- Research Article
- Open Access

# Approximation of Common Fixed Points of a Countable Family of Relatively Nonexpansive Mappings

- Daruni Boonchari
^{1}and - Satit Saejung
^{2}Email author

**2010**:407651

https://doi.org/10.1155/2010/407651

© Boonchari and S. Saejung. 2010

**Received:**22 June 2009**Accepted:**21 November 2009**Published:**7 December 2009

## Abstract

We introduce two general iterative schemes for finding a common fixed point of a countable family of relatively nonexpansive mappings in a Banach space. Under suitable setting, we not only obtain several convergence theorems announced by many authors but also prove them under weaker assumptions. Applications to the problem of finding a common element of the fixed point set of a relatively nonexpansive mapping and the solution set of an equilibrium problem are also discussed.

## Keywords

- Banach Space
- Convex Function
- Equilibrium Problem
- Nonexpansive Mapping
- Real Banach Space

## 1. Introduction and Preliminaries

Let be a nonempty subset of a Banach space , and let be a mapping from into itself. When is a sequence in , we denote strong convergence of to by and weak convergence by . We also denote the weak convergence of a sequence to in the dual by . A point is an asymptotic fixed point of if there exists in such that and . We denote and by the set of fixed points and of asymptotic fixed points of , respectively. A Banach space is said to be strictly convex if for and . It is also said to be uniformly convex if for each , there exists such that for and . The space is said to be smooth if the limit

exists for all . It is also said to be uniformly smooth if the limit exists uniformly in .

Many problems in nonlinear analysis can be formulated as a problem of finding a fixed point of a certain mapping or a common fixed point of a family of mappings. This paper deals with a class of nonlinear mappings, so-called relatively nonexpansive mappings introduced by Matsushita and Takahashi [1]. This type of mappings is closely related to the resolvent of maximal monotone operators (see [2–4]).

Let be a smooth, strictly convex and reflexive Banach space and let be a nonempty closed convex subset of . Throughout this paper, we denote by the function defined by

where is the normalized duality mapping from to the dual space given by the following relation:

We know that if is smooth, strictly convex, and reflexive, then the duality mapping is single-valued, one-to-one, and onto. The duality mapping is said to be weakly sequentially continuous if implies that (see [5] for more details).

Following Matsushita and Takahashi [6], a mapping is said to be relatively nonexpansive if the following conditions are satisfied:

(R1) is nonempty;

(R2) for all , ;

(R3) .

If satisfies (R1) and (R2), then is called relatively quasi-nonexpansive [7]. Obviously, relative nonexpansiveness implies relative quasi-nonexpansiveness but the converse is not true. Relatively quasi-nonexpansive mappings are sometimes called hemirelatively nonexpansive mappings. But we do prefer the former name because in a Hilbert space setting, relatively quasi-nonexpansive mappings are nothing but quasi-nonexpansive.

In [2], Alber introduced the generalized projection from onto as follows:

If is a Hilbert space, then and becomes the metric projection of onto . Alber's generalized projection is an example of relatively nonexpansive mappings. For more example, see [1, 8].

In 2004, Masushita and Takahashi [1, 6] also proved weak and strong convergence theorems for finding a fixed point of a single relatively nonexpansive mapping. Several iterative methods, as a generalization of [1, 6], for finding a common fixed point of the family of relatively nonexpansive mappings have been further studied in [7, 9–14].

Recently, a problem of finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a relatively nonexpansive mapping is studied by Takahashi and Zembayashi in [15, 16]. The purpose of this paper is to introduce a new iterative scheme which unifies several ones studied by many authors and to deduce the corresponding convergence theorems under the weaker assumptions. More precisely, many restrictions as were the case in other papers are dropped away.

First, we start with some preliminaries which will be used throughout the paper.

Lemma 1.1 (see [7, Lemma ]).

Let be a nonempty closed convex subset of a strictly convex and smooth Banach space and let be a relatively quasi-nonexpansive mapping from into itself. Then is closed and convex.

Lemma 1.2 (see [17, Proposition ]).

for all and .

Lemma 1.3 (see [17]).

for all .

Lemma 1.4 (see [17, Proposition ]).

Let be a smooth and uniformly convex Banach space and let and be sequences of such that either or is bounded. If , then .

Lemma 1.5 (see [2]).

Lemma 1.6 (see [18]).

for all and .

We next prove the following three lemmas which are very useful for our main results.

Lemma 1.7.

Proof.

The proof of this lemma can be extracted from that of Lemma 1.8; so it is omitted.

If has a stronger assumption, we have the following lemma.

Lemma 1.8.

and , such that .

Proof.

Lemma 1.9.

for all and , such that . Then, the following hold:

(1) ,

(2) is relatively quasi-nonexpansive.

Proof.

- (2)
It follows directly from the above discussion.

## 2. Weak Convergence Theorem

Theorem 2.1.

for any , for all , such that for all . Then converges strongly to , where is the generalized projection of onto .

Proof.

Therefore exists. This implies that and are bounded for all .

Let . From (2.3) and , we have

Since is a convergent sequence, it follows from the properties of that is a Cauchy sequence. Since is closed, there exists such that .

We first establish weak convergence theorem for finding a common fixed point of a countable family of relatively quasi-nonexpansive mappings. Recall that, for a family of mappings with , we say that satisfies the NST-condition [19] if for each bounded sequence in ,

where denotes the set of all weak subsequential limits of a sequence .

Theorem 2.2.

for any , for all , such that for all , for all . If is weakly sequentially continuous, then converges weakly to , where .

Proof.

for all . Since is bounded, there exists a subsequence of such that . Since is relatively nonexpansive, for all .

We show that . Let

It follows from Lemma 1.4 that . From (2.19) and , we have . Since satisfies NST-condition, we have . Hence .

Let . From Lemma 1.5 and , we have

Since is strictly convex, . This implies that and hence .

We next apply our result for finding a common element of a fixed point set of a relatively nonexpansive mapping and the solution set of an equilibrium problem. This problem is extensively studied in [11, 14–16]. Let be a subset of a Banach space and let be a bifunction. The equilibrium problem for a bifunction is to find such that for all . The set of solutions above is denoted by , that is

To solve the equilibrium problem, we usually assume that a bifunction satisfies the following conditions ( is closed and convex):

(A1) for all ;

(A2) is monotone, that is, ;

(A3)for all , ;

(A4)for all , is convex and lower semicontinuous.

The following lemma gives a characterization of a solution of an equilibrium problem.

Lemma 2.3.

Let be a nonempty closed convex subset of a Banach space . Let be a bifunction from satisfying (A1)–(A4). Suppose that . Then if and only if for all .

Proof.

Let , then for all . From (A2), we get that for all .

Conversely, assume that for all . For any , let

Hence .

Takahashi and Zembayashi proved the following important result.

Lemma 2.4 (see [15, Lemma ]).

for all . Then, the following hold:

(1) is single-valued;

(3) ;

(4) is closed and convex.

We now deduce Takahashi and Zembayashi's recent result from Theorem 2.2.

Corollary 2.5 (see [15, Theorem ]).

for every , satisfying and for some . If is weakly sequentially continuous, then converges weakly to , where .

Proof.

Thus for all . From Lemma 2.3, we have . Then satisfies the NST-condition. From Theorem 2.2 where , converges weakly to , where .

Using the same proof as above, we have the following result.

Corollary 2.6 (see [11, Theorem ]).

Assume that , and are three sequences in satisfying the following restrictions:

(a) ;

(b) , ;

(c) for some .

If is weakly sequentially continuous, then converges weakly to , where .

The following result also follows from Theorem 2.2.

Corollary 2.7 (see [9, Theorem ]).

Then the following hold:

(1)the sequence is bounded and each weak subsequential limit of belongs to ;

(2)if the duality mapping from into is weakly sequentially continuous, then converges weakly to the strong limit of .

Proof.

Thus the conclusions of this corollary follow.

## 3. Strong Convergence Theorem

In this section, we prove strong convergence of an iterative sequence generated by the hybrid method in mathematical programming. We start with the following useful common tools.

Lemma 3.1.

where for all and satisfy for all , for all and . Then the following statements hold:

(1) for all ,

(2) ,

(3) ,

(4)if , then for all ,

(5)if , then and .

Proof.

( ) Let . Using (3.1) and the relative quasi-nonexpansiveness of each , we have

( ) Since

we have .

( ) Assume that . From , we get that . Since is uniformly norm-to-norm continuous on bounded sets, we have

for all , as desired.

( ) Assume that . From the assumption and ( ), we have

Hence and .

Lemma 3.2 (see [21, Lemma ]).

Let be a closed convex subset of a strictly convex, smooth and reflexive Banach space satisfying Kadec-Klee property. Let and be a sequence in such that and for all . Then .

Recall that a Banach space satisfies Kadec–Klee property if whenever is a sequence in with and , it follows that .

### 3.1. The CQ-Method

Theorem 3.3.

for every , for all and satisfying for all , for all . Then converges strongly to .

Proof.

The proof is broken into 4 steps.

Step 1 ( is well defined).

then is closed and convex. Thus is closed and convex.

We next show that . Let . Then, from Lemma 1.7,

Thus . Hence for all .

Next, we show by induction that for all . Since , we have

for all . Hence , and we also have . So, we have for all and hence the sequence is well defined.

Step 2 ( ).

Step 3 ( ).

and . It follows from NST-condition that .

Step 4 ( ).

From Steps 2 and 3, we have . The conclusion follows by Lemma 3.2 and (3.23).

We apply Theorem 3.3 and the proof of Corollary 2.5 and then obtain the following result.

Corollary 3.4.

for every , satisfying and for some . Then, converges strongly to , where is the generalized projection of onto .

Remark 3.5.

Corollary 3.4 improves the restriction on of [15, Theorem ]. In fact, it is assumed in [15, Theorem ] that .

### 3.2. The Monotone CQ-Method

Let be a closed subset of a Banach space . Recall that a mapping is closed if for each in , if and , then . A family of mappings with is said to satisfy the -condition if for each bounded sequence in ,

- (1)
If satisfies NST-condition, then satisfies -condition.

- (2)
If and is closed, then satisfies -condition.

Theorem 3.7.

for every , satisfying and for all . Then converges strongly to .

Proof.

Step 1 ( is well defined).

This step is almost the same as Step 1 of the proof of Theorem 3.3, so it is omitted.

Step 2 ( is a Cauchy sequence in ).

Since exists, we have that is a Cauchy sequence. Therefore, for some .

Step 3 ( ).

for all . Since each is closed, .

Step 4 ( ).

Let . From Lemma 3.1(2), we have and . It follows from -condition that .

Step 5 ( ).

From Steps 3 and 4, we have . The conclusion follows by Lemma 3.2 and (3.23).

Letting identity and yield the following result.

Corollary 3.8 (see [12, Theorem ]).

Then converges strongly to .

Letting identity and yield the following result.

Corollary 3.9 (see [13, Theorem ]).

with the conditions: with and

(1) ;

(2) ;

(3) for some .

Then converges strongly to .

Remark 3.10.

Using Theorem 3.7, we can show that the conclusion of Corollary 3.9 remains true under the more general restrictions on and :

(1) are arbitrary;

(2) and .

### 3.3. The Shrinking Projection Method

Theorem 3.11.

for every , for all and satisfies for all , for all . Then converges strongly to .

Proof.

The proof is almost the same as the proofs of Theorems 3.3 and 3.7; so it is omitted.

In particular, applying Theorem 3.11 gives the following result.

Corollary 3.12.

for every , where is the duality mapping on . Assume that satisfies and for some . Then converges strongly to , where is the generalized projection of onto .

Remark 3.13.

Corollary 3.12 improves the restriction on of [16, Theorem ]. In fact, it is assumed in [16, Theorem ] that .

Corollary 3.14 (see [11, Theorem ]).

Assume that , and are three sequences in satisfying the restrictions:

(a) ;

(b) , ;

(c) for some .

Then converges strongly to .

Remark 3.15.

The conclusion of Corollary 3.14 remains true under the more general assumption; that is, we can replace (b) by the following one:

(b′) and .

We also deduce the following result.

Corollary 3.16 (see [14, Theorem ]).

Assume that , and are three sequences in satisfying the following restrictions:

(a) ;

(b) for all and ;

(c) , ;

(d) for some .

Then and converge strongly to .

Remark 3.17.

The conclusion of Corollary 3.16 remains true under the more general restrictions; that is, we replace (b) and (c) by the following one:

(b′) and .

Corollary 3.18 (see [10, Theorem ]).

where satisfies the following restrictions:

(i) for all and ;

(ii) for all , for all . If

(a)either for all or

(b) and for all .

then the sequence converges strongly to .

Remark 3.19.

The conclusion of Corollary 3.18 remains true under the more general restrictions on :

(1) are arbitrary.

(2) for all .

## Declarations

### Acknowledgments

The authors would like to thank the referee for their comments on the manuscript. The first author is supported by grant fund under 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 Thailand Research Fund under Grant MRG5180146.

## Authors’ Affiliations

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