Abstract and Applied Analysis

Volume 2012 (2012), Article ID 437430, 13 pages

http://dx.doi.org/10.1155/2012/437430

## A New Approach to the Approximation of Common Fixed Points of an Infinite Family of Relatively Quasinonexpansive Mappings with Applications

College of Statistics and Mathematics, Yunnan University of Finance and Economics, Yunnan, Kunming 650221, China

Received 23 June 2012; Revised 16 August 2012; Accepted 21 August 2012

Academic Editor: Simeon Reich

Copyright © 2012 Wei-Qi Deng. 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

By using a specific way of choosing the indexes,
we propose an iteration algorithm generated by the monotone CQ method
for approximating common fixed points of an infinite family of relatively
quasinonexpansive mappings. A strong convergence theorem without the
stronger assumptions of the AKTT condition and the ^{∗}AKTT condition
imposed on the involved mappings is established in the framework of Banach
space. As application, an iterative solution to a system of equilibrium
problems is studied. The result is more applicable than those of other
authors with related interest.

#### 1. Introduction

Let be a nonempty and closed convex subset of a real Banach space . A mapping is said to be nonexpansive if A mapping is said to be quasi-nonexpansive if and It is easy to see that if is nonexpansive with , then it is quasi-nonexpansive. There are many methods for approximating fixed points of quasi-nonexpansive mappings. In 1953, Mann [1] introduced the iteration as follows: a sequence is defined by where the initial element is arbitrary and is a sequence of real numbers in . Approximation of fixed points of nonexpansive mappings via Mann's algorithm has extensively been investigated. One of the fundamental convergence results was proved by Reich [2]. In infinite-dimensional Hilbert spaces, Mann iteration can yield only weak convergence (see [3, 4]).

Attempts to modify the Mann iteration method (1.3) for strong convergence have recently been made. Nakajo and Takahashi [5] proposed the following modification of Mann iteration method (1.3) for a nonexpansive mapping from into itself in a Hilbert space: from an arbitrary , where denotes the metric projection from a Hilbert space onto a closed convex subset of . They proved that the sequence converges strongly to .

Recently, Su and Qin [6] introduced a monotone CQ method for nonexpansive mapping, defined as follows: from an arbitrary , and it proved that the sequence converges strongly to .

We now recall some definitions concerning relatively quasi-nonexpansive mappings. Let be a real smooth Banach space with norm and let be the dual of . The normalized duality mapping from to is defined by where denotes the pairing between and . Readers are directed to [7] (and its review [8]), where the properties of the duality mapping and several related topics are presented. The function is defined by

Let be a mapping from into . A point in is said to be an *asymptotic fixed point* [9] of if contains a sequence which converges weakly to and . The set of asymptotic fixed points of is denoted by .

We say that the mapping is *relatively nonexpansive* (see [10]) if the following conditions are satisfied:(R1); (R2), ;(R3).If satisfies (R1) and (R2), then is called *relatively quasi-nonexpansive*.

Several articles have provided methods for approximating fixed points of relatively quasi-nonexpansive mappings [11–16]. Employing the ideas of Su and Qin [6], and of Aoyama et al. [17], in 2008, Nilsrakoo and Saejung [18] used the following iterations to obtain strong convergence theorems for common fixed points of a countable family of relatively quasi-nonexpansive mappings in a Banach space However, the results were obtained under two stronger assumption conditions, namely, the -condition and the -condition imposed on the involved mappings.

Inspired and motivated by those studies mentioned above, in this paper, we use a modified type of the iteration scheme (1.8) for approximating common fixed points of an infinite family of relatively quasi-nonexpansive mappings; without stronger assumptions imposed on the involved mappings, a strong convergence theorem in Banach spaces is obtained for solving a system of equilibrium problems. The results improve those of other authors with related interest.

#### 2. Preliminaries

Throughout the paper, let be a real Banach space. We say that is *strictly convex* if the following implication holds for :
It is also said to be *uniformly convex* if for any , there exists a such that
It is known that if is uniformly convex Banach space, then is reflexive and strictly convex. A Banach space is said to be *smooth* if
exists for each . In this case, the norm of is said to be *Gâteaux differentiable*. The space is said to have *uniformly Gâteaux differentiable norm* if for each ; the limit (2.3) is attained uniformly for . The norm of is said to be *Fréchet differentiable* if for each ; the limit (2.3) is attained uniformly for . The norm of is said to be *uniformly Fréchet differentiable* (and is said to be *uniformly smooth*) if the limit (2.3) is attained uniformly for .

We also know the following properties (see, e.g., [19] for details). (1) (, resp.) is uniformly convex (, resp.) is uniformly smooth.(2) for each .(3)If is reflexive, then is a mapping from onto .(4)If is strictly convex, then as .(5)If is smooth, then is single-valued.(6)If has a Fréchet differentiable norm, then is norm-to-norm continuous.(7)If is uniformly smooth, then is uniformly norm-to-norm continuous on each bounded subset of .(8)If is a Hilbert space, then is the identity operator.

Let be a smooth Banach space. The function is defined by It is obvious from the definition of the function that Moreover, we know the following results.

Lemma 2.1 (see [13]). *Let be a strictly convex and smooth Banach space, then if and only if . *

Lemma 2.2 (see [11]). *Let be a uniformly convex and smooth Banach space and let . Then there exists a continuous, strictly increasing, and convex function such that and
**
for all . *

Let be a nonempty and closed convex subset of . Suppose that is reflexive, strictly convex, and smooth. It is known in [20] that for any , there exists a unique point such that
Following Alber [21], we denote such an by . The mapping is called the *generalized projection* from onto . It is easy to see that in a Hilbert space, the mapping coincides with the metric projection . What follows are the well-known facts concerning the generalized projection.

Lemma 2.3 (see [20]). *Let be a nonempty closed convex subset of a smooth Banach space and let . Then
*

Lemma 2.4 (see [20]). *Let be a reflexive, strictly convex, and smooth Banach space, let be a nonempty closed convex subset of , and let . Then
*

Dealing with the generalized projection from onto the fixed point set of a relatively quasi-nonexpansive mapping, we have the following result.

Lemma 2.5 (see [18]). *Let be a strictly convex and smooth Banach space, let be a nonempty and closed convex subset of , and let be a relatively quasi-nonexpansive mapping from into . Then is closed and convex. *

Let be a subset of a Banach space and let be a family of mappings from into . For a subset of , we say that (i) satisfies AKTT-condition if (ii) satisfies AKTT-condition if

#### 3. Main Results

Recall that an operator in a Banach space is *closed* if and as , then .

Theorem 3.1. *Let be a uniformly convex and uniformly smooth Banach space, a nonempty and closed convex subset of . Let be a sequence of closed and relatively quasi-nonexpansive mappings with . Starting from an arbitrary , the sequence is define by
**
where and is a sequence in with ; is the solution to the positive integer equation: , that is, for each , there exists a unique such that
**
Then converges strongly to . *

*Proof. *We first claim that both and are closed and convex. This follows from the fact that is equivalent to the following:

It is clear that . Next, we show that
Suppose that for some . Letting , we then have
This implies that . It follows from and Lemma 2.3 that
Particularly,
and hence , which yields that
By induction, (3.4) holds. This implies that is well defined. It follows from the definition of and Lemma 2.3 that . Since , we have
Therefore, is nondecreasing. Using and Lemma 2.4, we have
for all and for all , that is, is bounded. Then
In particular, by (2.5), the sequence is bounded. This implies that is bounded. Note again that and for any positive integer , . By Lemma 2.4,
By Lemma 2.2, we have, for any positive integers with ,
where is a continuous, strictly increasing, and convex function with . Then the properties of the function yield that is a Cauchy sequence in , so there exists an such that
In view of and the definition of , we also have
This implies that
It follows from Lemma 2.2 that
Since is uniformly norm-to-norm continuous on bounded sets, we have
On the other hand, we have, for each ,
and hence
From (3.18) and , we obtain that
Since is uniformly norm-to-norm continuous on bounded sets, we have
It follows from (3.17) that, as ,
Now, set for each . Note that whenever . For example, by the definition of , we have and . Then it follows from (3.23) that
Since is a subsequence of , (3.14) implies that as . It immediately follows from (3.24) and the closedness of that for each , and hence . Furthermore, by (3.10),
This implies that . The proof is completed.

*Remark 3.2. * Note that the algorithm (3.1) is based on the projection onto an intersection of two closed and convex sets. An example [22] of how to compute such a projection is given as follows.

*Dykstra's Algorithm *

Let be closed and convex subsets of . For any and , the sequences are defined by the following recursive formulae:
for with initial values and for . If , then converges to , where .

#### 4. Applications

The so-called *convex feasibility problem* for a family of mappings is to find a point in the nonempty intersection , which exactly illustrates the importance of finding fixed points of infinite families. The following example also clarifies the same thing.

*Example 4.1. *Let be a smooth, strictly convex, and reflexive Banach space, a nonempty and closed convex subset of , and a countable family of bifunctions satisfying the conditions: for each , ();() is monotone, that is, ;();() the mapping is convex and lower semicontinuous. A *system of equilibrium problems* for is to find an such that
whose set of common solutions is denoted by , where denotes the set of solutions to the equilibrium problem for . It will be shown in Theorem 4.3 that such a system of problems can be reduced to approximation of some fixed points of a countable family of nonexpansive mappings.

*Example 4.2 (see [23]). *Let . Define a countable family of mappings as follows:
Then we have that(1) is a sequence of single-valued mappings;(2) is a sequence of closed relatively quasi-nonexpansive mappings;(3).

Now, we have the following result.

Theorem 4.3. * Let , and be the same as those in Theorem 3.1. Let be a countable family of bifunctions satisfying the conditions ()–(). Let be a countable family of mappings defined by (4.2). Let be the sequence generated by
**
where satisfies the positive integer equation: . If , then strongly converges to which is a common solution of the system of equilibrium problems for .*

*Proof. *Since each is single-valued, for all . In addition, we have pointed out in Example 4.2 that and is a sequence of closed relatively quasi-nonexpansive mappings. Hence, (4.3) can be rewritten as follows:
Therefore, this conclusion can be obtained immediately from Theorem 3.1.

#### Acknowledgments

The author is greatly grateful to the referees for their useful suggestions by which the contents of this article are improved. This study was supported by the General Project of Scientific Research Foundation of Yunnan University of Finance and Economics (YC2012A02).

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