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International Journal of Mathematics and Mathematical Sciences

Volume 2011 (2011), Article ID 560671, 18 pages

http://dx.doi.org/10.1155/2011/560671

## A New Composite General Iterative Scheme for Nonexpansive Semigroups in Banach Spaces

Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand

Received 1 February 2011; Accepted 19 March 2011

Academic Editor: Yonghong Yao

Copyright © 2011 Pongsakorn Sunthrayuth and Poom Kumam. 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.

#### Abstract

We introduce a new general composite iterative scheme for finding a common fixed point of nonexpansive semigroups in the framework of Banach spaces which admit a weakly continuous duality mapping. A strong convergence theorem of the purposed iterative approximation method is established under some certain control conditions. Our results improve and extend announced by many others.

#### 1. Introduction

Throughout this paper we denoted by and the set of all positive integers and all positive real numbers, respectively. Let be a real Banach space, and let be a nonempty closed convex subset of . A mapping of into itself is said to be *nonexpansive *if for each . We denote by the set of fixed points of . We know that is nonempty if is bounded; for more detail see [1]. A one-parameter family from of into itself is said to be a *nonexpansive semigroup* on if it satisfies the following conditions: (i);(ii) for all ;(iii)for each the mapping is continuous;(iv) for all and .

We denote by the set of all common fixed points of , that is, . We know that is nonempty if is bounded; see [2]. Recall that a self-mapping is a *contraction* if there exists a constant such that for each . As in [3], we use the notation to denote the collection of all contractions on , that is, . Note that each has a unique fixed point in .

In the last ten years, the iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, [3–5]. Let be a real Hilbert space, whose inner product and norm are denoted by and , respectively. Let be a strongly positive bounded linear operator on : that is, there is a constant with property A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space : where is the fixed point set of a nonexpansive mapping on and is a given point in .

In 2003, Xu [3] proved that the sequence generated by converges strongly to the unique solution of the minimization problem (1.2) provided that the sequence satisfies certain conditions. Using the viscosity approximation method, Moudafi [6] introduced the iterative process for nonexpansive mappings (see [3, 7] for further developments in both Hilbert and Banach spaces) and proved that if is a real Hilbert space, the sequence generated by the following algorithm: where is a contraction mapping with constant and satisfies certain conditions, converges strongly to a fixed point of in which is unique solution of the variational inequality:

In 2006, Marino and Xu [8] combined the iterative method (1.3) with the viscosity approximation method (1.4) considering the following general iterative process: where . They proved that the sequence generated by (1.6) converges strongly to a unique solution of the variational inequality: which is the optimality condition for the minimization problem: where is the fixed point set of a nonexpansive mapping and is a potential function for (i.e., for ). Kim and Xu [9] studied the sequence generated by the following algorithm: and proved strong convergence of scheme (1.9) in the framework of uniformly smooth Banach spaces. Later, yao, et al. [10] introduced a new iteration process by combining the modified Mann iteration [9] and the viscosity approximation method introduced by Moudafi [6]. Let be a closed convex subset of a Banach space, and let be a nonexpansive mapping such that and . Define in the following way: where and are two sequences in . They proved under certain different control conditions on the sequences and that converges strongly to a fixed point of . Recently, Chen and Song [11] studied the sequence generated by the algorithm in a uniformly convex Banach space, as follows: and they proved that the sequence defined by (1.11) converges strongly to the unique solution of the variational inequality: In 2010, Sunthrayuth and Kumam [12] introduced the a general iterative scheme generated by for the approximation of common fixed point of a one-parameter nonexpansive semigroup in a Banach space under some appropriate control conditions. They proved strong convergence theorems of the iterative scheme which solve some variational inequality. Very recently, Kumam and Wattanawitoon [13] studied and introduced a new composite explicit viscosity iteration method of fixed point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces. They proved strong convergence theorems of the composite iterative schemes which solve some variational inequalities under some appropriate conditions. In the same year, Sunthrayuth et al. [14] introduced a general composite iterative scheme for nonexpansive semigroups in Banach spaces. They established some strong convergence theorems of the general iteration scheme under different control conditions.

In this paper, motivated by Yao et al. [10], Sunthrayuth, and Kumam [12] and Kumam and Wattanawitoon [13] we introduce a new general iterative algorithm (3.23) for finding a common point of the set of solution of some variational inequality for nonexpansive semigroups in Banach spaces which admit a weakly continuous duality mapping and then proved the strong convergence theorem generated by the proposed iterative scheme. The results presented in this paper improve and extend some others from Hilbert spaces to Banach spaces and some others as special cases.

#### 2. Preliminaries

Throughout this paper, we write (resp., ) to indicate that the sequence weakly (resp., weak*) converges to ; as usual will symbolize strong convergence; also, a mapping denote the identity mapping. Let be a real Banach space, and let be its dual space. Let . A Banach space is said to be *uniformly convex* if, for each , there exists a such that for each , implies . It is known that a uniformly convex Banach space is reflexive and strictly convex (see also [15]). A Banach space is said to be *smooth* if the limit exists for each . It is also said to be *uniformly smooth* if the limit is attained uniformly for .

Let be a continuous strictly increasing function such that and as . This function is called a *gauge function *. The duality mapping associated with a gauge function is defined by
where denotes the generalized duality paring. In particular, the duality mapping with the gauge function , denoted by , is referred to as the normalized duality mapping. Clearly, there holds the relation for each (see [16]).

Browder [16] initiated the study of certain classes of nonlinear operators by means of the duality mapping . Following Browder [16], we say that Banach space has a *weakly continuous duality mapping* if there exists a gauge function for which the duality mapping is single-valued and continuous from the weak topology to the topology; that is, for each with , the sequence converges weakly* to . It is known that has a weakly continuous duality mapping with a gauge function for all . Set , ; then , where denotes the subdifferential in the sense of convex analysis (recall that the subdifferential of the convex function at is the set ).

In a Banach space having a weakly continuous duality mapping with a gauge function , we defined an operator is to be strongly positive (see [17]) if there exists a constant with the property If is a real Hilbert space, then the inequality (2.2) reduces to (1.1).

The first part of the next lemma is an immediate consequence of the subdifferential inequality and the proof of the second part can be found in [18].

Lemma 2.1 (see [18]). *Assume that a Banach space has a weakly continuous duality mapping with gauge . *(i)* For all , the following inequality holds:
In particular, for all , *(ii)* Assume that a sequence in converges weakly to a point . Then the following identity holds: *

Lemma 2.2 (see [17]). *Assume that a Banach space has a weakly continuous duality mapping with gauge . Let be a strongly positive linear bounded operator on with a coefficient and . Then .*

Lemma 2.3 (see [11]). * Let be a closed convex subset of a uniformly convex Banach space and let be a nonexpansive semigroup on such that . Then, for each and ,
*

Lemma 2.4 (see [19]). *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

Let be a Banach space which admits a weakly continuous duality mapping with gauge such that is invariant on , and let be a nonempty closed convex subset of such that . Let be a nonexpansive semigroup from into itself, let be a contraction mapping with a coefficient , let be a strongly positive linear bounded operator with a coefficient such that , and let such that which satisfies . Define the mapping by to be a contraction mapping. Indeed, for each , Thus, by Banach contraction mapping principle, there exists a unique fixed point , that is,

*Remark 3.1. *We note that space has a weakly continuous duality mapping with a gauge function for all . This shows that is invariant on .

Lemma 3.2. *Let be a uniformly convex Banach space which admits a weakly continuous duality mapping with gauge such that is invariant on , and let be a nonempty closed convex subset of such that . Let be a nonexpansive semigroup from into itself such that , let be a contraction mapping with a coefficient , let be a strongly positive linear bounded operator with a coefficient such that , and let such that which satisfies . Then the net defined by (3.3) with is a positive real divergent sequence, converges strongly as to a common fixed point , in which , and is the unique solution of the variational inequality:
*

*Proof. *Firstly, we show the uniqueness of a solution of the variational inequality (3.4). Suppose that are solutions of (3.4); then
Adding up (3.5), we obtain
which is a contradiction, we must have , and the uniqueness is proved. Here in after, we use to denote the unique solution of the variational inequality (3.4).

Next, we show that is bounded. Indeed, for each , we have
It follows that
Hence, is bounded, so are and .

Next, we show that as . We note that
Moreover, we note that
for all . Define the set ; then is a nonempty bounded closed convex subset of which is -invariant for each . Since and is bounded, there exists such that , and it follows by Lemma 2.3 that
for each . From (3.9)-(3.10), letting and noting (3.11) then, for each , we obtain
Assume that is such that as . Put and . We will show that contains a subsequence converging strongly to . Since is bounded sequence and Banach space is a uniformly convex, hence it is reflexive, and there exists a subsequence of which converges weakly to some as . Again, since is weakly sequentially continuous, we have by Lemma 2.1 that
Let , for all .

It follows that , . From (3.12), we have
On the other hand, we note that
Combining (3.14) with (3.15), we obtain . This implies that , that is, . In fact, since , and is the gauge function, then for , and
By Lemma 2.1, we have
This implies that
In particular, we have
Since the mapping is single-valued and weakly continuous, it follows from (3.19) that as . This implies that as .

Next, we show that solves the variational inequality (3.4), for each . From (3.3), we derive that
Now, we observe that
It follows from (3.20) that
Now, replacing and with and , respectively, in (3.22), and letting , and we notice that for , we obtain that . That is, is a solution of the variational inequality (3.4). By uniqueness, as , we have shown that each cluster point of the net is equal to . Then, we conclude that as . This proof is complete.

Theorem 3.3. *Let be a uniformly convex Banach space which admits a weakly continuous duality mapping with the gauge function such that is invariant in , and let be a nonempty closed convex subset of such that . Let be a nonexpansive semigroup from into itself such that , let be a contraction mapping with a coefficient , and let be a strongly positive linear bounded operator with a coefficient such that . Let be the sequences in and let be a positive real divergent sequence. Assume that the following conditions hold: *(C1)* and , *(C2)*,
*(C3)*,
** Then the sequence defined by
**
converges strongly to the common fixed point that is obtained in Lemma 3.2.*

*Proof. *From the condition (C1), we may assume, with no loss of generality, that for each . From Lemma 2.2, we have .

Firstly, we show that is bounded. Let ; we get
It follows that
By induction on , we have
Thus, is bounded. Since is bounded, then and and are also bounded.

Next, we show that , . From (3.23), we note that
By the conditions (C1)–(C3), then (3.27), we obtain
Moreover, we note that
Define the set . Then is a nonempty bounded closed convex subset of , which is —invariant for each and contains ; it follows from Lemma 2.3 that
Then, for all , from (3.28) and (3.30), into (3.29), we obtain , and hence
Next, we show that . We can take subsequence such that
By the assumption that is uniformly convex, hence it is reflexive and is bounded; then there exists a subsequence which converges weakly to some as . Since is weakly continuous, from Lemma 2.1, we have
Let , .

It follows that , .

From (3.31), we have
On the other hand, we note that
Combining (3.34) with (3.35), we obtain .

This implies that ; that is, .

Since the duality map is single-valued and weakly continuous, we get that
as required. Hence,
Since , by condition (C3), we obtain that . It follows from (3.37), that
Finally, we show that as . Now, from Lemma 2.1, we have
On the other hand, we note that
It follows from (3.40) that
where .

Put and . Then (3.41) reduces to formula . By conditions (C1) and (C3) and noting (3.38), it is easy to see that and . Applying Lemma 2.4, we obtain as this implies that as . This completes the proof.

Taking in (3.23), we can get the following corollary easily.

Corollary 3.4. * Let be a uniformly convex Banach space which admits a weakly continuous duality mapping with the gauge function such that invariant in , be a nonempty closed convex subset of such that . Let be a nonexpansive semigroup from into itself such that , be a contraction mapping with a coefficient and be a strongly positive linear bounded operator with a coefficient such that . Let be the sequences in and be a positive real divergent sequence. Assume the following conditions are hold: *(C1)* and ; *(C2)*.** Then the sequence defined by
**
converges strongly to the common fixed point , in which is the unique solution of the variational inequality:
*

A strong mean convergence theorem for nonexpansive mapping was first established by Baillon [20] and it was generalized to that for nonlinear semigroups by Reich et al. [21–23]. It is clear that Theorem 3.3 are valid for nonexpansive mappings. Thus, we have the following mean ergodic theorem of viscosity iteration process for nonexpansive mappings in Hilbert spaces.

Corollary 3.5. *Let be a real Hilbert space, and let be a nonempty closed convex subset of such that . Let be a nonexpansive mapping from into itself such that , be a contraction mapping with a coefficient , and let be a strongly positive linear bounded operator with a coefficient such that . Let be the sequences in and let be a positive real divergent sequence. Assume that the following conditions are hold: *(C1)* and ; *(C2)*;
*(C3)*.** Then the sequence defined by
**
converges strongly to the common fixed point , in which is the unique solution of the variational inequality:
*

#### Acknowledgments

The authors are grateful for the reviewers for the careful reading of the paper and for the suggestions which improved the quality of this work. They would like to thank the National Research University Project of Thailand's Office of the Higher Education Commission for financial support under NRU-CSEC project no. 54000267.

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