Volume 2011, Article ID 276859, 10 pagesdoi:10.1155/2011/276859 Research Article An Implicit Iteration Method for Variational Inequalities over the Set of Common Fixed Points for a Finit
Trang 1Volume 2011, Article ID 276859, 10 pages
doi:10.1155/2011/276859
Research Article
An Implicit Iteration Method for Variational
Inequalities over the Set of Common Fixed Points for a Finite Family of Nonexpansive Mappings in Hilbert Spaces
1 Vietnamese Academy of Science and Technology, Institute of Information Technology, 18,
Hoang Quoc Viet, Cau Giay, Ha Noi 122100, Vietnam
2 Department of Information Technology, Thai Nguyen National University,
Thainguye 842803, Vietnam
Correspondence should be addressed to Nguyen Buong,nbuong@ioit.ac.vn
Received 17 December 2010; Accepted 7 March 2011
Academic Editor: Jong Kim
Copyrightq 2011 N Buong and N T Quynh Anh 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 implicit iteration method for finding a solution for a variational inequality involving Lipschitz continuous and strongly monotone mapping over the set of common fixed points for a finite family of nonexpansive mappings on Hilbert spaces
1 Introduction
Let C be a nonempty closed and convex subset of a real Hilbert space H with inner product
·, · and norm · , and let F : H → H be a nonlinear mapping The variational inequality problem is formulated as finding a point p∗∈ C such that
p∗
Variational inequalities were initially studied by Kinderlehrer and Stampacchia in 1 and ever since have been widely investigated, since they cover as diverse disciplines as partial differential equations, optimal control, optimization, mathematical programming, mechanics, and financesee 1 3
Trang 2It is well known that if F is an L-Lipschitz continuous and η-strongly monotone, that
is, F satisfies the following conditions:
F x − F
y ≤ Lx − y,
F x − Fy
≥ ηx − y2
,
1.2
where L and η are fixed positive numbers, then 1.1 has a unique solution It is also known that1.1 is equivalent to the fixed-point equation
where P C denotes the metric projection from x ∈ H onto C and μ is an arbitrarily fixed
positive constant
Let {T i}N
i1 be a finite family of nonexpansive self-mappings of C For finding an element p ∈ ∩ N
i1FixTi, Xu and Ori introduced in 4 the following implicit iteration process
For x0∈ C and {β k}∞k1 ⊂ 0, 1, the sequence {x k} is generated as follows:
T1x1,
T2x2,
T1x N1 ,
1.4
The compact expression of the method is the form
where T n T n mod N , for integer n ≥ 1, with the mod function taking values in the set {1, 2, , N} They proved the following result.
Theorem 1.1 Let H be a real Hilbert space and C a nonempty closed convex subset of H Let {Ti}N
i1
i1FixTi / ∅, where FixT i {x ∈ C : T i x x} Let
x0∈ C and {β k}∞k1 be a sequence in 0, 1 such that lim k → ∞ β k 0 Then, the sequence {x k } defined
i1
Trang 3Further, Zeng and Yao introduced in 5 the following implicit method For an
arbitrary initial point x0∈ H, the sequence {x k}∞
k1is generated as follows:
T1x1− λ1μF T1x1,
T2x2− λ2μF T2x2,
1.6
The scheme is written in a compact form as
T k x k
They proved the following result
Theorem 1.2 Let H be a real Hilbert space and F : H → H a mapping such that for some constants
i1FixTi / ∅ Let μ ∈ 0, 2η/L2, and let x0∈ H, with {λ k}∞
k1 ⊂ 0, 1
and {β k}∞k1 ⊂ 0, 1 satisfying the conditions: ∞k1 λ k < ∞ and α ≤ β k ≤ β, k ≥ 1, for some
Recently, Ceng et al.6 extended the above result to a finite family of asymptotically self-maps
Clearly, from ∞
convergence without the condition ∞
implicit algorithm:
0T N t · · · T t
where T t
i are defined by
t
t } ⊂ 0, 1 for all t ∈ 0, 1 satisfy the following conditions: λ t → 0 as t → 0 and 0 < lim inf t → 0 β i t≤ lim supt → 0 β i t < 1, i
1, , N.
Trang 42 Main Result
We formulate the following facts for the proof of our results
Lemma 2.1 see 7 i x y2 ≤ x2 2y, x y and for any fixed t ∈ 0, 1,
ii 1 − tx ty2 1 − tx2 ty2− 1 − ttx − y2
, for all x, y ∈ H.
Put T λ x Tx − λμFTx, x ∈ H, λ ∈ 0, 1; for any nonexpansive mapping T of H,
we have the following lemma
Lemma 2.2 see 8 T λ x − T λ y ≤ 1 − λτx − y, for all x, y ∈ H and for a fixed number
Lemma 2.3 Demiclosedness Principle 9 Assume that T is a nonexpansive self-mapping of a
closed convex subset K of a Hibert space H If T has a fixed point, then I − T is demiclosed; that is,
strongly converges to some y, it follows that I − Tx y.
Now, we are in a position to prove the following result
Theorem 2.4 Let H be a real Hilbert space and F : H → H a mapping such that for some constants
i1FixTi / ∅ Let μ ∈ 0, 2η/L2 and let t ∈ 0, 1, {λ t }, {β i
t } ⊂ 0, 1,
such that
λ t −→ 0, as t −→ 0, 0 < lim inf
t → 0 β i t≤ lim sup
t → 0
0, that is, T I, we have that
T t x − T t y ≤ 1 − λ t τT t
N · · · T t
1x − T N t · · · T t
1y
≤ 1 − λ t τT t
i · · · T t
1x − T i t · · · T t
1y
≤ 1 − λ t τT t
2.2
So, T t is a contraction in H By Banach’s Contraction Principle, there exists a unique element
x t ∈ H such that x t T t x t for all t ∈ 0, 1.
Trang 5Next, we show that{x t } is bounded Indeed, for a fixed point p ∈ C, we have that
T t
x t − p T t x t − p T t x t − T t
N · · · T t
1p
T t
N · · · T t
T t
N · · · T t
≤ 1 − λ t τT t
N · · · T t
1x t − T t
N · · · T t
1p λ t μF
p
≤ 1 − λ t τT t
N−1 · · · T t
1x t − T t N−1 · · · T t
1p λ t μF
p
≤ 1 − λ t τT t
i · · · T t
1x t − T t
i · · · T t
1p λ t μF
p
≤ 1 − λ t τT t
1x t − T t
1p λ t μF
p
≤ 1 − λ t τx t − p λ t μF
p.
2.3
Therefore,
x t − p ≤ μ
τF
that implies the boundedness of{x t } So, are the nets {Fy N
t }, {y i
t }, i 1, , N.
Put
t
x t β1
t T1x t ,
t
y1t β2
t T2y1t ,
t
y i−1 t β i
t T i y t i−1 ,
t
t T N y N−1 t
2.5
Then,
Trang 6x t − p2
y N t − p2
y N
t − p2
− 2λ t μ F y N t
, y N t − p λ2
t μ2F y N
t 2
≤y N−1
t − p2
− 2λ t μ F y t N
, y N t − p λ2
t μ2F y N
t 2
≤y1
t − p2
− 2λ t μ F y N t
, y N t − p λ2
t μ2F y N
t 2
≤x t − p2− 2λ t μ F y t N
, y N t − p λ2
t μ2F y N
t 2
.
2.7
Thus,
ηy N
t − p2
F
p
2
F y N t 2
Further, for the sake of simplicity, we put y0t x tand prove that
y i−1
as t → 0 for i 1, , N.
Let{t k } ⊂ 0, 1 be an arbitrary sequence converging to zero as k → ∞ and x k: xt k
We have to prove that y i−1
k − T i y k i−1 → 0, where y i
k are defined by2.5 with t t k and
y i k y i
t k Let{x l } be a subsequence of {x k} such that
lim sup
k → ∞
y i−1
k − T i y i−1 k lim
l → ∞
y i−1
l − T i y i−1 l . 2.10
Let{x k j } be a subsequence of {x l} such that
lim sup
k → ∞
x k − p lim
j → ∞
Trang 7
From2.6 andLemma 2.1, it implies that
x k j − p2
k j μF
k j − p2
≤y N
k j − p2
− 2λ k j μ F y k N j
, x k j − p
1− β N
k j y k N−1 j − p β N
k j T N y k N−1 j − T N p 2
− 2λ k j μ F y k N j
, x k j − p
≤ 1− β N
k j
y N−1
k j − p2
β N
k j
T N y N−1 k j − T N p2
− 2λ k j μ F y k N j
, x k j − p
≤y N−1
k j − p2
− 2λ k j μ F y N k j
, x k j − p
≤ · · · ≤y1
k j − p2
− 2λ k j μ F y N k j
, x k j − p
≤x
k j − p2
− 2λ k j μ F y N k j
, x k j − p.
2.12
Hence,
lim
j → ∞
x k j − p lim
j → ∞
y i
k j − p, i 1, , N. 2.13
ByLemma 2.1,
y i
k j − p2
1− β i
k j
y i−1
k j − p2
β i
k j
T i y i−1
k j − p2
− β i
k j 1− β i
k j
y i−1
k j − T i y i−1 k j 2
≤ 1− β i
k j
y i−1
k j − p2
β i
k j
y i−1
k j − p2
− β i
k j 1− β i
k j
y i−1
k j − T i y i−1 k j 2
y i−1
k j − p2
− β i
k j 1− β i
k j
y i−1
k j − T i y i−1 k j 2
≤ · · · y0
k j − p2
− β i
k j 1− β i
k j
y i−1
k j − T i y i−1 k j 2
x
k − p2
− β i
k 1− βi
k y i−1
k − T i y i−1
k 2
2.14
Trang 8Without loss of generality, we can assume that α ≤ β i t ≤ β for some α, β ∈ 0, 1 Then, we have
1− βy i−1
k j − T i y k i−1 j 2
≤x
k j − p2
−y i
k j − p2
This together with2.13 implies that
lim
j → ∞
y i−1
k j − T i y k i−1 j 2
0, i 1, , N. 2.16
It means thaty i−1
t − T i y i−1
t → 0 as t → 0 for i 1, , N.
Next, we show thatx t − T i x t → 0 as t → 0 In fact, in the case that i 1 we have
y0t x t So,x t − T1x t → 0 as t → 0 Further, since
y1
t − T1x t 1− β1
t
andx t − T1x t → 0, we have that y1
t − T1x t → 0 Therefore, from
x t − y1
t ≤ x
t − T1x t T
1x t − y1
it follows thatx t − y1
t → 0 as t → 0 On the other hand, since
y2
t − T2y1t 1− β2
t y1
t − T2y t1 −→ 0,
y2
t − x t ≤ 1− β2
t y1
t − x t β2
tT
2y t1− x t
≤ 1− β2
t y1
t − x t β2
tT
2y t1− y1
t y1
t − x t,
2.19
we obtain thaty2
t − x t → 0 as t → 0 Now, from
x t − T2x t ≤x
t − y2
t y2
t − T2y t1 T
2y t1− T2x t
≤x
t − y2
t y2
t − T2y t1 y1
t − x t, 2.20
andx t − y2
t , y2
t − T2y1
t , y1
t − x t → 0, it follows that x t − T2x t → 0 Similarly, we obtain thatx t − T i x t → 0, for i 1, , N and y N
t − x t → 0 as t → 0.
Let {x k } be any sequence of {x t } converging weakly to p as k → ∞ Then, x k −
T i x k → 0, for i 1, , N and {y N
k } also converges weakly to p ByLemma 2.3, we have
p ∈ C ∩ N
i1FixTi and from 2.8, it follows that
p
, p − p
Trang 9Since p, p ∈ C, by replacing p by tp 1 − t p in the last inequality, dividing by t and taking
t → 0 in the just obtained inequality, we obtain
p, p − p
The uniqueness of p∗in1.1 guarantees that p p∗ Again, replacing p in 2.8 by p∗, we obtain the strong convergence for{x t} This completes the proof
3 Application
Recall that a mapping S : H → H is called a γ-strictly pseudocontractive if there exists a constant γ ∈ 0, 1 such that
, ∀x, y ∈ H. 3.1
It is well known10 that a mapping T : H → H by Tx αx1−αSx with a fixed α ∈ γ, 1 for all x ∈ H is a nonexpansive mapping and FixT FixS Using this fact, we can extend our result to the case C ∩ N i1FixSi , where S i is γ i-strictly pseudocontractive as follows
Let α i ∈ γ i , 1 be fixed numbers Then, C ∩ N i1Fix Ti with T i y α i y 1 − α i S i y, a
nonexpansive mapping, for i 1, , N, and hence
T t
i y 1− β i
t
y β i t T i y
1− β i
t 1 − α i y β i t 1 − α i S i y, i 1, , N.
3.2
So, we have the following result
Theorem 3.1 Let H be a real Hilbert space and F : H → H a mapping such that for some
i1FixSi / ∅ Let α i ∈ γ i , 1, μ ∈ 0, 2η/L2 and
let t ∈ 0, 1, {λ t }, {β i
t } ⊂ 0, 1, such that
λ t −→ 0, as t −→ 0, 0 < lim inf
t → 0 β i t≤ lim sup
t → 0
x t T t x t , T t: Tt
0T t
N · · · T t
It is known in11 that Fix S C where S N
i1 ξ i S i with ξ i > 0 and N
i1 ξ i 1
for N γ i-strictly pseudocontractions{S i}N
i1 Moreover, S is γ-strictly pseudocontractive with
Trang 10Theorem 3.2 Let H be a real Hilbert space and F : H → H a mapping such that for some
i1FixSi / ∅ Let α ∈ γ, 1, where
λ t −→ 0, as t −→ 0, 0 < lim inf
t → 0 β t≤ lim sup
t → 0
x t T t x t , T t: Tt
0
1− β t 1 − αI β t 1 − αN
i1
ξ i S i
, t ∈ 0, 1, 3.6
0 I − λ t μF, ξ i > 0, and N
i1 ξ i 1, converges strongly to the unique element p∗in1.1.
Acknowledgment
This work was supported by the Vietnamese National Foundation of Science and Technology Development
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