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Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2010, Article ID 596952, 12 pages
doi:10.1155/2010/596952

Research Article
On Some Geometric Constants and the Fixed Point
Property for Multivalued Nonexpansive Mappings
Jingxin Zhang1 and Yunan Cui2
1
2

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
Department of Mathematics, Harbin University of Science and Technology, Harbin 150080, China

Correspondence should be addressed to Jingxin Zhang, zhjx
Received 30 July 2010; Accepted 5 October 2010
Academic Editor: L. Gorniewicz
´
Copyright q 2010 J. Zhang and Y. Cui. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We show some geometric conditions on a Banach space X concerning the Jordan-von Neumann
constant, Zb˘ ganu constant, characteristic of separation noncompact convexity, and the
a
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 T : C → K C where C 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.


2

Fixed Point Theory and Applications

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
WCS X
4

CNJ X < 1

2

1.1

implies property D .
3 Satit Saejung 5 proved that the condition ε0 X < WCS X implies property D .
4 Gavira 6 showed that the condition

J X <1

1
R 1, X

1.2

implies DL condition.
In 2007, Dom´nguez Benavides and Gavira 7 have established FFP for multivalued
ı
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, Dom´nguez Benavides and Gavira 9 have given a survey of this
ı
subject and presented the main known results and current research directions.
In this paper, in terms of the Jordan-von Neumann constant, Zb˘ ganu constant, εβ X
a
and the coefficient R 1, X , 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 X be a Banach space and C be a nonempty subset of X; we denote all nonempty bounded
closed subsets of X by CB X and all nonempty compact convex subsets of X by KC X .
A multivalued mapping T : C → CB X is said to be nonexpansive if the inequality
H T x, T y ≤ x − y

2.1

holds for every x, y ∈ C, where H ·, · is the Hausdorff distance on CB X , that is,
H A, B : max sup inf x − y , sup inf x − y
x∈A y∈B

y∈B x∈A

,

A, B ∈ CB X .

2.2


Fixed Point Theory and Applications

3

Let C ⊂ X be a nonempty bounded closed convex subset and {xn } ∈ X a bounded
sequence; we use r C, {xn } and A C, {xn } to denote the asymptotic radius and the

asymptotic center of {xn } in C, respectively, that is,
inf lim sup xn − x : x ∈ C ,

r C, {xn }
A C, {xn }

n

2.3

x ∈ C : lim sup xn − x
n

r C, {xn }

.

It is known that A C, {xn } is a nonempty weakly compact convex as C is.
Let {xn } and C be as above; then {xn } is called regular relative to C if r C, {xn }
r C, {xni } for all subsequence {xni } of {xn }; further, {xn } is called asymptotically uniform
A C, {xni } for all subsequence {xni } of {xn }. In Banach spaces,
relative to C if A C, {xn }
we have the following results:
there always exists a subsequence of {xn } which is

1

Goebel 10 and Lim 2
regular relative to C;

2

Kirk 11 if C is separable, then {xn } contains a subsequence which is asymptotically uniform relative to C.

If D is a bounded subset of X, the Chebyshev radius of D relative to C is defined by
rC D

inf sup x − y .
x∈C y∈D

In 2006, Dhompongsa et al. 3 introduced the Domnguez-Lorenzo condition
condition, in short in the following way.

2.4

DL

Definition 2.1 see 3 . We say that a Banach space X satisfies the DL condition if there
exists λ ∈ 0, 1 such that for every weakly compact convex subset C of X and for every
bounded sequence {xn } in C which is regular with respect to C,
rC A C, {xn }

≤ λr C, {xn } .

2.5

The DL condition implies weak normal structure 3 . We recll that a Banach space X
is said to have a weak normal structure w-NS if for every weakly compact convex subset C
of X with diam C : sup{ x − y : x, y ∈ C} > 0 there exist x ∈ C such that sup{ x − y : y ∈
C} < diam C .

The DL condition also implies the existence of fixed points for multivalued
nonexpansive mappings.
Theorem 2.2 see 3 . Let C be a weakly compact convex subset of Banach space X; if C satisfies
(DL) condition, then multivalued nonexpansive mapping T : C → KC C has a fixed point.


4

Fixed Point Theory and Applications

Definition 2.3 see 4 . A Banach space X is said to have property D if there exists λ ∈ 0, 1
such that for every weakly compact convex subset C of X and for every sequence {xn } ⊂ C
and for every {yn } ⊂ A C, {xn } which are regular asymptotically uniform relative to C,
r C, yn

≤ λr C, {xn } .

2.6

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 C be a weakly compact convex subset of Banach space X; if C satisfies
property (D), then multivalued nonexpansive mapping T : C → KC C has a fixed point.
Before going to the results, let us recall some more definitions. Let X be a Banach space.
The Benavides coefficient R 1, X is defined by Dom´nguez Benavides [12] as
ı
R 1, X

sup lim inf{ xn

n→∞

x } ,

2.7

where the supremum is taken over all x ∈ X with X ≤ 1 and all weakly null sequence {xn } in BX
such that
D xn

: lim sup lim sup xn − xm ≤ 1.
n→∞

m→∞

2.8

Obviously, 1 ≤ R 1, X ≤ 2.
The weakly convergent sequence coefficient WCS X is equivalently defined by see
13

WCS X

inf

limn / m xn − xm
lim supn xn

,

2.9

where the infimum is taken over all weakly not strongly null sequences {xn } with
limn / m xn − xm 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 F be a filter on an index set N
and let X be a Banach space. A sequence xn in X convergers to x with respect to F, denoted by
limF xn x, if for each neighborhood U of x, {i ∈ I : xi ∈ U} ∈ F. A filter U on N 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 {A ⊂ N, i0 ∈ A} for some fixed i0 ∈ N; otherwise, it is called nontrivial. Let l∞ X
denote the subspace of the product space Πi∈N Xi equipped with the norm
xn

: sup xn < ∞.
n∈N

2.10


Fixed Point Theory and Applications

5

Let U be an ultrafilter on N and let
xn ∈ l∞ X : lim xn

NU

U

0 .

2.11

The ultrapower of X, denoted by X, is the quotient space l∞ X /NU equipped with
the quotient norm. Write xn U to denote the elements of ultrapower. It follows from the
definition of the quotient norm that
xn

lim xn .

U

2.12

U

Note that if U is nontrivial, then X can be embedded into X isometrically. For more
details see 14 .

3. Main Results
We first give some sufficient conditions which imply DL condition. The Jordan-von
Neumann constant CNJ X was defined in 1937 by Clarkson 15 as

CNJ X

⎧
⎨ x y
sup

⎩ 2 x

2

x−y

2

y

2

2

: x, y ∈ X, x

⎫
⎬
y /0 .
⎭

3.1

Theorem 3.1. Let X be a Banach space and C a weakly compact convex subset of X. Assume that
{xn } is a bounded sequence in C which is regulary relative to C. Then
rC A C, {xn }

≤

R 1, X 2CNJ X

r C, {xn } .
R 1, X 1

3.2

Proof. Denote r r C, {xn } and A A C, {xn }. We can assume that r > 0. Since {xn } ⊂ C is
bounded and C is a weakly compact set, by passing through a subsequence if necessary, we
can also assume that xn converges weakly to some element in x ∈ C and d limn / m xn − xm
r C, {yn } for any subsequence {yn } of
exists. We note that since {xn } is regular, r C, {xn }
{xn }. Observe that, since the norm is weak lower semicontinuity, we have
lim inf xn − x ≤ lim inf lim inf xn − xm
n

n

m

lim inf xn − xm
n/m

d.

3.3

Let η > 0; taking a subsequence if necessary, we can assume that xn − x < d η for all n.
r and x −z ≤ lim infn xn −z ≤ r. Denote
Let z ∈ A. Then we have lim supn xn −z
R R 1, X ; by definition, we have
R ≥ lim inf

n

xn − x
d η

z−x
r

lim inf
n

xn − x x − z
.
−
d η
r

3.4


6

Fixed Point Theory and Applications
R−1 / R 1 x

On the other hand, observe that the convexity of C implies
1 z ∈ C; since the norm is weak lower semicontinuity, we have
lim inf
n

n

lim inf
n

1
r

1
R d η

1
1
−
x
r Rr

≥
≥

1 xn − x x − z
−
R d η
r

1
xn − z
r
lim inf

1
r

1
r

1
Rr

1
R d η

xn −

2
z−
Rr

1
z
Rr

1
r

1
1
−
z
r Rr

x−

R−1
x
R 1

1
Rr

2
R

1

z−z

1
rC A ,
Rr

3.5

1
1 xn − x x − z
−
xn − z −
r
R d η
r

lim inf

1
1
−
r R d η

1
r

z−x ≥

n

≥

2/ R

1
Rr

1
r

1
Rr

1
r

xn − x −

z−x

1
rC A .
Rr

In the ultrapower X of X, we consider
u

1
{xn − z}U ∈ SX ,
r

1 xn − x x − z
−
R d η
r

v

U

∈ BX .

3.6

Using the above estimates, we obtain
u

v

u−v

lim
U

1 xn − x x − z
−
R d η
r

1
xn − z
r

≥

1
rC A ,
Rr

≥

1
1 xn − x x − z
−
lim
xn − z −

U
r
R d η
r

1
r
1
r

1
rC A .
Rr

3.7

Therefore, we have
CNJ X ≥

≥

u

v

2

u

2

2

u−v
v

2

2

2 1/r

1/ Rr 2 rC A
21 1

1 1
2 r

1
Rr

2

rC A 2 .

2

3.8


Fixed Point Theory and Applications

7

Since Jordan-von Neumann constant CNJ X of X equals to CNJ X of X, we obtain
CNJ X ≥

1 1
2 r

2

1
Rr

rC A 2 .

3.9

Hence we deduce the desired inequality.
By Theorems 2.2 and 3.1, we have the following result.
Corollary 3.2. Let C be a nonempty bounded closed convex subset of a Banach space X such that
1 2 /2 and T : C → KC C a nonexpansive mapping. Then T has a fixed
CNJ X < 1/R 1, X
point.
1 2 /2, then we have CNJ X < 2 which
Proof. since R 1, X ≥ 1, if CNJ X < 1/R 1, X
implies that X is uniformly nonsquare; hence X is reflexive. Thus by Theorems 2.2 and 3.1,
the result follows.
Remark 3.3. Note that J X 2 /2 ≤ CNJ X ; it is easy to see that Theorem 3.1 includes
6, Theorem 3 and Corollary 3.2 includes 6, Corollary 2 .

To characterize Hilbert space, Zb˘ ganu defined the following Zb˘ ganu constant: see
a
a
16

CZ X

x

sup

x−y

y
x

2

y

: x, y ∈ X, x

2

y >0 .

3.10

We first give the following tool.
Proposition 3.4. CZ X

CZ X .

Proof. Clearly, CZ X ≤ CZ X . To show CZ X ≤ CZ X , suppose x, y ∈ X are not all zero.
Without loss of generality, we assume x
a > 0.
a and
Let us choose η ∈ 0, a . Since x
limU xn
c:

x

y x y
x 2
y 2

lim

xn

yn
xn

U

2

xn − yn
: limcn ,

U
yn 2

3.11

the set A : {n ∈ N : |cn − c| < η and| xn − a| < η} belongs to U. In particular, noticing that
xn / 0 for n ∈ A, there exists n such that
x

y
x

2

x
y

y
2

<

xn

xn − yn

yn
xn

≤ CZ X

2

yn

2

η

η.

Hence, the inequality CZ X ≤ CZ X follows from the arbitrariness of η.

3.12


8

Fixed Point Theory and Applications

Theorem 3.5. Let X be a Banach space and C a weakly compact convex subset of X. Assume that
{xn } is a bounded sequence in C which is regulary relative to C. Then
rC A C, {xn }

R 1, X 2CZ X
r C, {xn } .
R 1, X 1

≤

3.13

Proof. Let u, v be as in Theorem 3.1. Then
u

v ≥

1
r

1
rC A ,
Rr

1
r

u−v ≥

1
rC A .
Rr

3.14

Therefore, by the definition of Zb˘ ganu constant, we have
a
CZ X ≥

tv u − tv

u

u

1 1
≥
2 r
Since Zb˘ ganu constant CZ X
a

2

2

v

2

1
Rr

3.15
rC A 2 .

of X equals to CZ X of X, we obtain

CZ X ≥

1 1

2 r

1
Rr

2

rC A 2 .

3.16

Hence we deduce the desired inequality.
Using Theorem 2.2, we obtain the following corollary.
Corollary 3.6. Let C be a nonempty weakly compact convex subset of a Banach space X such that
CZ X < 1 1/R 1, X 2 /2 and let T : C → KC C be a nonexpansive mapping. Then T 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 X be a Banach space. If CZ X < WCS X ; then X has property (D).
Proof. Let C be a weakly compact convex subset of X; suppose that {xn } ⊂ C and {yn } ⊂
A C, {xn } are regular and asymptotically uniform relative to C. Passing to a subsequence of
w
{yn }, still denoted by {yn }, we may assume that yn − y0 ∈ C and d limn / m yn −ym exists.
→
Let r r C, {xn } . Again passing to a subsequence of {xn }, still denoted by {xn }, we
assume in addition that
lim xn − y2n

n→∞

lim xn − y2n

n→∞

1

lim

n→∞

xn −

1
y2n
2

y2n

1

r.

3.17


Fixed Point Theory and Applications

9

Let us consider an ultrapower X of X. Put

1
xn − y2n
r

u

U

,

1
xn − y2n
r

v

1 U;

3.18

2,

3.19

then we know that u ∈ SX , v ∈ SX . We see that
u
u−v

v

lim
U

lim
U

xn − y2n
r

xn − y2n
r

xn − y2n xn − y2n
−
r
r

1

lim
U

1

y2n − y2n
r

1

d

.
r

3.20

Thus, By the definition of Zb˘ ganu constant, we have
a
CZ X ≥

v u−v

u
u

2

v

2

≥

d
.
r

3.21

Since the Zb˘ ganu constants of X and of X are the same, we obtain CZ X ≥ d/r. Now
a

we estimate d as follows:
lim yn − ym

d

n/m

lim

n/m

yn − y0 − ym − y0

≥ WCS X lim sup yn − y0

3.22

n

≥ WCS X r C, yn

.

Hence r C, {yn } ≤ CZ X /WCS X r C, {xn } 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 C be a nonempty bounded closed convex subset of a reflexive Banach space X such
that CZ X < WCS X and let T : C → KC C be a nonexpansive mapping. Then T has a fixed
point.
The separation measure of noncompactness is defined by

β B

sup ε : there exists a sequence {xn }in B such that sep {xn } ≥ ε

3.23

for any bounded subset B of a Banach space X, where
sep {xn }

inf{ xn − xm : n / m}.

3.24

The modulus of noncompact convexity associated to β is defined in the following way:
ΔX,β ε

inf 1 − d 0, A : A ⊂ BX is convex, β A ≥ ε .

3.25


10

Fixed Point Theory and Applications

The characteristic of noncompact convexity of X associated with the measure of
noncompactness β is defined by
εβ X

sup ε ≥ 0 : ΔX,β ε

0 .

3.26

When X is a reflexive Banach space, we have the following alternative expression for
the modulus of noncompact convexity associated with β,
εβ X

w − limxn , sep {xn } ≥ ε .

inf 1 − x : {xn } ⊂ BX , x

It is known that X is NUC if and only if εβ X
properties can be found in 17 .

n

3.27

0. The above-mentioned definitions and

Theorem 3.9. Let X be a reflexive Banach space. If εβ X < WCS X , then X has property (D).
Proof. Let C be a weakly compact convex subset of X; suppose that {xn } ⊂ C and {yj } ⊂
A C, {xn } are regular and asymptotically uniform relative to C. Passing to a subsequence of
w
{yj }, still denoted by {yj }, we may assume that yj − y0 ∈ C and d limk / l yk − yl exists.
→
Let r r C, {xn } .
Since {y0 , yj } ⊂ A C, {xn } , we have

lim sup xn − y0

lim sup xn − yj

r,

n

r,

∀j ∈ N.

n

3.28

So for any η ≥ 0, there exists N ∈ N such that xN − y0 ≥ r − η and xN − yi ≤ r η, for all
j ∈ N.
Without loss of generality, we suppose that yk − yl ≥ d − η for all k / l. Now we
consider sequence { xN − yj / r η } ⊂ BX ; notice that
xN − yj
r η

β

≥

xN − yj w xN − y0
.
−

→
r η
r η

d−η
,
r η

3.29

By the definition of ΔX,β · , we have
ΔX,β

d−η
r η

≤1−

xN − y0
r η

≤1−

r−η
.
r η

Since the last inequality is true for any η > 0, we obtain ΔX,β d/r
Now we estimate d as follows:
d

lim yk − yl

k/l

≥ WCS X lim sup yn − y0
n

≥ WCS X r C, yn

0; thus εβ X ≥ d/r.

yk − y0 − yl − y0

lim

k/l

3.30

.

3.31


Fixed Point Theory and Applications

11

Hence,

r C, yn

≤

εβ X
r C, {xn } .
WCS X

3.32

Remark 3.10. Since εβ X ≤ ε0 X , Theorem 3.9 implies the 5, Theorem 3 . Furthermore,
ε0 X 2 /4 ≥ 1
εβ X 2 /4; then Theorem 3.9 also includes
it is easy to see CNJ X ≥ 1
4, Theorem 3.7 .
By Theorem 3.9, we obtain the following Corollary.
Corollary 3.11. Let C be a nonempty bounded closed convex subset of a reflexive Banach space X
such that εβ X < WCS X and let T : C → KC C be a nonexpansive mapping. Then T has a fixed
point.
Noticing WCS X ≥ 1, obviously, Corollary 3.11 extends the following well-known
result.
Theorem 3.12 see 18, Theorem 3.5 . Let C be a nonempty bounded closed convex subset of a
reflexive Banach space X such that εβ X < 1 and let T : C → KC C be a nonexpansive mapping.
Then T has a fixed point.

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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ı
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12

Fixed Point Theory and Applications

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a
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