Volume 2009, Article ID 531308, 20 pagesdoi:10.1155/2009/531308 Research Article An Iterative Method for Generalized Equilibrium Problems, Fixed Point Problems and Variational Inequality
Trang 1Volume 2009, Article ID 531308, 20 pages
doi:10.1155/2009/531308
Research Article
An Iterative Method for Generalized
Equilibrium Problems, Fixed Point Problems
and Variational Inequality Problems
1 State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, Sichuan 610500, China
2 Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
Correspondence should be addressed to Nan-jing Huang,nanjinghuang@hotmail.com
Received 11 January 2009; Accepted 28 May 2009
Recommended by Fabio Zanolin
We introduce an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of generalized equilibrium problems, the set of common fixed points
of infinitely many nonexpansive mappings, and the set of solutions of the variational inequality
for α-inverse-strongly monotone mappings in Hilbert spaces We give some strong-convergence
theorems under mild assumptions on parameters The results presented in this paper improve and generalize the main result of Yao et al.2007
Copyrightq 2009 Qing-you Liu 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
1 Introduction
Let C be a nonempty closed convex subset of a real Hilbert space H and let Φ : C × C → R
be a bifunction, whereR is the set of real numbers Let Ψ : C → H be a nonlinear mapping.
The generalized equilibrium problemGEP for Φ : C × C → R and Ψ : C → H is to find
u ∈ C such that
Φu, v Ψu, v − u ≥ 0 ∀v ∈ C. 1.1 The set of solutions for the problem1.1 is denoted by Ω, that is,
Ω {u ∈ C : Φu, v Ψu, v − u ≥ 0, ∀v ∈ C}. 1.2
Trang 2IfΨ 0 in 1.1, then GEP1.1 reduces to the classical equilibrium problem EP and
Ω is denoted by EPΦ, that is,
EPΦ {u ∈ C : Φu, v ≥ 0, ∀v ∈ C} 1.3
IfΦ 0 in 1.1, then GEP1.1 reduces to the classical variational inequality and Ω is denoted by VIΨ, C, that is,
VIΨ, C {u∗∈ C : Ψu∗, v − u∗ ≥ 0, ∀v ∈ C}. 1.4
It is well known that GEP1.1 contains as special cases, for instance, optimization problems, Nash equilibrium problems, complementarity problems, fixed point problems, and variational inequalitiessee, e.g., 1 6
A mapping A : C → H is called α-inverse-strongly monotone 7
positive real number α such that
Ax − Ay, x − y ≥ αAx − Ay2 1.5
for all x, y ∈ C It is obvious that any α-inverse-strongly monotone mapping A is monotone and Lipschitz continuous A mapping S : C → C is called nonexpansive if
Sx − Sy ≤ x − y 1.6
for all x, y ∈ C We denote by FS the set of fixed points of S, that is, FS {x ∈ C : x
Sx} If C ⊂ H is bounded, closed and convex and S is a nonexpansive mappings of C into
itself, then FS is nonempty see 8
In 1997, Fl˚am and Antipin 9
approximation to initial data when EPΦ is nonempty and proved a strong convergence theorem In 2003, Iusem and Sosa 10
librium problems in finite-dimensional spaces They have also established the convergence
of the algorithms Recently, Huang et al 11
the equilibrium problem and proved the strong convergence theorem for approximating the solutions of the equilibrium problem
On the other hand, for finding an element of FS ∩ VIA, C, Takahashi and Toyoda
12
xn1 α nxn 1 − α n SP C x n − λ nAxn , n 0, 1, 2, , 1.7
where x0 ∈ C, P C is metric projection of H onto C, {α n } is a sequence in 0, 1 and {λ n} is
a sequence in0, 2α Further, Iiduka and Takahashi 13
scheme:
xn1 α nu βnxn γ nSPC x n − λ nAxn , 1.8
where u, x0 ∈ C, and proved the strong convergence theorems for iterative scheme 1.8 under some conditions on parameters In 2007, S Takahashi and W Takahashi 14
Trang 3iterative scheme by the viscosity approximation method for finding a common element of the set of an equilibrium problem and the set of fixed points of a nonexpansive mapping
in Hilbert spaces They also proved a strong convergence theorem which is connected with Combettes and Hirstoaga’s result 3 15
16
of solutions of the EPΦ and the set of common fixed points of nonexpansive mapping and obtained the weak convergence of the Mann type iterative algorithm Yao et al 17
introduced an iteration process for finding a common element of the set of solutions of the EPΦ and the set of common fixed points of infinitely many nonexpansive mappings
in Hilbert spaces They proved a strong-convergence theorem under mild assumptions
on parameters Very recently, Moudafi 18
common element ofΩ∩FS, where Ψ : C → H is an α-inverse-strongly monotone mapping,
and obtained a weak convergence theorem There are some related works, we refer to 19–22
and the references therein
Inspired and motivated by the works mentioned above, in this paper, we introduce an iterative process for finding a common element of the set of common fixed points of infinitely many nonexpansive mappings, the set of solutions of GEP1.1, and the solution set of the
variational inequality problem for an α-inverse-strongly monotone mapping in real Hilbert
spaces We give some strong-convergence theorems under mild assumptions on parameters The results presented in this paper improve and generalize the main result of Yao et al 17
2 Preliminaries
Let H be a real Hilbert space with inner product ·, · and norm · , and let C be a closed convex subset of H Then, for any x ∈ H, there exists a unique nearest point in C, denoted by
PC x, such that
x − P C x ≤ x − y ∀y ∈ C. 2.1
PC is called the metric projection of H onto C It is well known that P C is a nonexpansive mapping and satisfies
P Cx − PC y, x − y ≥PCx − PC y2 2.2
for all x, y ∈ H Furthermore, P C x ∈ C is characterized by the following properties:
x − P C x, y − PCx ≤ 0,
x − P C x 2y − PC x2 ≤x − y2 2.3
for all x ∈ H and y ∈ C It is easy to see that
where λ > 0 is a parameter in R.
Trang 4A set-valued mapping T : H → 2 H is called monotone if for all x, y ∈ H, p ∈ Tx and
GT of T is not properly contained in the graph of any other monotone mappings It is known
that a monotone mapping T is maximal if and only if for x, p ∈ H × H, x − y, p − q ≥ 0
for ally, q ∈ GT implies p ∈ Tx Let A : C → H be a monotone, L-Lipschitz continuous mappings and let N C u be the normal cone to C at u ∈ C, that is, NC u {w ∈ H : u − v, w ≥
0, ∀v ∈ C} Define
Tu
⎧
⎨
⎩
Au NCu, u ∈ C,
Then T is the maximal monotone and 0 ∈ Tu if and only if u ∈ VIA, C; see 23
Let{T n}∞n1 be a sequence of nonexpansive mappings of C into itself and let {π n}∞n1be
a sequence of nonnegative numbers in n of C into
itself as follows:
Un,n π nTnUn,n1 1 − π n I,
Un,k π kTkUn,k1 1 − π k I,
Un,2 π2 T2Un,3 1 − π2I,
Sn U n,1 π1 T1Un,2 1 − π1I.
2.6
Such a mapping S n is called the S-mapping generated by T n, Tn−1, , T1and π n, πn−1, , π1 see 24 n is nonexpansive and if x T nx then x Un,k S nx.
Lemma 2.1 see 24
n1 FTn / ∅ and let
subset of C, then for > 0, there exists n0 ≥ 1 such that for all n > n0
sup
x∈D
Trang 5UsingLemma 2.1, one can define a mapping S of C into itself as follows:
Sx lim
n → ∞ Snx lim
for every x ∈ C Such a mapping S is called the S-mapping generated by T1 , T2, and
sequence in C, then we put D {x n : n ≥ 0} Hence, it is clear fromRemark 2.2that for an
arbitrary > 0 there exists N0 ∈ N such that for all n > N0
S nxn − Sx n U n,1xn − U1 xn ≤ sup
x∈D
U n,1x − U1x ≤ . 2.9 This implies that
lim
Since T i and U n,i are nonexpansive, we deduce that, for each n ≥ 1,
S n1xn − S nxn π1 T1Un1,2xn − π1 T1Un,2xn
≤ π1 U n1,2xn − U n,2xn
π1 π2 T2Un1,3xn − π2 T2Un,3xn
≤ π1 π2 Un1,3xn − U n,3xn
≤n
i1
πi U n1,n1xn − U n,n1xn
≤ Mn
i1 πi
2.11
for some constant M ≥ 0.
Lemma 2.3 see 24
{T n}∞
n1 FTn / ∅, and let
n1 FTn .
For solving the generalized equilibrium problem, we assume that the bifunctionΦ :
C × C → R satisfies the following conditions:
a1 Φu, u 0 for all u ∈ C;
a2 Φ is monotone, that is, Φu, v Φv, u ≤ 0 for all u, v ∈ C;
a3 for each u, v, w ∈ C, lim t↓0 Φtw 1 − tu, v ≤ Φu, v;
a4 for each u ∈ C, v → Φu, v is convex and lower semicontinuous.
Trang 6The following lemma appears implicitly in 1
Lemma 2.4 see 1
C × C into R satisfying (a1)–(a4) Let r > 0 and x ∈ H Then, there exists u ∈ C such that
Φu, v 1
r v − u, u − x ≥ 0 ∀v ∈ C. 2.12 The following lemma was also given in 3
Lemma 2.5 see 3
Tr : H → C as follows:
Tr x
r v − u, u − x ≥ 0, ∀v ∈ C
2.13
for all x ∈ H Then, the following hold:
b1 T r is single-valued;
b2 T r is firmly nonexpansive, that is, for any x, y ∈ H, Trx − Tr y 2 ≤ T r x − Tr y, x − y;
b3 FT r EPΦ;
b4 EPΦ is closed and convex.
Φu, v Ψx, v − u 1
r v − u, u − x ≥ 0 ∀v ∈ C. 2.14 The following lemmas will be useful for proving the convergence result of this paper
Lemma 2.7 see 25 n } and {z n } be bounded sequences in Banach space E, and let {β n } be
a sequence in
zn β nxn 2.15
for all integers n ≥ 1 If
0 < lim inf
n → ∞ βn≤ lim sup
n → ∞ βn < 1,
lim sup
n → ∞ z n1 − z n − x n1 − x n ≤ 0, 2.16
Lemma 2.8 see 26 n } is a sequence of nonnegative real numbers such that
an1 ≤ 1 − α n a n δ n, n ≥ 1, 2.17
Trang 7where {α n } is a sequence in (0,1) and {δ n } is a sequence in R such that
1 ∞n1 αn ∞;
2 lim supn → ∞ δ n/αn ≤ 0 or ∞
n1 |δ n | < ∞.
3 Main Results
In this section, we deal with an iterative scheme by the approximation method for finding
a common element of the set of common fixed points of infinitely many nonexpansive mappings, the set of solutions of GEP1.1, and the solution set of the variational inequality
problem for an α-inverse-strongly monotone mapping in real Hilbert spaces.
Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H Let Φ be a
bifunction from C × C into R satisfying (a1)–(a4), Ψ : C → H an inverse-strongly monotone mapping with constant φ > 0, A : C → H an inverse-strongly monotone mapping with constant
n1 FTn ∩ Ω ∩ VIA, C / ∅, where sequence {T n } is
{y n }, and {u n } are generated by
Φu n, v Ψx n, v − un 1
rn v − u n, un − x n ≥ 0, ∀v ∈ C,
xn1 α nf x n β nxn γ nSnyn
3.1
some 0 < b < 2 and {rn
i α n β n γ n 1;
ii limn → ∞αn 0 and ∞n1 αn ∞;
iii 0 < lim inf n → ∞βn≤ lim supn → ∞ βn < 1;
iv lim infn → ∞λn > 0 and limn → ∞ |λ n1 − λ n | 0;
v lim infn → ∞rn > 0 and limn → ∞ |r n1 − r n | 0.
Then {x n }, {y n }, and {u n } converge strongly to the point z0 ∈ ∞n1 FTn ∩ Ω ∩ VIA, C, where
n1 FT n ∩Ω∩VIA,C fz0.
Proof For any x, y ∈ C and r ∈ 0, 2φ, we have
I − rΨx − I − rΨy2 x − y
− r Ψx − Ψy 2
,
3.2
Trang 8which implies that I − rΨ is nonexpansive.Remark 2.6implies that the sequences{u n} and
{x n} are well defined In view of the iterative sequence 3.1, we have
0≤ Φu n, v Ψx n, v − un 1
rn v − u n, un − x n
Φu n, v 1
rn v − u n, un − x n − r n Ψx n .
3.3
It follows fromLemma 2.5that u n T r n x n − r n Ψx n for all n ≥ 1 Let z∗ ∈∞n1 FTn ∩ Ω ∩ VIA, C For each n ≥ 1, we have z∗ S n z∗ T r n z∗− r n Ψz∗ ByLemma 2.5,
u n − z∗ 2 T r n x n − r n Ψx n − T r n z∗− r n Ψz∗ 2
≤ u n − z∗, x n − r n Ψx n − z∗− r n Ψz∗
1
2 u n − z∗ 2 x n − r n Ψx n − z∗− r n Ψz∗ 2
− u n − z∗ − x n − r n Ψx n − z∗− r n Ψz∗ 2
3.4
and so3.2 implies that
u n − z∗ 2≤ x n − r n Ψx n − z∗− r n Ψz∗ 2− u n − x n − r n Ψz∗− Ψx n 2
≤ x n − z∗ 2− u n − x n − r n Ψz∗− Ψx n 2
≤ x n − z∗ 2
.
3.5
For z∗ ∈ VIA, C, we have z∗ P C z∗ − λ nAz∗ from 2.4 Since P C is a nonexpansive
mapping and A is an inverse-strongly monotone mapping with constant > 0, by 3.1, we have
yn − z∗2
P C u n − λ nAun − P C z∗− λ nAz∗ 2
≤ u n − λ nAun − z∗− λ nAz∗ 2
≤ u n − z∗ 2 λ n λn − 2 Au n − Az∗ 2
≤ u n − z∗ 2
.
3.6
Thus,3.5 and 3.6 imply that
y n − z∗ ≤ u n − z∗ ≤ x n − z∗ , 3.7
Trang 9and so
x n1 − z∗ α nf x n β nxn γ nSnyn − z∗
≤ α n fx n − z∗ β n x n − z∗ γ n S nyn − z∗
≤ α n fx n − fz∗ fz∗ − z∗ β n x n − z∗ γ n S nyn − S nz∗
≤ α n α xn − z∗ fz∗ − z∗ β n x n − z∗ γ n y n − z∗
≤ α n α xn − z∗ fz∗ − z∗ β n x n − z∗ γ n x n − z∗
1 − α n n − z∗ α n 1 − α fz∗ − z∗
1− α
≤ max
x1 − z∗ , fz∗ − z∗
1− α
.
3.8
This implies that{x n } is bounded Therefore, {u n }, {y n }, {Ψx n }, {Au n }, and {S nyn} are also bounded
From u n T r n x n − r n Ψx n and u n1 T r n1 x n1 − r n1 Ψx n1, we have
Φu n, v Ψx n, v − un 1
rn v − u n, un − x n ≥ 0 ∀v ∈ C, 3.9
Φu n1, v Ψx n1, v − un1 1
rn1 v − u n1, un1 − x n1 ≥ 0 ∀v ∈ C. 3.10
Putting v u n1in3.9 and v u nin3.10, we get
Φu n, un1 Ψx n, un1 − u n 1
rn u n1 − u n, un − x n ≥ 0, Φu n1, un Ψx n1, un − u n1 1
rn1 u n − u n1, un1 − x n1 ≥ 0.
3.11
Adding the above two inequalities, the monotonicity ofΦ implies that
Ψx n1 − Ψx n, un − u n1
rn
≥ 0, 3.12 and so
0≤
un − u n1, rn Ψx n1 − Ψx n rn
rn1 u n1 − x n1 − u n − x n
1− rn
rn1
un1 x n1 − r n Ψx n1 − x n − r n Ψx n − x n1 rn
un1 − u n, un − u n1
1− rn
rn1
u n1 − x n1 x n1 − r n Ψx n1 − x n − r n Ψx n
.
3.13
Trang 10It follows from3.2 that
u n1 − u n 2≤ u n1 − u n
1 −rn1 rn u n1 − x n1 x n1 − x n
, 3.14
and hence
u n1 − u n ≤
1 −rn1 rn u n1 − x n1 x n1 − x n 3.15 From3.1,
y n1 − y n P C u n1 − λ n1Aun1 − P C u n − λ nAun
≤ u n1 − λ n1Aun1 − u n − λ nAun
≤ u n1 − λ n1Aun1 − u n − λ nAun1 |λ n1 − λ n | Au n
≤ u n1 − u n |λ n1 − λ n | Au n
3.16
Putting
1− β n f x n γn
we have
Obviously, we get
z n1 − z n
1αn1 − β n1 f x n1 γn1
αn
1− β n f x n γn
1− β n Snyn
≤ αn1
1− β n1 fx n1 − fx n
1αn1 − β n1 − αn
1− β n
fx n
γn1
1− β n1 S n1yn1 − S nyn
1− β γn1 n1− γn
1− β n
S nyn
≤ αn1
1− β αn1 n1− αn
1− β n
fx n
1− β αn n − αn1
1− β n1
S nyn
1− αn1
1− β n1
S n1yn1 − S nyn
3.19
Trang 11From2.11 and 3.16, we have
S n1yn1 − S nyn ≤ S n1yn1 − S n1yn S n1yn − S nyn
≤ y n1 − y n Mn
i1 πi
≤ u n1 − u n |λ n1 − λ n | Au n Mn
i1 πi
3.20
for some constant M ≥ 0 Combining 3.15, 3.19, and 3.20, we deduce
z n1 − z n − x n1 − x n
≤ αn1 α − 1
1− β n1 x n1 − x n
1αn1 − β n1 − αn
1− β n
fx n S nyn
γn1
1− β n1
M n
i1
.
3.21
It is easy to check that
lim
n → ∞
1αn1 − β n1 − αn
1− β n
0, lim
n → ∞
n
i1
n → ∞ |r n1 − r n | 0, 3.22
and so
lim sup
n → ∞ z n1 − z n − x n1 − x n ≤ 0. 3.23 Thus, byLemma 2.7, we obtain limn → ∞ z n − x n 0 It then follows that
lim
n → ∞ x n1 − x n lim
n → ∞ 1− β n
z n − x n 0. 3.24
By3.15 and 3.16, we have
lim
n → ∞ y n1 − y n lim
n → ∞ u n1 − u n 0. 3.25
... − z∗ , 3.7 Trang 9and so
x n1 − z∗... nyn
3.19
Trang 11From2.11 and 3.16, we have
S n1yn1...
,
3.2
Trang 8which implies that I − rΨ is nonexpansive.Remark 2.6implies that