Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 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
Nguyen Buong1 and Nguyen Thi Quynh Anh2
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
Copyright q 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
∗
F p , p − p∗ ≥ 0,
∀p ∈ C.
1.1
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 finance see 1–3.
2
Fixed Point Theory and Applications
It is well known that if F is an L-Lipschitz continuous and η-strongly monotone, that
is, F satisfies the following conditions:
Fx − F y ≤ Lx − y,
2
Fx − F y , x − y ≥ ηx − y ,
1.2
where L and η are fixed positive numbers, then 1.1 has a unique solution. It is also known
that 1.1 is equivalent to the fixed-point equation
p PC p − μF p ,
1.3
where PC denotes the metric projection from x ∈ H onto C and μ is an arbitrarily fixed
positive constant.
Let {Ti }N
i1 be a finite family of nonexpansive self-mappings of C. For finding an
element p ∈ ∩N
i1 FixTi , Xu and Ori introduced in 4 the following implicit iteration process.
For x0 ∈ C and {βk }∞
k1 ⊂ 0, 1, the sequence {xk } is generated as follows:
x1 β1 x0 1 − β1 T1 x1 ,
x2 β2 x1 1 − β2 T2 x2 ,
..
.
xN βN xN−1 1 − βN TN xN ,
xN1 βN1 xN 1 − βN1 T1 xN1 ,
1.4
..
.
The compact expression of the method is the form
xk βk xk−1 1 − βk Tk xk ,
k ≥ 1,
1.5
where Tn Tn 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
be N nonexpansive self-maps of C such that ∩N
/ ∅, where FixTi {x ∈ C : Ti x x}. Let
i1 FixTi x0 ∈ C and {βk }∞
k1 be a sequence in 0, 1 such that limk → ∞ βk 0. Then, the sequence {xk } defined
implicitly by 1.5 converges weakly to a common fixed point of the mappings {Ti }N
i1 .
Fixed Point Theory and Applications
3
Further, Zeng and Yao introduced in 5 the following implicit method. For an
arbitrary initial point x0 ∈ H, the sequence {xk }∞
k1 is generated as follows:
x1 β1 x0 1 − β1 T1 x1 − λ1 μFT1 x1 ,
x2 β2 x1 1 − β2 T2 x2 − λ2 μFT2 x2 ,
..
.
xN βN xN−1 1 − βN TN xN − λN μFTN xN ,
xN1 βN1 xN 1 − βN1 T1 xN1 − λN1 μFT1 xN1 ,
1.6
..
.
The scheme is written in a compact form as
xk βk xk−1 1 − βk Tk xk − λk μF Tk xk ,
k ≥ 1.
1.7
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
L, η > 0, F is L-Lipschitz continuous and η-strongly monotone. Let {Ti }N
i1 be N nonexpansive self2
FixT
∅.
Let
μ
∈
0,
2η/L
,
and
let
x
∈
H, with {λk }∞
maps of H such that C ∩N
/
i
0
k1 ⊂ 0, 1
i1
∞
∞
and {βk }k1 ⊂ 0, 1 satisfying the conditions: k1 λk < ∞ and α ≤ βk ≤ β, k ≥ 1, for some
α, β ∈ 0, 1. Then, the sequence {xk } defined by 1.7 converges weakly to a common fixed point of
the mappings {Ti }N
i1 . The convergence is strong if and only if lim infk → ∞ dxk , C 0.
Recently, Ceng et al. 6 extended the above result to a finite family of asymptotically
self-maps.
have that λk → 0 as k → ∞. To obtain strong
Clearly, from ∞
k1 λk < ∞ we
convergence without the condition ∞
k1 λk < ∞, in this paper we propose the following
implicit algorithm:
xt T t xt ,
t
T t : T0t TN
· · · T1t ,
t ∈ 0, 1,
1.8
where Tit are defined by
Tit x 1 − βti x βti Ti x,
i 1, . . . , N,
T0t y I − λt μF y,
x, y ∈ H,
1.9
I denotes the identity mapping of H, and the parameters {λt }, {βti } ⊂ 0, 1 for all t ∈ 0, 1
satisfy the following conditions: λt → 0 as t → 0 and 0 < lim inft → 0 βti ≤ lim supt → 0 βti < 1, i 1, . . . , N.
4
Fixed Point Theory and Applications
2. 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 T x − λμFT x, 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
μ ∈ 0, 2η/L2 , where τ 1 −
1 − μ2η − μL2 ∈ 0, 1.
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,
whenever {xk } is a sequence in K weakly converging to some x ∈ K and the sequence {I − T xk }
strongly converges to some y, it follows that I − T x 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
L, η > 0, F is L-Lipschitz continuous and η-strongly monotone. Let {Ti }N
i1 be N nonexpansive self2
FixT
∅.
Let
μ
∈
0,
2η/L
and
let
t
∈
0, 1, {λt }, {βti } ⊂ 0, 1,
maps of H such that C ∩N
/
i
i1
such that
λt −→ 0,
as t −→ 0,
0 < lim inf βti ≤ lim sup βti < 1,
t→0
t→0
i 1, . . . , N.
2.1
Then, the net {xt } defined by 1.8-1.9 converges strongly to the unique element p∗ in 1.1.
Proof. By using Lemma 2.2 with T λ T0t , that is, T I, we have that
t
T x − T t y ≤ 1 − λt τT t · · · T t x − T t · · · T t y
1
1
N
N
..
.
≤ 1 − λt τTit · · · T1t x − Tit · · · T1t y
2.2
..
.
≤ 1 − λt τT1t x − T1t y ≤ 1 − λt τx − y
∀x, y ∈ H.
So, T t is a contraction in H. By Banach’s Contraction Principle, there exists a unique element
xt ∈ H such that xt T t xt for all t ∈ 0, 1.
Fixed Point Theory and Applications
5
Next, we show that {xt } is bounded. Indeed, for a fixed point p ∈ C, we have that
Tit p p for i 1, . . . , N, and hence
xt − p T t xt − p T t xt − T t · · · T t p
1
N
t
t
I − λt μF TN
· · · T1t xt − I − λt μF TN
· · · T1t p − λt μF p t
t
· · · T1t xt − TN
· · · T1t p λt μF p ≤ 1 − λt τTN
t
t
≤ 1 − λt τTN−1
· · · T1t xt − TN−1
· · · T1t p λt μF p ..
.
≤ 1 − λt τTit · · · T1t xt − Tit · · · T1t p λt μF p 2.3
..
.
≤ 1 − λt τT1t xt − T1t p λt μF p ≤ 1 − λt τxt − p λt μF p .
Therefore,
xt − p ≤ μ F p τ
2.4
that implies the boundedness of {xt }. So, are the nets {FytN }, {yti }, i 1, . . . , N.
Put
yt1 1 − βt1 xt βt1 T1 xt ,
yt2 1 − βt2 yt1 βt2 T2 yt1 ,
..
.
yti 1 − βti yti−1 βti Ti yti−1 ,
2.5
..
.
ytN 1 − βtN ytN−1 βtN TN ytN−1 .
Then,
xt I − λt μF ytN .
2.6
6
Fixed Point Theory and Applications
Moreover,
2
xt − p2 I − λt μF ytN − p
2
2
ytN − p − 2λt μ F ytN , ytN − p λ2t μ2 F ytN 2
2
≤ ytN−1 − p − 2λt μ F ytN , ytN − p λ2t μ2 F ytN ..
.
2.7
2
2
≤ yt1 − p − 2λt μ F ytN , ytN − p λ2t μ2 F ytN 2
2
≤ xt − p − 2λt μ F ytN , ytN − p λ2t μ2 F ytN .
Thus,
2 λt μ 2
ηytN − p F p , ytN − p ≤
F ytN .
2
2.8
Further, for the sake of simplicity, we put yt0 xt and prove that
i−1
yt − Ti yti−1 −→ 0,
2.9
as t → 0 for i 1, . . . , N.
Let {tk } ⊂ 0, 1 be an arbitrary sequence converging to zero as k → ∞ and xk : xtk .
We have to prove that yki−1 − Ti yki−1 → 0, where yki are defined by 2.5 with t tk and
yki ytik . Let {xl } be a subsequence of {xk } such that
lim sup yki−1 − Ti yki−1 lim yli−1 − Ti yli−1 .
l→∞
k→∞
2.10
Let {xkj } be a subsequence of {xl } such that
lim sup xk − p lim xkj − p.
k→∞
j →∞
2.11
Fixed Point Theory and Applications
7
From 2.6 and Lemma 2.1, it implies that
2 2
xkj − p I − λkj μF ykNj − p
2
≤ ykNj − p − 2λkj μ F ykNj , xkj − p
2
N
N−1
T
−
p
β
y
−
T
p
1 − βkNj ykN−1
N
N
k
k
j
j
j
− 2λkj μ F ykNj , xkj − p
2
2
N
N−1
−
p
β
y
−
T
p
≤ 1 − βkNj ykN−1
T
N kj
N kj
j
2.12
− 2λkj μ F ykNj , xkj − p
2
N
,
x
−
p
−
2λ
μ
F
y
−
p
≤ ykN−1
k
k
j
j
kj
j
2
≤ · · · ≤ yk1j − p − 2λkj μ F ykNj , xkj − p
2
≤ xkj − p − 2λkj μ F ykNj , xkj − p .
Hence,
lim xkj − p lim yki j − p,
j →∞
j →∞
i 1, . . . , N.
2.13
By Lemma 2.1,
2 2
2
i
i i−1
−
p
β
y
−
p
ykj − p 1 − βki j yki−1
T
i kj
kj
j
2
− βki j 1 − βki j yki−1
− Ti yki−1
j
j
2
2
≤ 1 − βki j yki−1
− p βki j yki−1
− p
j
j
2
− βki j 1 − βki j yki−1
− Ti yki−1
j
j
2.14
2
2
yki−1
− p − βki j 1 − βki j yki−1
− Ti yki−1
j
j
j
2
2
≤ · · · yk0j − p − βki j 1 − βki j yki−1
− Ti yki−1
j
j
2
2
xkj − p − βki j 1 − βki j yki−1
− Ti yki−1
,
j
j
i 1, . . . , N.
8
Fixed Point Theory and Applications
Without loss of generality, we can assume that α ≤ βti ≤ β for some α, β ∈ 0, 1. Then, we have
2 2 2
α 1 − β yki−1
− Ti yki−1
≤ xkj − p − yki j − p .
j
j
2.15
This together with 2.13 implies that
2
i−1 lim yki−1
−
T
y
i
kj 0,
j
j →∞
i 1, . . . , N.
2.16
It means that yti−1 − Ti yti−1 → 0 as t → 0 for i 1, . . . , N.
Next, we show that xt − Ti xt → 0 as t → 0. In fact, in the case that i 1 we have
yt0 xt . So, xt − T1 xt → 0 as t → 0. Further, since
1
yt − T1 xt 1 − βt1 xt − T1 xt ,
2.17
and xt − T1 xt → 0, we have that yt1 − T1 xt → 0. Therefore, from
xt − yt1 ≤ xt − T1 xt T1 xt − yt1 ,
2.18
it follows that xt − yt1 → 0 as t → 0. On the other hand, since
2
yt − T2 yt1 1 − βt2 yt1 − T2 yt1 −→ 0,
2
yt − xt ≤ 1 − βt2 yt1 − xt βt2 T2 yt1 − xt 2.19
≤ 1 − βt2 yt1 − xt βt2 T2 yt1 − yt1 yt1 − xt ,
we obtain that yt2 − xt → 0 as t → 0. Now, from
xt − T2 xt ≤ xt − yt2 yt2 − T2 yt1 T2 yt1 − T2 xt ≤ xt − yt2 yt2 − T2 yt1 yt1 − xt ,
2.20
and xt − yt2 , yt2 − T2 yt1 , yt1 − xt → 0, it follows that xt − T2 xt → 0. Similarly, we obtain
that xt − Ti xt → 0, for i 1, . . . , N and ytN − xt → 0 as t → 0.
Let {xk } be any sequence of {xt } converging weakly to p as k → ∞. Then, xk −
Ti xk → 0, for i 1, . . . , N and {ykN } also converges weakly to p. By Lemma 2.3, we have
p ∈ C ∩N
i1 FixTi and from 2.8, it follows that
F p , p − p ≥ 0 ∀p ∈ C.
2.21
Fixed Point Theory and Applications
9
Since 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
F p , p − p ≥ 0 ∀p ∈ C.
2.22
The uniqueness of p∗ in 1.1 guarantees that p p∗ . Again, replacing p in 2.8 by p∗ , we
obtain the strong convergence for {xt }. 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
Sx − Sy2 ≤ x − y2 γ I − Sx − I − Sy2 ,
∀x, y ∈ H.
3.1
It is well known 10 that a mapping T : H → H by T x αx 1− 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
i1 FixSi , where Si is γi -strictly pseudocontractive as follows.
Let αi ∈ γi , 1 be fixed numbers. Then, C ∩N
i1 FixTi with Ti y αi y 1 − αi Si y, a
nonexpansive mapping, for i 1, . . . , N, and hence
Tit y 1 − βti y βti Ti y
1−
βti 1
3.2
− αi y βti 1
− αi Si y,
i 1, . . . , N.
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
constants L, η > 0, F is L-Lipschitz continuous and η-strongly monotone. Let {Si }N
i1 be N γi -strictly
FixS
∅.
Let
α
∈
γ
,
1,
μ
∈ 0, 2η/L2 and
pseudocontractive self-maps of H such that C ∩N
i /
i
i
i1
i
let t ∈ 0, 1, {λt }, {βt } ⊂ 0, 1, such that
λt −→ 0,
as t −→ 0,
0 < lim inf βti ≤ lim sup βti < 1,
t→0
t→0
i 1, . . . , N.
3.3
Then, the net {xt } defined by
xt T t xt ,
t
T t : T0t TN
· · · T1t ,
t ∈ 0, 1,
3.4
where Tit , for i 1, . . . , N, are defined by 3.2 and T0t x I − λt μFx, converges strongly to the
unique element p∗ in 1.1.
N
C where S N
It is known in 11 that FixS
i1 ξi Si with ξi > 0 and
i1 ξi 1
for N γi -strictly pseudocontractions {Si }N
i1 . Moreover, S is γ-strictly pseudocontractive with
γ max{γi : 1 ≤ i ≤ N}. So, we also have the following result.
10
Fixed Point Theory and Applications
Theorem 3.2. Let H be a real Hilbert space and F : H → H a mapping such that for some
constants L, η > 0, F is L-Lipschitz continuous and η-strongly monotone. Let {Si }N
i1 be N γi FixS
∅.
Let
α
∈
γ,
1, where
strictly pseudocontractive self-maps of H such that C ∩N
/
i
i1
γ max{γi : 1 ≤ i ≤ N}, μ ∈ 0, 2η/L2 , and let t ∈ 0, 1, {λt }, {βt } ⊂ 0, 1, such that
λt −→ 0,
as t −→ 0,
0 < lim inf βt ≤ lim sup βt < 1.
t→0
t→0
3.5
Then, the net {xt }, defined by
t
xt T xt ,
t
T :
T0t
N
1 − βt 1 − α I βt 1 − α ξi Si ,
t ∈ 0, 1,
3.6
i1
where T0t I − λt μF, ξi > 0, and
N
i1 ξi
1, converges strongly to the unique element p∗ in 1.1.
Acknowledgment
This work was supported by the Vietnamese National Foundation of Science and Technology
Development.
References
1 D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications,
vol. 88 of Pure and Applied Mathematics, Academic Press, New York, NY, USA, 1980.
2 R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer Series in Computational
Physics, Springer, New York, NY, USA, 1984.
3 E. Zeidler, Nonlinear Functional Analysis and Its Applications. III, Springer, New York, NY, USA, 1985.
4 H.-K. Xu and R. G. Ori, “An implicit iteration process for nonexpansive mappings,” Numerical
Functional Analysis and Optimization, vol. 22, no. 5-6, pp. 767–773, 2001.
5 L.-C. Zeng and J.-C. Yao, “Implicit iteration scheme with perturbed mapping for common fixed points
of a finite family of nonexpansive mappings,” Nonlinear Analysis. Theory, Methods & Applications, vol.
64, no. 11, pp. 2507–2515, 2006.
6 L.-C. Ceng, N.-C. Wong, and J.-C. Yao, “Fixed point solutions of variational inequalities for a
finite family of asymptotically nonexpansive mappings without common fixed point assumption,”
Computers & Mathematics with Applications, vol. 56, no. 9, pp. 2312–2322, 2008.
7 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.
8 Y. Yamada, “The hybrid steepest-descent method for variational inequalities problems over the
intesectionof the fixed point sets of nonexpansive mappings,” in Inhently Parallel Algorithms in
Feasibility and Optimization and Their Applications, D. Butnariu, Y. Censor, and S. Reich, Eds., pp. 473–
504, North-Holland, Amsterdam, Holland, 2001.
9 K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in Advanced
Mathematics, Cambridge University Press, Cambridge, UK, 1990.
10 H. Zhou, “Convergence theorems of fixed points for κ-strict pseudo-contractions in Hilbert spaces,”
Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 2, pp. 456–462, 2008.
11 G. L. Acedo and H.-K. Xu, “Iterative methods for strict pseudo-contractions in Hilbert spaces,”
Nonlinear Analysis. Theory, Methods & Applications, vol. 67, no. 7, pp. 2258–2271, 2007.