sup
n →∞
u
γf −
B
q, Sz
n
− q
lim
i →∞
u
γf −
B
lim sup
n →∞
u
γf −
B
q, y
n
− q
≤ lim sup
n →∞
u
γf −
B
q, y
n
− Sz
n
lim sup
n →∞
u
n →∞
u
γf −
B
q, Sz
n
− q
≤ 0.
3.40
16 Fixed Point Theory and Applications
Step 7. We show that lim
n →∞
x
n
− q 0 and lim
n →∞
u
n
− q 0, where q is a solution of
the optimization problem OP1. Indeed from IS and Lemma 2.2, we have
x
n1
− q
B
Sz
n
− q
≤
I − α
n
B
Sz
n
− q
2
2α
n
n
γ
f
x
n
− f
q
,y
n
− q
2α
n
u γf
q
−
Bq,y
n
− q
≤
− q
2α
n
u
γf −
B
q, y
n
− q
≤
1 −
1 μ
γα
n
2
x
n
− q
2α
n
u
γf −
B
q, y
n
− q
1 − 2
1 μ
γ − γk
α
n
x
n
− q
y
n
− x
n
2α
n
u
γf −
B
q, y
n
− q
,
3.41
that is,
x
n1
− q
2
4
2α
n
γk
y
n
− x
n
M
4
2α
n
u
γf −
B
q, y
n
− q
1 −
α
1 μ
γ
2
M
2
4
2γk
y
n
− x
n
M
4
2
u
γf − B
q, y
n
− q
. 3.43
x
1
x ∈ C,
y
n
α
n
u γf
x
n
I − α
n
I μB
Sx
n
,
x
n1
1 − β
n
1
2
x − u
2
− h
x
,
OP2
where h is a potential function for γf.
Corollary 3.3. Let H, C, A, B, f, u, γ,
γ,k, and μ be as in Theorem 3.1.Let{x
n
} be a sequence
generated by
x
1
x ∈ C,
y
n
α
n
u γf
x
n
n
P
C
y
n
− λ
n
Ay
n
,n≥ 1,
3.45
where {λ
n
}⊂0, 2α, {α
n
}⊂0, 1, and {β
n
}⊂0, 1.Let{α
n
}, {λ
n
} and {β
n
} satisfy the conditions
(i), (ii), (iii), and (iv) in Theorem 3.1.Then{x
n
} converges strongly to q ∈ VIC, A,whichisa
solution of the optimization problem
A mapping T : C → C is called strictly pseudocontractive if there exists α with 0 ≤ α<1
such that
Tx − Ty
2
≤
x − y
2
α
I − T
x −
I − T
y
2
4.1
.
4.2
On the other hand, since H is a real Hilbert space, we have
I − A
x −
I − A
y
2
x − y
2
Ax − Ay
2
x
1
x ∈ C,
y
n
α
n
u γf
x
n
I − α
n
I μB
S
1 − λ
n
x
n
λ
n
where {λ
n
}⊂0, 1 − α, {α
n
}⊂0, 1, and {β
n
}⊂0, 1.Let{α
n
}, {λ
n
}, and {β
n
} satisfy the
conditions (i), (ii), (iii), and (iv) in Theorem 3.1.Then{x
n
} converges strongly to q ∈ FS ∩ FT,
which is a solution of the optimization problem
min
x∈F
S
∩F
T
μ
2
Bx,x
n
. Thus, the desired result follows from Theorem 3.1.
Using Theorem 3.1, we also obtain the following result.
Fixed Point Theory and Applications 19
Theorem 4.2. Let H be a real Hilbert space. Let A be an α-inverse-strongly monotone mapping of
Hinto H and S a nonexpansive mapping of H into itself such that FS ∩ A
−1
0
/
∅.Letu ∈ H, and
let B be a strongly positive bounded linear operator on H with constant
γ>0 and f : H → H
a contraction with constant k ∈ 0, 1. Assume that μ>0 and 0 <γ<1 μ
γ/k.Let{x
n
} be a
sequence generated by
x
1
x ∈ H,
y
n
α
n
u γf
x
n
y
n
− λ
n
Ay
n
,n≥ 1,
4.6
where {λ
n
}⊂0, 2α, {α
n
}⊂0, 1, and {β
n
}⊂0, 1.Let{α
n
}, {λ
n
}, and {β
n
} satisfy the
conditions (i), (ii), (iii), and (iv) in Theorem 3.1.Then{x
n
} converges strongly to q ∈ FS ∩ A
−1
0,
which is a solution of the optimization problem
min
∞
n1
|α
n1
− α
n
| < ∞ on the control
parameter {α
n
} by the condition α
n
∈ 0, 1 for n ≥ 1, lim
n →∞
α
n
/α
n1
1 12, 13 or by the
perturbed control condition |α
n1
− α
n
| <oα
n1
σ
n
,
∞
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