# Strong Convergence Theorem for a New General System of Variational Inequalities in Banach Spaces

- S Imnang
^{1, 2}and - S Suantai
^{2, 3}Email author

**2010**:246808

https://doi.org/10.1155/2010/246808

© S. Imnang and S. Suantai. 2010

**Received: **26 July 2010

**Accepted: **30 December 2010

**Published: **4 January 2011

## Abstract

We introduce a new system of general variational inequalities in Banach spaces. The equivalence between this system of variational inequalities and fixed point problems concerning the nonexpansive mapping is established. By using this equivalent formulation, we introduce an iterative scheme for finding a solution of the system of variational inequalities in Banach spaces. Our main result extends a recent result acheived by Yao, Noor, Noor, Liou, and Yaqoob.

## 1. Introduction

*uniformly convex*if for each there exists a constant such that for any ,

*uniformly Frechet differentiable norm*if the limit (1.2) is attained uniformly for and in this case is said to be

*uniformly smooth*. We define a function , called the

*modulus of smoothness*of , as follows:

*-uniformly smooth*if there exists a constant such that for all . For , the generalized duality mapping is defined by

In particular, if
, the mapping
is called the *normalized duality mapping* and usually, we write
. If
is a Hilbert space, then
. Further, we have the following properties of the generalized duality mapping
:

It is known that if
is smooth, then
is single-valued, which is denoted by
. Recall that the duality mapping
is said to be *weakly sequentially continuous* if for each
with
weakly, we have
weakly-
. We know that if
admits a weakly sequentially continuous duality mapping, then
is smooth. For the details, see the work of Gossez and Lami Dozo in [1].

*accretive*if

for all
. A mapping
is said to be *nonexpansive* if
for all
. The fixed point set of
is denoted by
.

whenever
for
and
. A mapping
is called a *retraction* if
for all
. Furthermore,
is a *sunny nonexpansive retraction* from
onto
if
is a retraction from
onto
which is also sunny and nonexpansive.

A subset of is called a sunny nonexpansive retraction of if there exists a sunny nonexpansive retraction from onto . It is well known that if is a Hilbert space, then a sunny nonexpansive retraction is coincident with the metric projection from onto .

where is a Hilbert space and is a mapping from into . The variational inequality has been widely studied in the literature; see, for example, the work of Chang et al. in [2], Zhao and He [3], Plubtieng and Punpaeng [4], Yao et al. [5] and the references therein.

*a general system of variational inequalities*, where and are two constants. In particular, if , then problem (1.10) reduces to finding such that

which is defined by Verma [7] and is called *the new system of variational inequalities*. Further, if we add up the requirement that
, then problem (1.11) reduces to the classical variational inequality
.

The problem (1.12) is very interesting as it is connected with the fixed point problem for nonlinear mapping and the problem of finding a zero point of an accretive operator in Banach spaces, see [9–11] and the references therein.

where , are two sequences in and .

*the system of general variational inequalities in a real Banach space*. They proved a strong convergence theorem by using the following iterative algorithm:

where , , and are three sequences in and .

In this paper, motivated and inspired by the idea of Yao et al. [5] and Cheng et al. [6]. First, we introduce the following system of variational inequalities in Banach spaces.

*a new general system of variational inequalities in Banach spaces*, where for all . In particular, if , , and for , then problem (1.17) reduces to problem (1.15). Further, if , , then problem (1.17) reduces to the problem (1.10) in a real Hilbert space. Second, we introduce iteration process for finding a solution of a new general system of variational inequalities in a real Banach space. Starting with arbitrary points and let , , and be the sequences generated by

where for all and , are two sequences in . Using the demiclosedness principle for nonexpansive mapping, we will show that the sequence converges strongly to a solution of a new general system of variational inequalities in Banach spaces under some control conditions.

## 2. Preliminaries

In this section, we recall the well known results and give some useful lemmas that will be used in the next section.

Lemma 2.1 (see [15]).

for all , where is the -uniformly smooth constant of .

The following lemma concerns the sunny nonexpansive retraction.

The first result regarding the existence of sunny nonexpansive retractions on the fixed point set of a nonexpansive mapping is due to Bruck [18].

Remark 2.3.

If is strictly convex and uniformly smooth and if is a nonexpansive mapping having a nonempty fixed point set , then there exists a sunny nonexpansive retraction of onto .

Lemma 2.4 (see [19]).

where is a sequence in and is a sequence such that

Lemma 2.5 (see [20]).

Let and be bounded sequences in a Banach space and let be a sequence in with . Suppose for all integers and . Then, .

Lemma 2.6 (see [21]).

Let be a uniformly convex Banach space, a nonempty closed convex subset of , and be an nonexpansive mapping. Then is demiclosed at 0, that is, if weakly and strongly, then .

## 3. Main Results

In this section, we establish the equivalence between the new general system of variational inequalities (1.17) and some fixed point problem involving a nonexpansive mapping. Using the demiclosedness principle for nonexpansive mapping, we prove that the iterative scheme (1.18) converges strongly to a solution of a new general system of variational inequalities (1.17) in a Banach space under some control conditions. In order to prove our main result, the following lemmas are needed.

The next lemmas are crucial for proving the main theorem.

Lemma 3.1.

where is the 2-uniformly smooth constant of . In particular, if , then is a nonexpansive mapping.

Proof.

It is clear that, if , then is a nonexpansive mapping.

Lemma 3.2.

If for all , then is nonexpansive.

Proof.

From Lemma 3.1, we have is nonexpansive which implies by (3.4) that is nonexpansive.

Lemma 3.3.

Let be a nonempty closed convex subset of a real smooth Banach space . Let be the sunny nonexpansive retraction from onto . Let be three nonlinear mappings. For given , is a solution of problem (1.17) if and only if , and , where is the mapping defined as in Lemma 3.2.

Proof.

It is easy to see that (3.6) is equivalent to , and .

From now on we denote by the set of all fixed points of the mapping . Now we prove the strong convergence theorem of algorithm (1.18) for solving problem (1.17).

Theorem 3.4.

Let be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping. Let be the sunny nonexpansive retraction from onto . Let the mappings be -inverse strongly accretive with , for all and . For given , let the sequence be generated iteratively by (1.18). Suppose the sequences and are two sequences in such that

Then converges strongly to where is the sunny nonexpansive retraction of onto .

Proof.

It follows from Lemma 2.4, (3.24), and (3.26) that converges strongly to . This completes the proof.

Letting and for in Theorem 3.4, we obtain the following result.

Corollary 3.5 (see [5, Theorem 3.1]).

where , are two sequences in such that

Then converges strongly to where is the sunny nonexpansive retraction of onto .

## Declarations

### Acknowledgments

The authors wish to express their gratitude to the referees for careful reading of the manuscript and helpful suggestions. The authors would like to thank the Commission on Higher Education, the Thailand Research Fund, the Thaksin university, the Centre of Excellence in Mathematics, and the Graduate School of Chiang Mai University, Thailand for their financial support.

## Authors’ Affiliations

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