- Research Article
- Open Access

# A General Iterative Process for Solving a System of Variational Inclusions in Banach Spaces

- Uthai Kamraksa
^{1}and - Rabian Wangkeeree
^{1, 2}Email author

**2010**:190126

https://doi.org/10.1155/2010/190126

© Uthai Kamraksa and Rabian Wangkeeree. 2010

**Received:**6 April 2010**Accepted:**14 June 2010**Published:**5 July 2010

## Abstract

The purpose of this paper is to introduce a general iterative method for finding solutions of a general system of variational inclusions with Lipschitzian relaxed cocoercive mappings. Strong convergence theorems are established in strictly convex and 2-uniformly smooth Banach spaces. Moreover, we apply our result to the problem of finding a common fixed point of a countable family of strict pseudo-contraction mappings.

## Keywords

- Banach Space
- Variational Inequality
- Nonexpansive Mapping
- Common Fixed Point
- Nonempty Closed Convex Subset

## 1. Introduction

where . It is known that 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 constant such that for all .

From [1], we know the following property.

The best constant in the above inequality is called the -uniformly smoothness constant of (see [1] for more details).

In particular, is called the normalized duality mapping. It is known that for all . If is a Hilbert space, then is the identity. Note the following.

(1) is a uniformly smooth Banach space if and only if is single-valued and uniformly continuous on any bounded subset of .

(2) All Hilbert spaces, (or ) spaces ( ), and the Sobolev spaces ( ) are -uniformly smooth, while (or ) and spaces ( ) are -uniformly smooth.

(3) Typical examples of both uniformly convex and uniformly smooth Banach spaces are , where . More precisely, is -uniformly smooth for any .

Further, we have the following properties of the generalized duality mapping :

(i) for all with ,

(ii) for all and ,

(iii) for all .

It is known that, if is smooth, then is single valued. Recall that the duality mapping is said to be weakly sequentially continuous if, for each sequence with weakly, we have weakly- . We know that, if admits a weakly sequentially continuous duality mapping, then is smooth. For the details, see [2].

Let be a nonempty closed convex subset of a smooth Banach space . Recall the following definitions of a nonlinear mapping , the following are mentioned.

Definition 1.1.

Given a mapping .

for all .

for all .

*-inverse-strongly accretive*or

*-cocoercive*if there exists a constant such that

for all .

for all .

for all .

Remark 1.2.

( ) Every -strongly accretive mapping is an accretive mapping.

( ) Every -strongly accretive mapping is a -relaxed cocoercive mapping for any positive constant but the converse is not true in general. Then the class of relaxed cocoercive operators is more general than the class of strongly accretive operators.

( ) Evidently, the definition of the inverse-strongly accretive operator is based on that of the inverse-strongly monotone operator in real Hilbert spaces (see, e.g., [3]).

( ) The notion of the cocoercivity is applied in several directions, especially for solving variational inequality problems using the auxiliary problem principle and projection methods [4]. Several classes of relaxed cocoercive variational inequalities have been studied in [5, 6].

Next, we consider a system of quasivariational inclusions as follows.

where and are nonlinear mappings for each

As special cases of problem (1.11), we have the following.

(1) If and , then problem (1.11) is reduced to the following.

(2) Further, if in problem (1.12), then problem (1.12) is reduced to the following

In 2006, Aoyama et al. [7] considered the following problem.

where is a constant and is a sunny nonexpansive retraction from onto , see the definition below.

whenever for and . A mapping of into itself is called a retraction if . If a mapping of into itself is a retraction, then for all , where is the range of . A subset of is called a sunny nonexpansive retract of if there exists a sunny nonexpansive retraction from onto .

The following results describe a characterization of sunny nonexpansive retractions on a smooth Banach space.

Proposition 1.3 (see [8]).

Let be a smooth Banach space and a nonempty subset of . Let be a retraction and the normalized duality mapping on . Then the following are equivalent:

(1) is sunny and nonexpansive,

(2)

for all . That is, is nonexpansive if and only if is -strict pseudocontractive. We denote by the set of fixed points of .

Proposition 1.4 (see [9]).

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

Definition 1.5.

*family of uniformly*

*-strict pseudocontractions*if there exists a constant such that

It is unclear, in general, what the behavior of is as even if has a fixed point. However, in the case of having a fixed point, Ceng et al. [12] proved that, if is a Hilbert space, then converges strongly to a fixed point of . Reich [11] extended Browder's result to the setting of Banach spaces and proved that, if is a uniformly smooth Banach space, then converges strongly to a fixed point of , and the limit defines the (unique) sunny nonexpansive retraction from onto .

Reich [11] showed that, if is uniformly smooth and is the fixed point set of a nonexpansive mapping from into itself, then there is a unique sunny nonexpansive retraction from onto and it can be constructed as follows.

Proposition 1.6 (see [11]).

Notation 1.

We use to denote strong convergence to of the net as .

Definition 1.7 (see [13]).

is called the resolvent operator associated with , where is any positive number and is the identity mapping.

Recently, many authors have studied the problems of finding a common element of the set of fixed points of a nonexpansive mapping and one of the sets of solutions to the variational inequalities (1.11)–(1.14) by using different iterative methods (see, e.g., [7, 14–16]).

Very recently, Qin et al. [16] considered the problem of finding the solutions of a general system of variational inclusion (1.11) with -inverse strongly accretive mappings. To be more precise, they obtained the following results.

Lemma 1.8 (see [16]).

Theorem QCCK (see [16, Theorem ]).

where , , , and and are sequences in . If the control consequences and satisfy the following restrictions:

(C1) ,

(C2) and ,

then converges strongly to , where is the sunny nonexpansive retraction from onto and , where , is a solution to problem (1.11).

On the other hand, we recall the following well-known definitions and results.

*strongly positive*[17] if there exists a constant with property

where is the identity mapping and is the normalized duality mapping.

where is a potential function for for ).

where is nonself- -strict pseudo-contraction, is a contraction, and is a strongly positive bounded linear operator on a Hilbert space . They proved, under certain appropriate conditions imposed on the sequences and , that defined by (1.33) converges strongly to a fixed point of , which solves some variational inequality.

In this paper, motivated by Qin et al. [16], Moudafi [18], Marino and Xu [20], and Qin et al. [21], we introduce a general iterative approximation method for finding common elements of the set of solutions to a general system of variational inclusions (1.11) with Lipschitzian and relaxed cocoercive mappings and the set common fixed points of a countable family of strict pseudocontractions. We prove the strong convergence theorems of such iterative scheme for finding a common element of such two sets which is a unique solution of some variational inequality and is also the optimality condition for some minimization problems in strictly convex and -uniformly smooth Banach spaces. The results presented in this paper improve and extend the corresponding results announced by Qin et al. [16], Moudafi [18], Marino and Xu [20], Qin et al. [21], and many others.

## 2. Preliminaries

Now we collect some useful lemmas for proving the convergence result of this paper.

Lemma 2.1 (see [22]).

The resolvent operator associated with is single valued and nonexpansive for all .

Lemma 2.2 (see [13]).

where denotes the set of solutions to problem (1.13).

Lemma 2.3 (see [23]).

where is a constant in . Then is nonexpansive and .

Lemma 2.4 (see [24]).

Let be a nonempty closed convex subset of reflexive Banach space which satisfies Opial's condition, and suppose that is nonexpansive. Then the mapping is demiclosed at zero, that is, imply that .

Lemma 2.5 (see [25]).

where is a sequence in and is a sequence such that

(a) ,

(b) or

Then

Lemma 2.6 (see [26]).

Then

Definition 2.7 (see [27]).

Remark 2.8.

The example of the sequence of mappings satisfying AKTT-condition is supported by Example 3.11.

Lemma 2.9 (see [27, Lemma ]).

Then for each bounded subset of ,

Lemma 2.10 (see [28]).

Let be a real -uniformly smooth Banach space and a -strict pseudocontraction. Then is nonexpansive and .

Lemma 2.11 (see [29]).

Lemma 2.12 (see [17, Lemma ]).

Assume that is a strongly positive linear bounded operator on a smooth Banach space with coefficient and Then

## 3. Main Results

In this section, we prove that the strong convergence theorem for a countable family of uniformly -strict pseudocontractions in a strictly convex and -uniformly smooth Banach space admits a weakly sequentially continuous duality mapping. Before proving it, we need the following theorem.

Theorem 3.1 (see [17, Lemma ]).

Lemma 3.2.

If , then is nonexpansive.

Proof.

Hence (3.3) is proved. Assume that . Then, we have . This together with (3.3) implies that is nonexpansive.

Lemma 3.3.

where is defined as in Lemma 1.8. Assume that . Let be an -contraction; let be a strongly positive linear bounded self-adjoint operator with coefficient with . Then the following hold.

and is a solution of general system of variational inequality problem (1.11) such that .

Proof.

and is a solution of problem (1.11), where . This completes the proof of (ii).

Theorem 3.4.

where , and and are sequences in . Suppose that satisfies AKTT-condition. Let be the mapping defined by for all and suppose that . If the control consequences and satisfy the following restrictions

(C1) ,

(C2) and ,

and is a solution of general system of variational inequality problem (1.11) such that .

Proof.

First, we show that sequences , , and are bounded.

By the control condition (C2), we may assume, with no loss of generality, that .

This shows that the sequence is bounded, and so are , , and .

Apply Lemma 2.5 to (3.52) to conclude that as . This completes the proof.

Setting , and , we have the following result.

Theorem 3.5.

where , and and are sequences in . Suppose that satisfies AKTT-condition. Let be the mapping defined by for all and suppose that . If the control consequences and satisfy the following restrictions

(C1) ,

(C2) and ,

and is a solution of general system of variational inequality problem (1.11) such that .

Remark 3.6.

Theorem 3.4 mainly improves Theorem of Qin et al. [16], in the following respects:

(a)from the class of inverse-strongly accretive mappings to the class of Lipchitzian and relaxed cocoercive mappings,

(b)from a -strict pseudocontraction to the countable family of uniformly -strict pseudocontractions,

(c)from a uniformly convex and -uniformly smooth Banach space to a strictly convex and -uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping.

Further, if is a countable family of nonexpansive mappings, then Theorem 3.4 is reduced to the following result.

Theorem 3.7.

where , and and are sequences in . Suppose that satisfies AKTT-condition. Let be the mapping defined by for all and suppose that . If the control consequences and satisfy the following restrictions

(C1) ,

(C2) and ,

and is a solution of general system of variational inequality problem (1.11) such that .

Remark 3.8.

As in [27, Theorem ], we can generate a sequence of nonexpansive mappings satisfying AKTT-condition; that is, for any bounded subset of by using convex combination of a general sequence of nonexpansive mappings with a common fixed point. To be more precise, they obtained the following lemma.

Lemma 3.9 (see [27]).

where is a family of nonnegative numbers with indices with such that

(i) for all ,

(ii) for every ,

(iii) .

Then the following are given.

()Each is a nonexpansive mapping.

() satisfies AKTT-condition.

()If is defined by

then and .

Theorem 3.10.

where satisfies conditions (i)–(iii) of Lemma 3.9, , and and are sequences in . Suppose that satisfies AKTT-condition. Let be the mapping defined by for all and suppose that . If the control consequences and satisfy the following restrictions:

(C1) ,

(C2) and ,

and is a solution of general system of variational inequality problem (1.11) such that .

Proof.

where is defined by (3.59). It is clear that each mapping is nonexpansive. By Theorem 3.7 and Lemma 3.9, the conclusion follows.

The following example appears in [27] shows that there exists satisfying the conditions of Lemma 3.9.

Example 3.11.

## Declarations

### Acknowledgments

The first author is supported under grant from the program Strategic Scholarships for Frontier Research Network for the Ph.D. Program Thai Doctoral degree from the Office of the Higher Education Commission, Thailand and the second author is supported by the "Centre of Excellence in Mathematics" under the Commission on Higher Education, Ministry of Education, Thailand. Finally, The authors would like to thank the referees for reading this paper carefully, providing valuable suggestions and comments, and pointing out a major error in the original version of this paper.

## Authors’ Affiliations

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