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Some Sufficient Conditions for Fixed Points of Multivalued Nonexpansive Mappings
Fixed Point Theory and Applications volume 2009, Article number: 319804 (2010)
Abstract
We show some sufficient conditions on a Banach space concerning the generalized James constant, the generalized Jordan-von Neumann constant, the generalized Zbagănu constant, the coefficient
, the weakly convergent sequence coefficient WCS(
), and the coefficient of weak orthogonality, which imply the existence of fixed points for multivalued nonexpansive mappings. These fixed point theorems improve some previous results in the recent papers.
1. Introduction
In 1969, Nadler [1] established the multivalued version of Banach contraction principle. Since then the metric fixed point theory of multivalued mappings has been rapidly developed. Some classical fixed point theorems for singlevalued nonexpansive mappings have been extended to multivalued nonexpansive mappings. However, many questions remain open, for instance, the possibility of extending the well-known Kirk's theorem [2], that is, "Do Banach spaces with weak normal structure have the fixed point property (FPP) for multivalued nonexpansive mappings?"
Since weak normal structure is implied by different geometric properties of Banach spaces, it is natural to study whether those properties imply the FPP for multivalued mappings. Dhompongsa et al. [3, 4] introduced the DL condition 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 which imply either the DL condition or property (D). In this setting the following results have been obtained.
(i)Kaewkhao [5] proved that a Banach space with

satisfies the DL condition. He also showed that the condition

implies the DL condition [6].
(ii)Saejung [7] showed that a Banach space has property (D) whenever
.
In this paper, we show some sufficient conditions on a Banach space concerning the generalized James constant, the generalized Jordan-von Neumann constant, the generalized Zbăganu constant, the coefficient
, the weakly convergent sequence coefficient
, and the coefficient of weak orthogonality, which imply the existence of fixed points for multivalued nonexpansive mappings. These theorems improve the above results.
2. Preliminaries
Before going to the result, let us recall some concepts and results which will be used in the following sections. Let be a Banach space with the unit ball
and the unit sphere
. The two constants of a Banach space

are called the von Neumann-Jordan [8] and James constants [9], respectively, and are widely studied by many authors [10–20]. Recently, both constants are generalized in the following ways for (see [12, 13]):

It is clear that and
.
Recently, Gao and Saejung in [6] define a new constant for :

which is inspired by Zbăganu paper [21]. It is clear that

The modulus of convexity of (see [22]) is a function
defined by

The function is strictly increasing on
. Here
is the characteristic of convexity of
, and the space is called uniformly nonsquare if
.
In [23] the author introduces a modulus that scales the 3-dimensional convexity of the unit ball: he considers the number

and defines the function by

He also considers the coefficient corresponding to this modulus:

It is evident that for all
and in consequence
. Moreover this last inequality can be strict, since it was shown in [23] the existence of Banach spaces with
which are not uniformly nonsquare.
The weakly convergent sequence coefficient of
is defined as follows:
where the infimum is taken over all weakly null sequences
in
such that
and
exist.
The WORTH property was introduced by Sims in [24] as follows. A Banach space has the WORTH property if

for all and all weakly null sequences
. In [25], Jiménez-Melado and Llorens-Fuster defined the coefficient of weak orthogonality, which measures the degree of WORTH wholeness, by

where the infimum is taken over all and all weakly null sequence
. It is known that
has the WORTH property if and only if
.
Let be a nonempty subset of a Banach space
. We shall denote by
the family of all nonempty closed bounded subsets of
and by
the family of all nonempty compact convex subsets of
. A multivalued mapping
is said to be nonexpansive if

where denotes the Hausdorff metric on
defined by

Let be a bounded sequence in
. The asymptotic radius
and the asymptotic center
of
in
are defined by


respectively. It is known that is a nonempty weakly compact convex set whenever
is.
The sequence is called regular with respect to
if
for all subsequences
of
, and
is called asymptotically uniform with respect to
if
for all subsequences
of
.
Lemma 2.1.
-
(i)
(See Goebel [26] and Lim [27]) There always exists a subsequence of
which is regular with respect to
.
-
(ii)
(See Kirk [28]) If
is separable, then
contains a subsequence which is asymptotically uniform with respect to
.
If is a bounded subset of
, then the Chebyshev radius of
relative to
is defined by

Dhompongsa et al. [4] introduced the property (D) if there exists such that for any nonempty weakly compact convex subset
of
, any sequence
which is regular asymptotically uniform relative to
, and any sequence
which is regular asymptotically uniform relative to
we have

The Domínguez-Lorenzo condition, DL condition in short form, introduced in [3] is defined as follows: if there exists such that for every weakly compact convex subset
of
and for every bounded sequence
in
which is regular with respect to
we have,

It is clear from the definition that property (D) is weaker than the DL condition. The next results show that property (D) is stronger than weak normal structure and also implies the existence of fixed points for multivalued nonexpansive mappings [4].
Theorem 2.2.
Let be a Banach space satisfying property (D). Then
has weak normal structure.
Theorem 2.3.
Let be a nonempty weakly compact convex subset of a Banach space
which satisfies the property (D). Let
be a nonexpansive mapping, then
has a fixed point.
3. Main Results
Theorem 3.1.
Let be a weakly compact convex subset of a Banach space
and let
be a bounded sequence in
regular with respect to
. Then for every
,

Proof.
Denote and
. We can assume that
. By passing to a subsequence if necessary, we can also assume that
is weakly convergent to a point
. Since
is regular with respect to
, then passing through a subsequence does not have any effect to the asymptotic radius of the whole sequence
. Let
, then we have that

Denote . By the definition of
we have that

Convexity of implies that
and thus, we obtain

On the other hand, by the weak lower semicontinuity of the norm, we have that

For every there exists
such that
(1),
(2)
(3)
(4)
Now, put ,
and
. Using the above estimates, we obtain
,
and

Thus,

By the weak lower semicontinuity of the norm again, we conclude that and hence,

Therefore Since the above inequality is true for every
and every
, we obtain

and therefore,

Corollary 3.2.
Let be a nonempty bounded closed convex subset of a Banach space
such that
and let
be a nonexpansive mapping. Then
has a fixed point.
Proof.
When , then
satisfies the DL condition by Theorem 3.1. So
has a fixed point by Theorem 2.3.
Remark 3.3.
In particular, when , we get the result of Kaewkhao; a Banach space
with

satisfies the DL condition.
Theorem 3.4.
Let be a weakly compact convex subset of a Banach space
and let
be a bounded sequence in
regular with respect to
. Then for every
,

Proof.
Let , and
be as in the proof of the previous theorem. Thus,

Since and by the definition of
, we obtain

On the other hand, by the weak lower semicontinuity of the norm, we have that

For every , there exists
such that
(1)
(2)
(3)
(4)
Now, put ,
and
and use the above estimates to obtain
,
,
, and
, so that

By the definition of , we get

Let ; we obtain that
. Then we have

This holds for arbitrary ; hence, we have that

Corollary 3.5.
Let be a nonempty bounded closed convex subset of a Banach space
such that
and let
be a nonexpansive mapping. Then
has a fixed point.
Proof.
When , then
satisfies the DL condition by Theorem 3.4. So
has a fixed point by Theorem 2.3.
Remark 3.6.
In particular, when , we get the result of Kaewkhao; a Banach space
with

satisfies the DL condition.
Repeating the arguments in the proof of Theorem 3.4, we can easily get the following conclusion.
Theorem 3.7.
Let be a nonempty bounded closed convex subset of a Banach space
such that
and let
be a nonexpansive mapping. Then
has a fixed point.
Remark 3.8.
In particular, when , we get that

satisfies the DL condition which improves the result of Kaewkhao; a Banach space with

satisfies the DL condition.
Theorem 3.9.
A Banach space has property (D) whenever
.
Proof.
Let be a nonempty weakly compact convex subset of
. Suppose that
and
are regular asymptotically uniforms relative to
. Passing to a subsequence, we may assume that
is weakly convergent to a point
and
exists. Let
. Again, passing to a subsequence of
, still denoted by
, we assume in addition that

for all . Now, put

It is easy to see that ,
,
, and
. This implies that
or
Now we estimate
as follows:

Hence,

Remark 3.10.
-
(1)
Theorem 3.9 strengthens the result of Saejung [7] and
has property (D) whenever
.
-
(2)
Theorem 3.9 also improves the result
implying that the Banach space
has normal structure from Theorem 2.2.
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Zuo, Z., Cui, Y. Some Sufficient Conditions for Fixed Points of Multivalued Nonexpansive Mappings. Fixed Point Theory Appl 2009, 319804 (2010). https://doi.org/10.1155/2009/319804
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DOI: https://doi.org/10.1155/2009/319804
Keywords
- Banach Space
- Nonexpansive Mapping
- Compact Convex Subset
- Weak Lower Semicontinuity
- Asymptotically Uniform