#### Abstract

We suggest a modified extragradient method for solving the generalized variational inequalities in a Banach space. We prove some strong convergence results under some mild conditions on parameters. Some special cases are also discussed.

#### 1. Introduction

The well-known variational inequality problem is to find such that where is a nonempty closed convex subset of a real Hilbert space and is a nonlinear operator. This problem has been researched extensively due to its applications in industry, finance, economics, optimization, medical sciences, and pure and applied sciences; see, for instance, [1–19] and the reference contained therein. For solving the above variational inequality, Korpelevičh [20] introduced the following so-called extragradient method: for every , where is the metric projection from onto and . He showed that the sequences and generated by (1.2) converge to the same point . Since some methods related to extragradient methods have been considered in Hilbert spaces by many authors, please see, for example, [3, 5, 7, 14].

This naturally brings us to the following questions.

*Question 1. *Could we extend variational inequality from Hilbert spaces to Banach spaces?

*Question 2. *Could we extend the extragradient methods from Hilbert spaces to Banach spaces?

For solving Question 1, very recently, Aoyama et al. [21] first considered the following generalized variational inequality problem in a Banach space.

*Problem 1. **Let ** be a smooth Banach space and ** a nonempty closed convex subset of **. Let ** be an accretive operator of ** into **. Find a point ** such that *

This problem is connected with the fixed point problem for nonlinear mapping, the problem of finding a zero point of an accretive operator, and so on. For the problem of finding a zero point of an accretive operator by the proximal point algorithm, please consult [22]. In order to find a solution of Problem 1, Aoyama et al. [21] introduced the following iterative scheme for an accretive operator in a Banach space :
for every , where is a sunny nonexpansive retraction from onto . Then, they proved a weak convergence theorem in a Banach space which is generalized simultaneously by theorems of [4, 23] as follows.

Theorem 1.1. *Let be a uniformly convex and 2-uniformly smooth Banach space, and let be a nonempty closed convex subset of . Let be a sunny nonexpansive retraction from onto , let , and let be an -inverse-strongly accretive operator of into with . If and are chosen so that for some and for some with , then defined by (1.4) converges weakly to some element of , where is the 2-uniformly smoothness constant of .*

In this paper, motivated by the ideas in the literature, we first introduce a new iterative method in a Banach space as follows.

For fixed and arbitrarily given , define a sequence iteratively by for every , where is a sunny nonexpansive retraction from onto , , , and are three sequences in , and is a sequence of real numbers. We prove some strong convergence results under some mild conditions on parameters.

#### 2. Preliminaries

Let be a real Banach space, and let denote the dual of . Let be a nonempty closed convex subset of . A mapping of into is said to be accretive if there exists such that for all , where is called the duality mapping. A mapping of into is said to be -strongly accretive if, for , for all . A mapping of into is said to be -inverse-strongly accretive if, for , for all .

*Remark 2.1. * Evidently, the definition of the inverse strongly accretive mapping is based on that of the inverse strongly monotone mapping.

If is an -strongly accretive and -Lipschitz continuous mapping of into , then
from which it follows that must be -inverse-strongly accretive mapping.

Let . A Banach space is said to be uniformly convex if, for each , there exists such that, for any , It is known that a uniformly convex Banach space is reflexive and strictly convex. A Banach space is said to be smooth if the limit exists for all . It is also said to be uniformly smooth if the limit (2.6) is attained uniformly for . The norm of is said to be Frechet differentiable if, for each , the limit (2.6) is attained uniformly for . And we define a function called the modulus of smoothness of as follows: It is known that is uniformly smooth if and only if . Let be a fixed real number with . Then a Banach space is said to be -uniformly smooth if there exists a constant such that for all .

*Remark 2.2. *Takahashi et al. [24] remind us of the following fact: no Banach space is -uniformly smooth for . So, in this paper, we study a strong convergence theorem in a 2-uniformly smooth Banach space.

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

Lemma 2.3 (see [25]). *Let be a given real number with , and let be a -uniformly smooth Banach space. Then,
**
for all , where is the -uniformly smoothness constant of and is the generalized duality mapping from into defined by
**
for all .*

Let be a subset of , and let be a mapping of into . Then, is said to be sunny if whenever for and . A mapping of into itself is called a retraction if . If a mapping of into itself is a retraction, then for every , where is the range of . A subset of is called a sunny nonexpansive retract of if there exists a sunny nonexpansive retraction from onto . We know the following lemma concerning sunny nonexpansive retraction.

Lemma 2.4 (see [26]). * Let be a closed convex subset of a smooth Banach space , a nonempty subset of , and a retraction from onto . Then, is sunny and nonexpansive if and only if
**
for all and .*

*Remark 2.5. * It is well known that, if is a Hilbert space, then a sunny nonexpansive retraction is coincident with the metric projection from onto .

Let be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space , and let be a nonexpansive mapping of into itself with . Then, the set is a sunny nonexpansive retract of .

The following lemma is characterized by the set of solution Problem AIT by using sunny nonexpansive retractions.

Lemma 2.6 (see [21]). * Let be a nonempty closed convex subset of a smooth Banach space . Let be a sunny nonexpansive retraction from onto , and let be an accretive operator of into . Then, for all ,
**
where .*

Lemma 2.7 (see [27]). * Let be a nonempty bounded closed convex subset of a uniformly convex Banach space , and let be nonexpansive mapping of into itself. If is a sequence of such that weakly and strongly, then is a fixed point of .*

Lemma 2.8 (see [28]). * Let and be bounded sequences in a Banach space , and let be a sequence in which satisfies the following condition:
**
Suppose that
**
Then, .*

Lemma 2.9 (see [26]). * Assume that is a sequence of nonnegative real numbers such that
**
where is a sequence in and is a sequence in such that *(i)*, *(ii)* or . **Then, .*

#### 3. Main Results

In this section, we obtain a strong convergence theorem for finding a solution of Problem AIT for an -strongly accretive and -Lipschitz continuous mapping in a uniformly convex and 2-uniformly smooth Banach space. First, we assume that is a constant, a Lipschitz constant of , and the 2-uniformly smoothness constant of appearing in the following.

In order to obtain our main result, we need the following lemma concerning -inverse-strongly accretive mapping.

Lemma 3.1. *Let be a uniformly convex and 2-uniformly smooth Banach space, and let be a nonempty closed convex subset of . Let be a sunny nonexpansive retraction from onto , and let be an -inverse-strongly accretive mapping of into with . For given , let the sequence be generated iteratively by (1.5), where , , and are three sequences in and is a real number sequence in for some with satisfying the following conditions: *(i)*; *(ii)*; *(iii)*; *(iv)*. **Then we have and .*

*Proof. *First, we observe that is nonexpansive. Indeed, for all , from Lemma 2.3, we have
If , then is a nonexpansive mapping.

Letting , it follows from Lemma 2.6 that . Setting , from (3.1), we have
By (1.5) and (3.2), we have
Therefore, is bounded. Hence , , and are also bounded. We observe that
Setting for all we obtain
Combining (3.4) and (3.5), we have
this together with (ii) and (iv) implies that
Hence, by Lemma 2.8, we obtain as . Consequently,
From (1.5), we can write and note that and . It follows from (3.8) that
For , from (3.1) and (3.2), we obtain
Therefore, we have
Since and as , from (3.11), we obtain
From the definition of and (3.1), we also have
From the above results and assumptions, we note that , , are bounded, and as . Therefore, from (3.13), we have
which implies that
It follows from (3.12) and (3.15) that
This completes the proof.

Now we state and study our main result.

Theorem 3.2. *Let be a uniformly convex and 2-uniformly smooth Banach space with weakly sequentially continuous duality mapping, and let be a nonempty closed convex subset of . Let be a sunny nonexpansive retraction from onto , and let be an -strongly accretive and L-Lipschitz continuous mapping of into with . Let , , and be three sequences in and a real number sequence in for some with satisfying the following conditions: *(i)*; *(ii)* and ; *(iii)*; *(iv)*. **Then defined by (1.5) converges strongly to , where is a sunny nonexpansive retraction of onto .*

* Proof. *From Remark 2.1(2), we have that is an -inverse-strongly accretive mapping. Then, from Lemma 3.1, we have
On the other hand, we note that
which implies that
that is,
Next, we show that
To show (3.21), since is bounded, we can choose a sequence of that converges weakly to such that
We first prove . Since is in , it follows that is bounded, and so there exists a subsequence of which converges to . We may assume, without loss of generality, that as . Since is nonexpansive, it follows that
which implies that (noting that (3.20))
By Lemma 2.7 and (3.24), we have , and it follows from Lemma 2.6 that .

Now, from (3.22) and Lemma 2.4, we have

Finally, from (1.5) and (3.2), we have
which implies that
Finally, by Lemma 2.9 and (3.27), we conclude that converges strongly to . This completes the proof.

*Remark 3.3. *From (3.1), we know that is nonexpansive. If , it follows that there exists a sunny nonexpansive retraction of onto .

#### 4. Application

In this section, we prove a strong convergence theorem in a uniformly convex and 2-uniformly smooth Banach space by using Theorem 3.2. We study the problem of finding a fixed point of a strictly pseudocontractive mapping.

A mapping of into itself is said to be strictly pseudocontractive if there exists such that for all there exists such that This inequality can be written in the following form Now we give an application concerning a strictly pseudocontractive mapping.

Theorem 4.1. *Let be a uniformly convex and 2-uniformly smooth Banach space with weakly sequentially continuous duality mapping, and let be a nonempty closed convex subset and a sunny nonexpansive retract of . Let be a strictly pseudocontractive and L-Lipschitz continuous mapping of into itself with . Let , , and be three sequences in and a real number sequence in for some with satisfying the following conditions: *(i)*; *(ii)* and ; *(iii)*; *(iv)*. **For fixed and arbitrarily given , define a sequence iteratively by
**
for every . Then converges strongly to a fixed point of .*

*Proof. *Putting , we have from (4.2) that is -strongly accretive. At the same time, since is -Lipschitz continuous, then we have
for all , that is, is (+1)-Lipschitz continuous mapping. It follows from Remark 2.1 (2) that is -inverse-strongly accretive mapping. It is easy to show that . Therefore, using Theorem 3.2, we can obtain the desired conclusion. This completes the proof.

#### Acknowledgment

The paper is partially supported by the Program TH-1-3, Optimization Lean Cycle, of Sub-Projects TH-1 of Spindle Plan Four in Excellence Teaching and Learning Plan of Cheng Shiu University. The second author was partially supported by the Program TH-1-3, Optimization Lean Cycle, of Sub-Projects TH-1 of Spindle Plan Four inExcellence Teaching and Learning Plan of Cheng Shiu University and wassupported in part by NSC 100-2221-E-230-012.