Strong Convergence Results for Hierarchical Circularly Iterative Method about Hierarchical Circularly Optimization ()
1. Introduction
For a given nonlinear operator
, the following classical variational inequality problem is formulated as finding a point
such that
![](https://www.scirp.org/html/2-5300491\b7c97ea9-516b-4cb6-9feb-0735cdc92171.jpg)
Variational inequalities were initially studied by Stampacchia [1] and ever since have been widely studied, since they cover as diverse disciplines as partial differential equations, optimal control, optimization, mathematical programming, mechanics, and finance. On the other hand, a number of mathematical programs and iterative algorithms have been developed to resolve complex real world problems.
The concept of variational inequalities plays an important role in structural analysis, mechanics and economics. Recently, the hierarchical variational inequalities and hierarchical iterative sequence problems have attached many authors’ attention (see [2-11]).
2. Preliminaries and Lemma
It is well-known that, for any
, there exists a unique nearest point in
, denoted by
, such that
.
Such a mapping
from
onto
is called the metric projection.
Lemma 2.1 (see [12]) The metric projection
has the following basic properties:
1)
is firmly nonexpansive, i.e.,
![](https://www.scirp.org/html/2-5300491\965a7584-520f-4983-a6f2-c7c04ae3539f.jpg)
and so
is nonexpansive.
2)
, for all
and
.
Definition 2.2
1) A mapping
is said to be
-inversestrongly monotone if there exists
such that
.
2) A mapping
is said to be
-Lipschitzian if
.
3) A mapping
is said to be quasinonexpansive if
and
.
4) A mapping
is said to be strongly quasi-nonexpansive if
is quasi-nonexpansive and
, whenever
is a bounded sequence in H and
for some
.
5) (see [13]) A mapping
is said to be
-demicontractive if
and
.
Obviously, the above inequality is equivalent to
and it is clear from the preceding definitions that every quasi-nonexpansive mapping is 0-demicontractive.
Lemma 2.3 (see [14]) For
and
, we have the following statements:
a)
;
b)
;
![](https://www.scirp.org/html/2-5300491\1c2f9374-4f1c-4ce4-9ba0-0dbe904639a1.jpg)
For prove our result, we give the following lemma.
Lemma 2.4 ([11]) Let
be a sequence of real numbers such that there exists a subsequence
of
such that
for all
. Then there exists a nondecreasing
, such that ![](https://www.scirp.org/html/2-5300491\2b9f58bf-a0a0-41fc-8d50-833248533e60.jpg)
and the following properties are satisfied for all (sufficiently large) numbers sequence
:
![](https://www.scirp.org/html/2-5300491\6025f316-d65f-484e-826e-3e74b587cb11.jpg)
In fact,
.
Lemma 2.5 ([11]) Assume that
is a sequence of nonnegative real numbers such that
where
is a sequence in
and
is a sequence such that (a)
,
, (b)
. Then
.
Lemma 2.6 ([11]) Let
,
![](https://www.scirp.org/html/2-5300491\da1b78f2-1da2-442b-aeba-77fd77afece6.jpg)
and
, such that
•
is a bounded sequence;
•
, for all
;
• whenever
is a subsequence of
satisfying
, it follows that
;
•
,
.
Then
.
In [11], the existence and uniqueness of solutions of some related hierarchical optimization problems had been discussed.
Inspired by these results in the literature, a circularly iterative method in this paper is introduced for solving a system of variational inequalities with fixed-point set constraints. Under suitable conditions, strong convergence results are proved in the setting of Hilbert spaces. Our scheme can be regarded as a more general variant of the algorithm proposed by Maingé. The results presented in the paper improve and extend the corresponding results in [11] and other.
3. Main Results
First, we discuss the existence and uniqueness of solutions of some related hierarchical optimization problems.
Theorem 3.1 Let
be quasi-nonexpansive mappings and
be contractions
. Then there exists a unique element
such that the following inequalities,
(1)
Proof. The proof is a consequence of the well-known Banach’s contraction principle but it is given here for the sake of completeness. It is known that both sets
are closed and convex, and hence the projections
are well defined. It is clear that the mapping
![](https://www.scirp.org/html/2-5300491\660d8162-e399-4b83-8575-bbb9d610f513.jpg)
is a contraction. Hence, there exists a unique element
such that
.
Put
and
.
Then
,
and
.
Suppose that there is an element
such that the following inequalities,
![](https://www.scirp.org/html/2-5300491\e2fbbddf-70ee-477a-96b2-1d1c7d050c46.jpg)
Then
and
.
Hence,
.
This implies that
and hence
.
This completes the proof. ![](https://www.scirp.org/html/2-5300491\1557c4ba-dd78-4360-95fa-546c37e78f01.jpg)
For mappings
, suppose
we define the iterative sequences
by
(2)
where
satisfy
,
.
Theorem 3.2 For every
, let
be strongly quasi-nonexpansive mappings such that
are demiclosed at zero and let
be contractions with the coefficient
. Then the iterative sequences
by (2) strong converge to
, respectively, where
is the unique element in
verifying (1).
Recall that a mapping
is demiclosed at zero iff
whenever
and
.
We split the proof of Theorem 3.2 into the following lemmas.
Lemma 3.3 The sequences
are bounded.
Proof. Since
be strongly quasi-nonexpansive mappings,
be contractions with the coefficient
. Then we have
![](https://www.scirp.org/html/2-5300491\aa509e77-f6d5-42e8-b1a5-b872cfac09bc.jpg)
Similarly, we also have
![](https://www.scirp.org/html/2-5300491\a16e93a5-ffca-4630-a0d2-761be152a690.jpg)
It implies that
![](https://www.scirp.org/html/2-5300491\2d3360c3-3747-4c19-b6fb-3955d029d35a.jpg)
By induction, we have
![](https://www.scirp.org/html/2-5300491\f14a7d8c-f757-406a-a792-d08ef78d8999.jpg)
for all
. In particular, sequences
are bounded. Consequently, the sequences
are also bounded.
Lemma 3.4 For each
, the following inequality holds:
(3)
Proof. Since
![](https://www.scirp.org/html/2-5300491\1246c282-e028-4b86-8d07-a2d36364950c.jpg)
Similarly, we also have
![](https://www.scirp.org/html/2-5300491\ff331bb8-1840-4f25-9185-55efcaac5694.jpg)
By Lemma 3.3, we give following result,
(4)
Lemma 3.5 If there exists a subsequence
of
such that
then
(5)
Proof. In fact, we first consider the following assertion:
![](https://www.scirp.org/html/2-5300491\0ce476d2-c166-4125-b630-cb21ef2e8d90.jpg)
By Lemma 3.3, the sequences
![](https://www.scirp.org/html/2-5300491\c8e38242-5979-408f-95e0-665d730af9c4.jpg)
are bounded. So we have
.
Since
are strongly quasi-nonexpansive,
.
by the iteration scheme (1), we have
.
It follows from the boundedness of
that there exists a subsequence
of
such that
and
![](https://www.scirp.org/html/2-5300491\5027d845-10b5-4ac0-bfaf-b48432299217.jpg)
Since
is demiclosed at zero, it follows that
. It follows from (1), we get
![](https://www.scirp.org/html/2-5300491\54fadbde-765f-4a36-be14-fcaa9066dc9f.jpg)
Consequently,
.
By using the same argument, we have
![](https://www.scirp.org/html/2-5300491\6d9ed748-724a-4008-8c7e-856b08721d91.jpg)
Therefore, we obtain the desired inequality (4).
Next, we prove Theorem 3.2. Denote
![](https://www.scirp.org/html/2-5300491\dc45ef40-6a85-4c64-9a48-4833c4a2e600.jpg)
![](https://www.scirp.org/html/2-5300491\47cd73f3-fb17-4b95-b222-61688f24b01b.jpg)
Since
![](https://www.scirp.org/html/2-5300491\f8fbf7e5-3ee4-4fe6-a7bf-dbc4084d97bb.jpg)
We have the following statements from Lemma (3.3), Lemma(3.4) and Lemma(3.5):
•
is a bounded sequence;
•
, for all
;
• whenever
is a subsequence of
satisfying
, it follows that
.
Hence, it follows from Lemma 2.6 that
, It implies that
.
This means that
.
The proof of Theorem 3.2 is completed.![](https://www.scirp.org/html/2-5300491\b7be1d18-0d6e-4bd7-9c1a-9982e3dd84f5.jpg)