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On Some Geometric Constants and the Fixed Point Property for Multivalued Nonexpansive Mappings
Fixed Point Theory and Applications volume 2010, Article number: 596952 (2010)
Abstract
We show some geometric conditions on a Banach space X concerning the Jordan-von Neumann constant, Zbăganu constant, characteristic of (separation) noncompact convexity, and the coefficient R(1, X), the weakly convergent sequence coefficient, which imply the existence of fixed points for multivalued nonexpansive mappings.
1. Introduction
Fixed point theory for multivalued mappings has many useful applications in Applied Sciences, in particular, in game theory and mathematical economics. Thus it is natural to try of extending the known fixed point results for single-valued mappings to the setting of multivalued mappings.
In 1969, Nadler [1] established the multivalued version of Banach's contraction principle. One of the most celebrated results about multivalued mappings was given by Lim [2] in 1974. Using Edelstein's method of asymptotic centers, he proved the existence of a fixed point for a multivalued nonexpansive self-mapping where
is a nonempty bounded closed convex subset of a uniformly convex Banach space. Since then the metric fixed point theory of multivalued mappings has been rapidly developed. Some other classical fixed point theorems for single-valued mappings have been extended to multivalued mappings. However, many questions remain open, for instance, the possibility of extending the well-known Kirk's theorem, that is, do Banach spaces with weak normal structure have the fixed point property (FPP, in short) for multivalued nonexpansive mappings?
Since weak normal structure is implied by different geometrical properties of Banach spaces, it is natural to study if those properties imply the FPP for multivalued mappings. Dhompongsa et al. [3, 4] introduced the Domnguez-Lorenzo condition ((DL) condition, in short) and property (D) which imply the FPP for multivalued nonexpansive mappings. A possible approach to the above problem is to look for geometric conditions in a Banach space X which imply either the (DL) condition or property (D). In this setting the following results have been obtained.
-
(1)
Dhompongsa et al. [3] proved that uniformly nonsquare Banach spaces with property WORTH satisfy the (DL) condition.
-
(2)
Dhompongsa et al. [4] showed that the condition
(1.1)
implies property (D).
(3)Satit Saejung [5] proved that the condition implies property (D).
(4)Gavira [6] showed that the condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ2_HTML.gif)
implies (DL) condition.
In 2007, DomÃnguez Benavides and Gavira [7] have established FFP for multivalued nonexpansive mappings in terms of the modulus of squareness, universal infinite-dimensional modulus, and Opia modulus. Attapol Kaewkhao [8] has established FFP for multivalued nonexpansive mappings in terms of the James constant, the Jordan-von Neumann Constants, weak orthogonality.
Besides, In 2010, DomÃnguez Benavides and Gavira [9] have given a survey of this subject and presented the main known results and current research directions.
In this paper, in terms of the Jordan-von Neumann constant, Zbganu constant,
and the coefficient
, the weakly convergent sequence coefficient, we show some geometrical properties which imply the property (D) or (DL) condition and so the FPP for multivalued nonexpansive mappings.
2. Preliminaries
Let be a Banach space and
be a nonempty subset of
; we denote all nonempty bounded closed subsets of
by
and all nonempty compact convex subsets of
by
.
A multivalued mapping is said to be nonexpansive if the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ3_HTML.gif)
holds for every , where
is the Hausdorff distance on
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ4_HTML.gif)
Let be a nonempty bounded closed convex subset and
a bounded sequence; we use
and
to denote the asymptotic radius and the asymptotic center of
in
, respectively, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ5_HTML.gif)
It is known that is a nonempty weakly compact convex as
is.
Let and
be as above; then
is called regular relative to
if
for all subsequence
of
; further,
is called asymptotically uniform relative to
if
for all subsequence
of
. In Banach spaces, we have the following results:
(1)(Goebel [10] and Lim [2]) there always exists a subsequence of which is regular relative to
;
-
(2)
(Kirk [11]) if
is separable, then
contains a subsequence which is asymptotically uniform relative to
.
If is a bounded subset of
, the Chebyshev radius of
relative to
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ6_HTML.gif)
In 2006, Dhompongsa et al. [3] introduced the Domnguez-Lorenzo condition ((DL) condition, in short) in the following way.
Definition 2.1 (see [3]).
We say that a Banach space satisfies the (DL) condition if there exists
such that for every weakly compact convex subset
of
and for every bounded sequence
in
which is regular with respect to
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ7_HTML.gif)
The (DL) condition implies weak normal structure [3]. We recll that a Banach space is said to have a weak normal structure (w-NS) if for every weakly compact convex subset
of
with
there exist
such that
.
The (DL) condition also implies the existence of fixed points for multivalued nonexpansive mappings.
Theorem 2.2 (see [3]).
Let be a weakly compact convex subset of Banach space
; if
satisfies (DL) condition, then multivalued nonexpansive mapping
has a fixed point.
Definition 2.3 (see [4]).
A Banach space is said to have property (D) if there exists
such that for every weakly compact convex subset
of
and for every sequence
and for every
which are regular asymptotically uniform relative to
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ8_HTML.gif)
It was observed that property (D) is weaker than the (DL) condition and stronger than weak normal structure, and Dhompongsa et al. [4] proved that property (D) implies the w-MFPP.
Theorem 2.4 (see [4]).
Let be a weakly compact convex subset of Banach space
; if
satisfies property (D), then multivalued nonexpansive mapping
has a fixed point.
Before going to the results, let us recall some more definitions. Let be a Banach space.
The Benavides coefficient is defined by DomÃnguez Benavides [12] as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ9_HTML.gif)
where the supremum is taken over all with
and all weakly null sequence
in
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ10_HTML.gif)
Obviously, .
The weakly convergent sequence coefficient is equivalently defined by (see [13])
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ11_HTML.gif)
where the infimum is taken over all weakly (not strongly) null sequences with
existing.
The ultrapower of a Banach space has proved to be useful in many branches of mathematics. Many results can be seen more easily when treated in this setting.
First we recall some basic facts about ultrapowers. Let be a filter on an index set
and let
be a Banach space. A sequence
in
convergers to
with respect to
, denoted by
, if for each neighborhood
of
,
. A filter
on
is called an ultrafilter if it is maximal with respect to the set inclusion. An ultrafilter is called trivial if it is of the form
for some fixed
; otherwise, it is called nontrivial. Let
denote the subspace of the product space
equipped with the norm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ12_HTML.gif)
Let be an ultrafilter on
and let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ13_HTML.gif)
The ultrapower of , denoted by
, is the quotient space
equipped with the quotient norm. Write
to denote the elements of ultrapower. It follows from the definition of the quotient norm that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ14_HTML.gif)
Note that if is nontrivial, then
can be embedded into
isometrically. For more details see [14].
3. Main Results
We first give some sufficient conditions which imply (DL) condition. The Jordan-von Neumann constant was defined in 1937 by Clarkson [15] as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ15_HTML.gif)
Theorem 3.1.
Let be a Banach space and
a weakly compact convex subset of
. Assume that
is a bounded sequence in
which is regulary relative to
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ16_HTML.gif)
Proof.
Denote and
. We can assume that
. Since
is bounded and
is a weakly compact set, by passing through a subsequence if necessary, we can also assume that
converges weakly to some element in
and
exists. We note that since
is regular,
for any subsequence
of
. Observe that, since the norm is weak lower semicontinuity, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ17_HTML.gif)
Let ; taking a subsequence if necessary, we can assume that
for all
.
Let . Then we have
and
. Denote
; by definition, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ18_HTML.gif)
On the other hand, observe that the convexity of implies
; since the norm is weak lower semicontinuity, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ19_HTML.gif)
In the ultrapower of
, we consider
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ20_HTML.gif)
Using the above estimates, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ21_HTML.gif)
Therefore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ22_HTML.gif)
Since Jordan-von Neumann constant of
equals to
of
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ23_HTML.gif)
Hence we deduce the desired inequality.
By Theorems 2.2 and 3.1, we have the following result.
Corollary 3.2.
Let be a nonempty bounded closed convex subset of a Banach space
such that
and
a nonexpansive mapping. Then
has a fixed point.
Proof.
since , if
, then we have
which implies that
is uniformly nonsquare; hence
is reflexive. Thus by Theorems 2.2 and 3.1, the result follows.
Remark 3.3.
Note that ; it is easy to see that Theorem 3.1 includes [6, Theorem
] and Corollary 3.2 includes [6, Corollary
].
To characterize Hilbert space, Zbganu defined the following Zb
ganu constant: (see [16])
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ24_HTML.gif)
We first give the following tool.
Proposition 3.4.
.
Proof.
Clearly, . To show
, suppose
are not all zero. Without loss of generality, we assume
.
Let us choose . Since
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ25_HTML.gif)
the set belongs to
. In particular, noticing that
for
, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ26_HTML.gif)
Hence, the inequality follows from the arbitrariness of
.
Theorem 3.5.
Let be a Banach space and
a weakly compact convex subset of
. Assume that
is a bounded sequence in
which is regulary relative to
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ27_HTML.gif)
Proof.
Let be as in Theorem 3.1. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ28_HTML.gif)
Therefore, by the definition of Zbganu constant, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ29_HTML.gif)
Since Zbganu constant
of
equals to
of
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ30_HTML.gif)
Hence we deduce the desired inequality.
Using Theorem 2.2, we obtain the following corollary.
Corollary 3.6.
Let be a nonempty weakly compact convex subset of a Banach space
such that
and let
be a nonexpansive mapping. Then
has a fixed point.
In the following, we present some properties concerning geometrical constants of Banach spaces which also imply the property (D).
Theorem 3.7.
Let be a Banach space. If
; then
has property (D).
Proof.
Let be a weakly compact convex subset of
; suppose that
and
are regular and asymptotically uniform relative to
. Passing to a subsequence of
, still denoted by
, we may assume that
and
exists.
Let . Again passing to a subsequence of
, still denoted by
, we assume in addition that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ31_HTML.gif)
Let us consider an ultrapower of
. Put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ32_HTML.gif)
then we know that . We see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ33_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ34_HTML.gif)
Thus, By the definition of Zbganu constant, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ35_HTML.gif)
Since the Zbganu constants of
and of
are the same, we obtain
. Now we estimate
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ36_HTML.gif)
Hence and the assertion follows by the definition of property (D).
Using Theorems 2.4 and 3.7, we obtain the follwing corollary.
Corollary 3.8.
Let be a nonempty bounded closed convex subset of a reflexive Banach space
such that
and let
be a nonexpansive mapping. Then
has a fixed point.
The separation measure of noncompactness is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ37_HTML.gif)
for any bounded subset of a Banach space
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ38_HTML.gif)
The modulus of noncompact convexity associated to is defined in the following way:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ39_HTML.gif)
The characteristic of noncompact convexity of associated with the measure of noncompactness
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ40_HTML.gif)
When is a reflexive Banach space, we have the following alternative expression for the modulus of noncompact convexity associated with
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ41_HTML.gif)
It is known that is NUC if and only if
. The above-mentioned definitions and properties can be found in [17].
Theorem 3.9.
Let be a reflexive Banach space. If
, then
has property (D).
Proof.
Let be a weakly compact convex subset of
; suppose that
and
are regular and asymptotically uniform relative to
. Passing to a subsequence of
, still denoted by
, we may assume that
and
exists. Let
.
Since , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ42_HTML.gif)
So for any , there exists
such that
and
for all
.
Without loss of generality, we suppose that for all
. Now we consider sequence
; notice that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ43_HTML.gif)
By the definition of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ44_HTML.gif)
Since the last inequality is true for any , we obtain
; thus
. Now we estimate
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ45_HTML.gif)
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F596952/MediaObjects/13663_2010_Article_1311_Equ46_HTML.gif)
Remark 3.10.
Since , Theorem 3.9 implies the [5, Theorem
]. Furthermore, it is easy to see
; then Theorem 3.9 also includes [4, Theorem
].
By Theorem 3.9, we obtain the following Corollary.
Corollary 3.11.
Let be a nonempty bounded closed convex subset of a reflexive Banach space
such that
and let
be a nonexpansive mapping. Then
has a fixed point.
Noticing , obviously, Corollary 3.11 extends the following well-known result.
Theorem 3.12 (see [18, Theorem ]).
Let be a nonempty bounded closed convex subset of a reflexive Banach space
such that
and let
be a nonexpansive mapping. Then
has a fixed point.
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Acknowledgments
The authors would like to thank the anonymous referee for providing some suggestions to improve the manuscript. This work was supported by China Natural Science Fund under grant 10571037.
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Zhang, J., Cui, Y. On Some Geometric Constants and the Fixed Point Property for Multivalued Nonexpansive Mappings. Fixed Point Theory Appl 2010, 596952 (2010). https://doi.org/10.1155/2010/596952
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DOI: https://doi.org/10.1155/2010/596952