Research Article | Open Access

# Hybrid Iteration Method for Common Fixed Points of a Finite Family of Nonexpansive Mappings in Banach Spaces

**Academic Editor:**Joaquim J. Júdice

#### Abstract

Let be a real uniformly convex Banach space, and let be nonexpansive mappings from into itself with , where . From an arbitrary initial point , hybrid iteration scheme is defined as follows: , , where is an -Lipschitzian mapping, , , , , , and for some . Under some suitable conditions, the strong and weak convergence theorems of to a common fixed point of the mappings are obtained. The results presented in this paper extend and improve the results of Wang (2007) and partially improve the results of Osilike, Isiogugu, and Nwokoro (2007).

#### 1. Introduction

Let be a Banach space endowed with the norm . A mapping is said to be nonexpansive if for any . is said to be -Lipschitzian if there exists constant such that for any .

Let be a Hilbert space with inner product and associated with norm , is said to be -strong monotone if there exists such that

The interest and importance of construction of fixed points of nonexpansive mappings stem mainly from the fact that it may be applied in many areas, such as imagine recovery and signal processing (see, e.g., [1–3]). Especially, numerous problems in physics, optimization, economics, traffic analysis, and mechanics reduce to find a solution of equilibrium problem. The equilibrium problem is to find

where is a nonempty closed convex subset of a Hilbert space , is a bifunction from to , and is the set of real numbers.

It has been shown by Blum and Oettli [4] and Noor and Oettli [5] that variational inequalities and mathematical programming problems can be viewed as a special realization of the abstract equilibrium problems. Given a mapping , let for all . It is well-known that is a solution of (1.2) if and only if for all . Very recently, Yao et al. [6] find a common element of the set of solutions of equilibrium problem (1.2) and the set of common fixed points of a finite family of nonexpansive mappings by using an iterative scheme of a finite family of nonexpansive mappings. See the references therein for more details. Therefore, the topic on construction of fixed points of nonexpansive mappings is useful for equilibrium problems in physics, optimization, traffic analysis, and so forth.

Motivated by earlier results of Xu and Kim [7] and Yamada [8], some authors [9–14] further extended hybrid iteration method used this method to approximate fixed points of nonexpansive mappings, and obtained some strong and weak convergence theorems for nonexpansive mappings.

Recently, Wang [12] introduced an explicit hybrid iteration method for nonexpansive mappings and obtained the following convergence theorem.

Theorem 1.1 ([12]). *Let be a Hilbert space, let be a nonexpansive mapping with , and let be a -strong monotone and mapping. For any given , is defined by
**
where . If and satisfy the following conditions: (1) for some ; (2) ; (3) , then,*(1)* converges weakly to a fixed point of .*(2)* converges strongly to a fixed point of if only if .*

Very recently, Osilike et al. [11] extended Wang's results to arbitrary Banach spaces without the strong monotonicity assumption imposed on the hybrid operator and obtained the following result.

Theorem 1.2 ([11]). *Let be an arbitrary Banach space endowed with the norm , let be a nonexpansive mapping with , and let be an mapping. Let be the sequence generated from an arbitrary by
**
where , and and satisfy the following conditions: (1) for all and some ; (2) ; (3) , then,*(1)* exists for each ,*(2)*,*(3)* converges strongly to a fixed point of if and only if .*

Motivated by above work, we obtain the strong and weak convergence theorems for a finite family of nonexpansive mappings in uniformly convex Banach space by using hybrid iteration method. The results presented in this paper extend and improve the results of Wang [12] and partially improve the results of Osilike et al. [11].

#### 2. Preliminaries

Throughout this paper, we denote .

A mapping is said to be demicompact if, for any sequence in such that there exists subsequence of such that converges strongly to

For studying the strong convergence of fixed points of a nonexpansive mapping, Senter and Dotson [15] introduced condition . Later on, Maiti and Ghosh [16], Tan and Xu [17] studied condition and pointed out that Condition is weaker than the requirement of demicompactness for nonexpansive mappings. A mapping with is said to satisfy condition if there exists a nondecreasing function with and for all such that for all , where .

A family of mappings from into itself with is said to satisfy condition (B) if there exists a nondecreasing function with and for all such that for all .

A Banach space is said to satisfy Opial's condition if, for any sequence in , implies that for all with , where denotes that converges weakly to .

A mapping with domain and range in is said to be demi-closed at if whenever is a sequence in such that converges weakly to and converges strongly to , then .

In the coming Lemma we will use the following well-known results.

Lemma 2.1 ([18]). *Let and be two nonnegative sequences satisfying
**
If and then exists.*

Lemma 2.2 (see [19]). *Let be a real uniformly convex Banach space and let , be two constant with . Suppose that is a real sequence and , are two sequences in . Then the conditions
**
imply that where is a constant.*

Lemma 2.3 (see [20]). *Let be a real uniformly convex Banach space, let be a nonempty closed convex subset of , and let be a nonexpansive mapping. Then is demiclosed at zero.*

#### 3. Main Results

Theorem 3.1. *Let be a real uniformly convex Banach space endowed with the norm , let , be nonexpansive mappings from into itself with , and let be an -Lipschitzian mapping. For any given , is defined by
**
where , , , . If and satisfy the following conditions:*(1)* for all and some ,*(2)*, **then *(1)* exists for each *(2)* for each , *(3)* converges strongly to a common fixed point of if and only if .*

*Proof. * (1) For any , we have
Since , it follows from Lemma 2.1 that exists.

(2) Since exists for any , is bounded. So are and . Thus we may assume that , that is,
Since , and
we have
Thus, it follows from Lemma 2.2 that

In addition,
so
Therefore, it follows from (3.6) that

On the other hand, since
we have
Thus, it follows from (3.9) that
From (3.9) and (3.12), we can obtain
Further, for any positive integer , we also have

For each ,
It follows from (3.9) and (3.14) that . This implies that for each .

(3) Suppose that converges strongly to a common fixed point of the mappings , then Since , we have

Conversely, suppose that Since is bounded, there exists constant such that . From (3.2), for any , we obtain
where Furthermore, we have
It follows from Lemma 2.1 that exists. Since , we obtain that . We now show that is a Cauchy sequence.

For arbitrary , there exists positive integer such that for all . In addition, since , there exists positive integer such that for all . Taking , for any , from (3.16), we have
Taking the infimum in above inequalities for all , we obtain
This implies that is a Cauchy sequence. Therefore there exists such that converges strongly to . Since for each , it follows from Lemma 2.3 that . This completes the proof.

From Lemma 2.3 and for each , using routine method, we can easily show the following weak convergence theorem, whose proof is omitted.

Theorem 3.2. *Let be a real uniformly convex Banach space satisfying Opial's condition, let be nonexpansive mappings from into itself with and let be an -Lipschitzian mapping. For any given , is defined as in Theorem 3.1, and and satisfy the conditions appeared in Theorem 3.1. Then converges weakly to a common fixed point of the mappings of .*

*Example 3.3. *Let be endowed with standard norm , where is real number set. Define and by and for all , respectively. Obviously, and are nonexpansive mappings, and is a common fixed point of and . Let be defined by for all . We now chose parameters , and as follows:
It is easy to see that is a -Lipschitzian mapping and , , and satisfy the conditions of Theorem 3.2. Then is generated by
where and . It follows from Theorem 3.2 that converges strongly to the common fixed point of and . As , by using Mathematical 5.0 to compute , we know that , , and . This example shows that the algorithm is efficient for approximating common fixed points of nonexpansive mappings.

*Remark 3.4. *By using Theorem 3.1 and Lemma 2.3, we can easily prove that converges strongly to a common fixed point of the mappings of if one of the mappings is demicompact or satisfies condition (B). Therefore the results presented in this paper improve and extend the results of Wang [12] and partially improve the results of Osilike et al. [11].

*Remark 3.5. *We do not know how to overcome the constraint condition when we try to extend Theorem 3.1 to arbitrary Banach spaces.

#### Acknowledgments

The authors are extremely grateful to the referees for their useful suggestions that improved the content of the paper. This work was supported by Scientific Research Foundation of Yunnan University of Finance and Economics.

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#### Copyright

Copyright © 2009 Lin Wang et al. 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.