# On the Fixed-Point Set of a Family of Relatively Nonexpansive and Generalized Nonexpansive Mappings

- Weerayuth Nilsrakoo
^{1}and - Satit Saejung
^{2, 3}Email author

**2010**:414232

**DOI: **10.1155/2010/414232

© W. Nilsrakoo and S. Saejung. 2010

**Received: **13 November 2009

**Accepted: **18 January 2010

**Published: **10 March 2010

## Abstract

We prove that the set of common fixed points of a given countable family of relatively nonexpansive mappings is identical to the fixed-point set of a single strongly relatively nonexpansive mapping. This answers Kohsaka and Takahashi's question in positive. We also introduce the concept of strongly generalized nonexpansive mappings and prove the analogue version of the result above for Ibaraki-Takahashi's generalized nonexpansive mappings. The duality theorem for two classes of strongly relatively nonexpansive mappings and of strongly generalized nonexpansive mappings is proved.

## 1. Introduction

Let
be a subset of a Banach space
. A mapping
is *nonexpansive* if
for all
In this paper, the fixed-point set of the mapping
is denoted by
, that is,
In 1973, Bruck [1] proved that for a given countable family of nonexpansive mappings in a strictly convex Banach space there exists a single nonexpansive mapping whose fixed-point set is identical to the set of common fixed points of the family. More precisely, the following is obtained.

Theorem 1.1.

Then is nonexpansive and

Recall that
is *strictly convex* if whenever
and
are norm-one elements in
satisfying
it follows that
It is worth mentioning that Bruck's result above remains true for the class of quasi-nonexpansive mappings, that is, the set of common fixed points of a countable family of quasi-nonexpansive mappings is identical to the fixed-point set of a single quasi-nonexpansive mapping. A mapping
:
is *quasi-nonexpansive* if
and
for all
and

In 2004, Matsushita and Takahashi [2–4] introduced the so-called relatively nonexpansive mappings in Banach spaces. This class of mappings includes the resolvent of a maximal monotone operator and Alber's generalized projection. For more examples, we refer to [2–6]. Recently, Kohsaka and Takahashi [7] proved an analogue version of Bruck's result for a family of relatively nonexpansive mappings and they asked the following question.

Question 1.

For a given countable family of relatively nonexpansive mappings, is there a single *strongly relatively nonexpansive mapping* such that its fixed-point set is identical to the set of common fixed points of the family?

A positive answer to this question is given in [7] for a finite family of mappings. The purpose of this paper is to give the answer of Kohsaka and Takahashi's question in positive. We also introduce a concept of strongly generalized nonexpansive mappings and present the analogue version of the result above for Ibaraki-Takahashi's generalized nonexpansive mappings. Finally, inspired by [8], we prove the duality theorem for two classes of strongly relatively nonexpansive mappings and of strongly generalized nonexpansive mappings.

## 2. Preliminaries

We collect together some definitions and preliminaries which are needed in this paper. The strong and weak convergences of a sequence
in a Banach space
to an element
are denoted by
and
, respectively. A Banach space
is *uniformly convex* if whenever
and
are sequences in
satisfying
and
it follows that
It is known that if
is uniformly convex, then it is reflexive and strictly convex. We say that
is *uniformly smooth* if the dual space
of
is uniformly convex. A Banach space
is *smooth* if the limit
exists for all norm-one elements
and
in
. It is not hard to show that if
is reflexive, then
is smooth if and only if
is strictly convex. The value of
at
is denoted by
The *duality mapping*
is defined by

for all The following facts are known (e.g., see [9, 10]).

(a)If is smooth, then is single valued.

(b)If is strictly convex, then is one-to-one, that is, implies that .

(c)If is reflexive, then is onto.

(d) If is uniformly smooth, then is uniformly norm-to-norm continuous on each bounded subset of .

For a smooth Banach space , Alber [5] considered the functional defined by

Using this functional, Matsushita and Takahashi [2–4] studied and investigated the following mappings in Banach spaces. Suppose that
is a subset of a smooth Banach space
. A mapping
is *relatively nonexpansive* if the following properties are satisfied.

(R1)

(R2) for all and

(R3) is demiclosed at zero, that is; whenever a sequence in converges weakly to and converges strongly to it follows that

In a Hilbert space , the duality mapping is an identity mapping and for all . Hence, if is relatively nonexpansive, then it is quasi-nonexpansive and is demiclosed at zero.

Recently, Kohsaka and Takahashi [7] proved an analogue version of Bruck's result for a family of relatively nonexpansive mappings. More precisely, they obtained the following.

Theorem 2.1 (see [7, Theorem 3.4]).

Then is strongly relatively nonexpansive and

Recall that a relatively nonexpansive mapping
[6] is *strongly relatively nonexpansive* if whenever
is a bounded sequence in
such that
for some
it follows that

To obtain the result for a countable family of relatively nonexpansive mappings, the same authors proved the following result.

Theorem 2.2 (see [7, Theorem 3.3]).

Then is relatively nonexpansive and .

Remark 2.3.

They also asked the question of whether the mapping in Theorem 2.2 is strongly relatively nonexpansive (see [7, Problem 3.5]).

The following lemmas are needed in proving the result.

Lemma 2.4 (see [11, Theorem 2]).

for all and

Lemma 2.5.

for all with and .

Proof.

This completes the proof.

Lemma 2.6 (see [12, Lemma 2.10]).

then is a constant sequence.

Lemma 2.7 (see [13, Proposition 2]).

Let be a smooth and uniformly convex Banach space. Suppose that either or is a bounded sequence in and Then

## 3. Relatively Nonexpansive Mappings and Quasi-Nonexpansive Mappings

We first start with some observation which is a tool for proving Theorem 3.2.

Theorem 3.1.

Let be a bounded sequence in Then the following are equivalent.

(a)

(b) for each .

In particular,

Proof.

In particular, for all and Hence, for each the series converges (absolutely). This implies that the mapping is well defined.

Assertion follows immediately from (3.12) and (3.14).

Conversely, we assume that for each Since is uniformly norm-to-norm continuous on each bounded set,

By the uniform norm-to-norm continuity of on each bounded set, we can conclude assertion from (3.16). This completes the proof.

Theorem 3.2.

Then is relatively nonexpansive and

Proof.

To show that is relatively nonexpansive, we prove only that is demiclosed at zero. Suppose that is a sequence in such that and . From Theorem 3.1, we have for each . Since each is demiclosed at zero, . Consequently, as desired.

We now give an answer of Kohsaka and Takahashi's question in positive.

Theorem 3.3.

The mapping in Theorem 2.2 is strongly relatively nonexpansive.

Proof.

is strongly relatively nonexpansive.

Using the same idea as in Theorem 3.1, we also have the following result whose proof is left to the reader to verify.

Theorem 3.4.

Then is demi-closed at zero if and only if each mapping is demi-closed at zero.

## 4. Ibaraki-Takahashi's Generalized Nonexpansive Mappings

Let
be a subset of a smooth Banach space
. In 2007, Ibaraki and Takahashi [14] introduced the following mapping. A mapping
is *generalized nonexpansive* if the following properties are satisfied:

(G1)

(G2) for all and .

A mapping
satisfies *property (G3)* if whenever
is a sequence in
such that
and
it follows that
Here
denotes the
convergence in the dual space.

The generalized resolvent of the maximal monotone operator where is a smooth and uniformly convex Banach space, and the sunny generalized nonexpansive retraction from a strictly convex, smooth, and reflexive Banach space onto its closed subset are examples of generalized nonexpansive mappings satisfying property (G3) (see [15]). The relation between two classes of relatively nonexpansive mappings and of generalized nonexpansive mappings is recently obtained in [8].

The property of the mapping and the demiclosedness of are related as shown in the following remark.

Remark 4.1.

Let be a subset of a smooth Banach space and Then the following assertions hold true.

If is uniformly smooth, the duality mapping is weakly sequentially continuous, and satisfies property then is demiclosed at zero.

If is uniformly convex, is weakly sequentially continuous, and is demi-closed at zero, then satisfies property

Theorem 4.2.

Then the mapping is well defined and the following assertions hold true.

If is strictly convex, then and is generalized nonexpansive.

If is uniformly convex and is a bounded sequence in then the following statements are equivalent:

(a)

(b) for each

The mapping is demi-closed at zero if and only if each mapping is demi-closed at zero.

Suppose that is uniformly convex and uniformly smooth. Then the mapping satisfies property if and only if each mapping satisfies property

Proof.

It follows from Lemma 2.6 that is a constant sequence, and hence for all This implies that , that is, . Now Again, using the convexity of we can show that satisfies property and hence it is generalized nonexpansive, as desired.

(ii)Since the proof of this assertion is very similar to that of Theorem 3.1, it is omitted.

(iii) and (iv) follow directly from .

Remark 4.3.

Theorem 4.2 generalizes [16, Theorem 3.3] from a finite family to a countable one.

Following Reich [6], we introduced the following concept. A generalized nonexpansive mapping
is *strongly generalized nonexpansive* if whenever
is a bounded sequence in
such that
for some
it follows that
.

Lemma 4.4.

Then . If, in addition, is uniformly convex, then is strongly generalized nonexpansive.

Proof.

Consequently, and hence . This implies that . Since is strongly generalized nonexpansive, It follows from Lemma 2.7 that and hence This implies that and is strongly generalized nonexpansive, as desired.

The following is an analogue version of Kohsaka and Takahashi's question for a countable family of generalized nonexpansive mappings.

Theorem 4.5.

Let be a closed convex subset of a smooth and uniformly convex Banach space and let be a countable family of generalized nonexpansive mappings such that Then there exists a strongly generalized nonexpansive mapping such that

Proof.

Notice that is generalized nonexpansive and by Theorem 4.2 Moreover, by Lemma 4.4 and the fact that the identity is strongly generalized nonexpansive, the conclusion is satisfied by the mapping

## 5. Duality between Strongly Relatively Nonexpansive Mappings and Strongly Generalized Nonexpansive Mappings

Let
be a subset of a smooth, strictly convex and reflexive Banach space
and let
be a mapping. We can define the *duality*
of
by (see [8])

We now consider a functional from into still denoted by , by

where is the duality mapping from onto It is clear that Then, whenever are elements in and are elements in satisfying and it follows that

Remark 5.1.

The following assertions hold (see [8]).

If and then In particular, Moreover, if is a sequence in and then

(i) if and only if

(ii) if and only if

If and then

The following duality theorem is proved in [8].

Theorem 5.2.

Let be a subset of a smooth, strictly convex and reflexive Banach space and let be a mapping. Suppose that is the duality of . Then the following assertions hold true.

(1)If is relatively nonexpansive, then is generalized nonexpansive with property

(2) If is generalized nonexpansive with property then is relatively nonexpansive.

We now prove the duality theorem for strongly relatively nonexpansive mappings and strongly generalized nonexpansive mappings.

Theorem 5.3.

Let be a subset of a smooth, strictly convex and reflexive Banach space and let : be a mapping. Suppose that : is the duality of . Then the following assertions hold true.

(1)If is strongly relatively nonexpansive, then is strongly generalized nonexpansive with property

(2)If is strongly generalized nonexpansive with property then is strongly relatively nonexpansive.

Proof.

We prove only and leave for the reader to verify. Suppose that is a bounded sequence in such that for some We assume that is a sequence in such that and is a point in such that Clearly, is bounded. Moreover, by Remark 5.1, we have and Consequently, It follows from the strongly relative nonexpansiveness that This completes the proof.

## Declarations

### Acknowledgment

The corresponding author was supported by the Centre of Excellence in Mathematics, the Commission on Higher Education of Thailand.

## Authors’ Affiliations

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