Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2011, Article ID 976505, 23 pages
doi:10.1155/2011/976505
Research Article
Generalized Systems of Variational Inequalities
and Projection Methods for Inverse-Strongly
Monotone Mappings
Wiyada Kumam,1, 2, 3 Prapairat Junlouchai,1 and Poom Kumam2, 3
1
Department of Mathematics, Faculty of Science and Technology, Rajamangala University of Technology
Thanyaburi (RMUTT), Thanyaburi, Pathumthani 12110, Thailand
2
Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi
(KMUTT), Bangmod, Bangkok 10140, Thailand
3
Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand
Correspondence should be addressed to Poom Kumam, poom.kum@kmutt.ac.th
Received 20 February 2011; Accepted 3 May 2011
Academic Editor: Jianshe Yu
Copyright q 2011 Wiyada Kumam 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 an iterative sequence for finding a common element of the set of fixed points of a
nonexpansive mapping and the solutions of the variational inequality problem for three inversestrongly monotone mappings. Under suitable conditions, some strong convergence theorems for
approximating a common element of the above two sets are obtained. Moreover, using the above
theorem, we also apply to find solutions of a general system of variational inequality and a zero of
a maximal monotone operator in a real Hilbert space. As applications, at the end of the paper we
utilize our results to study some convergence problem for strictly pseudocontractive mappings.
Our results include the previous results as special cases extend and improve the results of Ceng et
al., 2008 and many others.
1. Introduction
Variational inequalities are known to play a crucial role in mathematics as a unified
framework for studying a large variety of problems arising, for instance, in structural
analysis, engineering sciences and others.Roughly speaking, they can be recast as fixed-point
problems, and most of the numerical methods related to this topic are based on projection
methods. Let H be a real Hilbert space with inner product ·, · and · , and let E be a
nonempty, closed, convex subset of H. A mapping A : E → H is called α-inverse-strongly
monotone if there exists a positive real number α > 0 such that
2
Ax − Ay, x − y ≥ αAx − Ay ,
∀x, y ∈ E
1.1
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see 1, 2. It is obvious that every α-inverse-strongly monotone mapping A is monotone
and Lipschitz continuous. A mapping S : E → E is called nonexpansive if
Sx − Sy ≤ x − y,
∀x, y ∈ E.
1.2
We denote by FS the set of fixed points of S and by PE the metric projection of H onto E.
Recall that the classical variational inequality, denoted by V IA, E, is to find an x∗ ∈ E such
that
Ax∗ , x − x∗ ≥ 0,
∀x ∈ E.
1.3
The set of solutions of V IA, E is denoted by Γ. The variational inequality has been widely
studied in the literature; see, for example, 3–6 and the references therein.
For finding an element of FS ∩ Γ, Takahashi and Toyoda 7 introduced the following
iterative scheme:
xn1 αn xn 1 − αn SPE xn − λn Axn ,
1.4
for every n 0, 1, 2, . . ., where x0 x ∈ E, {αn } is a sequence in 0, 1, and {λn } is a sequence
in 0, 2α. On the other hand, for solving the variational inequality problem in the finitedimensional Euclidean space Rn , Korpelevich 1976 8 introduced the following so-called
extragradient method:
x0 x ∈ E,
yn PE xn − λn Axn ,
xn1 PE xn − λn Ayn ,
1.5
for every n 0, 1, 2, . . ., where λn ∈ 0, 1/k. Many authors using extragradient method for
approximating common fixed points and variational inequality problems see also 9, 10.
Recently, Nadezhkina and Takahashi 11 and Zeng et al. 12 proposed some iterative
schemes for finding elements in FS ∩ Γ by combining 1.4 and 1.5. Further, these iterative
schemes are extended in Y. Yao and J. C. Yao 13 to develop a new iterative scheme for
finding elements in FS ∩ Γ.
Consider the following problem of finding x∗ , y∗ ∈ E × E such that see cf. Ceng et al.
14:
λAy∗ x∗ − y∗ , x − x∗ ≥ 0,
μBx∗ y∗ − x∗ , x − y∗ ≥ 0,
∀x ∈ E,
∀x ∈ E,
1.6
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which is called general system of variational inequalities GSVI, where λ > 0 and μ > 0 are two
constants. In particular, if A B, then problem 1.6 reduces to finding x∗ , y∗ ∈ E × E such
that
λAy∗ x∗ − y∗ , x − x∗ ≥ 0,
μAx∗ y∗ − x∗ , x − y∗ ≥ 0,
∀x ∈ E,
∀x ∈ E,
1.7
which is defined by 15, 16, and is called the new system of variational inequalities. Further,
if x∗ y∗ , then problem 1.7 reduces to the classical variational inequality VIA, E, that is, find
x∗ ∈ E such that Ax∗ , x − x∗ ≥ 0, for all x ∈ E.
We can characteristic problem, if x∗ ∈ FS ∩ V IA, E, then it follows that x∗ Sx∗ ∗
PE x − ρAx∗ , where ρ > 0 is a constant.
In 2008, Ceng et al. 14 introduced a relaxed extragradient method for finding
solutions of problem 1.6. Let the mappings A, B : E → H be α-inverse-strongly monotone
and β-inverse-strongly monotone, respectively. Let S : E → E be a nonexpansive mapping.
Suppose x1 u ∈ E and {xn } is generated by
xn1
yn PE xn − μBxn ,
αn u βn xn γn SPE yn − λn Ayn ,
1.8
where λ ∈ 0, 2α, μ ∈ 0, 2β, and {αn }, {βn }, {γn } are three sequences in 0, 1 such that
αn βn γn 1, for all n ≥ 1. First, problem 1.6 is proven to be equivalent to a fixed point
problem of a nonexpansive mapping.
In this paper, motivated by what is mentioned above, we consider generalized system
of variational inequalities as follows.
Let E be a nonempty, closed, convex subset of a real Hilbert space H. Let A, B, C : E →
H be three mappings. We consider the following problem of finding x∗ , y∗ , z∗ ∈ E × E × E
such that
λAy∗ x∗ − y∗ , x − x∗ ≥ 0,
∀x ∈ E,
μBz∗ y∗ − z∗ , x − y∗ ≥ 0,
∀x ∈ E,
τCx∗ z∗ − x∗ , x − z∗ ≥ 0,
∀x ∈ E,
1.9
which is called a general system of variational inequalities where λ > 0, μ > 0 and τ > 0 are three
constants.
In particular, if A B C, then problem 1.9 reduces to finding x∗ , y∗ , z∗ ∈ E × E × E
such that
λAy∗ x∗ − y∗ , x − x∗ ≥ 0,
∀x ∈ E,
μAz∗ y∗ − z∗ , x − y∗ ≥ 0,
∀x ∈ E,
τAx∗ z∗ − x∗ , x − z∗ ≥ 0,
∀x ∈ E.
1.10
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Discrete Dynamics in Nature and Society
Next, we consider some special classes of the GSVI problem 1.9 reduce to the following
GSVI.
i If τ 0, then the GSVI problems 1.9 reduce to GSVI problem 1.6.
ii If τ μ 0, then the GSVI problems 1.9 reduce to classical variational inequality
VIA,E problem.
The above system enters a class of more general problems which originated mainly from the
Nash equilibrium points and was treated from a theoretical viewpoint in 17, 18. Observe at
the same time that, to construct a mathematical model which is as close as possible to a real
complex problem, we often have to use constraints which can be expressed as one several
subproblems of a general problem. These constrains can be given, for instance, by variational
inequalities, by fixed point problems, or by problems of different types.
This paper deals with a relaxed extragradient approximation method for solving a
system of variational inequalities over the fixed-point sets of nonexpansive mapping. Under
classical conditions, we prove a strong convergence theorem for this method. Moreover, the
proposed algorithm can be applied for instance to solving the classical variational inequality
problems.
2. Preliminaries
Let E be a nonempty, closed, convex subset of a real Hilbert space H. For every point x ∈ H,
there exists a unique nearest point in E, denoted by PE x, such that
x − PE x ≤ x − y,
∀y ∈ E.
2.1
PE is call the metric projection of H onto E.
Recall that, PE x is characterized by following properties: PE x ∈ E and
x − PE x, y − PE x ≤ 0,
x − y2 ≥ x − PE x2 PE x − y2 ,
2.2
for all x ∈ H and y ∈ E.
Lemma 2.1 see cf. Zhang et al. 19. The metric projection PE has the following properties:
i PE : H → E is nonexpansive;
ii PE : H → E is firmly nonexpansive, that is,
PE x − PE y2 ≤ PE x − PE y, x − y ,
∀x, y ∈ H;
2.3
∀y ∈ E.
2.4
iii for each x ∈ H,
z PE x ⇐⇒ x − z, z − y ≥ 0,
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Lemma 2.2 see Osilike and Igbokwe 20. Let E, ·, · be an inner product space. Then for all
x, y, z ∈ E and α, β, γ ∈ 0, 1 with α β γ 1, one has
αx βy γz2 αx2 βy2 γz2 − αβx − y2 − αγx − z2 − βγ y − z2 .
2.5
Lemma 2.3 see Suzuki 21. 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 integers n ≥ 0 and lim supn → ∞ yn1 − yn − xn1 − xn ≤ 0. Then,
limn → ∞ yn − xn 0.
Lemma 2.4 see Xu 22. Assume {an } is a sequence of nonnegative real numbers such that
an1 ≤ 1 − αn an δn ,
n ≥ 0,
2.6
where {αn } is a sequence in 0, 1 and {δn } is a sequence in R such that
i
∞
n1
αn ∞,
ii lim supn → ∞ δn /αn ≤ 0 or
∞
n1
|δn | < ∞.
Then, limn → ∞ an 0.
Lemma 2.5 Goebel and Kirk 23. Demiclosedness Principle. Assume that T is a nonexpansive
self-mapping of a nonempty, closed, convex subset E of a real Hilbert space H. If T has a fixed point,
then I − T is demiclosed; that is, whenever {xn } is a sequence in E converging weakly to some x ∈ E
(for short, xn x ∈ E), and the sequence {I − T xn } converges strongly to some y (for short,
I − T xn → y), it follows that I − T x y. Here, I is the identity operator of H.
The following lemma is an immediate consequence of an inner product.
Lemma 2.6. In a real Hilbert space H, there holds the inequality
x y2 ≤ x2 2 y, x y ,
∀x, y ∈ H.
2.7
Remark 2.7. We also have that, for all u, v ∈ E and λ > 0,
I − λAu − I − λAv2 u − v − λAu − Av2
u − v2 − 2λu − v, Au − Av
λ2 Au − Av2
≤ u − v2 λλ − 2αAu − Av2 .
So, if λ ≤ 2α, then I − λA is a nonexpansive mapping from E to H.
2.8
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Discrete Dynamics in Nature and Society
3. Main Results
In this section, we introduce an iterative precess by the relaxed extragradient approximation
method for finding a common element of the set of fixed points of a nonexpansive mapping
and the solution set of the variational inequality problem for three inverse-strongly monotone
mappings in a real Hilbert space. We prove that the iterative sequence converges strongly to
a common element of the above two sets.
In order to prove our main result, the following lemmas are needed.
Lemma 3.1. For given x∗ , y∗ , z∗ ∈ E × E × E, x∗ , y∗ , z∗ is a solution of problem 1.9 if and only if
x∗ is a fixed point of the mapping G : E → E defined by
Gx PE PE PE x − τCx − μBPE x − τCx
−λAPE PE x − τCx − μBPE x − τCx ,
∀x ∈ E,
3.1
where y∗ PE z∗ − μBz∗ and z∗ PE x∗ − τCx∗ .
Proof.
λAy∗ x∗ − y∗ , x − x∗ ≥ 0,
μBz∗ y∗ − z∗ , x − y∗ ≥ 0,
τCx∗ z∗ − x∗ , x − z∗ ≥ 0,
∀x ∈ E,
∀x ∈ E,
3.2
∀x ∈ E,
⇔
−y∗ λAy∗ x∗ , x − x∗ ≥ 0,
∗
−z μBz∗ y∗ , x − y∗ ≥ 0,
−x∗ τCx∗ z∗ , x − z∗ ≥ 0,
∀x ∈ E,
∀x ∈ E,
3.3
∀x ∈ E,
⇔
y∗ − λAy∗ − x∗ , x∗ − x ≥ 0,
∗
z − μBz∗ − y∗ , y∗ − x ≥ 0,
x∗ − τCx∗ − z∗ , z∗ − x ≥ 0,
∀x ∈ E,
∀x ∈ E,
3.4
∀x ∈ E,
⇔
x∗ PE y∗ − λAy∗ ,
y∗ PE z∗ − μBz∗ ,
z∗ PE x∗ − τCx∗ ,
⇔ x∗ PE PE z∗ − μBz∗ − λAPE z∗ − μBz∗ .
3.5
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Thus,
x∗ PE PE PE x∗ − τCx∗ − μBPE x∗ − τCx∗ −λAPE PE x∗ −τCx∗ − μBPE x∗ − τCx∗ .
3.6
Lemma 3.2. The mapping G defined by Lemma 3.1 is nonexpansive mappings.
Proof. For all x, y ∈ E,
Gx − G y PE PE PE x − τCx − μBPE x − τCx
−λAPE PE x − τCx − μBPE x − τCx
− PE PE PE y − τCy − μBPE y − τCy
−λAPE PE y − τCy − μBPE y − τCy
≤ PE x − τCx − μBPE x − τCx
− λAPE PE x − τCx − μBPE x − τCx
− PE y − τCy − μBPE y − τCy
−λAPE PE y − τCy − μBPE y − τCy I − λA PE x − τCx − μBPE x − τCx
−I − λA PE y − τCy − μBPE y − τCy ≤ PE x − τCx − μBPE x − τCx − PE y − τCy − μBPE y − τCy I − μB PE x − τCx − I − μB PE y − τCy ≤ PE x − τCx − PE y − τCy ≤ x − τCx − y − τCy I − τCx − I − τC y ≤ x − y.
3.7
This shows that G : E → E is a nonexpansive mapping.
Throughout this paper, the set of fixed points of the mapping G is denoted by Υ.
Now, we are ready to proof our main results in this paper.
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Theorem 3.3. Let E be a nonempty, closed, convex subset of a real Hilbert space H. Let the mapping
A, B, C : E → H be α-inverse-strongly monotone, β-inverse-strongly monotone, and γ-inversestrongly monotone, respectively. Let S be a nonexpansive mapping of E into itself such that FS ∩
Υ
/ ∅. Let f be a contraction of H into itself and given x1 ∈ H arbitrarily and {xn } is generated by
zn PE xn − τCxn ,
xn1
yn PE zn − μBzn ,
αn fxn βn xn γn SPE yn − λAyn ,
3.8
n ≥ 0,
where λ ∈ 0, 2α, μ ∈ 0, 2β, τ ∈ 0, 2γ, and {αn }, {βn }, {γn } are three sequences in 0, 1 such
that
i αn βn γn 1,
ii limn → ∞ αn 0 and
∞
n1
αn ∞,
iii 0 < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1.
Then, {xn } converges strongly to x ∈ FS ∩ Γ, where x PFS∩Γ fx and x, y, z is a solution of
problem 1.9, where
y PE z − μBz ,
z PE x − τCx.
3.9
Proof. Let x∗ ∈ FS ∩ Γ. Then, x∗ Sx∗ and x∗ Gx∗ , that is,
x∗ PE PE PE x∗ − τCx∗ − μBPE x∗ − τCx∗ − λAPE PE x∗ − τCx∗ − μBPE x∗ − τCx∗ .
3.10
Put x∗ PE y∗ −λAy∗ and tn PE yn −λAyn . Then, x∗ PE PE z∗ −μBz∗ −λAPE z∗ −μBz∗ implies that y∗ PE z∗ − μBz∗ , where z∗ PE x∗ − τCx∗ . Since I − λA, I − μB and I − τC are
nonexpansive mappings. We obtain that
tn − x∗ PE yn − λAyn − x∗ PE yn − λAyn − PE y∗ − λAy∗ ≤ yn − λAyn − y∗ − λAy∗ I − λAyn − I − λAy∗ ≤ yn − y∗ yn − PE z∗ − μBz∗ PE zn − μBzn − PE z∗ − μBz∗ 3.11
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≤ I − μB zn − I − μB z∗ ≤ zn − z∗ ,
3.12
zn − z∗ PE xn − τCxn − PE x∗ − τCx∗ ≤ xn − τCxn − x∗ − τCx∗ I − τCxn − I − τCx∗ 3.13
≤ xn − x∗ .
Substituting 3.13 into 3.12, we have
tn − x∗ ≤ xn − x∗ ,
3.14
yn − y∗ ≤ xn − x∗ .
3.15
and by 3.11 we also have
Since xn1 αn fxn βn xn γn Stn and by Lemma 2.2, we compute
xn1 − x∗ αn fxn βn xn γn Stn − x∗ αn fxn − x∗ βn xn − x∗ γn Stn − x∗ ≤ αn fxn − x∗ βn xn − x∗ γn Stn − x∗ ≤ αn fxn − x∗ βn xn − x∗ γn tn − x∗ ≤ αn fxn − x∗ βn xn − x∗ γn xn − x∗ αn fxn − x∗ 1 − αn xn − x∗ αn fxn − fx∗ fx∗ − x∗ 1 − αn xn − x∗ ≤ αn fxn − fx∗ αn fx∗ − x∗ 1 − αn xn − x∗ ≤ αn kxn − x∗ αn fx∗ − x∗ 1 − αn xn − x∗ 10
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αn k 1 − αn xn − x∗ αn fx∗ − x∗ 1 − αn 1 − kxn − x∗ αn fx∗ − x∗ ∗
1 − αn 1 − k xn − x αn 1 − k
fx∗ − x∗ 1 − k
.
3.16
By induction, we get
xn1 − x∗ ≤ M,
3.17
where M max{x0 − x∗ 1/1 − kfx∗ − x∗ }, n ≥ 0. Therefore, {xn } is bounded.
Consequently, by 3.11, 3.12 and 3.13, the sequences {tn }, {Stn }, {yn }, {Ayn }, {zn }, {Bzn },
{Cxn }, and {fxn } are also bounded. Also, we observe that
zn1 − zn PE xn1 − τCxn1 − PE xn − τCxn ≤ I − τCxn1 − I − τCxn 3.18
≤ xn1 − xn ,
tn1 − tn PE yn1 − λAyn1 − PE yn − λAyn ≤ yn1 − λAyn1 − yn − λAyn I − λAyn1 − I − λAyn ≤ yn1 − yn 3.19
PE zn1 − μBzn1 − PE zn − μBzn ≤ zn1 − zn ≤ xn1 − xn .
Let xn1 1 − βn wn βn xn . Thus, we get
xn1 − βn xn αn fxn γn SPC yn − λn Ayn
αn u γn Stn
wn 1 − βn
1 − βn
1 − βn
3.20
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It follows that
wn1 − wn αn1 fxn1 γn1 Stn1 αn fxn γn Stn
−
1 − βn1
1 − βn
αn1 fxn1 γn1 Stn1 αn1 fxn αn1 fxn αn fxn γn Stn
−
−
−
1 − βn1
1 − βn1
1 − βn1
1 − βn1
1 − βn
1 − βn
γn1 Stn1 γn Stn
αn1 αn
αn1
fxn −
−
fxn1 − fxn 1 − βn1
1 − βn1 1 − βn
1 − βn1
1 − βn
αn1 αn
αn1
−
fxn fxn1 − fxn 1 − βn1
1 − βn1 1 − βn
γn1 Stn1 γn1 Stn
γn1 Stn
γn Stn
−
−
1 − βn1
1 − βn1 1 − βn1 1 − βn
αn1 αn
αn1
−
fxn fxn1 − fxn 1 − βn1
1 − βn1 1 − βn
γn
γn1
γn1
−
Stn
Stn1 − Stn 1 − βn1
1 − βn1 1 − βn
αn1 αn
αn1
−
fxn fxn1 − fxn 1 − βn1
1 − βn1 1 − βn
γn1
αn
αn1
−
Stn Stn1 − Stn 1 − βn1 1 − βn
1 − βn1
αn
αn1 αn1
−
fxn Stn
fxn1 − fxn 1 − βn1
1 − βn1 1 − βn
γn1
Stn1 − Stn .
1 − βn1
3.21
Combining 3.19 and 3.21, we obtain
αn1 fxn1 − fxn αn1 − αn fxn Stn wn1 − wn − xn1 − xn ≤ 1 − βn1
1 − βn1 1 − βn
γn1 Stn1 − Stn − xn1 − xn 1 − βn1 αn1 kxn1 − xn αn1 − αn fxn Stn ≤ 1 − βn1
1 − βn1 1 − βn
γn1 tn1 − tn − xn1 − xn 1 − βn1 αn1 αn1
αn fxn Stn kxn1 − xn ≤
−
1 − βn1
1 − βn1 1 − βn
12
γn1
1−β
αn1
1−β
n1
n1
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xn1 − xn − xn1 − xn kxn1 − xn αn1
1 − β
n1
−
αn fxn Stn 1 − βn
γn1 − 1 βn1 xn1 − xn 1 − βn1
αn1 kxn1 − xn αn1 − αn fxn Stn 1 − βn1
1 − βn1 1 − βn
αn1 xn1 − xn .
1 − βn1 3.22
This together with i, ii, and iii implies that
lim supwn1 − wn − xn1 − xn ≤ 0.
n→∞
3.23
Hence, by Lemma 2.3, we have
lim wn − xn 0.
3.24
lim xn1 − xn lim 1 − βn wn − xn 0.
3.25
n→∞
Consequently,
n→∞
n→∞
From 3.18 and 3.19, we also have zn1 − zn → 0tn1 − tn → 0 and yn1 − yn → 0 as
n → ∞. Since
xn1 − xn αn fxn βn xn γn Stn − xn αn fxn − xn γn Stn − xn ,
3.26
it follows by ii and 3.25 that
lim xn − Stn 0.
n→∞
3.27
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13
Since x∗ ∈ FS ∩ Γ, from 3.15 and Lemma 2.2, we get
2
xn1 − x∗ 2 αn fxn βn xn γn Stn − x∗ 2
≤ αn fxn − x∗ βn xn − x∗ 2 γn Stn − x∗ 2
2
≤ αn fxn − x∗ βn xn − x∗ 2 γn tn − x∗ 2
2
2
αn fxn − x∗ βn xn − x∗ 2 γn PE yn − λAyn − PE y∗ − λAy∗ 2
2
≤ αn fxn − x∗ βn xn − x∗ 2 γn yn − λAyn − y∗ − λAy∗ 2
2
αn fxn − x∗ βn xn − x∗ 2 γn yn − y∗ − λ Ayn − Ay∗ 2
αn fxn − x∗ βn xn − x∗ 2
2 γn yn − y∗ 2 − 2λ yn − y∗ , Ayn − Ay∗ λ2 Ayn − Ay∗ 2
≤ αn fxn − x∗ βn xn − x∗ 2
2
2
2 γn yn − y∗ − 2λαAyn − Ay∗ λ2 Ayn − Ay∗ 2
2
2 αn fxn − x∗ βn xn − x∗ 2 γn yn − y∗ λλ − 2αAyn − Ay∗ 2
2 ≤ αn fxn − x∗ βn xn − x∗ 2 γn xn − x∗ 2 λλ − 2αAyn − Ay∗ 2
2
αn fxn − x∗ βn xn − x∗ 2 γn xn − x∗ 2 γn λλ − 2αAyn − Ay∗ 2 2
αn fxn − x∗ βn γn xn − x∗ 2 γn λλ − 2αAyn − Ay∗ 2
2
αn fxn − x∗ 1 − αn xn − x∗ 2 γn λλ − 2αAyn − Ay∗ 2
2
≤ αn fxn − x∗ xn − x∗ 2 γn λλ − 2αAyn − Ay∗ .
3.28
Therefore, we have
2
2
−γn λλ − 2αAyn − Ay∗ ≤ αn fxn − x∗ xn − x∗ 2 − xn1 − x∗ 2
2
αn fxn − x∗ xn − x∗ xn1 − x∗ × xn − x∗ − xn1 − x∗ 2
≤ αn fxn − x∗ xn − x∗ xn1 − x∗ xn − xn1 .
From ii, iii, and xn1 − xn → 0, as n → ∞, we get Ayn − Ay∗ → 0 as n → ∞.
3.29
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Since x∗ ∈ FS ∩ Υ, from 3.11 and Lemma 2.2, we get
2
xn1 − x∗ 2 αn fxn βn xn γn Stn − x∗ 2
≤ αn fxn − x∗ βn xn − x∗ 2 γn tn − x∗ 2
2
2
≤ αn fxn − x∗ βn xn − x∗ 2 γn yn − y∗ 2
2
αn fxn − x∗ βn xn − x∗ 2 γn PE zn − μBzn − PE z∗ − μBz∗ 2
2
≤ αn fxn − x∗ βn xn − x∗ 2 γn zn − μBzn − z∗ − μBz∗ 2
2
αn fxn − x∗ βn xn − x∗ 2 γn zn − z∗ − μBzn − μBz∗ 2
≤ αn fxn − x∗ βn xn − x∗ 2 γn zn − z∗ 2 μ μ − 2β Bzn − Bz∗ 2
2
≤ αn fxn − x∗ xn − x∗ 2 γn μ μ − 2β Bzn − Bz∗ 2 .
3.30
Thus, we also have
2
−γn μ μ − 2β Bzn − Bz∗ 2 ≤ αn fxn − x∗ xn − x∗ 2 − xn1 − x∗ 2
2
αn fxn − x∗ xn − x∗ xn1 − x∗ × xn − x∗ − xn1 − x∗ 2
≤ αn fxn − x∗ xn − x∗ xn1 − x∗ xn − xn1 .
3.31
By again ii, iii, and 3.25, we also get Bzn − Bz∗ → 0 as n → ∞.
Let x∗ ∈ FS ∩ Υ; again from 3.12, 3.13 and Lemma 2.2, we get
2
xn1 − x∗ 2 αn fxn βn xn γn Stn − x∗ 2
≤ αn fxn − x∗ βn xn − x∗ 2 γn tn − x∗ 2
2
≤ αn fxn − x∗ βn xn − x∗ 2 γn zn − z∗ 2
2
≤ αn fxn − x∗ βn xn − x∗ 2 γn xn − τCxn − x∗ − τCx∗ 2
2
≤ αn fxn − x∗ βn xn − x∗ 2 γn xn − x∗ 2 τ τ − 2γ Cxn − Cx∗ 2
2
≤ αn fxn − x∗ xn − x∗ 2 γn τ τ − 2γ Cxn − Cx∗ 2 .
3.32
Discrete Dynamics in Nature and Society
15
Again, we have
2
−γn τ τ − 2γ Cxn − Cx∗ 2 ≤ αn fxn − x∗ xn − x∗ 2 − xn1 − x∗ 2
2
αn fxn − x∗ xn − x∗ xn1 − x∗ × xn − x∗ − xn1 − x∗ 2
≤ αn fxn − x∗ xn − x∗ xn1 − x∗ xn − xn1 .
3.33
Similarly again by ii, iii, and xn − xn1 → 0 as n → ∞, and from 3.33, we also that
Cxn − Cx∗ → 0.
On the other hand, we compute that
zn − z∗ 2 PE xn − τCxn − PE x∗ − τCx∗ 2
≤ xn − τCxn − x∗ − τCx∗ , PE xn − τCxn − PE x∗ − τCx∗ xn − τCxn − x∗ − τCx∗ , zn − z∗ 1
xn − τCxn − x∗ − τCx∗ 2 zn − z∗ 2
2
−xn − τCxn − x∗ − τCx∗ − zn − z∗ 2
1
I − τCxn − I − τCx∗ 2 zn − z∗ 2
2
−xn − τCxn − x∗ − τCx∗ − zn − z∗ 2
3.34
1
xn − x∗ 2 zn − z∗ 2 − xn − zn − τCxn − Cx∗ − x∗ − z∗ 2
2
1
xn − x∗ 2 zn − z∗ 2 − xn − zn − x∗ − z∗ − τCxn − Cx∗ 2
2
1
xn − x∗ 2 zn − z∗ 2 − xn − zn − x∗ − z∗ 2
2
2τxn − zn − x∗ − z∗ , Cxn − Cx∗ − τ 2 Cxn − Cx∗ 2 .
≤
So, we obtain
zn − z∗ 2 ≤ xn − x∗ 2 − xn − zn − x∗ − z∗ 2
2τxn − zn − x∗ − z∗ , Cxn − Cx∗ − τ 2 Cxn − Cx∗ 2 .
Hence, by 3.12, it follows that
2
xn1 − x∗ 2 αn fxn βn xn γn Stn − x∗ 2
≤ αn fxn − x∗ βn xn − x∗ 2 γn Stn − x∗ 2
3.35
16
Discrete Dynamics in Nature and Society
≤ αn kxn − x∗ 2 βn xn − x∗ 2 γn tn − x∗ 2
≤ αn kxn − x∗ 2 βn xn − x∗ 2 γn zn − z∗ 2
≤ αn kxn − x∗ 2 βn xn − x∗ 2 γn xn − x∗ 2 − γn xn − zn − x∗ − z∗ 2
2τγn xn − zn − x∗ − z∗ , Cxn − Cx∗ − τ 2 γn Cxn − Cx∗ 2
αn kxn − x∗ 2 1 − αn xn − x∗ 2 − γn xn − zn − x∗ − z∗ 2
2τγn xn − zn − x∗ − z∗ , Cxn − Cx∗ − τ 2 γn Cxn − Cx∗ 2
≤ αn kxn − x∗ 2 xn − x∗ 2 − γn xn − zn − x∗ − z∗ 2
2τγn xn − zn − x∗ − z∗ , Cxn − Cx∗ ≤ αn kxn − x∗ 2 xn − x∗ 2 − γn xn − zn − x∗ − z∗ 2
2τγn xn − zn − x∗ − z∗ Cxn − Cx∗ ,
3.36
which implies that
γn xn − zn − x∗ − z∗ 2 ≤ αn kxn − x∗ 2 xn − x∗ 2 − xn1 − x∗ 2
2τγn xn − zn − x∗ − z∗ Cxn − Cx∗ ∗ 2
∗
∗
∗
3.37
≤ αn kxn1 − x 2γn τxn − zn − x − z Cxn − Cx xn − xn1 xn − x∗ xn1 − x∗ .
By ii, iii, xn − xn1 → 0, and Cxn − Cx∗ → 0 as n → ∞, from 3.37 and we get
xn − zn − x∗ − z∗ → 0 as n → ∞. Now, observe that
2
zn − tn x∗ − z∗ 2 zn − PE yn − λAyn PE y∗ − λAy∗ − z∗ zn − PE yn − λAyn PE y∗ − λAy∗
2
−z∗ μBzn − μBzn μBz∗ − μBz∗ zn − μBzn − z∗ − μBz∗
2
− PE yn − λAyn − PE y∗ − λAy∗ μBzn − Bz∗ Discrete Dynamics in Nature and Society
17
2
≤ zn − μBzn − z∗ − μBz∗ − PE yn − λAyn − PE y∗ − λAy∗ 2μ Bzn − Bz∗ , zn − μBzn − z∗ − μBz∗
− PE yn − λAyn − PE y∗ − λAy∗ μBzn − Bz∗ 2
zn − μBzn − z∗ − μBz∗ − PE yn − λAyn − PE y∗ − λAy∗ 2μBzn − Bz∗ , zn − tn x∗ − z∗ 2
≤ zn − μBzn − z∗ − μBz∗ 2
− PE yn − λn Ayn − PE y∗ − λn Ay∗ 2μBzn − Bz∗ zn − tn x∗ − z∗ 2
≤ zn − μBzn − z∗ − μBz∗ 2
− SPE yn − λn Ayn − SPE y∗ − λn Ay∗ 2μBzn − Bz∗ zn − tn x∗ − z∗ 2
zn − μBzn − z∗ − μBz∗ − Stn − Sx∗ 2
2μBzn − Bz∗ zn − tn x∗ − z∗ ≤ zn − μBzn − z∗ − μBz∗ − Stn − x∗ × zn − μBzn − z∗ − μBz∗ Stn − x∗ 2μBzn − Bz∗ zn − tn x∗ − z∗ .
3.38
Since Stn − xn → 0, xn − zn − x∗ − z∗ → 0, and Bzn − Bz∗ → 0, as n → ∞, it follows
that
zn − tn x∗ − z∗ → 0,
as n → ∞.
3.39
Since
Stn − tn ≤ Stn − xn xn − zn − x∗ − z∗ zn − tn x∗ − z∗ ,
3.40
we obtain
lim Stn − tn 0.
n→∞
3.41
18
Discrete Dynamics in Nature and Society
Next, we show that
lim sup fx − x, xn − x ≤ 0,
n→∞
3.42
where x PFS∩Υ fx.
Indeed, since {tn } and {Stn } are two bounded sequence in E, we can choose a
subsequence {tni } of {tn } such that tni of tn such that tni z ∈ E and
lim sup fx − x, Stn − x lim fx − x, Stni − x .
i→∞
n→∞
3.43
Since limn → ∞ Stn − tn 0, we obtain Stni z as i → ∞. Now, we claim that z ∈ FS ∩ Υ.
First by Lemma 2.5, it is easy to see that z ∈ FS.
Since Stn − tn → 0, Stn − xn → 0, and
tn − xn tn − Stn Stn − xn ≤ tn − Stn Stn − xn 3.44
Stn − tn Stn − xn ,
we conclude that tn − xn → 0 as n → ∞. Furthermore, by Lemma 3.2 that G is
nonexpansive, then
tn − Gtn Gxn − Gtn ≤ xn − tn .
3.45
Thus limn → ∞ tn − Gtn 0. According to Lemma 2.5, we obtain z ∈ Υ. Therefore, there
holds z ∈ FS ∩ Υ.
On the other hand, it follows from 2.2 that
lim sup fx − x, xn − x lim sup fx − x, Stn − x
n→∞
n→∞
lim fx − x, Stni − x
n→∞
fx − x, z − x
≤ 0.
3.46
Discrete Dynamics in Nature and Society
19
Finally, we show that xn → x, and by 3.14 that
2
xn1 − x2 αn fxn βn xn γn Stn − x
2
≤ βn xn − x γn Stn − x 2αn fxn − x, xn1 − x
2
≤ βn xn − x γn Stn − x 2αn fxn − fx, xn1 − x
2αn fx − x, xn1 − x
≤ βn xn − x2 γn Stn − x2 2αn fxn − fxxn1 − x
2αn fx − x, xn1 − x
≤ βn xn − x2 γn tn − x2 2αn αxn − xxn1 − x
3.47
2αn fx − x, xn1 − x
≤ 1 − αn 2 xn − x2 αn α xn − x2 xn1 − x2
2αn fx − x, xn1 − x
which implies that
xn1 − x2 ≤
α2n
21 − ααn
1−
xn − x2 xn − x2
1 − ααn
1 − ααn
2αn fx − x, xn1 − x
1 − ααn
: 1 − σn xn − x2 δn ,
3.48
n ≥ 0,
where σn 21−ααn /1−ααn and δn α2n /1−ααn xn −x2 2αn /1−ααn fx−x, xn1 −x.
Therefore, by 3.46 and Lemma 2.4, we get that {xn } converges to x, where x PFS∩Υ fx.
This completes the proof.
Setting A B C, we obtain the following corollary.
Corollary 3.4. Let E be a nonempty, closed, convex subset of a real Hilbert space H. Let the mapping
A : E → H be α-inverse-strongly monotone. Let S be a nonexpansive mapping of E into itself such
that FS ∩ Υ /
∅. Let f be a contraction of H into itself and given x0 ∈ H arbitrarily and {xn } is
generated by
xn1
zn PE xn − τAxn ,
yn PE zn − μAzn ,
αn fxn βn xn γn SPE yn − λAyn ,
3.49
n ≥ 1,
where λ, μ, τ ∈ 0, 2α and {αn }, {βn }, {γn } are three sequences in 0, 1 such that
20
Discrete Dynamics in Nature and Society
i αn βn γn 1,
ii limn → ∞ αn 0 and
∞
n1
αn ∞,
iii 0 < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1.
Then, {xn } converges strongly to x ∈ FS ∩ Υ, where x PFS∩Υ fx and x, y, z is a solution of
problem 1.10, where
y PE z − μAz ,
3.50
z PE x − τAx.
Setting A ≡ B ≡ 0 the zero operators, we obtain the following corollary for solving
the foxed points problem and the classical variational inequality problems.
Corollary 3.5. Let E be a nonempty, closed, convex subset of a real Hilbert space H. Let the mapping
A : E → H be α-inverse-strongly monotone. Let S be a nonexpansive mapping of E into itself such
that FS ∩ V IA, E / ∅. Let f be a contraction of H into itself and given x0 ∈ H arbitrarily and
{xn } is generated by
xn1 αn fxn βn xn γn SPE xn − λAxn ,
n ≥ 1,
3.51
where λ ∈ 0, 2α and {αn }, {βn }, {γn } are three sequences in 0, 1 such that
i αn βn γn 1,
ii limn → ∞ αn 0 and
∞
n1
αn ∞,
iii 0 < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1.
Then, {xn } converges strongly to x ∈ FS ∩ V IA, E, where x PFS∩V IA,E fx.
4. Some Applications
We recall that a mapping T : E → E is called strictly pseudocontractive if there exists some k
with 0 ≤ k < 1 such that
T x − T y2 ≤ x − y2 kI − T x − I − T y2 ,
∀x, y ∈ E.
4.1
For recent convergence result for strictly pseudocontractive mappings, put A I − T . Then,
we have
I − Ax − I − Ay2 ≤ x − y2 kAx − Ay2 .
4.2
I − Ax − I − Ay2 ≤ x − y2 Ax − Ay2 − 2 x − y, Ax − Ay .
4.3
On the other hand,
Discrete Dynamics in Nature and Society
21
Hence, we have
1 − k
Ax − Ay2 .
x − y, Ax − Ay ≥
2
4.4
Consequently, if T : E → E is a strictly pseudocontractive mapping with constant k, then the
mapping A I − T is 1 − k/2-inverse-strongly monotone.
Setting A I − T , B I − V , and C I − W, we obtain the following corollary.
Theorem 4.1. Let E be a nonempty, closed, convex subset of a real Hilbert space H. Let T, V, W
be strictly pseudocontractive mappings with constant k of C into itself, and let S be a nonexpansive
mapping of E into itself such that FS ∩ Υ / ∅. Let f be a contraction of H into itself and given
x0 ∈ H arbitrarily and {xn } is generated by
zn I − τxn τWxn ,
yn I − μ zn μV zn ,
xn1 αn fxn βn xn γn S 1 − λyn λT yn ,
4.5
n ≥ 1,
where λ ∈ 0, 2α, μ ∈ 0, 2β, and τ ∈ 0, 2γ and {αn }, {βn }, {γn } are three sequences in 0, 1
such that
i αn βn γn 1,
ii limn → ∞ αn 0 and
∞
n1
αn ∞,
iii 0 < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1.
Then {xn } converges strongly to x ∈ FS ∩ Υ, where x PFS∩Υ fx and x, y, z is a solution of
problem 1.9, where
y PE z − μBz ,
z PE x − τCx.
4.6
Proof. Since A I − T , B I − V , and C I − W, we have
PE xn − τCxn I − τxn τWxn ,
PE yn − λAyn I − λyn λT yn ,
4.7
PE zn − μBzn I − μ zn μV zn .
Thus, the conclusion follows immediately from Theorem 3.3.
If fx x0 , for all x ∈ E, and T V W in Theorem 4.1, we obtain the following
corollary.
22
Discrete Dynamics in Nature and Society
Corollary 4.2. Let E be a nonempty, closed, convex subset of a real Hilbert space H. Let T be strictly
pseudocontractive mappings with constant k of C into itself, and let S be a nonexpansive mapping of
E into itself such that FS ∩ Υ /
∅. Given x0 ∈ H arbitrarily and {xn } is generated by
xn1
zn I − τxn τT xn ,
yn I − μ zn μT zn ,
αn x0 βn xn γn S 1 − λyn λT yn ,
4.8
n ≥ 1,
where λ ∈ 0, 2α, μ ∈ 0, 2β, and τ ∈ 0, 2γ and {αn }, {βn }, {γn } are three sequences in 0, 1
such that
αn βn γn 1,
lim αn 0 and
n→∞
∞
αn ∞,
n1
4.9
0 < lim inf βn ≤ lim sup βn < 1.
n→∞
n→∞
Then, {xn } converges strongly to x ∈ FS ∩ Υ, where x PFS∩Υ x and x, y, z is a solution of
problem 1.10, where
y PE z − μAz
z PE x − τAx.
4.10
Acknowledgments
W. Kumam would like to give thanks to the Centre of Excellence in Mathematics, the
Commission on Higher Education, Thailand, for their financial support for Ph.D. program
at KMUTT. Moreover, the authors also would like to thank the Higher Education Research
Promotion and National Research University Project of Thailand, Office of the Higher
Education Commission for financial support under CSEC Project no. 54000267 and P.
Kumam was supported by the Commission on Higher Education and the Thailand Research
Fund under Grant no. MRG5380044.
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