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Ergodic Retractions for Families of Asymptotically Nonexpansive Mappings
Fixed Point Theory and Applications volume 2010, Article number: 281362 (2010)
Abstract
We prove some theorems for the existence of ergodic retractions onto the set of common fixed points of a family of asymptotically nonexpansive mappings. Our results extend corresponding results of Benavides and RamÃrez (2001), and Li and Sims (2002).
1. Introduction
Let 
be a Banach space and 
a nonempty closed and convex subset of 
. We recall some definitions.
Definition 1.1.
A mapping 
is said to be
(i)nonexpansive if
(ii)asymptotically nonexpansive if there exists a sequence 
of positive numbers satisfying the property 
and
(iii)of asymptotically nonexpansive type if for each 
in 
, we have
(iv)weakly asymptotically nonexpansive if it satisfies the condition
(v)retraction if 
. A subset 
of 
is called a nonexpansive retract of 
if either 
or there exists a retraction of 
onto 
which is a nonexpansive mapping.
Definition 1.2.
We say that a nonempty closed convex subset 
of 
satisfies property 
with respect to
(i)a mapping 
if 
for every 
where
(ii)a semigroup of mappings 
if 
for every 
where
Obviously, 
itself verifies 
.
Definition 1.3.
-
(i)
A mapping
is said to satisfy the 
-fixed point property (
-fpp) if 
has a fixed point in every nonempty closed convex subset 
of 
which satisfies 
with respect to 
. -
(ii)
A semigroup
is said to satisfy the 
-fpp if 
has a common fixed point in every nonempty closed convex subset 
of 
which satisfies 
with respect to the semigroup 
. -
(iii)
A family
is said to satisfy the 
-fpp if 
has a common fixed point in every nonempty closed convex subset 
of 
which satisfies 
with respect to each 
.
is said to satisfy the 
-fixed point property (
-fpp) if 
has a fixed point in every nonempty closed convex subset 
of 
which satisfies 
with respect to 
.
is said to satisfy the 
-fpp if 
has a common fixed point in every nonempty closed convex subset 
of 
which satisfies 
with respect to the semigroup 
.
is said to satisfy the 
-fpp if 
has a common fixed point in every nonempty closed convex subset 
of 
which satisfies 
with respect to each 
.In 1965, Kirk [1] proved that if 
is a weakly compact convex subset of a Banach space with normal structure, then every nonexpansive mapping 
has a fixed point. (A nonempty convex subset 
of a normed linear space is said to have normal structure if each bounded convex subset 
of 
consisting of more than one point contains a nondiametral point). Goebel and Kirk [2] proved that if 
is assumed to be uniformly convex, then every asymptotically nonexpansive self-mapping 
of 
has a fixed point. This was extended to mappings of asymptotically nonexpansive type by Kirk in [3]. However, whether normal structure implies the existence of fixed points for mappings of asymptotically nonexpansive type is a natural and still open question. Li and Sims [4] proved the following fixed point result in the case that 
has uniform normal structure (It is known that a space with uniform normal structure is reflexive and that all uniformly convex or uniformly smooth Banach spaces have uniform normal structure).
Theorem 1.4.
Suppose 
is a Banach space with uniform normal structure; 
is a nonempty bounded subset of 
. Then
(i)every continuous and asymptotically nonexpansive type mapping 
satisfies 
-fpp;
(ii)every semigroup 
of asymptotically nonexpansive type mappings on 
such that 
is continuous on 
for each 
satisfies 
-fpp.
On the other hand, Bruck [5] initiated the study of the structure of the fixed point set 
in a general Banach space 
: if 
is a weakly compact convex subset of 
and 
is nonexpansive and satisfies a conditional fixed point property, then 
is a nonexpansive retract of 
. The same author [6] used this fact to derive the existence of fixed points for a commuting family of nonexpansive mappings. See, for example, [7, 8] for some related results.
Benavides and RamÃrez [9] studied the structure of the set of fixed points for (weakly) asymptotically nonexpansive mappings.
Theorem 1.5.
Let 
be a Banach space and 
a nonempty weakly compact convex subset of 
. Assume that every asymptotically nonexpansive self-mapping of 
satisfies the 
-fpp. Then for any commuting family 
of asymptotically nonexpansive self-mappings of 
, the common fixed point set of 
, 
, is a nonempty nonexpansive retract of 
.
In this paper, we prove some theorems to guarantee the existence of nonexpansive retractions onto the common fixed points of some families of (weakly) asymptotically nonexpansive (type) mappings. The results obtained in this paper extend in some sense, for example, Theorems 1.4 and 1.5, above.
2. Nonexpansive Retractions for Families of Weakly Asymptotically Nonexpansive Mappings
Theorem 2.1.
Let 
be a nonempty weakly compact convex subset of a Banach space 
, and 
a family of weakly asymptotically nonexpansive mappings on 
such that 
. Assume one of the following assumptions is satisfied:
(a)
satisfies the 
-fpp;
(b)
is a nonexpansive retract of 
.
Then for each 
, there exists a nonexpansive retraction 
from 
onto 
, the common fixed points of 
, such that 
, and every closed convex 
-invariant subset of 
is also 
-invariant.
Proof.
Consider 
with the product topology induced by the weak topology on 
. Now, consider an 
and define
By applying an argument similar to that in the proof of [9, Theorem 
], it follows that 
is compact (the topology on 
is that of weak pointwise convergence) and there is a minimal element 
in the following sense:
First, we assume the case (a). We shall prove that 
for all 
. For a given 
, consider the set 
Then 
is a nonempty weakly compact convex subset of 
, because 
is convex and compact. We will show that for all 
, 
satisfies property 
with respect to 
. Fix 
and take 
and 
such as
, for some 
. There exists 
such that 
. Consider a subnet 
of 
such that 
exists for every 
. Now, taking 
, we have 
. Since 
is nonexpansive, 
, and 
, it follows that 
and then 
. Thus 
satisfies the property 
with respect to 
. Since, 
satisfies the 
-fpp (by (a)), it follows that 
. So, there exists 
with 
. Let 
. Then 
, and by using the minimality of 
, we have
So, we get 
. Since this is so for each 
and 
belongs to 
, it follows that 
and 
.
Now, we assume the case (b). From (b), there is a nonexpansive retraction 
from 
onto 
. Put 
. Since 
, we can replace 
by 
in the above assertions to obtain a minimal element 
in the sense (*), where 
ia defined here as
We note that 
, (
). Since 
, every closed convex 
-invariant subset of 
is also 
-invariant and consequently 
-invariant, (
). So it is easy to see that 
, (
). Therefore, for every 
, the set 
is an 
-invariant subset of 
. So, considering the fact that 
, we obtain 
. Now, we can repeat the argument used in the last paragraph to get the desired result.
A nonexpansive retraction satisfying the thesis of Theorem 2.1 is usually called an ergodic retraction (see e.g., [10, 11]).
Combining Theorem 1.5 [9, Theorem 
] and Theorem 2.1(a), we get the following improvement of Theorem 1.5.
Corollary 2.2.
Let 
be a Banach space and 
a nonempty weakly compact convex subset of 
. Assume that every asymptotically nonexpansive self-mapping of 
satisfies 
-fpp. Then for any commuting family 
of asymptotically nonexpansive self-mappings of 
and for each 
, there exists a nonexpansive retraction 
from 
onto 
, such that 
, and every closed convex 
-invariant subset of 
is also 
-invariant.
3. Ergodic Retractions for a Semigroup of Asymptotically Nonexpansive Type
Assume that 
is a semigroup and 
is the space of all bounded real-valued functions defined on 
with supremum norm. For 
and 
, we define elements 
and 
in 
by 
and 
for each 
, respectively. An element 
of 
is said to be a mean on 
if 
. We often write 
instead of 
for 
and 
. A mean 
is said to be invariant if 
for each 
and 
. 
is said to be amenable if there is an invariant mean on 
. As is well known, 
is amenable when it is a commutative semigroup [12].
The following result which we need is well known (see [13]).
Lemma 3.1.
Let 
be a function of a semigroup 
into 
such that the weak closure of 
is weakly compact. Then, for any 
, there exists a unique element 
in 
such that 
for all 
. Moreover, if 
is a mean, then 
We can write 
by 
. As a direct consequence of Lemma 3.1, we have the following lemma.
Lemma 3.2.
Let 
be a nonempty closed convex subset of a Banach space 
, 
a semigroup of weakly asymptotically nonexpansive mappings on 
such that weak closure of 
is weakly compact for each 
, and 
a mean on 
.
If we write 
instead of 
, then the following hold.
(i)
for each 
.
(ii)
for each 
.
(iii)If 
is invariant, then 
for each 
and 
is a nonexpansive mapping from 
into itself.
Proof.
We only need to prove that 
is nonexpansive: consider 
and 
. Then for each 
, we have
Therefore, 
, for every 
. Consequently, we get
The following is our main result which is an improvement of Theorem 1.4 [4, Theorem 
].
Theorem 3.3.
Suppose 
is a Banach space with uniform normal structure; 
is a nonempty bounded closed and convex subset of 
; 
is a semigroup of asymptotically nonexpansive type mappings on 
such that 
is continuous on 
for each 
. Then there exists a nonexpansive retraction 
from 
onto 
, such that 
for each 
, and every closed convex 
-invariant subset of 
is also 
-invariant.
Proof.
Consider 
with the product topology induced by the weak topology on 
. Now, define
We note that 
, because the mapping 
in Lemma 3.2 belongs to 
. By applying an argument similar to that in the proof of [9, Theorem 
] (see also the proof of [7, Lemma 
]), it follows that 
is compact and there is a minimal element 
in the following sense:
We will prove that 
for all 
. For a given 
, consider the set 
Then 
is a nonempty weakly compact convex subset of 
, because 
is convex and compact. Take 
and 
such as
, for some
. There exists 
, such that 
. Consider a subnet 
of 
such that 
exists for every 
. Now, taking 
, we have 
. Since 
is nonexpansive, 
and 
for every 
, it follows that 
and then 
. Thus 
satisfies the property 
with respect to the semigroup 
. Now, from Theorem 1.4, it follows that 
. From this and the argument used in the proof of Theorem 2.1, we obtain 
. Since this holds for each 
, 
.
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Acknowledgments
The author would like to thank the referee for useful comments and for pointing out an oversight regarding an earlier draft of this paper. This paper is dedicated to Professor William Art Kirk. This research was in part supported by a Grant from IPM (no. 88470021).
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Saeidi, S. Ergodic Retractions for Families of Asymptotically Nonexpansive Mappings. Fixed Point Theory Appl 2010, 281362 (2010). https://doi.org/10.1155/2010/281362
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DOI: https://doi.org/10.1155/2010/281362