A General Iterative Approach to Variational Inequality Problems and Optimization Problems

Abstract

We introduce a new general iterative scheme for finding a common element of the set of solutions of variational inequality problem for an inverse-strongly monotone mapping and the set of fixed points of a nonexpansive mapping in a Hilbert space and then establish strong convergence of the sequence generated by the proposed iterative scheme to a common element of the above two sets under suitable control conditions, which is a solution of a certain optimization problem. Applications of the main result are also given.

1. Introduction

Let be a real Hilbert space with inner product and induced norm . Let be a nonempty closed convex subset of and be self-mapping on . We denote by the set of fixed points of and by the metric projection of onto .

Let be a nonlinear mapping of into . The variational inequality problem is to find a such that

(1.1)

We denote the set of solutions of the variational inequality problem (1.1) by . The variational inequality problem has been extensively studied in the literature; see [1–5] and the references therein.

Recently, in order to study the problem (1.1) coupled with the fixed point problem, many authors have introduced some iterative schemes for finding a common element of the set of the solutions of the problem (1.1) and the set of fixed points of nonexpansive mappings; see [6–9] and the references therein. In particular, in 2005, Iiduka and Takahashi [8] introduced an iterative scheme for finding a common point of the set of fixed points of a nonexapansive mapping and the set of solutions of the problem (1.1) for an inverse-strong monotone mapping : and

(1.2)

where and . They proved that the sequence generated by (1.2) strongly converges strongly to . In 2010, Jung [10] provided the following new composite iterative scheme for the fixed point problem and the problem (1.1): and

(1.3)

where is a contraction with constant ,,, and . He proved that the sequence generated by (1.3) strongly converges strongly to a point in , which is the unique solution of a certain variational inequality.

On the other hand, the following optimization problem has been studied extensively by many authors:

(1.4)

where , are infinitely many closed convex subsets of such that , , is a real number, is a strongly positive bounded linear operator on (i.e., there is a constant such that , for all ), and is a potential function for (i.e., for all ). For this kind of optimization problems, see, for example, Deutsch and Yamada [11], Jung [10], and Xu [12, 13] when and for a given point in .

In 2007, related to a certain optimization problem, Marino and Xu [14] introduced the following general iterative scheme for the fixed point problem of a nonexpansive mapping:

(1.5)

where and . They proved that the sequence generated by (1.5) converges strongly to the unique solution of the variational inequality

(1.6)

which is the optimality condition for the minimization problem

(1.7)

where is a potential function for . The result improved the corresponding results of Moudafi [15] and Xu [16].

In this paper, motivated by the above-mentioned results, we introduce a new general composite iterative scheme for finding a common point of the set of solutions of the variational inequality problem (1.1) for an inverse-strongly monotone mapping and the set of fixed points of a nonexapansive mapping and then prove that the sequence generated by the proposed iterative scheme converges strongly to a common point of the above two sets, which is a solution of a certain optimization problem. Applications of the main result are also discussed. Our results improve and complement the corresponding results of Chen et al. [6], Iiduka and Takahashi [8], Jung [10], and others.

2. Preliminaries and Lemmas

Let be a real Hilbert space and let be a nonempty closed convex subset of . We write to indicate that the sequence converges weakly to . implies that converges strongly to .

First we recall that a mapping is a contraction on if there exists a constant such that  , . A mapping is called nonexpansive if . We denote by the set of fixed points of .

For every point , there exists a unique nearest point in , denoted by , such that

(2.1)

for all . is called the metric projection of onto . It is well known that is nonexpansive and satisfies

(2.2)

for every . Moreover, is characterized by the properties:

(2.3)

In the context of the variational inequality problem for a nonlinear mapping , this implies that

(2.4)

It is also well known that satisfies the Opial condition, that is, for any sequence with , the inequality

(2.5)

holds for every with .

A mapping of into is called inverse-strongly monotone if there exists a positive real number such that

(2.6)

for all ; see [4, 7, 17]. For such a case, is called -inverse-strongly monotone. We know that if , where is a nonexpansive mapping of into itself and is the identity mapping of , then is -inverse-strongly monotone and . A mapping of into is called strongly monotone if there exists a positive real number such that

(2.7)

for all . In such a case, we say is -strongly monotone. If is -strongly monotone and -Lipschitz continuous, that is, for all , then is -inverse-strongly monotone. If is an -inverse-strongly monotone mapping of into , then it is obvious that is -Lipschitz continuous. We also have that for all and ,

(2.8)

So, if , then is a nonexpansive mapping of into . The following result for the existence of solutions of the variational inequality problem for inverse strongly-monotone mappings was given in Takahashi and Toyoda [9].

Proposition 2.1.

Let   be a bounded closed convex subset of a real Hilbert space and let be an -inverse-strongly monotone mapping of into . Then, is nonempty.

A set-valued mapping is called monotone if for all , , and imply . A monotone mapping is maximal if the graph of is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping is maximal if and only if for , for every implies . Let be an inverse-strongly monotone mapping of into and let be the normal cone to at , that is, , and define

(2.9)

Then is maximal monotone and if and only if ; see [18, 19].

We need the following lemmas for the proof of our main results.

Lemma 2.2.

In a real Hilbert space , there holds the following inequality:

(2.10)

for all .

Lemma 2.3 (Xu [12]).

Let be a sequence of nonnegative real numbers satisfying

(2.11)

where and satisfy the following conditions:

(i) and or, equivalently, ;

(ii) or ;

(iii).

Then .

Lemma 2.4 (Marino and Xu [14]).

Assume that is a strongly positive linear bounded operator on a Hilbert space with constant and . Then .

The following lemma can be found in [20, 21] (see also Lemma 2.2 in [22]).

Lemma 2.5.

Let be a nonempty closed convex subset of a real Hilbert space , and let be a proper lower semicontinunous differentiable convex function. If is a solution to the minimization problem

(2.12)

then

(2.13)

In particular, if solves the optimization problem

(2.14)

then

(2.15)

where is a potential function for .

3. Main Results

In this section, we present a new general composite iterative scheme for inverse-strongly monotone mappings and a nonexpansive mapping.

Theorem 3.1.

Let be a nonempty closed convex subset of a real Hilbert space such that . Let be an -inverse-strongly monotone mapping of into and a nonexpansive mapping of into itself such that . Let and let be a strongly positive bounded linear operator on with constant and a contraction of into itself with constant . Assume that and . Let be a sequence generated by

(IS)

where , , and . Let , , and satisfy the following conditions:

(i); ;

(ii) for all and for some ;

(iii) for some , with ;

(iv), , .

Then converges strongly to , which is a solution of the optimization problem

(OP1)

where is a potential function for .

Proof.

We note that from the control condition (i), we may assume, without loss of generality, that . Recall that if is bounded linear self-adjoint operator on , then

(3.1)

Observe that

(3.2)

which is to say that is positive. It follows that

(3.3)

Now we divide the proof into several steps.

Step 1.

We show that is bounded. To this end, let and for every . Let . Since is nonexpansive and from (2.4), we have

(3.4)

Similarly, we have

(3.5)

Now, set . Let . Then, from (IS) and (3.4), we obtain

(3.6)

From (3.5) and (3.6), it follows that

(3.7)

By induction, it follows from (3.7) that

(3.8)

Therefore, is bounded. So , , , , , , and are bounded. Moreover, since and , and are also bounded. And by the condition (i), we have

(3.9)

Step 2.

We show that and . Indeed, since and are nonexpansive and , we have

(3.10)

Similarly, we get

(3.11)

Simple calculations show that

(3.12)

So, we obtain

(3.13)

Also observe that

(3.14)

By (3.11), (3.13), and (3.14), we have

(3.15)

where , , and . From the conditions (i) and (iv), it is easy to see that

(3.16)

Applying Lemma 2.3 to (3.15), we obtain

(3.17)

Moreover, by (3.10) and (3.13), we also have

(3.18)

Step 3.

We show that and . Indeed,

(3.19)

which implies that

(3.20)

Obviously, by (3.9) and Step 2, we have as . This implies that

(3.21)

By (3.9) and (3.21), we also have

(3.22)

Step 4.

We show that and . To this end, let . Since and , we have

(3.23)

So we obtain

(3.24)

Since from the condition (i) and from Step 3, we have . Moreover, from (2.4) we obtain

(3.25)

and so

(3.26)

Thus

(3.27)

Then, we have

(3.28)

Since , and , we get . Also by (3.21)

(3.29)

Step 5.

We show that . In fact, since

(3.30)

from (3.9) and (3.29), we have .

Step 6.

We show that

(3.31)

where is a solution of the optimization problem (OP1). First we prove that

(3.32)

Since is bounded, we can choose a subsequence of such that

(3.33)

Without loss of generality, we may assume that converges weakly to .

Now we will show that . First we show that . Assume that . Since and , by the Opial condition and Step 5, we obtain

(3.34)

which is a contradiction. Thus we have .

Next, let us show that . Let

(3.35)

Then is maximal monotone. Let . Since and , we have

(3.36)

On the other hand, from , we have and hence

(3.37)

Therefore, we have

(3.38)

Since in Step 4 and is -inverse-strongly monotone, we have as . Since is maximal monotone, we have and hence .

Therefore, . Now from Lemma 2.5 and Step 5, we obtain

(3.39)

By (3.9) and (3.39), we conclude that

(3.40)

Step 7.

We show that and , where is a solution of the optimization problem (OP1). Indeed from (IS) and Lemma 2.2, we have

(3.41)

that is,

(3.42)

where , , and

(3.43)

From (i), in Steps 3, and 6, it is easily seen that , , and . Hence, by Lemma 2.3, we conclude as . This completes the proof.

As a direct consequence of Theorem 3.1, we have the following results.

Corollary 3.2.

Let , and be as in Theorem 3.1. Let be a sequence generated by

(3.44)

where and . Let and satisfy the conditions (i), (ii), and (iv) in Theorem 3.1. Then converges strongly to , which is a solution of the optimization problem

(OP2)

where is a potential function for .

Corollary 3.3.

Let , , , , , , , , , and be as in Theorem 3.1. Let be a sequence generated by

(3.45)

where , , and . Let , and satisfy the conditions (i), (ii), (iii), and (iv) in Theorem 3.1. Then converges strongly to , which is a solution of the optimization problem

(OP3)

where is a potential function for .

Remark 3.4.

• (1) Theorem 3.1 and Corollary 3.3 improve and develop the corresponding results in Chen et al. [6], Iiduka and Takahashi [8], and Jung [10].

• (2) Even though for , the iterative scheme (3.44) in Corollary 3.2 is a new one for fixed point problem of a nonexpansive mapping.

4. Applications

In this section, as in [6, 8, 10], we prove two theorems by using Theorem 3.1. First of all, we recall the following definition.

A mapping is called strictly pseudocontractive if there exists with such that

(4.1)

for every . If , then is nonexpansive. Put , where is a strictly pseudo-contractive mapping with constant . Then is -inverse-strongly monotone; see [2]. Actually, we have, for all ,

(4.2)

On the other hand, since is a real Hilbert space, we have

(4.3)

Hence we have

(4.4)

Using Theorem 3.1, we found a strong convergence theorem for finding a common fixed point of a nonexpansive mapping and a strictly pseudo-contractive mapping.

Theorem 4.1.

Let , , , , , , , , , and be as in Theorem 3.1. Let be an -strictly pseudo-contractive mapping of into itself such that . Let be a sequence generated by

(4.5)

where , , and . Let , , and satisfy the conditions (i), (ii), (iii), and (iv) in Theorem 3.1. Then converges strongly to , which is a solution of the optimization problem

(OP4)

where is a potential function for .

Proof.

Put . Then is -inverse-strongly monotone. We have and . Thus, the desired result follows from Theorem 3.1.

Using Theorem 3.1, we also obtain the following result.

Theorem 4.2.

Let be a real Hilbert space. Let be an -inverse-strongly monotone mapping of into and a nonexpansive mapping of into itself such that . Let ,  and let be a strongly positive bounded linear operator on with constant and a contraction with constant . Assume that and . Let be a sequence generated by

(4.6)

where , , and . Let , , and satisfy the conditions (i), (ii), (iii), and (iv) in Theorem 3.1. Then converges strongly to , which is a solution of the optimization problem

(OP5)

where is a potential function for .

Proof.

We have . So, putting , by Theorem 3.1, we obtain the desired result.

Remark 4.3.

• (1) Theorems 4.1 and 4.2 complement and develop the corresponding results in Chen et al. [6] and Jung [10].

• (2) In all our results, we can replace the condition on the control parameter by the condition for , [12, 13] or by the perturbed control condition , [23].