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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 604369, 8 pages
http://dx.doi.org/10.1155/2013/604369
Research Article

Approximation for the Hierarchical Constrained Variational Inequalities over the Fixed Points of Nonexpansive Semigroups

School of Information and Calculation, Beifang University of Nationalities, Yinchuan 750021, China

Received 22 December 2012; Accepted 6 February 2013

Academic Editor: Jen-Chih Yao

Copyright © 2013 Li-Jun Zhu. 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.

Abstract

The purpose of the present paper is to study the hierarchical constrained variational inequalities of finding a point such that , where is the set of the solutions of the following variational inequality: , where are two strongly positive bounded linear operators, is a -contraction, is a nonexpansive mapping, and is the fixed points set of a nonexpansive semigroup . We present a double-net convergence hierarchical to some elements in which solves the above hierarchical constrained variational inequalities.

1. Introduction

Let be a real Hilbert space with inner product and norm , respectively. Let be a nonempty closed convex subset of . Recall that a self-mapping of is said to be contractive if there exists a constant such that for all . A mapping is called nonexpansive if

We denote by the set of fixed points of ; that is, . A bounded linear operator is called strongly positive on if there exists a constant such that

It is well known that the variational inequality for an operator, , over a nonempty, closed, and convex set, , is to find a point with the property

The set of the solutions of the variational inequality (3) is denoted by VI. If the mapping is a monotone operator, then we say that VI(3) is monotone. It is well known that if is Lipschitzian and strongly monotone, then for small enough , the mapping is a contraction on and so the sequence of Picard iterates, given by , converges strongly to the unique solution of the VI(3). This sort of VI(3) where is strongly monotone and Lipschitzian is originated from Yamada [1]. However, if is only monotone (not strongly monotone), then their iterative methods do not apply to VI.

Many practical problems such as signal processing and network resource allocation are formulated as the variational inequality over the set of the solutions of some nonlinear mappings (e.g., the fixed point set of nonexpansive mappings), and algorithms to solve these problems have been proposed. Iterative algorithms have been presented for the convex optimization problem with a fixed point constraint along with proof that these algorithms strongly converge to the unique solution of problems with a strongly monotone operator. The strong monotonicity condition guarantees the uniqueness of the solution. For some related works on the variational inequalities, please see [223] and the references therein. Particulary, the variational inequality problems over the fixed points of nonexpansive mappings have been considered. The reader can consult [16, 24]. On the other hand, we note that in the literature, nonlinear ergodic theorems for nonexpansive semigroups have been considered by many authors; see, for example, [2532]. In this paper, we will consider a general variational inequality problem with the variational inequality constraint is the fixed points of nonexpansive semigroups.

The purpose of the present paper is to study the hierarchical constrained variational inequalities of finding a point such that where is the set of the solutions of the following variational inequality: where and are two strongly positive bounded linear operators, is a -contraction, is a nonexpansive mapping, and is the fixed points set of a nonexpansive semigroup . We present a double-net convergence hierarchical to some elements in which solves the above hierarchical constrained variational inequalities.

2. Preliminaries

Let be a nonempty closed convex subset of a real Hilbert space . The metric (or the nearest point) projection from onto is the mapping which assigns to each point the unique point satisfying the property

It is well known that is a nonexpansive mapping and satisfies

Moreover, is characterized by the following property:

Recall that a family of mappings of into itself is called a nonexpansive semigroup if it satisfies the following conditions:(S1) for all ;(S2) for all ;(S3) for all and ;(S4) for all , is continuous.

We denote by the set of fixed points of and by the set of all common fixed points of ; that is, . It is known that is closed and convex.

We need the following lemmas for proving our main results.

Lemma 1 (see [33]). Let be a nonempty bounded closed convex subset of a Hilbert space and let be a nonexpansive semigroup on . Then, for every ,

Lemma 2 (see [34]). Let be a closed convex subset of a real Hilbert space and let be a nonexpansive mapping. Then, the mapping is demiclosed. That is, if is a sequence in such that weakly and strongly, then .

Lemma 3 (see [35]). Let be a nonempty closed convex subset of a real Hilbert space . Assume that a mapping is monotone and weakly continuous along segments (i.e., weakly, as , whenever for ). Then the variational inequality is equivalent to the dual variational inequality

3. Main Results

Now we consider the following hierarchical variational inequality with the variational inequality constraint over the fixed points set of nonexpansive semigroups .

Problem 1. Let be a nonempty closed convex subset of a real Hilbert space . Let be a -contraction with coefficient and let be a nonexpansive mapping. Let be a nonexpansive semigroup on and let be two strongly positive bounded linear operators with coefficients and , respectively. Let be a constant satisfying .
Now, our objective is to find such that where is the set of the solutions of the following variational inequality:

We observe that is strongly monotone and Lipschitz continuous. In fact, we have

Hence, the existence and the uniqueness of the solution to Problem 1 are guaranteed.

In order to solve the above hierarchical constrainted variational inequality, we present the following double net.

Algorithm 4. Set . Then, . For each , we define a double net implicitly by

Note that this implicit manner algorithm is well defined. In fact, we define the mapping

Note that this self-mapping is a contraction. As a matter of fact, we have Since , . Hence, is a contraction. Therefore, by Banach’s Contraction Principle, has a unique fixed point which is denoted by .

Next we show the behavior of the net as and successively.

Theorem 5. Assume that . Then, for each fixed , the net defined by (15) converges in norm, as , to a solution . Moreover, as , the net converges in norm to the unique solution of Problem 1.

Proof. We first show that the sequence is bounded. Take . From (15), we have
Hence It follows that for each fixed , is bounded.
Next, we show that for all and consequently, as , the entire net converges in norm to .
For each fixed , we set . It is clear that for each fixed , . Notice that
Moreover, we observe that if , then
that is, is -invariant for all .
From (15), we deduce
Since is bounded, and are also bounded. Then, from Lemma 1, we deduce for all and fixed that
Set for all . We then have , and for any ,
Notice that
Thus, we have
So
Assume such that as . By (27), we obtain immediately that
Since is bounded, there exists a subsequence of such that converges weakly to a point . From (23) and Lemma 2, we get . We can substitute for in (28) to get
The weak convergence of to actually implies that strongly. This has proved the relative norm-compactness of the net as for each fixed .
In (28), we take the limit as to get
In particular, solves the following variational inequality:
Note that the mapping is monotone for all , since
By Lemma 3, (31) is equivalent to its dual VI:
Next we show that as , the entire net converges in norm to . We assume , where . Similarly, by the above proof, we deduce which solves the following variational inequality:
In (33), we take to get
In (34), we take to get
Adding up (35) and (36) yields
At the same time we note that
Therefore, by (37) and (38), we deduce Hence the entire net converges in norm to as .
As , the net converges to the unique solution of Problem 1.
In (33), we take any to deduce
By virtue of the monotonicity of and the fact that , we have
We can rewrite (40) as
It follows from (41) and (42) that
Hence
Therefore,
In particular,
which implies that is bounded.
We next prove that ; namely, if is a null sequence in such that weakly as , then . To see this, we use (33) to get
However, since is monotone,
Combining the last two relations yields
Letting as in (49), we get
The equivalent dual VI of (50) is
Namely, is a solution of VI(13); hence .
We further prove that , the unique solution of VI(12). As a matter of fact, we have by (45)
Therefore, the weak convergence to of right implies that in norm. Now we can let in (45) to get
which is equivalent to its dual VI
It turns out that solves VI(12). By uniqueness, we have . This is sufficient to guarantee that in norm, as . This completes the proof.

Corollary 6. For each , let be a double net defined by
for all . Then, for each fixed , the net defined by (55) converges in norm, as , to a solution . Moreover, as , the net converges in norm to which solves the following variational inequality: where is the set of the solutions of the following variational inequality:

Corollary 7. For each , let be a double net defined by
for all . Then, for each fixed , the net defined by (58) converges in norm, as , to a solution . Moreover, as , the net converges to the minimum norm solution of the following variational inequality:

Proof. In (55), we take and . Then (55) reduces to (58). Hence, the net defined by (58) converges in norm to which satisfies
This indicates that
Therefore, is the minimum norm solution of the VI(59). This completes the proof.

Acknowledgment

The author was supported in part by NSFC 71161001-G0105, NNSF of China (10671157 and 61261044), NGY2012097 and Beifang University of Nationalities scientific research project.

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