Abstract and Applied Analysis
Volume 2014 (2014), Article ID 589679, 10 pages
http://dx.doi.org/10.1155/2014/589679

Research Article

Existence and Algorithm for the Systems of Hierarchical Variational Inclusion Problems

Nopparat Wairojjana1,2 and Poom Kumam²

¹Faculty of Science and Technology, Valaya Alongkorn Rajabhat University under the Royal Patronage, 1 Phaholyothin Road, Klong Nueng, Klong Luang, Pathumthani 13180, Thailand
²Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand

Received 18 May 2014; Revised 3 September 2014; Accepted 4 September 2014; Published 23 December 2014

Academic Editor: Juan R. Torregrosa

Copyright © 2014 Nopparat Wairojjana and Poom Kumam. 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.

We study the existence and approximation of a solution for a system of hierarchical variational inclusion problems in Hilbert spaces. In this study, we use Maingé’s approach for finding the solutions of the system of hierarchical variational inclusion problems. Our result in this paper improves and generalizes some known corresponding results in the literature.

1. Introduction

Let be a real Hilbert space with inner product and norm being and , respectively, and let be a nonempty closed convex subset of . A mapping is called nonexpansive if

We use to denote the set of fixed points of ; that is, . It is well known that is a closed convex set, if is nonexpansive mappings.

A variational inclusion problem [1–3] is the problem of finding a point such that

where is a single-valued nonlinear mapping and is a multivalued mapping. We use to denote the set of solutions of the variational inclusion (2).

On the other hand, a hierarchical fixed point problem [4–11] is the problem of finding a point such that

If the set is replaced by the solution set of the variational inequality, then the hierarchical fixed point problems are called hierarchical variational inequality problems or hierarchical optimization problems. Many problems in mathematics, for example, the signal recovery [12], the power control problem [13], and the beamforming problem [14], can be considered in the framework of this kind of the hierarchical variational inequality problems.

Recently, Chang et al. [15] introduced bilevel hierarchical variational inclusion problems; that is, find such that, for given positive real numbers and , the following inequalities hold:

where are mappings, are multivalued mappings, and is the set of solutions to variational inclusion problem (2) with , for . They solved the convex programming problems and quadratic minimization problems by using Maingés scheme.

In this paper, we consider the following system of hierarchical variational inclusion problem: find , such that, for given positive real numbers , and , the following inequalities hold:

Some special cases of the system of hierarchical variational inclusion problem (5) are as follows.

(I)If , , where is a nonlinear mapping for each , in (5), then and the system of hierarchical variational inclusion problem (5) reduces to the following system of hierarchical optimization problem: find , such that

 which was studied by Li [16].(II)If for each , where is the metric projection from onto a nonempty closed convex subset in (6), then it is clear that the and the system of hierarchical optimization problem (6) reduces to the following system of optimization problem: find such that

(III)If , then the system of optimization problem (7) reduces to the following system of variational inequality problem: find such that

(IV)If , , and in (5) then the system of hierarchical variational inclusion problem (5) reduces to the following bilevel hierarchical variational inclusion problem: find such that

 which was studied by Chang et al. [15].(V)In (9), if , , for each , then bilevel hierarchical variational inclusion problem (9) reduces to the following bilevel hierarchical optimization problem: find such that

 which was studied by Maingé [17] and Kraikaew and Saejung [18].(VI)In (10), if for each , then bilevel hierarchical optimization problem (10) reduces to the following problem [19–21]: find such that

(VII)In (11), if then the problem (11) reduces to the following problem: find such that

(VIII)In (5), if , , , and then the system of hierarchical variational inclusion problem (5) reduces to the following hierarchical variational inclusion problem: find such that

(IX)In (13), if , then the hierarchical variational inclusion problem (13) reduces to the following hierarchical fixed point problem: find such that

(X)In (15), if then the hierarchical fixed point problem (15) reduces to the following classic variational inequality problem: find such that

Motivated and inspired by Chang et al. [15], we introduce the system of a hierarchical variational inclusion problem (5) and investigate a more general variant of the scheme proposed by Chang et al. [15] to solve the system of a hierarchical variational inclusion problem. Our analysis and method allow us to prove the existence and approximation of solutions to the system of a hierarchical variational inclusion problem (5). The results presented in this paper extend and improve the results of Chang et al. [15], Maingé [17], Kraikaew and Saejung [18], and some authors.

2. Preliminaries

This section collects some definitions and lemmas which can be used in the proofs for the main results in the next section. Some of them are known; others are not hard to derive. We use for strong convergence and for weak convergence.

Definition 1. Let be a mapping and let be a multivalued mapping.

(1)A mapping is called nonexpansive if

(2)A mapping is called quasinonexpansive if and

 It should be noted that is quasinonexpansive if and only if for all ,

(3)A mapping is called strongly quasinonexpansive if is quasinonexpansive and , whenever is a bounded sequence in and for some .(4)A mapping is called -Lipschitzian if there exists such that

(5)A mapping is called -strongly monotone if there exists such that

 It is easy to prove that if is a -Lipschitzian and -strongly monotone mapping and if , then the mapping is a contraction.(6)A mapping is called -inverse-strongly monotone if there exists such that

(7)A multivalued mapping is called monotone if for all , and imply that

(8)A multivalued mapping is called maximal monotone if it is monotone and for any ,

 for every (the graph of mapping ) implies that .

Lemma 2 (see [22]). Let be an -inverse-strongly monotone mapping. Then

(1) is an -Lipschitz continuous and monotone mapping;(2)for any constant , one has

(3)if , then is a nonexpansive mapping, where is the identity mapping on .

Lemma 3. Let and be any points. Then one has the following.

(1)That if and only if there holds the relation:

(2)That if and only if there holds the relation:

(3)There holds the relation:

 Consequently, is nonexpansive and monotone.

Definition 4. Let be a multivalued maximal monotone mapping. Then the mapping defined by

is called the resolvent operator associated with , where is any positive number and is the identity mapping.

Proposition 5 (see [22]). Let be a multivalued maximal monotone mapping, and let be an -inverse-strongly monotone mapping. Then the following conclusions hold.

(1)The resolvent operator associated with is single-valued and nonexpansive for all .(2)The resolvent operator is 1-inverse-strongly monotone; that is,

(3) is a solution of the variational inclusion (2) if and only if , for all ; that is, is a fixed point of the mapping . Therefore one has

 where is the set of solutions of variational inclusion problem (2).(4)If , then is a closed convex subset in .

Lemma 6 (see [23]). For and , the following statements hold:

(1);(2).

Lemma 7 (see [24]). Let be a sequence of real numbers, and there exists a subsequence of such that for all , where is the set of all positive integers. Then there exists a nondecreasing sequence of such that and the following properties are satisfied by all (sufficiently large) number :

In fact, is the largest number in the set such that holds.

Lemma 8 (see [18]). Let , , , and be such that

(1) is a bounded sequence;(2), for all ;(3)whenever is a subsequence of satisfying

 it follows that ;(4) and .

Then .

Lemma 9 (see [15]). Let be a multivalued maximal monotone mapping, let be an -inverse-strongly monotone mapping, and let be the set of solutions of variational inclusion problem (2) and . Then the following statements hold.

(1)If , then the mapping defined by

 is quasinonexpansive, where is the identity mapping and is the resolvent operator associated with .(2)The mapping is demiclosed at zero; that is, for any sequence , if and , then .(3)For any , the mapping defined by

 is a strongly quasinonexpansive mapping and .(4) is demiclosed at zero.

3. Main Results

Throughout this section, we always assume that the following conditions are satisfied:

(C1) is a multivalued maximal monotone mapping, is an -inverse-strongly monotone mapping, and is the set of solutions to variational inclusion problem (2) with , , and , for all ;(C2) and , , are the mappings defined by

 respectively.

Next, there are our main results.

3.1. An Existence Theorem

Theorem 10. Let , , , , and satisfy conditions (C1) and (C2), and let be contractions with a contractive constant , for all . Then there exists a unique element such that the following three inequalities are satisfied:

Proof. The proof is a consequence of Banach’s contraction principle but it is given here for the sake of completeness. By Proposition 5 and Lemma 9, , , and are nonempty closed and convex. Therefore the metric projection is well defined for each .
Since is a contraction mapping for each , then we have which is a contraction and also have

which is a contraction. Hence there exists a unique element such that

Putting and , then , , and .
Suppose that there is an element such that the following three inequalities are satisfied:

Then

Therefore

This implies that , , and . This completes the proof.

3.2. A Convergence Theorem

Theorem 11. Let , , , , and satisfy conditions (C1) and (C2), and let be contractions with a contractive constant , for all . Let , , and be three sequences defined by

where is a sequence in satisfying and . Then the sequences , , and generated to be (42) converge to , and , respectively, where is the unique element in verifying (36).

Proof. (i) First we prove that sequences , , and are bounded.
From Lemma 9, it follow that is strongly quasinonexpansive and for each . Since is contraction with the coefficient for each and , , and , it follows that

where . Similarly, we can also compute that

This implies that

By induction, we have

for all .
Hence , , and are bounded. Consequently, , , and are bounded.
(ii) Next we prove that for each the following inequality holds:

From (42) and Lemma 6, we have

Similarly, we can also prove that

Adding up inequalities (48) and (49), inequality (47) is proved.
(iii) Next, we prove that if there exists a subsequence such that

then

Since the norm is convex and , by (42), we have

This implies that

Since the sequences , , and are bounded, we have

By Lemma 9, , , and are strongly quasinonexpansive. We have

Consequently, we obtain that

It follows from the boundedness of and which is reflexive that there exists a subsequence of such that and

By Lemma 9, is demiclosed at zero, and so . Hence from (36) we have

Therefore

Similarly, we can also prove that

Hence, we have the desired inequality.
(iv) Finally, we prove that the sequences , , and generated to be (42) converge to , and , respectively.
It is clear that

Substituting (61) into (47), we have

Set

Then, we have the following statements.

(i)From (i), is bounded sequence. (ii)From (62), , for all . (iii)From (iii), whenever is a subsequence of satisfying

 it follows that .

By Lemma 8, we have

Hence, we obtain that

This completes the proof.

3.3. Consequence Results

Using Theorem 11, we can prove the following results.

Theorem 12. Let , , , , and satisfy conditions (C1) and (C2), and let be a -Lipschitzian and -strongly monotone mapping. Let , , and be three sequences defined by

where , , with , and is a sequence in satisfying and . Then the sequences , , and converge to , and , respectively, where is the unique element in such that the following three inequalities are satisfied:

Proof. It is easy to see that , , and are contraction mappings and all the conditions in Theorem 11 are satisfied. By Theorem 11, we have the sequences , , and which converge to such that the following three inequalities are satisfied:

Substituting , , and into (69), we obtain that the sequences , , and converge to such that the following three inequalities are satisfied:

This completes the proof

Setting in Theorem 11, we obtain the following corollary.

Corollary 13. Let , , , , and satisfy conditions (C1) and (C2), and let be contractions with a contractive constant , for all . Let , , and be three sequences defined by

where is a sequence in satisfying and . Then the sequences , , and generated to be (42) converge to , , and , respectively, where is the unique element in such that the following three inequalities are satisfied:

Corollary 14. Let , , , , and satisfy conditions (C1) and (C2), and let be -Lipschitzian and -strongly monotone mapping. Let , , and be three sequences defined by

where , , with , and is a sequence in satisfying and . Then the sequences , , and converge to , , and , respectively, where is the unique element in such that the following three inequalities are satisfied:

Setting , , and in Theorem 11, we obtain the following corollary.

Corollary 15. Let , , , , and satisfy conditions (C1) and (C2), and let be contractions with a contractive constant . Let be the sequences defined by

where is a sequence in satisfying and . Then the sequences converge to such that the following three inequalities are satisfied:

Corollary 16. Let , , , , and satisfy conditions (C1) and (C2), and let be -Lipschitzian and -strongly monotone mapping. Let be the sequences defined by

where and is a sequence in satisfying and . Then the sequences converge to such that the following three inequalities are satisfied:

Conflict of Interests

The authors declare that there is no conflict of interests regarding to the publication of this paper.

Acknowledgment

The authors would like to thank the National Research University Project of Thailand’s Office of the Higher Education Commission for financial support (NRU no. 57000621).

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