Research Article | Open Access

# Modified Relaxed Extragradient Method for a General System of Variational Inequalities and Nonexpansive Mappings in Banach Spaces

**Academic Editor:**Yonghong Yao

#### Abstract

The purpose of this paper is to introduce a new modified relaxed extragradient method and study for finding some common solutions for a general system of variational inequalities with inversestrongly monotone mappings and nonexpansive mappings in the framework of real Banach spaces. By using the demiclosedness principle, it is proved that the iterative sequence defined by the relaxed extragradient method converges strongly to a common solution for the system of variational inequalities and nonexpansive mappings under quite mild conditions.

#### 1. Introduction

Let be a real Hilbert space with inner product and norm , and be a nonempty closed convex subset of . Let be the projection of onto , it is known that projection operator is nonexpansive and satisfies the following: Moreover, is characterized by the properties and for all .

Let be a mapping. Recall that the classical variational inequality, denoted by , is to find such that

One can see that the variational inequality (1.2) is equivalent to a fixed point problem. element is a solution of the variational inequality (1.2) if and only if is a fixed point of the mapping , where is the identity mapping and is a constant.

Variational inequalities introduced by Stampacchia [1] in the early sixties have had a great impact and influence in the development of almost all branches of pure and applied sciences and have witnessed an explosive growth in theoretical advances, algorithmic development, and so forth; see, for example, [1–18] and the references therein.

For a monotone mapping , Noor [2] studied the following problem of finding such that: where are constants. If we add up the requirement that , then the problem (1.3) is reduced to the classical variational inequality (1.2). The problem of finding solutions of (1.3) by using iterative methods has been studied by many authors; see [2–9] and the references therein.

Recently, some authors also studied the problem of finding a common element of the fixed point set of nonexpansive mappings and the solution set of variational inequalities for -inversestrongly monotone mappings in the framework of real Hilbert spaces [10] and Banach spaces [11].

On the other hands, Ceng et al. [12] introduce the following general system of variational inequalities involving two different operators. For two given operators, consider the problem finding such that where are constants. To illustrate the applications of this system, we can refer to an example of related nonlinear optimization problem put forward by Zhu and Marcotte [13]. Very recently, Yao et al. [14] extend the system of variational inequality problems (1.4) to Banach spaces.

In the present paper, motivated and inspired by the methods of Ceng et al. [12], Iiduka and Takahashi [10], Qin et al. [11], and Yao et al. [14], we consider the following general system of variational inequalities in Banach spaces.

Let be a nonempty closed convex subset of a real smooth Banach space. Let be -inversestrongly accretive mapping and -inverse-strongly accretive mapping. Find such that where are constants, and is the normalized duality mapping. For more details of , one may see Li [15]. In a real Hilbert space, is the identity mapping, and the system (1.5) is reduced to (1.4). If we add up the requirement that , then the problem (1.4) is reduced to the generalized variational inequality (1.3), in particular.

And we consider the problem of finding a common element of the fixed point set of nonexpansive mappings and the solution set of the general system of variational inequalities for -inversestrongly monotone mappings in the framework of real Banach spaces. By using the demiclosedness principle, we prove that under quite mild conditions the iterative sequence defined by the relaxed extragradient method converges strongly to a common solution of this system of variational inequalities and nonexpansive mappings. Our results improve and extend the corresponding results announced by other authors, such as [2–4, 6, 8, 10–12, 14].

#### 2. Preliminaries

Let be a nonempty closed convex subset of a Banach space of . Let be the dual space of , and let denote the pairing between and . For , the generalized duality mapping is defined by for all . In particular, is called the normalized duality mapping. It is known that for all . If is a Hilbert space, then is the identity mapping. Further, we have the following properties of the generalized duality mapping : (1) for all with ,(2) for all and ,(3) for all .

Let . is said to be uniformly convex if, for any , there exists such that for any . implies .

It is known that a uniformly convex Banach space is reflexive and strictly convex, is said to be Gteaux differentiable if the limit exists for each . In this case, is said to be smooth. The norm of is said to be uniformly Gteaux differentiable if, for each , the limit (2.2) is attained uniformly for . The norm of is said to be Frchet differentiable, if, for each , the limit (2.2) is attained uniformly for . The norm of is said to be uniformly Frchet differentiable if the limit (2.2) is attained uniformly for . It is well-known that (uniform) Frchet differentiable of the norm of implies (uniform) Gteaux differentiability of the norm of .

The modulus of smoothness of is defined by where is a function. It is known that a Banach space is uniformly smooth if and only if . Let be a fixed real number with . A Banach space is said to be uniformly smooth if there exists a fixed constant such that , for all .

Next, we always assume that is a smooth Banach space. Let be a nonempty closed convex subsets of . Recall that an operator of into is said to be accretive if is said to be -inversestrongly accretive if there exists a constant such that

Let be a subset of and be a mapping of into . Then is said to be sunny if whenever for and . A subset of is called a sunny nonexpansive retract of if there exists a sunny nonexpansive retraction from onto .

In order to prove the main result, we also need the following lemmas. The following Lemma 2.2 describes characterization of sunny nonexpansive retraction on a smooth Banach space.

Lemma 2.1 (see [16]). * Let be a real 2 uniformly smooth Banach space with the best smooth constant . Then the following inequality holds:
*

Lemma 2.2 (see [17]). *Let be a closed convex subset of a smooth Banach space , let be a nonempty subset of , and let be a retraction form onto . Then is sunny and nonexpansive if and only if
**
for all and .*

Lemma 2.3 (see [18]). *Let and be bounded sequences in a Banach space and a sequence in with
**
Suppose that for all integers and
**
Then, .*

Lemma 2.4 (see [19]). *Let be a nonempty closed convex subset of a real uniformly smooth Banach space. Let and be two nonexpansive mappings from into itself with a common fixed point. Define a mapping by
**
where is a constant in . Then is nonexpansive and .*

Lemma 2.5 (see [20]). *Assume that is a sequence of nonnegative real numbers such that
**
where is a sequence in and is a sequence such that *(a)*,
*(b)*. **Then .*

Lemma 2.6. *Let be a nonempty closed convex subset of a real 2 uniformly smooth Banach space with the best smooth constant . Let the mappings be -inverse strongly accretive and -inverse strongly accretive, respectively, and then one has and are nonexpansive, where , . *

* Proof. * Indeed, for all , from Lemma 2.1, we have
This shows that is nonexpansive mapping, so is .

Lemma 2.7. *Let be a nonempty closed convex subset of a real 2 uniformly smooth Banach space . Let be the sunny nonexpansive retraction from onto . Let be -inverse strongly accretive mapping and -inverse strongly accretive mapping, respectively. Let be a mapping defined by
**
If , , then is nonexpansive. *

* Proof. *For all , from Lemma 2.6, we have
Therefore, from (2.15), we obtain immediately that the mapping is nonexpansive.

Lemma 2.8. *Let be a nonempty closed convex subset of a real smooth Banach space . Let be the sunny nonexpansive retraction from onto . Let be two possibly nonlinear mappings. For given , is a solution of problem (1.5) if and only if , where . *

* Proof. *We note that we can rewrite (1.5) as
From Lemma 2.2, we can deduce that (2.16) is equivalent to
This completes the proof.

*Remark 2.9. *From Lemma 2.8, we note that , which implies that is a fixed point of the mappings .

#### 3. Main Result

To solve the general system of variation inequality problem (1.5), now we are in a position to state and prove the main result in this paper.

Theorem 3.1. *Let be a uniformly convex and 2 uniformly smooth Banach space with the best smooth constant , a nonempty closed convex subset of , and be the sunny nonexpansive retraction from onto . Let be -inverse strongly accretive mapping and -inverse strongly accretive mapping, respectively, and a nonexpansive mapping with a fixed point. Assume that , where is defined as Lemma 2.7. Let be a sequence generated in the following manner:
**
where , , , and , and are three sequences in , and the following conditions are satisfied *(1)*,
*(2)*,
*(3)*,
*(4)*. ** Then the sequence defined by (3.1) converges strongly to , and is a solution of the problem (1.5), where , is a sunny nonexpansive retraction of onto . *

* Proof. *We divide the proof of Theorem 3.1 into six steps. *Step *1. First, we prove that is closed and convex. We know that is closed and convex. Next, we show that is closed and convex. From Lemma 3.1 and 3.2, we can see that are nonexpansive. This shows that is closed and convex. * Step *2. Now we prove that the sequences , and are bounded. Let , and from Remark 2.9, we obtain that
Putting , we see that
Putting for each , we arrive at
Hence, it follows that
Therefore, is bounded. Hence , and are bounded.

On the other hand, we have
where is an appropriate constant such that
*Step *3. We prove that .

Setting for each , we see that
Now, we compute from
Combining (3.6) and (3.9), we arrive at
It follows from the conditions (1.3), (1.4), and (1.5) that
Hence, by Lemma 2.3, we obtain that . Consequently,
On the other hand, it follows from the algorithm (3.1) that
From the condition (1.3) and formula (3.12), we see that
* Step *4. We prove that . Define a mapping by
where . From Lemma 2.4, we see that is a nonexpansive mapping with
On the other hand, we have
This implies that
It follows from the conditions (1.3), (1.4), (1.5), and (3.18) that
*Step* 5*.* Next, we show that .

Let be the fixed point of the contraction , where . That is, . It follows that
On the other hand, for any , we see that
It follows that
In view of (3.19), we see that
On the other hand, we see that and . It follows that as . Owing to the fact that is strong to weak* uniformly continuous on bounded subsets of , we see that
Hence, for any , there exists such that , the following inequality holds:
Since is arbitrary and (3.23), we see that
*Step* 6. Finally, we show that as . Observe that
which implies that
From the conditions (1.3) and (3.26) and applying Lemma 2.5 to (3.28), we obtain that
This completes the proof.

*Remark 3.2. *Since for all is uniformly convex and 2 uniformly smooth, we see that Theorem 3.1 is applicable to for all . There are a number of sequences satisfying the restrictions (C1)–(C3), for example, for each .

Corollary 3.3. *Let be a real Hilbert space and a nonempty closed convex subset of . Let be -inverse strongly monotone mapping and -inverse strongly monotone mapping, respectively, and nonexpansive mappings with a fixed point. Assume that , where . Suppose that is generated by
**
where , , and , and are three sequences in , and the following conditions are satisfied: *(1)*,
*(2)*,
*(3)*,
*(4)*. ** Then the sequence defined by (3.30) converges strongly to , and is a solution of the problem (1.4), where , is the projection of onto , and is the projection of onto . *

*Remark 3.4. *Theorem 3.1 and Corollary 3.3 improve and extend the corresponding results announced by other authors, such as [2–4, 6, 8, 10–12, 14].

#### Acknowledgments

The authors would like to thank editors and referees for many useful comments and suggestions for the improvement of the paper. This work was partially supported by the Natural Science Foundation of Zhejiang Province (Y6100696) and NSFC (11071169).

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#### Copyright

Copyright © 2012 Yuanheng Wang and Liu Yang. 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.