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

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

Let be a real Banach space, and be its dual space. Let denote the unit sphere of . is said to be uniformly convex if for each there exists a constant such that for any ,

(1.1)

The norm on is said to be Gâteaux differentiable if the limit

(1.2)

exists for each and in this case is said to have a 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:

(1.3)

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 . For , the generalized duality mapping is defined by

(1.4)

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 :

(1) for all with ,

(2) for all and ,

(3) for all .

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].

Let be a nonempty closed convex subset of a smooth Banach space . Recall that a mapping is said to be accretive if

(1.5)

for all . A mapping is said to be -strongly accretive if there exists a constant such that

(1.6)

for all . A mapping is said to be -inverse strongly accretive if there exists a constant such that

(1.7)

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

Let be a nonempty subset of . A mapping is said to be sunny if

(1.8)

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 .

Conveying an idea of the classical variational inequality, denoted by , is to find an such that

(1.9)

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.

Let be two mappings. In 2008, Ceng et al. [6] considered the following problem of finding such that

(1.10)

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

(1.11)

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 .

In 2006, Aoyama et al. [8] first considered the following generalized variational inequality problem in Banach spaces. Let be an accretive operator. Find a point such that

(1.12)

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.

Aoyama et al. [8] introduced the following iterative algorithm in Banach spaces:

(1.13)

where is a sunny nonexpansive retraction from onto . Then they proved a weak convergence theorem which is generalized simultaneously theorems of Browder and Petryshyn [12] and Gol'shteĭn and Tret'yakov [13]. In 2008, Hao [14] obtained a strong convergence theorem by using the following iterative algorithm:

(1.14)

where , are two sequences in and .

Very recently, in 2009, Yao et al. [5] introduced the following system of general variational inequalities in Banach spaces. For given two operators , they considered the problem of finding such that

(1.15)

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

(1.16)

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.

Let be a nonempty closed convex subset of a real Banach space . Let for all be three mappings. We consider the following problem of finding such that

(1.17)

which is called 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

(1.18)

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]).

Let be a -uniformly smooth Banach space with . Then

(2.1)

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

The following lemma concerns the sunny nonexpansive retraction.

Lemma 2.2 (see [16, 17]).

Let be a closed convex subset of a smooth Banach space . Let be a nonempty subset of and be a retraction. Then is sunny and nonexpansive if and only if

(2.2)

for all and .

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]).

Assume is a sequence of nonnegative real numbers such that

(2.3)

where is a sequence in and is a sequence such that

(i);

(ii) or .

Then .

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.

Let be a nonempty closed convex subset of a real 2-uniformly smooth Banach space . Let the mapping be -inverse strongly accretive. Then, we have

(3.1)

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

Proof.

Indeed, for all , from Lemma 2.1, we have

(3.2)

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

Lemma 3.2.

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 an -inverse strongly accretive mapping for . Let be a mapping defined by

(3.3)

If for all , then is nonexpansive.

Proof.

For all , we have

(3.4)

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.

Note that we can rewrite (1.17) as

(3.5)

From Lemma 2.2, we can deduce that (3.5) is equivalent to

(3.6)

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

(C1) and ;

(C2).

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

Proof.

Let and , it follows from Lemma 3.3 that

(3.7)

Put and . Then and

(3.8)

From Lemma 3.1, we have is nonexpansive. Therefore

(3.9)

It follows that

(3.10)

By induction, we have

(3.11)

Therefore, is bounded. Hence , , , , , and are also bounded. By nonexpansiveness of and , we have

(3.12)

Let , . Then for all and

(3.13)

By (3.12) and (3.13), we have

(3.14)

This together with (C1) and (C2), we obtain that

(3.15)

Hence, by Lemma 2.5, we get as . Consequently,

(3.16)

Since

(3.17)

therefore

(3.18)

Furthermore, by Lemma 3.2, we have is nonexpansive. Thus, we have

(3.19)

which implies as .

Since

(3.20)

therefore

(3.21)

Let be the sunny nonexpansive retraction of onto . Now we show that

(3.22)

To prove (3.22), since is bounded, we can choose a subsequence of which converges weakly to and

(3.23)

From Lemma 2.6 and (3.21), we obtain . Now, from Lemma 2.2, (3.23), and the weakly sequential continuity of the duality mapping , we have

(3.24)

From (3.9), we have

(3.25)

which implies that

(3.26)

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]).

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 , and let , be the sequences generated by

(3.27)

where , are two sequences in such that

(C1) and ;

(C2).

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