Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 472935, 14 pages
doi:10.1155/2012/472935
Research Article
Implicit Iterative Method for Hierarchical
Variational Inequalities
L.-C. Ceng,1 Q. H. Ansari,2, 3 N.-C. Wong,4 and J.-C. Yao5
1
Scientific Computing Key Laboratory of Shanghai Universities, Department of Mathematics,
Shanghai Normal University, Shanghai 200234, China
2
Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India
3
Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals,
Dhahran 31261, Saudi Arabia
4
Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan
5
Center for General Education, Kaohsiung Medical University, Kaohsiung 80708, Taiwan
Correspondence should be addressed to N.-C. Wong, wong@math.nsysu.edu.tw
Received 1 February 2012; Accepted 10 February 2012
Academic Editor: Yonghong Yao
Copyright q 2012 L.-C. Ceng 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 a new implicit iterative scheme with perturbation for finding the approximate
solutions of a hierarchical variational inequality, that is, a variational inequality over the common
fixed point set of a finite family of nonexpansive mappings. We establish some convergence
theorems for the sequence generated by the proposed implicit iterative scheme. In particular,
necessary and sufficient conditions for the strong convergence of the sequence are obtained.
1. Introduction
Let H be a real Hilbert space with inner product ·, · and norm · and C a nonempty closed
convex subset of H. For a given nonlinear operator A : C → H, the classical variational
inequality problem VIP 1 is to find x∗ ∈ C such that
Ax∗ , x − x∗ ≥ 0,
∀x ∈ C.
1.1
The set of solutions of VIP is denoted by VIC, A. If the set C is replaced by the set FixT of fixed points of a mapping T ; then the VIP is called a hierarchical variational inequality
problem HVIP. The signal recovery 2, the power control problem 3, and the beamforming
problem 4 can be written in the form of a hierarchical variational inequality problem. In the
2
Journal of Applied Mathematics
recent past, several authors paid their attention toward this kind of problem and developed
different kinds of solution methods with applications; see 2, 5–11 and the references therein.
Let F : H → H be η-strongly monotone i.e., if there exists a constant η > 0 such that
Fx − Fy, x − y ≥ ηx − y2 , for all x, y ∈ H and κ-Lipschitz continuous i.e., if there exists
a constant κ > 0 such that Fx − Fy ≤ κx − y, for all x, y ∈ H. Assume that C is the
intersection of the sets of fixed points of N nonexpansive mappings Ti : H → H. For an
arbitrary initial guess x0 ∈ H, Yamada 10 proposed the following hybrid steepest-descent
method:
xn1 : Tn1 xn − λn1 μFTn1 xn ,
∀n ≥ 0.
1.2
Here, Tk : Tk mod N , for every integer k > N, with the mod function taking values in the set
{1, 2, . . . , N}; that is, if k jN q for some integers j ≥ 0 and 0 ≤ q < N, then Tk TN if
q 0 and Tk Tq if 1 < q < N. Moreover, μ ∈ 0, 2η/κ2 and the sequence {λn } ⊂ 0, 1 of
parameters satisfies the following conditions:
i limn → ∞ λn 0;
ii ∞
n0 λn ∞;
∞
iii n0 |λn − λnN | is convergent.
Under these conditions, Yamada 10 proved the strong convergence of the sequence {xn } to
the unique element of VIC, F.
Xu and Kim 12 replaced the condition iii by the following condition:
iii limn → ∞ λn /λnN 1, or equivalently, limn → ∞ λn − λnN /λnN 0
and proved the strong convergence of the sequence {xn } to the unique element of VIC, F.
On the other hand, let K be a nonempty convex subset of H, and let {Ti }N
i1 be a finite
family of nonexpansive self-maps on K. Xu and Ori 13 introduced the following implicit
∞
iteration process: for x0 ∈ K and {αn }∞
n1 ⊂ 0, 1, the sequence {xn }n1 is generated by the
following process:
xn αn xn−1 1 − αn Tn xn ,
∀n ≥ 1,
1.3
where we use the convention Tn : Tn mod N . They also studied the weak convergence of
the sequence generated by the above scheme to a common fixed point of the mappings
{Ti }N
i1 under certain conditions. Subsequently, Zeng and Yao 14 introduced another implicit
iterative scheme with perturbation for finding the approximate common fixed points of a
finite family of nonexpansive self-maps on H.
Motivated and inspired by the above works, in this paper, we propose a new implicit
iterative scheme with perturbation for finding the approximate solutions of the hierarchical
variational inequalities, that is, variational inequality problem over the common fixed point
set of a finite family of nonexpansive self-maps on H. We establish some convergence
theorems for the sequence generated by the proposed implicit iterative scheme with
perturbation. In particular, necessary and sufficient conditions for strong convergence of the
sequence generated by the proposed implicit iterative scheme with perturbation are obtained.
Journal of Applied Mathematics
3
2. Preliminaries
Throughout the paper, we write xn x to indicate that the sequence {xn } converges weakly
to x in a Banach space E. Meanwhile, xn → x implies that {xn } converges strongly to x. For
a given sequence {xn } ⊂ E, ωw xn denotes the weak ω-limit set of {xn }, that is,
ωw xn : x ∈ E : xnj x for some subsequence nj of {n} .
2.1
A Banach space E is said to satisfy Opial’s property if
lim infxn − x < lim infxn − y
n→∞
n→∞
∀y ∈ E, y / x,
2.2
whenever a sequence xn x in E. It is well known that every Hilbert space H satisfies
Opial’s property; see for example 15.
A mapping A : H → H is said to be hemicontinuous if for any x, y ∈ H, the mapping
g : 0, 1 → H, defined by gt : Atx 1−ty for all t ∈ 0, 1, is continuous in the weak
topology of the Hilbert space H. The metric projection onto a nonempty, closed and convex set
C ⊆ H, denoted by PC , is defined by, for all x ∈ H, PC x ∈ C and x − PC x infy∈C x − y.
Proposition 2.1. Let C ⊆ H be a nonempty closed and convex set and A : H → H monotone and
hemicontinuous. Then,
a [1] VIC, A {x∗ ∈ C : Ay, y − x∗ ≥ 0, for all y ∈ C},
b [1] VIC, A /
∅ when C is bounded,
c [16, Lemma 2.24] VIC, A FixPC I −λA for all λ > 0, where I stands for the identity
mapping on H,
d [16, Theorem 2.31] VIC, A consists of one point if A is strongly monotone and Lipschitz
continuous.
On the other hand, it is well known that the metric projection PC onto a given nonempty closed and convex set C ⊆ H is nonexpansive with FixPC C 17, Theorem 3.1.4
i. The fixed point set of a nonexpansive mapping has the following properties.
Proposition 2.2. Let C ⊆ H be a nonempty closed and convex subset and T : C → C a nonexpansive
map.
a [18, Proposition 5.3] FixT is closed and convex.
b [18, Theorem 5.1] FixT /
∅ when C is bounded.
The following proposition provides an example of a nonexpansive mapping in which
the set of fixed points is equal to the solution set of a monotone variational inequality.
4
Journal of Applied Mathematics
Proposition 2.3 see 6, Proposition 2.3. Let C ⊆ H be a nonempty closed and convex set and
A : H → H an α-inverse-strongly monotone operator. Then, for any given λ ∈ 0, 2α, the mapping
Sλ : H → H, defined by
Sλ x : PC I − λAx,
∀x ∈ H,
2.3
is nonexpansive and FixSλ VIC, A.
The following lemmas will be used in the proof of the main results of this paper.
Lemma 2.4 see 18, Demiclosedness Principle. Assume that T is a nonexpansive self-mapping
on a closed convex subset K of a Hilbert space H. If T has a fixed point, then I − T is demiclosed, that
is, whenever {xn } is a sequence in K weakly converging to some x ∈ K and the sequence {I − T xn }
strongly converges to some y, it follows that I − T x y, where I is the identity operator of H.
∞
∞
Lemma 2.5 see 19, page 80. Let {an }∞
n1 , {bn }n1 , and {δn }n1 be sequences of nonnegative real
numbers satisfying the inequality
an1 ≤ 1 δn an bn ,
∀n ≥ 1.
2.4
∞
∞
If ∞
n1 δn < ∞ and
n1 bn < ∞, then limn → ∞ an exists. If in addition {an }n1 has a subsequence,
which converges to zero, then limn → ∞ an 0.
Let T : H → H be a nonexpansive mapping and F : H → H κ-Lipschitz continuous
and η-strongly monotone for some constants κ > 0, η > 0. For any given numbers λ ∈ 0, 1
and μ ∈ 0, 2η/κ2 , we define the mapping T λ : H → H by
T λ x : T x − λμFT x,
∀x ∈ H.
2.5
Lemma 2.6 see 12. If 0 ≤ λ < 1 and 0 < μ < 2η/κ2 , then
λ
T x − T λ y ≤ 1 − λτx − y,
where τ 1 −
∀x, y ∈ H,
2.6
1 − μ2η − μκ2 ∈ 0, 1.
3. An Iterative Scheme and Convergence Results
Let {Ti }N
i1 be a finite family of nonexpansive self-maps on H. Let A : H → H be an α-inversestrongly monotone mapping i.e., if there exists a constant α > 0 such that Ax − Ay, x − y ≥
αAx − Ay2 , for all x, y ∈ H. Let F : H → H be κ-Lipschitz continuous and η-strongly
∞
∞
monotone for some constants κ > 0, η > 0. Let {αn }∞
n1 ⊂ 0, 1, {βn }n1 ⊂ 0, 2α, {λn }n1 ⊂
2
0, 1, and take a fixed number μ ∈ 0, 2η/κ . We introduce the following implicit iterative
Journal of Applied Mathematics
5
scheme with perturbation F. For an arbitrary initial point x0 ∈ H, the sequence {xn }∞
n1 is
generated by the following process:
xn αn xn−1 1 − αn Tn xn − βn Axn − λn μF ◦ Tn xn − βn Axn ,
∀n ≥ 1.
3.1
Here, we use the convention Tn : Tn mod N . If A ≡ 0, then the implicit iterative scheme 3.1
reduces to the implicit iterative scheme studied in 14.
Let A : H → H be an α-inverse-strongly monotone mapping and s ∈ 0, 2α. By
Lemma 2.6, for every u ∈ H and t ∈ 0, 1, the mapping St : H → H defined by
St x : tu 1 − tT λ x
3.2
with x x − sAx,
satisfies
St x − St y 1 − t
T λ x − T λ y
≤ 1 − t1 − λτx − y
≤ 1 − tx − y
1 − tx − sAx − y − sAy 1 − t x − y − s Ax − Ay 2
2
≤ 1 − t x − y − s2α − sAx − Ay
≤ 1 − tx − y, ∀x, y ∈ H,
3.3
where 0 ≤ λ < 1, 0 < μ < 2η/κ2 , and τ 1 − 1 − μ2η − μκ2 ∈ 0, 1. By Banach’s contraction
principle, there exists a unique xt ∈ H such that
xt tu 1 − tT λ xt tu 1 − t T xt − sAxt − λμF ◦ T xt − sAxt .
3.4
This shows that the implicit iterative scheme 3.1 with perturbation F is well defined and
can be employed for finding the approximate solutions of the variational inequality problem
over the common fixed point set of a finite family of nonexpansive self-maps on H.
We now state and prove the main results of this paper.
Theorem 3.1. Let H be a real Hilbert space, A an α-inverse-strongly monotone mapping, and F :
H → H a κ-Lipschitz continuous and η-strongly monotone mapping for some constants κ, η > 0.
N nonexpansive self-maps on H with a nonempty common fixed point set N
Let {Ti }N
i1 be
i1 FixTi .
N
2
∅.
Denote
by
T
:
T
for
n
>
N.
Let
μ
∈
0,
2η/κ
, x0 ∈ H,
Suppose VI i1 FixTi , A /
n
n mod N
∞
∞
∞
⊂
0,
1,
{α
}
⊂
0,
1,
and
{β
}
⊂
0,
2α
be
such
that
λ
<
∞,
β
{λn }∞
n n1
n n1
n ≤ λn and
n1
n1 n
,
defined
by
a ≤ αn ≤ b, for all n ≥ 1, for some a, b ∈ 0, 1. Then, the sequence {xn }∞
n1
xn : αn xn−1 1 − αn Tnλn xn
αn xn−1 1 − αn Tn xn − βn Axn − λn μF ◦ Tn xn − βn Axn ,
converges weakly to an element of
N
i1
FixTi .
∀n ≥ 1,
3.5
6
Journal of Applied Mathematics
If, in addition, xn − Tn xn oβn , then {xn } converges weakly to an element of VI
FixT
i , A.
i1
N
Proof. Notice first that the following identity:
tx 1 − ty2 tx2 1 − ty2 − t1 − tx − y2
holds for all x, y ∈ H and all t ∈ 0, 1. Let x be an arbitrary element of
that
N
i1
3.6
FixTi . Observe
2
2 αn xn−1 1 − αn Tnλn xn − x
xn − x
2
2
αn xn−1 − x
2 1 − αn Tnλn xn − x − αn 1 − αn xn−1 − Tnλn xn .
3.7
Since A is α-inverse strongly monotone and {βn }∞
n1 ⊂ 0, 2α, we have
2
xn − x − βn Axn − Ax
xn − x
2 − 2βn Axn − Ax,
xn − x
βn2 Axn − Ax
2
≤ xn − x
2 − βn 2α − βn Axn − Ax
2
3.8
≤ xn − x
2.
By Lemma 2.6, we have
λn
λ
λ
λ
Tn xn − x Tn n xn − Tn n x Tn n x − x
≤ Tnλn xn − Tnλn x Tnλn x − x
≤ 1 − λn τxn − x
λn μFx
≤ 1 − λn τ xn − x − βn Axn − Ax
βn Ax
λn μFx
≤ 1 − λn τ xn − x
βn Ax
λn μFx
3.9
≤ 1 − λn τxn − x
βn Ax
λn μFx
≤ 1 − λn τxn − x
λn Ax
μFx
μFx
Ax
1 − λn τxn − x
λn τ ·
.
τ
It follows
2
2
μFx
Ax
λn
2
.
λn ·
Tn xn − x ≤ 1 − λn τxn − x
τ
3.10
Journal of Applied Mathematics
7
This together with 3.7 yields
2 μFx
Ax
≤ αn xn−1 − x
1 − αn 1 − λn τxn − x
λn ·
xn − x
τ
2
− αn 1 − αn xn−1 − Tnλn xn 2
μFx
Ax
2
2
≤ αn xn−1 − x
1 − αn xn − x
1 − αn λn ·
τ
2
λn
− αn 1 − αn xn−1 − Tn xn ,
2
2
2
3.11
and so,
2
2
μFx
λn Ax
− 1 − αn xn−1 − Tnλn xn ≤ xn−1 − x
1 − αn ·
xn − x
αn
τ
3.12
2
x
μF
x
A
− xn − xn−1 2 .
≤ xn−1 − x
2 λn ·
τa
2
2
Since ∞
μFx
2 /τa converges, by Lemma 2.5, limn → ∞ xn − x
exists. As
n1 λn · Ax
a consequence, the sequence {xn } is bounded. Moreover, we have
μFx
Ax
− xn − x
λn ·
xn − xn−1 ≤ xn−1 − x
τa
2
2
2
2
−→ 0
as n −→ ∞.
3.13
Therefore,
lim xn − xn−1 0.
3.14
n→∞
Obviously, it is easy to see that limn → ∞ xn − xni 0 for each i 1, 2, . . . , N. Now observe
that
1 − bxn−1 − Tnλn xn ≤ 1 − αn xn−1 − Tnλn xn xn − xn−1 −→ 0 as n −→ ∞.
3.15
Also note that the boundedness of {xn } implies that {Tn xn } and {FTn xn } are both bounded.
Thus, we have
xn−1 − Tn xn ≤ xn−1 − Tnλn xn Tnλn xn − Tn xn ≤ xn−1 − Tnλn xn λn μFTn xn −→ 0
as n −→ ∞.
3.16
This implies
xn − Tn xn ≤ xn − xn−1 xn−1 − Tn xn −→ 0 as n −→ ∞.
3.17
8
Journal of Applied Mathematics
Consequently,
xn − Tn xn ≤ xn − Tn xn Tn xn − Tn xn ≤ xn − Tn xn xn − xn 3.18
xn − Tn xn βn Axn ≤ xn − Tn xn λn Axn −→ 0
as n −→ ∞,
and hence, for each i 1, 2, . . . , N,
xn − Tni xn ≤ xn − xni xni − Tni xni Tni xni − Tni xn ≤ 2xn − xni xni − Tni xni −→ 0
as n −→ ∞.
3.19
This shows that limn → ∞ xn − Tni xn 0 for each i 1, 2, . . . , N. Therefore,
lim xn − Tl xn 0,
n→∞
for each l 1, 2, . . . , N.
3.20
On the other hand, since {xn } is bounded, it has a subsequence {xnj }, which converges
weakly to some x ∈ H, and so, we have limj → ∞ xnj − Tl xnj 0. From Lemma 2.4, it follows
that I − Tl is demiclosed at zero. Thus, x ∈ FixTl . Since l is an arbitrary element in the finite
set {1, 2, . . . , N}, we get x ∈ N
i1 FixTi .
Now, let x∗ be an arbitrary element of ωw xn . Then, there exists another subsequence
{xnk } of {xn }, which converges weakly to x∗ ∈ H. Clearly, by repeating the same argument,
∗
∗
x, then by the Opial’s property
we get x∗ ∈ N
i1 FixTi . We claim that x x. Indeed, if x /
of H, we conclude that
lim xn − x∗ lim infxnk − x∗ n→∞
k→∞
< lim infxnk − x lim xn − x lim infxnj − x
k→∞
n→∞
< lim infxnj − x∗ lim xn − x∗ .
j →∞
j →∞
3.21
n→∞
This leads to a contradiction, and so, we get x∗ x. Therefore, ωw xn is a singleton set.
Hence, {xn } converges weakly to a common fixed point of the mappings {Ti }N
i1 , denoted still
by x∗ .
Assume that xn − Tn xn oβn . Let y ∈ N
i1 FixTi be arbitrary but fixed. Then, it
follows from the nonexpansiveness of each Ti and the monotonicity of A that
Tn xn − y2 Tn xn − βn Axn − Tn y 2
2
≤ xn − y − βn Axn 2
xn − y 2βn Axn , y − xn βn2 Axn 2
2
xn − y 2βn Ay, y − xn Axn − Ay, y − xn βn2 Axn 2
2
≤ xn − y 2βn Ay, y − xn βn2 Axn 2 ,
3.22
Journal of Applied Mathematics
9
which implies that
1 xn − y2 − Tn xn − y2 2 Ay, y − xn βn Axn 2
βn
xn − y − Tn xn − y
xn − y Tn xn − y
2 Ay, y − xn βn Axn 2
βn
0≤
≤M
3.23
xn − Tn xn 2 Ay, y − xn M2 βn ,
βn
where M : sup{xn − y Tn xn − y Axn : n ≥ 1} < ∞. Note that λn → 0, βn ≤
λn , for all n ≥ 1, and xn − Tn xn oβn . Thus, for any ε > 0, there exists an integer m0 ≥ 1
such that Mxn − Tn xn /βn M2 βn ≤ ε for all n ≥ m0 . Consequently, 0 ≤ ε 2Ay, y − xn for all n ≥ m0 . Since xn x∗ , we have ε 2Ay, y − x∗ ≥ 0 as n → ∞. Therefore, from the
arbitrariness of ε > 0, we deduce that Ay, y − x∗ ≥ 0 for all y ∈ N
i1 FixTi . Proposition 2.1
a ensures that
∗
Ax , y − x∗ ≥ 0,
∀y ∈
N
3.24
FixTi ;
i1
that is, x∗ ∈ VI
N
i1
FixTi , A.
Corollary 3.2 see 14, Theorem 2.1. Let H be a real Hilbert space and F : H → Hκ-Lipschitz
continuous and η-strongly monotone for some constants κ > 0, η > 0. Let {Ti }N
i1 be N nonexpansive
2
FixT
∅.
Let
μ
∈
0,
2η/κ
,
x
∈
H,
{λn }∞
self-maps on H such that C N
i /
0
n1 ⊂ 0, 1, and
∞i1
∞
{αn }n1 ⊂ 0, 1 be such that n1 λn < ∞ and a ≤ αn ≤ b, for all n ≥ 1, for some a, b ∈ 0, 1.
Then, the sequence {xn }∞
n1 , defined by
xn : αn xn−1 1 − αn Tnλn xn αn xn−1 1 − αn Tn xn − λn μFTn xn ,
∀n ≥ 1,
3.25
converges weakly to a common fixed point of the mappings {Ti }N
i1 .
Proof. In Theorem 3.1, putting A ≡ 0, we can see readily that for any given positive number
α ∈ 0, ∞, A : H → H is an α-inverse-strongly monotone mapping. In this case, we have
VI
N
FixTi , A
i1
N
FixTi .
3.26
i1
Hence, for any given sequence {βn }∞
n1 ⊂ 0, 2α with βn ≤ λn ∀n ≥ 1, the implicit iterative
scheme 3.5 reduces to 3.25. Therefore, by Theorem 3.1, we obtain the desired result.
Lemma 3.3. In the setting of Theorem 3.1, we have
a limn → ∞ xn − x
exists for each x ∈ C,
b limn → ∞ dxn , C exists, where dxn , C infp∈C xn − p,
10
Journal of Applied Mathematics
c lim infn → ∞ xn − Tn xn 0,
where C : VI
N
i1
FixTi , A.
Proof. Conclusion a follows from 3.12, and conclusion c follows from 3.18. We prove
conclusion b. Indeed, for each x ∈ C,
2
2 ≤ Ax − Axn−1 Axn−1 μFx
− Fxn−1 μFxn−1 μFx
Ax
2
1
≤
Axn−1 μκxn−1 − x
μFxn−1 xn−1 − x
α
3.27
2
1
μκ xn−1 − x
Axn−1 μFxn−1 α
2
2
1
μκ xn−1 − x
≤2
2 2 Axn−1 μFxn−1 .
α
This together with 3.12 implies that
2
μFx
Ax
≤ xn−1 − x
λn ·
xn − x
τa
2
2
1
1
2
2
2
μκ xn−1 − x
λn ·
2 Axn−1 μFxn−1 ≤ xn−1 − x
τa
α
2 2 1/α μκ
2
2 2 λn ·
≤ 1 λn ·
xn−1 − x
Axn−1 μFxn−1 τa
τa
≤ 1 γn xn−1 − x
2 γn ,
3.28
2
2
and hence,
dxn , C2 ≤ 1 γn dxn−1 , C2 γn ,
3.29
where
2
2 1/α μκ
2
2 ,
,
γn λn · max
Axn−1 μFxn−1 τa
τa
∀n ≥ 1.
3.30
∞
Since ∞
n1 λn < ∞ and both {Axn−1 } and {Fxn−1 } are bounded, it is known that
n1 γn <
∞. On account of Lemma 2.5, we deduce that limn → ∞ dxn , C exists, that is, conclusion b
holds.
Finally, we give necessary and sufficient conditions for the strong convergence of the
sequence generated by the implicit iterative scheme 3.5 with perturbation F.
Journal of Applied Mathematics
11
Theorem 3.4. In the setting of Theorem 3.1, the sequence {xn } converges strongly to an element of
N
VI N
i1 FixTi , A if and only if lim infn → ∞ dxn , C 0 where C : VI i1 FixTi , A.
Proof. From 3.28, we derive for each n ≥ 1
2 ≤ 1 γn xn−1 − x
2 γn ,
xn − x
∀x ∈ C,
3.31
∞
∞
< ∞. Put M
n1 1 γn . Then 1 ≤ M < ∞.
Suppose that the sequence {xn } converges strongly to a common fixed point p of the
family {Ti }N
i1 . Then, limn → ∞ xn − p 0. Since
where
n1 γn
0 ≤ dxn , C ≤ xn − p,
3.32
we have lim infn → ∞ dxn , C 0.
Conversely, suppose that lim infn → ∞ dxn , C 0. Then, by Lemma 3.3 b, we deduce
that limn → ∞ dxn , C 0. Thus, for arbitrary ε > 0, there exists a positive integer N0 such that
dxn , C < !
Furthermore, the condition
that
∞
n1 γn
∞
"
jn
ε
8M
∀n ≥ N0 .
,
3.33
< ∞ implies that there exists a positive integer N1 such
ε2
γj < 8M
∀n ≥ N1 .
,
3.34
Choose N∗ max{N0 , N1 }. Observe that 3.31 yields
2 ≤ 1 γn 1 γn−1 xn−2 − x
2 1 γn γn−1 γn
xn − x
≤
n
n−1
"
#
2
γj
1 γj xN∗ − x
1 γj γn
n
#
jN∗ 1
⎡
⎣xN∗ − x
≤M
2
jN∗ 1
n
"
⎤
γj ⎦.
jN∗ 1
ij1
3.35
12
Journal of Applied Mathematics
!
2
and ∞
Note that dxN∗ , C < ε/ 8M
∈ C,
jN∗ γj < ε /8M. Thus, for all n, m ≥ N∗ and all x
we have from 3.35 that
xm − x
2
xn − xm 2 ≤ xn − x
≤ 2xn − x
2 2xm − x
2
⎤
⎡
⎤
⎡
n
m
"
"
2
2
⎣xN∗ − x
⎣xN∗ − x
≤ 2M
γj ⎦ 2M
γj ⎦
jN∗ 1
⎡
⎣xN∗ − x
≤ 4M
2
∞
"
jN∗ 1
⎤
3.36
γj ⎦
jN∗
ε2
2
.
< 4M xN∗ − x
8M
Taking the infimum over all x ∈ C, we obtain
2
dxN∗ , C ε
xn − xm ≤ 4M
8M
2
2
≤ 4M
ε2
ε2
8M
8M
ε2 ,
3.37
and hence, xn −xm ≤ ε. This shows that {xn }∞
n1 is a Cauchy sequence in H. Let xn → p ∈ H
as n → ∞. Then, we derive from 3.20 that for each l 1, 2, . . . , N,
p − Tl p ≤ p − xn xn − Tl xn Tl xn − Tl p
≤ 2xn − p xn − Tl xn −→ 0 as n −→ ∞.
3.38
Therefore, p ∈ FixTl for each l 1, 2, . . . , N, and hence, p ∈ N
i1 FixTi .
On the other hand, choose a positive sequence {εn }∞
n1 ⊂ 0, ∞ such that εn → 0 as
n → ∞. For each n ≥ 1, from the definition of dxn , C, it follows that there exists a point
pn ∈ C such that
xn − pn ≤ dxn , C εn .
3.39
Since dxn , C → 0 and εn → 0 as n → ∞, it is clear that xn − pn → 0 as n → ∞. Note that
pn − p ≤ pn − xn xn − p −→ 0
as n −→ ∞.
3.40
Hence, we get
lim pn − p 0.
3.41
n→∞
Furthermore, for each βn ∈ 0, 2α, the mapping Sβn : H → H is defined as follows:
Sβn x : PNi1 FixTi I − βn A x,
∀x ∈ H.
3.42
Journal of Applied Mathematics
13
From Proposition 2.3, we deduce that Sβn is nonexpansive and
N
Fix Sβn VI
FixTi , A C.
3.43
i1
From Proposition 2.2 a, we conclude that FixSβn is closed and convex. Thus, from the
condition VI N
/ ∅, it is known that C is a nonempty closed and convex set.
i1 FixTi , A Since {pn } lies in C and converges strongly to p, we must have p ∈ C.
Remark 3.5. Setting A 0 in Lemma 3.3 and Theorem 3.4 above, we shall derive Lemma 2.1
and Theorem 2.2 in 14 as direct consequences, respectively.
Acknowledgment
This paper is partially supported by the Taiwan NSC Grants 99-2115-M-110-007-MY3 and
99-2221-E-037-007-MY3.
References
1 D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications,
vol. 88, Academic Press, New York, NY, USA, 1980.
2 P. L. Combettes, “A block-iterative surrogate constraint splitting method for quadratic signal recovery,” IEEE Transactions on Signal Processing, vol. 51, no. 7, pp. 1771–1782, 2003.
3 H. Iiduka, “Fixed point optimization algorithmand its application to power control in CDMA data
networks,” Mathematical Programming. In press.
4 K. Slavakis and I. Yamada, “Robust wideband beamforming by the hybrid steepest descent method,”
IEEE Transactions on Signal Processing, vol. 55, no. 9, pp. 4511–4522, 2007.
5 H. Iiduka, “A new iterative algorithm for the variational inequality problem over the fixed point set
of a firmly nonexpansive mapping,” Optimization, vol. 59, no. 5-6, pp. 873–885, 2010.
6 H. Iiduka, “Iterative algorithm for solving triple-hierarchical constrained optimization problem,”
Journal of Optimization Theory and Applications, vol. 148, no. 3, pp. 580–592, 2011.
7 P.-E. Maingé and A. Moudafi, “Strong convergence of an iterative method for hierarchical fixed-point
problems,” Pacific Journal of Optimization, vol. 3, no. 3, pp. 529–538, 2007.
8 A. Moudafi, “Krasnoselski-Mann iteration for hierarchical fixed-point problems,” Inverse Problems,
vol. 23, no. 4, pp. 1635–1640, 2007.
9 H.-K. Xu, “Viscosity method for hierarchical fixed point approach to variational inequalities,”
Taiwanese Journal of Mathematics, vol. 14, no. 2, pp. 463–478, 2010.
10 I. Yamada, “The hybrid steepest descent method for the variational inequality problems over the
intersection of fixed point sets of nonexpansive mappings,” in Inherently Parallel Algorithms in
Feasibility and Optimization and Their Applications, D. Butnariu, Y. Censor, and S. Reich, Eds., vol. 8
of Studies in Computational Mathematics, pp. 473–504, North-Holland, Amsterdam, The Netherlands,
2001.
11 I. Yamada and N. Ogura, “Hybrid steepest descent method for variational inequality problem over
the fixed point set of certain quasi-nonexpansive mappings,” Numerical Functional Analysis and
Optimization, vol. 25, no. 7-8, pp. 619–655, 2004.
12 H. K. Xu and T. H. Kim, “Convergence of hybrid steepest-descent methods for variational
inequalities,” Journal of Optimization Theory and Applications, vol. 119, no. 1, pp. 185–201, 2003.
13 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.
14 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 and Applications, vol.
64, no. 11, pp. 2507–2515, 2006.
14
Journal of Applied Mathematics
15 Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,” Bulletin of the American Mathematical Society, vol. 73, pp. 591–597, 1967.
16 V. V. Vasin and A. L. Ageev, Ill-Posed Problems with a Priori Information, VSP, Utrecht, The Netherlands,
1995.
17 W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, Japan, 2000.
18 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.
19 M. O. Osilike, S. C. Aniagbosor, and B. G. Akuchu, “Fixed points of asymptotically demicontractive
mappings in arbitrary Banach spaces,” Panamerican Mathematical Journal, vol. 12, no. 2, pp. 77–88,
2002.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014