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Fixed Point Theorems for Suzuki Generalized Nonexpansive Multivalued Mappings in Banach Spaces
Fixed Point Theory and Applications volume 2010, Article number: 457935 (2010)
Abstract
In the first part of this paper, we prove the existence of common fixed points for a commuting pair consisting of a single-valued and a multivalued mapping both satisfying the Suzuki condition in a uniformly convex Banach space. In this way, we generalize the result of Dhompongsa et al. (2006). In the second part of this paper, we prove a fixed point theorem for upper semicontinuous mappings satisfying the Suzuki condition in strictly spaces; our result generalizes a recent result of DomÃnguez-Benavides et al. (2009).
1. Introduction
A mapping on a subset
of a Banach space
is said to be nonexpansive if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ1_HTML.gif)
In 2008, Suzuki [1] introduced a condition which is weaker than nonexpansiveness. Suzuki's condition which was named by him the condition reads as follows: a mapping
is said to satisfy the condition
on
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ2_HTML.gif)
He then proved some fixed point and convergence theorems for such mappings. We shall at times refer to this concept by saying that is a generalized nonexpansive mapping in the sense of Suzuki. Very recently, the current authors used a modified Suzuki condition for multivalued mappings and proved a fixed point theorem for multivalued mappings satisfying this condition in uniformly convex Banach spaces (see [2]).
In this paper, we first present a common fixed point theorem for commuting pairs consisting of a single-valued and a multivalued mapping both satisfying the Suzuki condition. This result extends a result of Dhompongsa et al. [3].
In the next part, we shall consider a recent result of DomÃnguez-Benavides et al. [4] on the existence of fixed points in an important class of spaces which are usually called strictly spaces. These spaces contain all Lebesgue function spaces
for
. In this paper, we also generalize results of DomÃnguez-Benavides et al. [4] to upper semicontinuous mappings satisfying the Suzuki condition.
2. Preliminaries
Given a mapping on a subset
of a Banach space
, the set of its fixed points will be denoted by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ3_HTML.gif)
We start by the following definition due to Suzuki.
Definition 2.1 (see [1]).
Let be a mapping on a subset
of a Banach space
. The mapping
is said to satisfy the Suzuki condition
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ4_HTML.gif)
As the following example shows, the Suzuki condition is weaker than nonexpansiveness. Therefore, it is natural to call these mappings as "generalized nonexpansive mappings". However, we shall at times refer to these mappings as those satisfying the condition .
Example 2.2.
Let be equipped with the usual metric
, and let
. We put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ5_HTML.gif)
The mapping is continuous and satisfies the condition
. However,
is not nonexpansive.
Lemma 2.3 (see [1, Lemma ]).
Let be a mapping defined on a closed subset
of a Banach space
. Assume that
satisfies the condition
. Then
is closed. Moreover, if
is strictly convex and
is convex, then
is also convex.
Theorem 2.4 (see [5, Theorem ]).
Let be a nonempty bounded closed convex subset of a uniformly convex Banach space
. Let
be a mapping satisfying the condition (
). Then
has a fixed point.
Let be a metric space. We denote by
the collection of all nonempty closed bounded subsets of
; we also write
to denote the collection of all nonempty compact convex subsets of
. Let
be the Hausdorff metric with respect to
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ6_HTML.gif)
for all where
Let be a multivalued mapping. An element
is said to be a fixed point of
provided that
Definition 2.5.
A multivalued mapping is said to be nonexpansive provided that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ7_HTML.gif)
Suzuki's condition can be modified to incorporate multivalued mappings. This was done by the current authors in [2]. We call these mappings generalized multivalued nonexpansive mappings in the sense of Suzuki or multivalued mappings satisfying the condition . We now state Suzuki's condition for multivalued mappings as follows.
Definition 2.6.
A multivalued mapping is said to satisfy the condition
provided that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ8_HTML.gif)
Example 2.7.
Define a mapping on
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ9_HTML.gif)
It is not difficult to see that satisfies the Suzuki condition; however,
is not nonexpansive.
The following lemma, proved by Goebel and Kirk [6], plays an important role in the coming discussions.
Lemma 2.8.
Let and
be two bounded sequences in a Banach space
, and let
. If for every natural number
we have
and
, then
.
Definition 2.9.
A multivalued mapping is said to be upper semicontinuous on
if
is open in
whenever
is open.
We recall that if is single valued, then
reduces to a continuous function.
3. Fixed Points in Uniformly Convex Banach Spaces
Let be a nonempty closed convex subset of a Banach space
. Assume that
is a bounded sequence in
. For each
, the asymptotic radius of
at
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ10_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ11_HTML.gif)
The number is known as the asymptotic radius of
relative to
. Similarly, the set
is called the asymptotic center of
relative to
. In the case that
is a reflexive Banach space and
is a nonempty closed convex bounded subset of
, the set
is always a nonempty closed convex subset of
. To see this, observe that by the definition of
, for each
, the set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ12_HTML.gif)
is nonempty. It is not difficult to see that each is closed and convex; hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ13_HTML.gif)
is closed and convex. Moreover, it follows from the weak compactness of that
is nonempty. It is easy to see that if
is uniformly convex and if
is a closed convex subset of
, then
consists of exactly one point.
A bounded sequence is said to be regular with respect to
if for every subsequence
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ14_HTML.gif)
It is also known that if is uniformly convex and if
is a nonempty closed convex subset of
, then for any
, there exists a unique point
such that
.
The following lemma was proved by Goebel and Lim.
Let be a bounded sequence in
and let
be a nonempty closed convex subset of
. Then
has a subsequence which is regular relative to
.
Definition 3.2.
Let be a nonempty closed convex bounded subset of a Banach space
, and let
and
be two mappings. Then
and
are said to be commuting mappings if for every
such that
and
, we have
.
Now the time is ripe to state and prove the main result of this section.
Theorem 3.3.
Let be a nonempty closed convex bounded subset of a uniformly convex Banach space
. Let
be a single-valued mapping, and let
be a multivalued mapping. If both
and
satisfy the condition (
) and if
and
are commuting, then they have a common fixed point, that is, there exists a point
such that
.
Proof.
By Theorem 2.4, the mapping has a nonempty fixed point set
which is a closed convex subset of
(by Lemma 2.3). We show that for
,
. To see this, let
; since
and
are commuting, we have
for each
. Therefore,
is invariant under
for each
. Since
is a bounded closed convex subset of the uniformly convex Banach space
, we conclude that
has a fixed point in
and therefore,
for
Now we find an approximate fixed point sequence for in
. Take
, since
, therefore, we can choose
. Define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ15_HTML.gif)
Since is a convex set, we have
. Let
be chosen in such a way that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ16_HTML.gif)
We see that Indeed, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ17_HTML.gif)
Since satisfies the condition
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ18_HTML.gif)
which contradicts the uniqueness of as the unique nearest point of
(note that
). Similarly, put
; again we choose
in such a way that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ19_HTML.gif)
By the same argument, we get In this way, we will find a sequence
in
such that
where
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ20_HTML.gif)
Therefore, for every natural number we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ21_HTML.gif)
from which it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ22_HTML.gif)
Our assumption now gives
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ23_HTML.gif)
hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ24_HTML.gif)
We now apply Lemma 2.8 to conclude that where
. Moreover, by passing to a subsequence we may assume that
is regular (see Lemma 3.1). Since
is uniformly convex,
is singleton, say
(note that
. Let
. For each
, we choose
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ25_HTML.gif)
On the other hand, there is a natural number such that for every
we have
. This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ26_HTML.gif)
and hence from our assumption we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ27_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ28_HTML.gif)
Moreover, for all natural numbers
Indeed, since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ29_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ30_HTML.gif)
Since and
, by the fact that the mappings
and
are commuting, we obtain
. Now, by the uniqueness of
as the nearest point to
, we get
Since is compact, the sequence
has a convergent subsequence
with
. Because
for all
, and
is closed, we obtain
. Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ31_HTML.gif)
and for we have
. This entails
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ32_HTML.gif)
Since is regular, this shows that
. And hence
.
As a consequence, we obtain the theorem already proved by Dhompongsa et al. (see [3, Theorem ]).
Corollary 3.4.
Let be a nonempty closed convex bounded subset of a uniformly convex Banach space
,
, and
a single-valued and a multivalued nonexpansive mapping, respectively. Assume that
and
are commuting mappings. Then there exists a point
such that
.
Corollary 3.5.
Let be a nonempty bounded closed convex subset of a uniformly convex Banach space
, and let
be a multivalued mapping satisfying the Suzuki condition (C). Then
has a fixed point.
Corollary 3.6.
Let be a nonempty closed convex bounded subset of a uniformly convex Banach space
, and let
be a nonexpansive multivalued mapping. Then
has a fixed point.
4. Strictly
Spaces
Definition 4.1.
Let be a Banach space and let
be a linear topology on
. We say that
is a strictly
space if there exists a function
satisfying the following
(i) is continuous;
(ii) is strictly increasing;
(iii), for every
(iv) is strictly increasing;
(v), for every
and for every bounded and
-null sequence
, where
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ33_HTML.gif)
In this case we also say that satisfies the strict property
with respect to
Example 4.2 (see [9]).
Let be a positive
-finite measure space. For every
, consider the Banach space
with the usual norm. Let
be the topology of convergence locally in measure (clm). Then
endowed with the clm-topology satisfies the property
with
Definition 4.3.
Let be a Banach space and let
be a linear topology on
which is weaker than the norm topology. Let
be a closed convex bounded subset of
; then for
we write
We say that
has property (
) if for every
the set
is a nonempty and norm-compact subset of
.
Theorem 4.4.
Let be a strictly
Banach space and let
be a nonempty closed convex bounded subset of
satisfying the property
. Suppose, in addition, that
is
-sequentially compact. If
satisfies the condition
, and if
is an upper semicontinuous mapping, then
has a fixed point.
Proof.
First, we find an approximate fixed point sequence. Choose and
. Define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ34_HTML.gif)
Let be chosen in such a way that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ35_HTML.gif)
Similarly, put ; again we choose
in such a way that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ36_HTML.gif)
In this way, we will find a sequence in
such that
, where
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ37_HTML.gif)
Therefore, for every natural number we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ38_HTML.gif)
from which it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ39_HTML.gif)
Our assumption now gives
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ40_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ41_HTML.gif)
We now apply Lemma 2.8 to conclude that where
. Since
is
-sequentially compact, by passing to a subsequence, we may assume that
is
-convergent to
Now we are going to show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ42_HTML.gif)
Taking any , by the compactness of
, we can find
such that
. On the other hand, there is a natural number
such that for every
we have
. This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ43_HTML.gif)
and hence from the assumption we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ44_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ45_HTML.gif)
Since is compact, the sequence
has a convergent subsequence
with
. It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ46_HTML.gif)
On the other hand, we have that Since
is strictly increasing, it follows that
Hence
and so
. Now we define the mapping
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F457935/MediaObjects/13663_2010_Article_1285_Equ47_HTML.gif)
by . From [10, Proposition
], we know that the mapping
is upper semicontinuous. Since
is a compact convex set, we can apply the Kakutani-Bohnenblust-Karlin Theorem (see [11]) to obtain a fixed point for
and hence for
.
Corollary 4.5.
Let be a strictly
Banach space and let
be a nonempty closed convex bounded subset of
satisfying the property
. Suppose, in addition, that
is
-sequentially compact. If
is a nonexpansive mapping, then
has a fixed point.
Corollary 4.6.
Let be a strictly
Banach space and let
be a nonempty closed convex and bounded subset of
satisfying the property
. Suppose, in addition, that
is
-sequentially compact. If
is a continuous mapping satisfying the condition (C), then
has a fixed point.
Finally we mention that by Example 2.2, this corollary generalizes the recent result of DomÃnguez-Benavides et al. [4].
References
Suzuki T: Fixed point theorems and convergence theorems for some generalized nonexpansive mappings. Journal of Mathematical Analysis and Applications 2008,340(2):1088–1095. 10.1016/j.jmaa.2007.09.023
Abkar A, Eslamian M: A fixed point theorem for generalized nonexpansive multivalued mapping. to appear in Fixed Point Theory
Dhompongsa S, Kaewcharoen A, Kaewkhao A: The DomÃnguez-Lorenzo condition and multivalued nonexpansive mappings. Nonlinear Analysis. Theory, Methods & Applications 2006,64(5):958–970. 10.1016/j.na.2005.05.051
DomÃnguez-Benavides T, GarcÃa-Falset J, Llorens-Fuster E, Lorenzo-RamÃrez P: Fixed point properties and proximinality in Banach spaces. Nonlinear Analysis. Theory, Methods & Applications 2009,71(5–6):1562–1571. 10.1016/j.na.2008.12.048
Dhompongsa S, Inthakon W, Kaewkhao A: Edelstein's method and fixed point theorems for some generalized nonexpansive mappings. Journal of Mathematical Analysis and Applications 2009,350(1):12–17. 10.1016/j.jmaa.2008.08.045
Goebel K, Kirk WA: Iteration processes for nonexpansive mappings. In Topological Methods in Nonlinear Functional Analysis. Volume 21. American Mathematical Society, Providence, RI, USA; 1983:115–123.
Goebel K: On a fixed point theorem for multivalued nonexpansive mappings. 1975, 29: 70–72.
Lim TC: A fixed point theorem for multivalued nonexpansive mappings in a uniformly convex Banach space. Bulletin of the American Mathematical Society 1974, 80: 1123–1126. 10.1090/S0002-9904-1974-13640-2
DomÃnguez-Benavides T, GarcÃa-Falset J, Japón-Pineda MA: The -fixed point property for nonexpansive mappings. Abstract and Applied Analysis 1998,3(3–4):343–362. 10.1155/S1085337598000591
Hu S, Papageorgiou NS: Handbook of Multivalued Analysis, Mathematics and Its Applications. Volume 1. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1997:xvi+964.
Goebel K, Kirk WA: Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, UK; 1990:viii+244.
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Abkar, A., Eslamian, M. Fixed Point Theorems for Suzuki Generalized Nonexpansive Multivalued Mappings in Banach Spaces. Fixed Point Theory Appl 2010, 457935 (2010). https://doi.org/10.1155/2010/457935
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DOI: https://doi.org/10.1155/2010/457935