The variational inequality problem is to find such that

The set of solutions of the variational inequality problem is denoted by . It is well known that the variational inequality theory has emerged as an important tool in studying a wide class of obstacle, unilateral, and equilibrium problems; which arise in several branches of pure and applied sciences in a unified and general framework. Several numerical methods have been developed for solving variational inequalities and related optimization problems, see [1–24] and the references therein. In particular, Korpelevich's extragradient method which was introduced by Korpelevič [4] in 1976 generates a sequence via the recursion

where is the metric projection from onto , is a monotone operator, and is a constant. Korpelevich [4] proved that the sequence converges strongly to a solution of . Note that the setting of the space is Euclid space .

Recently, hierarchical fixed point problems and hierarchical minimization problems have attracted many authors' attention due to their link with some convex programming problems. See [25–32]. Motivated and inspired by these results in the literature, in this paper we are devoted to solve the following hierarchical variational inequality :

For this purpose, in this paper, we first suggest and analyze an implicit extragradient method. It is shown that the net defined by this implicit extragradient method converges strongly to the unique solution of in Hilbert spaces. As a special case, we obtain the minimum norm solution of the variational inequality .

Let be a real Hilbert space with inner product and norm , and let be a closed convex subset of . Recall that a mapping is called -inverse strongly monotone if there exists a positive real number such that

A mapping is said to be -contraction if there exists a constant such that

It is well known that, for any , there exists a unique such that

We denote by , where is called the metric projection of onto . The metric projection of onto has the following basic properties:

(i) for all ;

(ii) for every ;

(iii) for all , ;

(iv) for all , .

Such properties of will be crucial in the proof of our main results. Let be a monotone mapping of into . In the context of the variational inequality problem, it is easy to see from property (iii) that

We need the following lemmas for proving our main result.

Lemma 2.1 (see [13]).

Let be a nonempty closed convex subset of a real Hilbert space . Let the mapping be -inverse strongly monotone, and let be a constant. Then, one has

In particular, if , then is nonexpansive.

Lemma 2.2 (see [32]).

Let be a nonempty closed convex subset of a real Hilbert space . Assume that the mapping is monotone and weakly continuous along segments, that is, weakly as . Then, the variational inequality

is equivalent to the dual variational inequality

3. Main Result

In this section, we will introduce our implicit extragradient algorithm and show its strong convergence to the unique solution of .

Algorithm 1.

Let be a closed convex subset of a real Hilbert space . Let be an -inverse strongly monotone mapping. Let be a (nonself) contraction with coefficient . For any , define a net as follows:

where is a constant.

Note the fact that is a possible nonself mapping. Hence, if we take , then (3.1) reduces to

Remark 3.1.

We notice that the net defined by (3.1) is well defined. In fact, we can define a self-mapping as follows:

From Lemma 2.1, we know that if , the mapping is nonexpansive.

For any , we have

This shows that the mapping is a contraction. By Banach contractive mapping principle, we immediately deduce that the net (3.1) is well defined.

Theorem 3.2.

Suppose the solution set of is nonempty. Then the net generated by the implicit extragradient method (3.1) converges in norm, as , to the unique solution of the hierarchical variational inequality . In particular, if one takes that , then the net defined by (3.2) converges in norm, as , to the minimum-norm solution of the variational inequality .

Proof.

Take that . Since , using the relation (2.4), we have . In particular, if we take , we obtain

From (3.1), we have

Noting that is nonexpansive, thus,

That is,

Therefore, is bounded and so are , . Since is -inverse strongly monotone, it is -Lipschitz continuous. Consequently, and are also bounded.

From (3.6),(2.5), and the convexity of the norm, we deduce

Therefore, we have

Hence

By the property (ii) of the metric projection , we have

where is some appropriate constant. It follows that

and hence (by (3.7))

which implies that

Since , we derive

Next, we show that the net is relatively norm-compact as . Assume that is such that as . Put and .

By the property (ii) of metric projection , we have

Hence

Therefore,

In particular,

Since is bounded, without loss of generality, we may assume that converges weakly to a point . Since , we have . Hence, also converges weakly to the same point .

Next we show that . We define a mapping by

Then is maximal monotone (see [33]). Let . Since and , we have . On the other hand, from , we have

that is,

Therefore, we have

Noting that , , and is Lipschitz continuous, we obtain . Since is maximal monotone, we have and hence .

Therefore we can substitute for in (3.20) to get

Consequently, the weak convergence of and to actually implies that strongly. This has proved the relative norm-compactness of the net as .

Now we return to (3.20) and take the limit as to get

In particular, solves the following VI

or the equivalent dual VI (see Lemma 2.2)

Therefore, . That is, is the unique solution in of the contraction . Clearly this is sufficient to conclude that the entire net converges in norm to as .

Finally, if we take that , then VI (3.28) is reduced to

Equivalently,

This clearly implies that

Therefore, is the minimum-norm solution of .This completes the proof.

Remark 3.3.

(1) Note that our Implicit Extragradient Algorithms (3.1) and (3.2) have strong convergence in an infinite dimensional Hilbert space.

(2) In many problems, it is needed to find a solution with minimum norm; see [34–38]. Our Algorithm (3.2) solves the minimum norm solution of .