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On the Fixed-Point Set of a Family of Relatively Nonexpansive and Generalized Nonexpansive Mappings
Fixed Point Theory and Applications volume 2010, Article number: 414232 (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.
Let be a closed convex subset of a strictly convex Banach space
and let
be a sequence of nonexpansive mappings such that
Suppose that
is a sequence in
such that
and
is defined by

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]).
Let be a closed convex subset of a uniformly convex and uniformly smooth Banach space
and let
be a finite family of relatively nonexpansive mappings such that
. Suppose that
and
are finite sequences such that
and
is defined by

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]).
Let and
be as in Theorem 2.1 and let
be a sequence of relatively nonexpansive mappings such that
. Suppose that
and
are sequences such that
and
is defined by

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]).
Let be a uniformly convex Banach space and let
Then there exists a strictly increasing, continuous and convex function
such that
and

for all and
Lemma 2.5.
Let be a uniformly convex Banach space and let
. Then there exists a strictly increasing, continuous and convex function
such that
and

for all with
and
.
Proof.
We note that both series and
converge. For
let
be a function satisfying the properties of Lemma 2.4. Using the convexity of
, we have

This completes the proof.
Lemma 2.6 (see [12, Lemma  2.10]).
Let be a strictly convex Banach space and let
with
If
is a sequence in
such that both series
and
converge, and

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 closed convex subset of a uniformly convex and uniformly smooth Banach space
and let
be a sequence of mappings such that
and

Suppose that is a sequence in
such that
and
is defined by

Let be a bounded sequence in
Then the following are equivalent.
(a)
(b) for each
.
In particular,
Proof.
For fixed and
we have

In particular, for all
and
Hence, for each
the series
converges (absolutely). This implies that the mapping
is well defined.
Let be a bounded sequence in
Suppose that

By the boundedness of we put

Then for all
We now consider the following estimates for each
such that
and for any
:

where is the function given in Lemma 2.5 associated with the uniform convexity of
and the number
. Notice that
. Consequently, for
,

This implies that

We next prove that

Let be given. We choose an integer
such that
Since
as
for all
, we now choose an integer
such that

for all and
Then, if

This implies that (3.9) holds. In particular, since is uniformly norm-to-norm continuous on each bounded set, we can conclude from (3.8) that

and from (3.9) that

This together with (3.4) gives

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,

We show that

Let Then there exist positive integers
such that
and

for all and
If
, then

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.
Let be a closed convex subset of a uniformly convex and uniformly smooth Banach space
and let
be a countable family of relatively nonexpansive mappings such that
Suppose that
is a sequence in
such that
and
is defined by

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.
The mapping can be rewritten as

where is the identity mapping,
for all
, and
It follows from Theorem 3.2 that
is relatively nonexpansive, where
Consequently, by Theorem 2.1 with
the mapping

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.
Let be a closed convex subset of a uniformly convex Banach space
and let
be a sequence of quasi-nonexpansive mappings such that
. Suppose that
is a sequence in
such that
and
is defined by

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.
Let be a closed convex subset of a smooth Banach space
and let
be a sequence of generalized nonexpansive mappings such that
Suppose that
is a sequence in
such that
and
is defined by

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.
Using some basic properties of the functional , we have
for all
,
Since
the sequence
is bounded for each
and, hence the series
converges (absolutely). This implies that
is well defined. For fixed
and
we have the following expressions:

(i)The inclusion is obvious. To see the reverse inclusion, let
By the convexity of
and the expressions of (4.2), we have

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.
Let be a closed convex subset of a strictly convex and smooth Banach space
. Suppose that
is a generalized nonexpansive mapping and a strongly generalized nonexpansive mapping, respectively, and suppose that
For
let the mapping
be defined by

Then . If, in addition,
is uniformly convex, then
is strongly generalized nonexpansive.
Proof.
The first assertion follows from Theorem 4.2 We now assume that
is uniformly convex. Suppose that
is a bounded sequence in
such that
for some
It is clear that the sequences
and
are both bounded. By the uniform convexity of
, we have

where is a function given by Lemma 2.4. Since
and
are generalized nonexpansive,

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.
Suppose that is a sequence in
such that
and
We define
by

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.
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Acknowledgment
The corresponding author was supported by the Centre of Excellence in Mathematics, the Commission on Higher Education of Thailand.
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Nilsrakoo, W., Saejung, S. On the Fixed-Point Set of a Family of Relatively Nonexpansive and Generalized Nonexpansive Mappings. Fixed Point Theory Appl 2010, 414232 (2010). https://doi.org/10.1155/2010/414232
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DOI: https://doi.org/10.1155/2010/414232
Keywords
- Banach Space
- Convex Function
- Nonexpansive Mapping
- Bounded Sequence
- Duality Theorem