- Open Access
On the hierarchical variational inclusion problems in Hilbert spaces
© Chang et al.; licensee Springer 2013
Received: 22 February 2013
Accepted: 18 June 2013
Published: 8 July 2013
The purpose of this paper is by using Maingé’s approach to study the existence and approximation problem of solutions for a class of hierarchical variational inclusion problems in the setting of Hilbert spaces. As applications, we solve the convex programming problems and quadratic minimization problems by using the main theorems. Our results extend and improve the corresponding recent results announced by many authors.
MSC: 47J05, 47H09, 49J25.
Throughout this paper, we assume that H is a real Hilbert space, C is a nonempty closed and convex subset of H and denote by the set of fixed points of a mapping .
A number of problems arising in structural analysis, mechanics and economics can be considered in the framework of this kind of variational inclusions (see, for example, ).
The set of solutions of the variational inclusion (1.1) is denoted by Ω.
which is called the mixed quasi-variational inequality.
Especially, if , then (1.2) is equivalent to the minimizing problem of ϕ on H, i.e., to find such that .
This problem is called Hartman-Stampacchia variational inequality problem.
Recently, hierarchical fixed point problems, hierarchical optimization problems and hierarchical minimization problems have attracted many authors’ attention due to their link with convex programming problems, optimization problems and monotone variational inequality problems etc. (see [5–21] and others).
The purpose of this paper is to introduce and study the following bi-level hierarchical variational inclusion problem in the setting of Hilbert spaces:
where are mappings and are multi-valued mappings, is the set of solutions to variational inclusion problem (1.1) with , for .
(III) If , and both sets and are nonempty closed and convex subsets of H, then bi-level hierarchical variational inclusion problem (1.5) reduces to the following (one-level) hierarchical variational inclusion problem:
However, in practice, both solution sets and (and hence the two projections) are not given explicitly.
To overcome this drawback, inspired by the method studied by Yamada et al. [25, 26], Maingé  and Kraikaew et al. , we investigate a more general variant of the scheme proposed by Maingé , Kraikaew et al.  to replace the projection by some suitable mappings with a nice fixed point set. This strategy also suggests an effective approximation process. Our analysis and method allow us to prove the existence and approximation of solutions to problem (1.5). As applications, we utilize the main results to study the quadratic minimization problems and convex programming problems in Hilbert spaces. The results presented in the paper extend and improve the corresponding results in [20, 21, 25, 26] and others.
For the sake of convenience, we first recall some definitions and lemmas for our main results.
for every (the graph of mapping M) implies that .
Lemma 2.2 
A is an -Lipschitz continuous and monotone mapping;
- (ii)For any constant , we have(2.1)
If , then is a nonexpansive mapping, where I is the identity mapping on H.
Such a mapping from H onto C is called the metric projection.
- (ii)is firmly nonexpansive, i.e.,
- (iii)For each ,(2.2)
is called the resolvent operator associated with M, where λ is any positive number and I is the identity mapping.
Proposition 2.5 
The resolvent operator associated with M is single-valued and nonexpansive for all .
- (ii)The resolvent operator is 1-inverse-strongly monotone, i.e.,
- (iii)is a solution of the variational inclusion (1.1) if and only if , , i.e., u is a fixed point of the mapping . Therefore we have(2.3)
where Ω is the set of solutions of variational inclusion problem (1.1).
If , then Ω is a closed convex subset in H.
In the sequel, we denote the strong and weak convergence of a sequence in H to an element by and , respectively.
Lemma 2.6 
Lemma 2.7 
In fact, is the largest number n in the set such that holds.
Lemma 2.8 
is a bounded sequence;
- (iii)whenever is a subsequence of satisfying
it follows that ;
- (i)A mapping is said to be nonexpansive if
- (ii)A mapping is said to be quasi-nonexpansive if andIt should be noted that T is quasi-nonexpansive if and only if ,(2.4)
- (iii)A mapping is said to be strongly quasi-nonexpansive if T is quasi-nonexpansive and(2.5)
whenever is a bounded sequence in H and for some .
- (i)If , then the mapping defined by(2.6)
is quasi-nonexpansive, where I is the identity mapping and is the resolvent operator associated with M.
The mapping is demiclosed at zero, i.e., for any sequence , if and , then .
- (iii)For any , the mapping defined by(2.7)
is a strongly quasi-nonexpansive mapping and .
, is demiclosed at zero.
Proof (i) Since , it follows from Lemma 2.2(iii) and Proposition 2.5 that the mapping K is nonexpansive and . This implies that K is quasi-nonexpansive.
(ii) Since K is a nonexpansive mapping on H, is demiclosed at zero.
(iii) It is obvious that and is quasi-nonexpansive.
Next we prove that , is a strongly quasi-nonexpansive mapping.
Now we prove that .
(iv) Since and is demi-closed at zero, hence is demi-closed at zero. This completes the proof. □
3 Main results
, , is a multi-valued maximal monotone mapping, is an α-inverse-strongly monotone mapping and is the set of solutions to variational inclusion problem (1.1) with , and ;
- (C2)and , , , are the mappings defined by(3.1)
We have the following result.
Proof (I) First we prove that (3.3) has a unique solution .
are the unique solution of (3.3).
(II) Now we prove that and are bounded.
This implies that and are bounded. Consequently, the sequences and both are bounded.
Adding up the last two inequalities, the inequality (3.4) is proved.
(IV) Next we prove the following fact.
The above conclusion can be proved as follows.
The desired inequality is proved.
(V) Finally we prove that the sequences and defined by (3.2) converge to and , respectively.
By (II), is a bounded sequence;
From (3.4), , ;
- (iii)By (IV), for any subsequence satisfying
it follows that .
Hence it follows from Lemma 2.8 that and . This completes the proof of Theorem 3.1. □
Remark 3.3 It is easy to prove that if is a μ-Lipschitzian and r-strongly monotone mapping and if , then the mapping is a contraction.
Now we are in a position to prove the following main result.
Then the sequence converges strongly to the unique solution of bi-level hierarchical variational inclusion problem (1.5).
This implies that the sequence converges strongly to , which is the unique solution of bi-level hierarchical variational inclusion problem (1.5). This completes the proof of Theorem 3.4. □
4 Some applications
In this section, we shall utilize Theorem 3.1 and Theorem 3.4 to study the convex mathematical programming problem and quadratic minimization problem.
(I) Applications to convex mathematical programming problems.
In (1.5) taking , and , then hierarchical variational inclusion problem (1.5) reduces to the following problem:
Thus problem (4.3) reduces to the convex mathematical programming problem on :
Hence, we have the following result.
where , . Then converges strongly to , which is the unique solution of convex mathematical programming problem (4.5).
(II) Applications to quadratic minimization problems.
The linear operator is μ-Lipschitzian and r-strongly monotone, where and .
If , then the linear operator is contractive with a contractive constant .
Conclusion (1) is proved.
Therefore, is contractive with a contractive constant . This completes the proof. □
From Theorem 3.4 and Lemma 4.2 we have the following result.
and . This completes the proof.
The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A2042138).
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