Volume 2011, Article ID 780764, 15 pagesdoi:10.1155/2011/780764 Research Article An Iteration Method for Common Solution of a System of Equilibrium Problems in Hilbert Spaces 1 Departmen
Trang 1Volume 2011, Article ID 780764, 15 pages
doi:10.1155/2011/780764
Research Article
An Iteration Method for Common Solution of
a System of Equilibrium Problems in Hilbert Spaces
1 Department of Mathematics Education, Kyungnam University, Masan Kyunganm 631-701,
Republic of Korea
2 Department of Mathematics, Vietnamse Academy of Science and Technology,
Institute of Information Technology, 18, Hoang Quoc Viet, q Cau Giay, Hanoi 122100, Vietnam
Correspondence should be addressed to Jong Kyu Kim,jongkyuk@kyungnam.ac.kr
Received 11 December 2010; Revised 3 March 2011; Accepted 4 March 2011
Academic Editor: Qamrul Hasan Ansari
Copyrightq 2011 J K Kim and N Buong 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
The strong convergence theorem is proved for finding a common solution for a system of
equilibrium problems: find u∗ ∈ S : ∩ N
i1EPFi , EPF i : {z ∈ C : F i z, v ≥ 0 ∀v ∈ C}, i
1, , N, where C is a closed convex subset of a Hilbert space H and F i are N bifunctions from
C × C into R given exactly or approximatively As an application, finding a common solution for a
system of variational inequality problems is given
1 Introduction
Let H be a real Hilbert space with the scalar product and the norm denoted by the symbols
·, · and · , respectively Let C be a nonempty closed convex subset of H, and let
F i i 1, , N be N bifunctions from C × C into R In this paper, we consider the system of
equilibrium problems:
find u∗∈ S : ∩ N
i1EPFi ,
Condition 1 The bifunction F satisfies the following conditions:
A1 Fu, u 0 for all u ∈ C.
A2 Fu, v Fv, u ≤ 0 for all u, v ∈ C × C.
Trang 2A3 For every u ∈ C, Fu, · : C → R is lower semicontinuous and convex.
A4 limt → 0 F1 − tu tz, v ≤ Fu, v for all u, z, v ∈ C × C × C.
Definition 1.1 A mapping A of C into H is called monotone if
Ax − Ay
for all x, y ∈ C.
Definition 1.2 A mapping T of C into H is called k-strictly pseudocontractive in the
Tx − Ty
2≤ x − y 2 k I − Tx − I − Ty
where I is the identity operator in H.
The above inequality is equivalent to
Ax − Ay
, x − y≥ λ Ax − Ay
where the operator A : I − T is λ 1 − k/2-inverse strongly monotone hence monotone and Lipschitz continuous with the Lipschitz constant 2/1 − k Clearly, when k 0, T is
nonexpansive, that is,
Tx − Ty
mappings strictly includes the class of nonexpansive mappings Denote by FT the set of fixed points of the operator T in C, that is,
problems, saddle point problems, variational inequality problems, minimization problems, Nash equilibria in noncooperative games, vector equilibrium problems, as well as certain fixed point problems
procedure approach in11–15
Trang 3If N > 1, then 1.1 is a problem of finding a common solution for a system of
F i i 1, , N are bounded, Fr´echet differentiable with respect to v and ∇ v F i u, u are
Lipschitz continuous, that is,
∇v F i x, x − ∇ v F i
y, y
where L is a positive constant.
With the case that
finding a solution of an equilibrium problem which is also a common fixed point for a system
of a finite family of strictly pseudocontractive mappings17–19
is a problem of finding an element which is a solution of a variational inequality problem and a common fixed point for a finite family of strictly pseudocontractive mappings and investigated intensively in 20–32 If all F i have the form 1.9, then 1.1 is a problem of
from C into H14,33–35
x0 x ∈ H,
u i
n ∈ C : F i
u i
n , v
u i
n − x n , v − u i
n
≥ 0, ∀v ∈ C, i 1, , N,
x n1 x n − β n
n
i1
x n − u i n
α n x n
,
1.10
As an application, we find a common solution for a system of N variational inequality
problems with monotone mappings
2 Main Results
We formulate the following facts which are necessary in the proof of our main results
Lemma 2.1 see 5 Let C be a nonempty closed convex subset of a Hilbert space H, and let F be a
bifunction of C × C into R satisfying the Condition 1 Let r > 0 and x ∈ H Then, there exists z ∈ C such that
Fz, v 1r z − x, v − z ≥ 0, ∀v ∈ C. 2.1
Trang 4Lemma 2.2 see 5 Assume that F : C × C → R satisfies the Condition 1 For r > 0 and x ∈ H, define a mapping T r : H → C as follows:
T r x z ∈ C : Fz, v 1r z − x, v − z ≥ 0, ∀v ∈ C
Then, the following hold:
i T r is single-valued;
ii T r is firmly nonexpansive, that is, for any x, y ∈ H,
T r x − T r
y
2 ≤T r x − T r
y
iii FT r EPF;
iv EPF is closed and convex.
Lemma 2.3 Let F h u, v be a bifunction approximating the bifunction Fu, v in the sense
where gt is a real positive function Then, for each r > 0 and x ∈ H, we have
T h
where
T h
r x z ∈ C : F h z, v 1
r z − x, v − z ≥ 0 ∀v ∈ C
Proof Let x be an arbitrary element of H By replacing v by z in 2.2 and by z in 2.6, we obtain
Fz, z F h z, z ≥ 1r x − z, z − z z − x, z − z. 2.7
Fz, z − F h z, z ≥ 1r z − z 2. 2.8 Consequently,
The proof is completed
Trang 5Lemma 2.4 see 36 Let {a n }, {b n }, and {c n } be sequences of positive numbers satisfying the
conditions:
i a n1 ≤ 1 − b n a n c n , b n < 1,
ii∞
n0 b n ∞, lim n → ∞ c n /b n 0.
Then, lim n → ∞ a n 0.
Lemma 2.5 see 37 Assume that T is a nonexpansive mapping of a closed convex subset C of a
Hilbert space H Then I − T is demiclosed at zero; that is whenever {x n } is a sequence in C weakly
converging to some x ∈ C and the sequence {I − Tx n } strongly converges to zero, it follows
I − Tx 0.
Lemma 2.6 see 17 Let A be a λ-inverse strongly monotone mapping from C into H such that
S A / ∅, where S A {x ∈ C : Ax 0} Then, S A VIC, A.
1, , N.
N
i1
A i
y n
result
Theorem 2.7. i For each α n > 0, problem 2.11 has a unique solution y n
ii limn → ∞ y n u∗, u∗∈ S, u∗ ≤ y , for all y ∈ S.
iii y n − y m ≤ |α n − α m |/α n u∗
Proof. i Since the mapping n
defined on H, it is maximal monotone Therefore,2.11 has a unique solution for each α n > 0
38
Thus, from2.11 it follows that
N
i1
A i
y n
− A i
y
, y n − y α n
Trang 6
property of A l , and A l y 0, l 1, , N, it implies that
1
2 y n k − T l
y n k
2≤A l
y n k
, y n k − y
i1
A i
y n k
, y n k − y
≤ −α n k
y n k , y n k − y
−α n k
y n k − y, y n k − y− α n k
y, y n k − y
≤ −α n k
y, y n k − y
≤ α n k2 y 2,
2.15
that is,
y n k − T l
y n k
Therefore,
lim
k → ∞ A l
y n k
ByLemma 2.5, A l y 0, that is, y ∈ FT l , l 1, , N It means that y ∈ S Because S is
2.14 and the weak convergence of {y n k } to y u∗, it also follows that y n k → u∗ , as
k → ∞ Moreover, the sequence {y n } converges strongly to u∗as n → ∞
iii From 2.11, 2.14, and the monotone property of A i, it follows
α n
y n , y n − y m
− α m
y m , y n − y m
or
y n − y m ≤ |α n − α m|
α n y m ≤ |α n − α m|
for each α n , α m > 0 The proof is completed.
Trang 7Theorem 2.8 Suppose that α n , β n satisfy the following conditions:
α n , β n > 0 α n ≤ 1, lim
n → ∞ α n lim
n → ∞
|α n − α n1|
α2
n β n 0,
∞
n0
α n β n ∞, lim
n → ∞ β n 2N α n2
α n < 1.
2.20
Then,
lim
n → ∞ x n u∗∈ S, 2.21
where x n is defined by1.10.
Proof Let y nbe a solution of2.11 Set Δn x n − y n Then,
Δn1 x n1 − y n1 ≤ x n1 − y n y n1 − y n ,
x n1 − y n
x n − y n − β n
N
i0
A i x n − A i
y n
α n
From the monotone and Lipschitz continuous properties of A i , i 1, , N, 2.11, and u i
n
T i x n, we can write
x n − y n − β n
N
i1
A i x n − A i
y n
α n
x n − y n
2
x n − y n 2 β2
n
N
i1
A i x n − A i
y n
α n
x n − y n
2
− 2β n
i1
A i x n − A i
y n
α n
x n − y n
, x n − y n
≤ x n − y n 2
1− 2β n α n β2
n 2N α n2
.
2.23
Hence,
x n1 − y n ≤ Δn
1− 2β n α n β2
n 2N α n21/2
Therefore,
Δn1≤ Δn
1− 2β n α n β2
n 2N α n21/2
|α n − α α n1|
n u∗
≤ Δn
1− α n β n1/2
|α n − α α n1|
n u∗
2.25
Trang 8Thus, applying the inequality
a b2≤ 1 ε
a2b ε2
ε > 0, ε α n β n
we obtain
0≤ Δ2
n1≤ Δ2
n
1− α n β n
2α n β n
n − α n1
α n u∗
2 2
α n β n
2α n β n
≤ a2
n
2α n β n−1
2
α n β n2
α n − α n1
α2
n β n u∗
2
2α n β n
2α n β n
.
2.27
Set
b n α n β n
1
2α n β n
,
c n
α n − α n1
α2
n β n u∗
2
2α n β n
2α n β n
.
2.28
large n Hence, lim n → ∞Δ2
n 0 Since limn → ∞ y n u∗, we have
lim
n → ∞ x n u∗∈ S. 2.29 Now, let F n i u, v : F h n
x0 x ∈ H,
u i
n ∈ C : F n
i
u i
n , v
u i
n − x n , v − u i
n
≥ 0 ∀v ∈ C, i 1, , N,
x n1 x n − β n
n
i1
x n − u i n
α n x n
,
2.30
We have the following result
Theorem 2.9 Suppose that α n , β n , and h n satisfy the conditions in Theorem 2.8 and
lim
n → ∞
h n h n1
α2
Trang 9Then, we have
lim
n → ∞ x n u∗∈ S, 2.32
where x n is defined by2.30.
Proof Let y nbe a solution of the following equation:
N
i1
A n i
y n
α n y n 0, A n
i I − T n
where each T i nis defined by
T n
i x z ∈ C : F n
Since
and limn → ∞ y n u∗, in order to prove that limn → ∞ x n u∗, it is necessary to prove that
lim
n → ∞ x n − y n lim
n → ∞ y n − y n 0. 2.36
A i x − A n
i x T i x − T n
Therefore, from2.11, 2.33, and the monotone property of A n
i it implies that
y n − y n 2 α1
n
N
i1
A n i
y n
− A i
y n
, y n − y n
α n
N
i1
A n i
y n
− A i
y n
, y n − y n
.
2.38
Consequently, we have
y n − y n ≤ α1
n
N
i1
A n i
y n
− A i
y n
≤ N h α n
n g
T i
y n .
2.39
Trang 10On the other hand,
T i
y n
T i
y n
− T i u∗ u∗
≤ y n − u∗ u∗
≤ y n 2 u∗
≤ 3 u∗
2.40
Therefore,
y n − y n ≤ C0N h n
where C0 sup{gt : 0 < t ≤ 3 u∗ } It means that limn → ∞ y n u∗because limn → ∞ h n /α n 0
Secondly, to prove
lim
n → ∞ x n − y n 0, 2.42
as in the proof ofTheorem 2.8, first we need to estimate the value y n1 − y n By the argument
N
i1
A n
i
y n
− A n1
i
y n1, y n − y n1 α n
y n , y n − y n1− α n1y n1 , y n − y n1 0 2.43
Thus,
y n − y n1 2 α n − α α n1
n
−y n1 , y n − y n1
α1
n
N
i1
A n1 i
y n1
− A n i
y n
, y n − y n1
≤ α n − α n1
α n
−y n , y n − y n1 1
α n
N
i1
A n1 i
y n1− A n
i
y n
, y n − y n1
≤ α n − α α n1
n y n y n − y n1 α1
n
N
i1
A n1 i
y n
− A n i
y n
, y n − y n1 .
2.44
Therefore,
y n − y n1 ≤ α n − α α n1
n y n α1
n
N
i1
A n1 i
y n
− A i
y n
A i
y n
− A n i
y n
≤ α n − α α n1
n y n N h n h α n1
n g
y n .
2.45
Trang 11Using2.14 and 2.41, we have
y n ≤ u∗ C0N h n
y n − y n1 ≤ α n − α n1
α n C NC1
h n h n1
where C1 sup{gt : 0 < t < C} Now, set Δ n x n − y n It is not difficult to verify that
x n1 − y n ≤ Δn
1− 2β n α n β2
n 2N α n21/2
,
Δn1≤ Δn
1− α n β n1/2|α n − α α n1|
n C NC1
h n h n1
α n .
2.48
Therefore, limn → ∞ Δn 0 The proof is completed
Remark The sequences α n 1 n −p , 0 < p < 1/2, and β n γ0α nwith
0 < γ0< 1
3 Applications
Hilbert space H into H.
Theorem 3.1 Let x0 x be an arbitrary element in H If {α n }, {β n } are chosen as in Theorem 2.8 , and the iteration sequence {x n } is defined as follows:
u i
n ∈ C,
A i
u i n
, v − u i
n u i
n − x n , v − u i
n
≥ 0, ∀v ∈ C, i 1, , N,
x n1 x n − β n
N
i1
x n − u i n
α n x n
,
3.2
then the sequence {x n } converges strongly to a common solution for 3.1.
Trang 12If C ≡ H, then we have a problem of finding a common zero for a system of monotone hemicontinous mappings A i , i 1, , N In this case, variational inequality in 3.2 has the
form A i u i
n u i
Theorem 3.2 Let A i , i 1, , N be N hemicontinuous monotone mappings defined on H Let x0
x be an arbitrary element in H, let {α n } and {β n } be the sequences that are chosen as in Theorem 2.8 , and, the iteration sequence {x n } be defined as follows:
u i
n : A i
u i n
u i
n x n ,
x n1 x n − β n
N
i1
x n − u i n
α n x n
Then the sequence {x n } converges strongly to an element u∗such that
in general, is ill-posed Some methods for finding a solution of each variational inequality in
3.1 are presented in 39
Here we show an iterative regularization method for finding a common solution of
A n
function Obviously, the bifunctions
F n u, v :A n
Theorem 3.3 Let x0 x be an arbitrary element in H If {α n }, {β n } are chosen as in Theorem 2.9 , and the iteration sequence {x n } is defined as follows:
u i
n ∈ C :A i
u i n
, v − u i n
u i
n − x n , v − u i
n
≥ 0 ∀v ∈ C, i 1, , N,
x n1 x n − β n
N
i1
x n − u i n
α n x n
,
3.7
then the sequence {x n } converges strongly to a common solution for 3.1.
A i , i 1, , N, could be found by the following.
Trang 13Theorem 3.4 Let A i , i 1, , N be N hemicontinuous monotone mappings defined on H Let x0
x be an arbitrary element in H, let {α n } and {β n } be the sequences that are chosen as in Theorem 2.9 , and the iteration sequence {x n } be defined as follows:
u i
n : A i
u i n
u i
n x n ,
x n1 x n − β n
N
i1
x n − u i n
α n x n
Then the sequence {x n } converges strongly to an element u∗such that
Acknowledgment
This work was supported by the Kyungnam University Research Fund, 2010
References
1 F E Browder and W V Petryshyn, “Construction of fixed points of nonlinear mappings in Hilbert
space,” Journal of Mathematical Analysis and Applications, vol 20, pp 197–228, 1967.
2 E Blum and W Oettli, “From optimization and variational inequalities to equilibrium problems,” The Mathematics Student, vol 63, no 1–4, pp 123–145, 1994.
3 W Oettli, “A remark on vector-valued equilibria and generalized monotonicity,” Acta Mathematica Vietnamica, vol 22, no 1, pp 213–221, 1997.
4 O Chadli, S Schaible, and J C Yao, “Regularized equilibrium problems with application to
noncoercive hemivariational inequalities,” Journal of Optimization Theory and Applications, vol 121,
no 3, pp 571–596, 2004
5 P L Combettes and S A Hirstoaga, “Equilibrium programming in Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol 6, no 1, pp 117–136, 2005.
6 J K Kim and Ng Buong, “Regularization inertial proximal point algorithm for monotone
hemicontinuous mapping and inverse strongly monotone mappings in Hilbert spaces,” Journal of Inequalities and Applications, vol 2010, Article ID 451916, 10 pages, 2010.
7 A S Stukalov, “A regularized extragradient method for solving equilibrium programming problems
in a Hilbert space,” Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, vol 45, no 9, pp 1538–
1554, 2005
8 I V Konnov and O V Pinyagina, “D-gap functions for a class of equilibrium problems in Banach
spaces,” Computational Methods in Applied Mathematics, vol 3, no 2, pp 274–286, 2003.
9 G Mastroeni, “Gap functions for equilibrium problems,” Journal of Global Optimization, vol 27, no 4,
pp 411–426, 2003
10 P N Anh and J K Kim, “A new method for solving monotone generalized variational inequalities,”
Journal of Inequalities and Applications, vol 2010, Article ID 657192, 20 pages, 2010.
11 A S Antipin, “Equilibrium programming: gradient-type methods,” Automation and Remote Control,
vol 58, pp 1337–1347, 1997
12 M Bounkhel and B R Al-Senan, “An iterative method for nonconvex equilibrium problems,” Journal
of Inequalities in Pure and Applied Mathematics, vol 7, no 2, article 75, pp 1–8, 2006.
13 O Chadli, I V Konnov, and J C Yao, “Descent methods for equilibrium problems in a Banach space,”
Computers & Mathematics with Applications, vol 48, no 3-4, pp 609–616, 2004.
14 G Marino and H.-K Xu, “Weak and strong convergence theorems for strict pseudo-contractions in
Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol 329, no 1, pp 336–346, 2007.
15 A Moudafi, “Second-order differential proximal methods for equilibrium problems,” Journal of Inequalities in Pure and Applied Mathematics, vol 4, no 1, article 18, pp 1–7, 2003.
...References
1 F E Browder and W V Petryshyn, “Construction of fixed points of nonlinear mappings in Hilbert
space,” Journal of Mathematical Analysis and Applications, vol 20, pp 197–228,... Optimization Theory and Applications, vol 121,
no 3, pp 571–596, 2004
5 P L Combettes and S A Hirstoaga, “Equilibrium programming in Hilbert spaces,” Journal of Nonlinear and Convex...
10 P N Anh and J K Kim, “A new method for solving monotone generalized variational inequalities,”
Journal of Inequalities and Applications, vol 2010, Article ID 657192, 20 pages,