Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 949141, 15 pages
doi:10.1155/2012/949141
Research Article
Iterative Algorithms Approach to Variational
Inequalities and Fixed Point Problems
Yeong-Cheng Liou,1 Yonghong Yao,2 Chun-Wei Tseng,1
Hui-To Lin,1 and Pei-Xia Yang2
1
2
Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan
Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China
Correspondence should be addressed to Yonghong Yao, yaoyonghong@yahoo.cn
Received 21 September 2011; Revised 16 November 2011; Accepted 17 November 2011
Academic Editor: Khalida Inayat Noor
Copyright q 2012 Yeong-Cheng Liou 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 consider a general variational inequality and fixed point problem, which is to find a point x∗
with the property that GVF: x∗ ∈ GVIC, A and gx∗ ∈ FixS where GVIC, A is the solution
set of some variational inequality FixS is the fixed points set of nonexpansive mapping S, and g
is a nonlinear operator. Assume the solution set Ω of GVF is nonempty. For solving GVF, we
suggest the following method gxn1 βgxn 1 − βSPC αn Fxn 1 − αn gxn − λAxn ,
n ≥ 0. It is shown that the sequence {xn } converges strongly to x∗ ∈ Ω which is the unique solution
of the variational inequality Fx∗ − gx∗ , gx − gx∗ ≤ 0, for all x ∈ Ω.
1. Introduction
Let A : C → H and g : C → C be two nonlinear mappings. We concern the following
generalized variational inequality of finding u ∈ C, gu ∈ C such that
gv − gu, Au ≥ 0,
∀gv ∈ C.
1.1
The solution set of 1.1 is denoted by GVIC, A, g. It has been shown that a large class
of unrelated odd-order and nonsymmetric obstacle, unilateral, contact, free, moving, and
equilibrium problems arising in regional, physical, mathematical, engineering, and applied
sciences can be studied in the unified and general framework of the general variational
inequalities 1.1, see 1–16 and the references therein. Noor 17 has introduced a new
type of variational inequality involving two nonlinear operators, which is called the general
variational inequality. It is worth mentioning that this general variational inequality is
2
Abstract and Applied Analysis
remarkably different from the so-called general variational inequality which was introduced
by Noor 18 in 1988. Noor 17 proved that the general variational inequalities are equivalent
to nonlinear projection equations and the Wiener-Hopf equations by using the projection
technique. Using this equivalent formulation, Noor 17 suggested and analyzed some
iterative algorithms for solving the special general variational inequalities and further proved
that these algorithms have strong convergence.
For g I, where I is the identity operator, problem 1.1 is equivalent to finding u ∈ C
such that
v − u, Au ≥ 0,
∀v ∈ C,
1.2
which is known as the classical variational inequality introduced and studied by Stampacchia
19 in 1964. This field has been extensively studied due to a wide range of applications in
industry, finance, economics, social, pure and applied sciences. For related works, please see
20–35. Our main purposes in the present paper is devoted to study this topic.
Motivated and inspired by the works in this field, in this paper, we consider a general
variational inequality and fixed point problem, which is to find a point x∗ with the property
that
x∗ ∈ GVIC, A,
gx∗ ∈ FixS,
GVF
where FixS is the fixed points set of nonexpansive mapping S. Assume the solution set Ω
of GVF is nonempty. For solving GVF, we suggest the following method
gxn1 βgxn 1 − β SPC αn Fxn 1 − αn gxn − λAxn ,
n ≥ 0.
1.3
It is shown that the sequence {xn } converges strongly to x∗ ∈ Ω which is the unique solution
of the following variational inequality
Fx∗ − gx∗ , gx − gx∗ ≤ 0,
∀x ∈ Ω.
1.4
Our results contain some interesting results as special cases.
2. Preliminaries
Let H be a real Hilbert space with inner product ·, · and norm · , respectively. Let C
be a nonempty closed convex subset of H. Recall that a mapping S : C → C is said to be
nonexpansive if
Sx − Sy ≤ x − y,
2.1
for all x, y ∈ C. We denote by FixS the set of fixed points of S. A mapping F : C → H is said
to be L-Lipschitz continuous, if there exists a constant L > 0 such that Fx−Fy
≤ L
x−y
Abstract and Applied Analysis
3
for all x, y ∈ C. A mapping A : C → H is said to be α-inverse strongly g-monotone if and
only if
2
Ax − Ay, gx − g y ≥ αAx − Ay ,
2.2
for some α > 0 and for all x, y ∈ C. A mapping g : C → C is said to be strongly monotone if
there exists a constant γ > 0 such that
2
gx − g y , x − y ≥ γ x − y ,
2.3
for all x, y ∈ C.
Let B be a mapping of H into 2H . The effective domain of B is denoted by domB, that
is, domB {x ∈ H : Bx /
∅}. A multivalued mapping B is said to be a monotone operator
on H if and only if
x − y, u − v ≥ 0,
2.4
for all x, y ∈ domB, u ∈ Bx, and v ∈ By. A monotone operator B on H is said to be maximal
if and only if its graph is not strictly contained in the graph of any other monotone operator
on H. Let B be a maximal monotone operator on H and let B−1 0 {x ∈ H : 0 ∈ Bx}.
It is well known that, for any u ∈ H, there exists a unique u0 ∈ C such that
u − u0 inf{
u − x
: x ∈ C}.
2.5
We denote u0 by PC u, where PC is called the metric projection of H onto C. The metric
projection PC of H onto C has the following basic properties:
i PC x − PC y
≤ x − y
for all x, y ∈ H;
ii x − y, PC x − PC y ≥ PC x − PC y
2 for every x, y ∈ H;
iii x − PC x, y − PC x ≤ 0 for all x ∈ H, y ∈ C.
It is easy to see that the following is true:
u ∈ GVI C, A, g ⇐⇒ gu PC gu − λAu ,
∀λ > 0.
2.6
We use the following notation:
i xn x stands for the weak convergence of xn to x;
ii xn → x stands for the strong convergence of xn to x.
We need the following lemmas for the next section.
Lemma 2.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let G : C → C be
a nonlinear mapping and let the mapping A : C → H be α-inverse strongly g-monotone. Then, for
any λ > 0, one has
PC gx − λAx − PC g y − λAy 2 ≤ gx − g y 2 λλ − 2αAx − Ay2 ,
x, y ∈ C.
2.7
4
Abstract and Applied Analysis
Proof. Consider the following:
PC gx − λAx − PC g y − λAy 2
2
≤ gx − g y − λ Ax − Ay 2
2
gx − g y − 2λ Ax − Ay, gx − g y λ2 Ax − Ay
2
2
2
≤ gx − g y − 2λαAx − Ay λ2 Ax − Ay
2
2
≤ gx − g y λλ − 2αAx − Ay .
2.8
If λ ∈ 0, 2α, we have
PC gx − λAx − PC g y − λAy ≤ gx − g y − λ Ax − Ay ≤ gx − g y .
2.9
Lemma 2.2 see 36. Let C be a closed convex subset of a Hilbert space H. Let S : C → C be a
nonexpansive mapping. Then FixS is a closed convex subset of C and the mapping I −S is demiclosed
at 0, that is, whenever {xn } ⊂ C is such that xn x and I − Sxn → 0, then I − Sx 0.
Lemma 2.3 see 37. Let {xn } and {yn } be bounded sequences in a Banach space X and let {βn } be
a sequence in 0, 1 with 0 < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1. Suppose xn1 1 − βn yn βn xn
for all n ≥ 0 and lim supn → ∞ yn1 − yn − xn1 − xn ≤ 0. Then, limn → ∞ yn − xn 0.
Lemma 2.4 see 38. Assume {an } is a sequence of nonnegative real numbers such that
an1 ≤ 1 − γn an δn γn ,
2.10
where {γn } is a sequence in 0, 1 and {δn } is a sequence such that
1 ∞
n1 γn ∞;
2 lim supn → ∞ δn ≤ 0 or ∞
n1 |δn γn | < ∞.
Then limn → ∞ an 0.
3. Main Results
In this section, we will prove our main results.
Theorem 3.1. Let C be a nonempty closed and convex subset of a real Hilbert space H. Let F :
C → H be an L-Lipschitz continuous mapping, g : C → C be a weakly continuous and γ-strongly
monotone mapping such that Rg C. Let A : C → H be an α-inverse strongly g-monotone
mapping and let S : C → C be a nonexpansive mapping. Suppose that Ω /
∅. Let β ∈ 0, 1 and
γ ∈ L, 2α. For given x0 ∈ C, let {xn } ⊂ C be a sequence generated by
gxn1 βgxn 1 − β SPC αn Fxn 1 − αn gxn − λAxn ,
n ≥ 0,
3.1
Abstract and Applied Analysis
5
where {αn } ⊂ 0, 1 satisfies C1: limn → ∞ αn 0 and C2: n αn ∞. Then the sequence
{xn } generated by 3.1 converges strongly to x∗ ∈ Ω which is the unique solution of the following
variational inequality:
Fx∗ − gx∗ , gx − gx∗ ≤ 0,
∀x ∈ Ω.
3.2
Proof. First, we show the solution set of variational inequality 3.2 is singleton. Assume x
∈
Ω also solves 3.2. Then, we have
− gx∗ ≤ 0,
Fx∗ − gx∗ , gx
Fx
− gx,
gx∗ − gx
≤ 0.
3.3
It follows that
≤0
Fx
− gx
− Fx∗ gx∗ , gx∗ − gx
2 ⇒ gx∗ − gx
≤ Fx∗ − Fx,
gx∗ − gx
2 ⇒ gx∗ − gx
≤ Fx∗ − Fx,
gx∗ − gx
≤ Fx∗ − Fx
gx∗ − gx
⇒ gx∗ − gx
≤ Fx∗ − Fx
.
3.4
Since g is γ-strongly monotone, we have
2 γ x − y ≤ gx − g y , x − y ≤ gx − g y x − y,
∀x, y ∈ C.
3.5
Hence,
γ x − y ≤ gx − g y ,
∀x, y ∈ C.
3.6
In particular, γ
x∗ − x
≤ gx∗ − gx
.
By 3.4, we deduce
≤ gx∗ − gx
γ
x∗ − x
≤ Fx∗ − Fx
≤ L
x∗ − x
,
3.7
which implies that x
x∗ because of L < γ by the assumption. Therefore, the solution of
variational inequality 3.2 is unique.
Pick up any u ∈ Ω. It is obvious that u ∈ GVIC, A, g and gu ∈ FixS. Set un PC αn Fxn 1 − αn gxn − λAxn , n ≥ 0. From 2.6, we know gu PC gu − μAu for
any μ > 0. Hence, we have
gu PC gu − 1 − αn λAu PC αn gu 1 − αn gu − λAu ,
∀n ≥ 0.
3.8
6
Abstract and Applied Analysis
From 3.6, 3.8, and Lemma 2.1, we get
un − gu PC αn Fxn 1 − αn gxn − λAxn
−PC αn gu 1 − αn gu − λAu ≤ αn Fxn − gu 1 − αn gxn − λAxn − gu − λAu ≤ αn Fxn − Fu
αn Fu − gu 1 − αn gxn − gu
≤ αn L
xn − u
αn Fu − gu 1 − αn gxn − gu
3.9
αn L gxn − gu αn Fu − gu 1 − αn gxn − gu
γ
L
1− 1−
αn gxn − gu αn Fu − gu.
γ
≤
It follows from 3.1 that
gxn1 − gu ≤ βgxn − gu 1 − β Sun − Sgu
≤ βgxn − gu 1 − β un − gu
L
αn gxn − gu
≤ β gxn − gu 1 − β 1 − 1 −
γ
1 − β αn Fu − gu
L 1− 1−
1 − β αn gxn − gu
γ
Fu − gu
L 1−
.
1 − β αn
γ
1 − L/γ
3.10
This indicates by induction that
Fu − gu
gxn1 − gu ≤ max gxn − gu,
.
1 − L/γ
3.11
Hence, {gxn } is bounded. By 3.6, we have xn − u
≤ 1/γ
gxn − gu
. This implies
that {xn } is bounded. Consequently, {Fxn }, {Axn }, {un }, and {Sun } are all bounded.
Note that we can rewrite 3.1 as gxn1 βgxn 1 − βSun for all n. Next, we will
use Lemma 2.3 to prove that xn1 − xn → 0. In fact, we firstly have
Sun − Sun−1 SPC αn Fxn 1 − αn gxn − λAxn
−SPC αn−1 Fxn−1 1 − αn−1 gxn−1 − λAxn−1 ≤ αn Fxn 1 − αn gxn − λAxn
− αn−1 Fxn−1 1 − αn−1 gxn−1 − λAxn−1 ≤ αn Fxn − Fxn−1 |αn − αn−1 |
Fxn−1 Abstract and Applied Analysis
7
1 − αn gxn − λAxn − gxn−1 − λAxn−1 |αn − αn−1 |gxn−1 − λAxn−1 ≤ αn L
xn − xn−1 1 − αn gxn − gxn−1 |αn − αn−1 | Fxn−1 gxn−1 − λAxn−1 L gxn − gxn−1 1 − αn gxn − gxn−1 γ
|αn − αn−1 | Fxn−1 gxn−1 − λAxn−1 L
1− 1−
αn gxn − gxn−1 γ
|αn − αn−1 | Fxn−1 gxn−1 − λAxn−1 .
≤ αn
3.12
It follows that
Sun − Sun−1 − gxn − gxn−1 ≤ |αn − αn−1 | Fxn−1 gxn−1 − λAxn−1 .
3.13
Since αn → 0 and the sequences {Fxn }, {gxn }, and {Axn } are bounded, we have
lim sup Sun − Sun−1 − gxn − gxn−1 ≤ 0.
n→∞
3.14
By Lemma 2.3, we obtain
lim Sun − gxn 0.
3.15
lim gxn1 − gxn lim 1 − β Sun − gxn 0.
3.16
n→∞
Hence,
n→∞
n→∞
This together with 3.6 imply that
lim xn1 − xn 0.
n→∞
By the convexity of the norm and 3.9, we have
gxn1 − gu2 β gxn − gu 1 − β Sun − Sgu 2
2
2 ≤ βgxn − gu 1 − β Sun − Sgu
2
2 ≤ βgxn − gu 1 − β un − gu
3.17
8
Abstract and Applied Analysis
2
≤ βgxn − gu
1 − β αn Fxn − gu
2
1 − αn gxn − λAxn − gu − λAu 2
≤ βgxn − gu
2
1 − β αn Fxn − gu
2 1 − αn gxn − λAxn − gu − λAu .
3.18
From Lemma 2.1, we derive
gxn − λAxn − gu − λAu 2 ≤ gxn − gu2 λλ − 2α
Axn − Au
2 .
3.19
Thus,
gxn1 − gu2 ≤ βgxn − gu2
2
1 − β αn Fxn − gu
2
1 − αn gxn − gu λλ − 2α
Axn − Au
2
2 2
1 − β αn Fxn − gu 1 − 1 − β αn gxn − gu
1 − β 1 − αn λλ − 2α
Axn − Au
2 .
3.20
So,
1 − β 1 − αn λ2α − λ
Axn − Au
2
2 2 2
≤ 1 − β αn Fxn − gu gxn − gu − gxn1 − gu
2
≤ 1 − β αn Fxn − gu
gxn − gu gxn1 − gu gxn1 − gxn .
3.21
Since αn → 0, gxn1 − gxn → 0 and lim infn → ∞ 1 − β1 − αn λ2α − λ > 0, we obtain
lim Axn − Au
0.
n→∞
3.22
Abstract and Applied Analysis
9
Set yn gxn − λAxn − gu − λAu for all n. By using the property of projection, we get
un − gu2 PC αn Fxn 1 − αn gxn − λAxn
2
−PC αn gu 1 − αn gu − λAu ≤ αn Fxn − gu 1 − αn yn , un − gu
≤
1 αn Fxn − gu 1 − αn yn 2 un − gu2
2
2 −αn Fxn − gu 1 − αn yn − un gu
2
2
2 1 αn Fxn − gu 1 − αn gxn − gu un − gu
2
2 −αn Fxn − gu − yn gxn − un − λAxn − Au
2
2
2 1 αn Fxn − gu 1 − αn gxn − gu un − gu
2
2
2
− gxn − un − λ2 Axn − Au
− α2n Fxn − gu − yn 2λαn Axn − Au, Fxn − gu − yn 2λ gxn − un , Axn − Au
−2αn gxn − un , Fxn − gu − yn .
3.23
It follows that
un − gu2 ≤ αn Fxn − gu2 1 − αn gxn − gu2 − gxn − un 2
2λαn Axn − Au
Fxn − gu − yn 2λgxn − un Axn − Au
3.24
2αn gxn − un Fxn − gu − yn .
From 3.18 and 3.24, we have
gxn1 − gu2 ≤ βgxn − gu2 1 − β un − gu2
2 2
≤ βgxn − gu 1 − β αn Fxn − gu
2 2
1 − αn 1 − β gxn − gu − 1 − β gxn − un 2λ 1 − β αn Axn − Au
Fxn − gu − yn 2λ 1 − β gxn − un Axn − Au
10
Abstract and Applied Analysis
2 1 − β αn gxn − un Fxn − gu − yn 2
2
2 ≤ gxn − gu αn Fxn − gu − 1 − β gxn − un 2λαn Axn − Au
Fxn − gu − yn 2λgxn − un Axn − Au
3.25
2αn gxn − un Fxn − gu − yn .
Then, we obtain
2 1 − β gxn − un ≤ gxn − gu gxn1 − gu gxn1 − gxn 2
αn Fxn − gu 2λαn Axn − Au
Fxn − gu − yn 2λgxn − un Axn − Au
2αn gxn − un Fxn − gu − yn .
3.26
Since limn → ∞ αn 0, limn → ∞ gxn1 − gxn 0 and limn → ∞ Axn − Au
0, we have
lim gxn − un 0.
n→∞
3.27
Next, we prove lim supn → ∞ Fx∗ − gx∗ , un − gx∗ ≤ 0 where x∗ is the unique solution of
3.2. We take a subsequence {uni } of {un } such that
lim sup Fx∗ − gx∗ , un − gx∗ lim Fx∗ − gx∗ , uni − gx∗ i→∞
n→∞
lim Fx∗ − gx∗ , gxni − gx∗ .
3.28
i→∞
Since {xni } is bounded, there exists a subsequence {xnij } of {xni } which converges weakly
to some point z ∈ C. Without loss of generality, we may assume that xni z. This implies
that gxni gz due to the weak continuity of g. Now, we show z ∈ Ω. First, we note
that from 3.15 and 3.27 that gxn − Sgxn → 0. Hence, limi → ∞ gxni − Sgxni 0.
By the demiclosedness principle of the nonexpansive mapping see Lemma 2.2, we deduce
gz ∈ FixS. Next, we only need to prove z ∈ GVIC, A, g. Set
Tv ⎧
⎨Av NC v,
v ∈ C,
⎩∅,
v∈
/ C.
3.29
By 39, we know that T is maximal g-monotone. Let v, w ∈ GT . Since w − Av ∈ NC v
and xn ∈ C, we have
gv − gxn , w − Av ≥ 0.
3.30
Abstract and Applied Analysis
11
From un PC αn Fxn 1 − αn gxn − λAxn , we get
gv − un , un − αn Fxn 1 − αn gxn − λAxn
≥ 0.
3.31
It follows that
un − gxn αn gv − un ,
Axn −
Fxn − gxn λAxn
λ
λ
≥ 0.
3.32
Then,
gv − gxni , w ≥ gv − gxni , Av
un − gxni ≥ gv − gxni , Av − gv − uni , i
λ
αn − gv − uni , Axni i gv − uni , Fxni − gxni λAxni
λ
gv − gxni , Av − Axni gv − gxni , −Axni
un − gxni − gv − uni , i
− gv − uni , Axni
λ
αni gv − uni , Fxni − gxni λAxni
λ
un − gxni − gxni − uni , Axni
≥ − gv − uni , i
λ
αni gv − uni , Fxni − gxni λAxni .
λ
3.33
Since gxni − uni → 0 and gxni gz, we deduce that gv − gz, w ≥ 0 by taking
i → ∞ in 3.33. Thus, z ∈ T −1 0 by the maximal g-monotonicity of T . Hence, z ∈ GVIC, A, g.
Therefore, z ∈ Ω. From 3.28, we obtain
lim sup Fx∗ − gx∗ , un − gx∗ lim Fx∗ − gx∗ , gxni − gx∗ n→∞
i→∞
Fx∗ − gx∗ , gz − gx∗ ≤ 0.
3.34
12
Abstract and Applied Analysis
We take u x∗ in 3.23 to get
un − gx∗ 2 ≤ αn Fxn − gx∗ , un − gx∗ 1 − αn gxn − λAxn − gx∗ − λAx∗ , un − gx∗ ≤ αn Fxn − Fx∗ , un − gx∗ αn Fx∗ − gx∗ , un − gx∗ 1 − αn gxn − λAxn − gx∗ − λAx∗ un − gx∗ ≤ αn L
xn − x∗ un − gx∗ αn Fx∗ − gx∗ , un − gx∗ 1 − αn gxn − gx∗ un − gx∗ L gxn − gx∗ un − gx∗ αn Fx∗ − gx∗ , un − gx∗ γ
1 − αn gxn − gx∗ un − gx∗ L
1− 1−
αn gxn − gx∗ un − gx∗ γ
αn Fx∗ − gx∗ , un − gx∗ 1 − 1 − L/γ αn gxn − gx∗ 2 1 un − gx∗ 2
2
2
αn Fx∗ − gx∗ , un − gu .
3.35
≤ αn
It follows that
un − gx∗ 2 ≤ 1 − 1 − L αn gxn − gx∗ 2 2αn Fx∗ − gx∗ , un − gx∗ . 3.36
γ
Therefore,
gxn1 − gx∗ 2 ≤ βgxn − gx∗ 2 1 − β un − gx∗ 2
2
L
∗ 2
αn gxn − gx∗ ≤ β gxn − gx 1 − β 1 − 1 −
γ
2 1 − β αn Fx∗ − gx∗ , un − gx∗ 2
L 1− 1−
1 − β αn gxn − gx∗ γ
2 1 − β αn Fx∗ − gx∗ , un − gx∗ 2
L 1− 1−
1 − β αn gxn − gx∗ γ
Abstract and Applied Analysis
L 2
1−
1 − β αn
Fx∗ − gx∗ , un − gx∗ γ
1 − L/γ
2
1 − γn gxn − gx∗ δn γn ,
13
3.37
where γn 1−L/γ1−βαn and δn 2/1−L/γFx∗ −gx∗ , un −gx∗ . From condition
C2, we have n γn ∞. By 3.34, we have lim supn → ∞ δn ≤ 0. We can therefore apply
Lemma 2.4 to conclude that gxn → gx∗ and xn → x∗ . This completes the proof.
Corollary 3.2. Let C be a nonempty closed and convex subset of a real Hilbert space H. Let F :
C → H be an L-contraction. Let A : C → H be an α-inverse strongly monotone mapping and let
S : C → C be a nonexpansive mapping. Suppose that Ω /
∅. Let β ∈ 0, 1 and γ ∈ L, 2α. For given
x0 ∈ C, let {xn } ⊂ C be a sequence generated by
xn1 βxn 1 − β SPC αn Fxn 1 − αn xn − λAxn ,
n ≥ 0,
3.38
where {αn } ⊂ 0, 1 satisfies C1: limn → ∞ αn 0 and C2: n αn ∞. Then the sequence {xn }
generated by 3.38 converges strongly to x∗ ∈ Ω which is the unique solution of the following
variational inequality:
Fx∗ − x∗ , x − x∗ ≤ 0,
∀x ∈ Ω.
3.39
Acknowledgments
The authors are very grateful to the referees for their comments and suggestions which
improved the presentation of this paper. The first author was partially supported by the
Program TH-1-3, Optimization Lean Cycle, of Subprojects TH-1 of Spindle Plan Four in
Excellence Teaching and Learning Plan of Cheng Shiu University and was supported in
part by NSC 100-2221-E-230-012. The second author was supported in part by Colleges and
Universities Science and Technology Development Foundation 20091003 of Tianjin, NSFC
11071279 and NSFC 71161001-G0105.
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