Fixed Point Theory and Applications

, 2009:319804 | Cite as

Some Sufficient Conditions for Fixed Points of Multivalued Nonexpansive Mappings

Open Access
Research Article

Abstract

We show some sufficient conditions on a Banach space Open image in new window concerning the generalized James constant, the generalized Jordan-von Neumann constant, the generalized Zbagănu constant, the coefficient Open image in new window , the weakly convergent sequence coefficient WCS( Open image in new window ), 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.

Keywords

Banach Space Nonexpansive Mapping Compact Convex Subset Weak Lower Semicontinuity Asymptotically Uniform 

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 Open image in new window 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 Open image in new window with
satisfies the DL condition. He also showed that the condition

implies the DL condition [6].

(ii)Saejung [7] showed that a Banach space Open image in new window has property (D) whenever Open image in new window .

In this paper, we show some sufficient conditions on a Banach space Open image in new window concerning the generalized James constant, the generalized Jordan-von Neumann constant, the generalized Zbăganu constant, the coefficient Open image in new window , the weakly convergent sequence coefficient Open image in new window , 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 Open image in new window be a Banach space with the unit ball Open image in new window and the unit sphere Open image in new window . 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, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. Recently, both constants are generalized in the following ways for Open image in new window (see [12, 13]):

It is clear that Open image in new window and Open image in new window .

Recently, Gao and Saejung in [6] define a new constant for Open image in new window :

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

The modulus of convexity of Open image in new window (see [22]) is a function Open image in new window defined by

The function Open image in new window is strictly increasing on Open image in new window . Here Open image in new window is the characteristic of convexity of Open image in new window , and the space is called uniformly nonsquare if Open image in new window .

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 Open image in new window by

He also considers the coefficient corresponding to this modulus:

It is evident that Open image in new window for all Open image in new window and in consequence Open image in new window . Moreover this last inequality can be strict, since it was shown in [23] the existence of Banach spaces with Open image in new window which are not uniformly nonsquare.

The weakly convergent sequence coefficient Open image in new window of Open image in new window is defined as follows: Open image in new window where the infimum is taken over all weakly null sequences Open image in new window in Open image in new window such that Open image in new window and Open image in new window exist.

The WORTH property was introduced by Sims in [24] as follows. A Banach space Open image in new window has the WORTH property if

for all Open image in new window and all weakly null sequences Open image in new window . 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 Open image in new window and all weakly null sequence Open image in new window . It is known that Open image in new window has the WORTH property if and only if Open image in new window .

Let Open image in new window be a nonempty subset of a Banach space Open image in new window . We shall denote by Open image in new window the family of all nonempty closed bounded subsets of Open image in new window and by Open image in new window the family of all nonempty compact convex subsets of Open image in new window . A multivalued mapping Open image in new window is said to be nonexpansive if

where Open image in new window denotes the Hausdorff metric on Open image in new window defined by

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

respectively. It is known that Open image in new window is a nonempty weakly compact convex set whenever Open image in new window is.

The sequence Open image in new window is called regular with respect to Open image in new window if Open image in new window for all subsequences Open image in new window of Open image in new window , and Open image in new window is called asymptotically uniform with respect to Open image in new window if Open image in new window for all subsequences Open image in new window of Open image in new window .

Lemma 2.1.
  1. (i)

    (See Goebel [26] and Lim [27]) There always exists a subsequence of Open image in new window which is regular with respect to Open image in new window .

     
  2. (ii)

    (See Kirk [28]) If Open image in new window is separable, then Open image in new window contains a subsequence which is asymptotically uniform with respect to Open image in new window .

     

If Open image in new window is a bounded subset of Open image in new window , then the Chebyshev radius of Open image in new window relative to Open image in new window is defined by

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

The Domínguez-Lorenzo condition, DL condition in short form, introduced in [3] is defined as follows: if there exists Open image in new window such that for every weakly compact convex subset Open image in new window of Open image in new window and for every bounded sequence Open image in new window in Open image in new window which is regular with respect to Open image in new window 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 Open image in new window be a Banach space satisfying property (D). Then Open image in new window has weak normal structure.

Theorem 2.3.

Let Open image in new window be a nonempty weakly compact convex subset of a Banach space Open image in new window which satisfies the property (D). Let Open image in new window be a nonexpansive mapping, then Open image in new window has a fixed point.

3. Main Results

Theorem 3.1.

Let Open image in new window be a weakly compact convex subset of a Banach space Open image in new window and let Open image in new window be a bounded sequence in Open image in new window regular with respect to Open image in new window . Then for every Open image in new window ,

Proof.

Denote Open image in new window and Open image in new window . We can assume that Open image in new window . By passing to a subsequence if necessary, we can also assume that Open image in new window is weakly convergent to a point Open image in new window . Since Open image in new window is regular with respect to Open image in new window , then passing through a subsequence does not have any effect to the asymptotic radius of the whole sequence Open image in new window . Let Open image in new window , then we have that
Denote Open image in new window . By the definition of Open image in new window we have that
Convexity of Open image in new window implies that Open image in new window and thus, we obtain
On the other hand, by the weak lower semicontinuity of the norm, we have that

For every Open image in new window there exists Open image in new window such that

(1) Open image in new window ,

(2) Open image in new window

(3) Open image in new window

(4) Open image in new window

By the weak lower semicontinuity of the norm again, we conclude that Open image in new window and hence,
Therefore Open image in new window Since the above inequality is true for every Open image in new window and every Open image in new window , we obtain
and therefore,

Corollary 3.2.

Let Open image in new window be a nonempty bounded closed convex subset of a Banach space Open image in new window such that Open image in new window and let Open image in new window be a nonexpansive mapping. Then Open image in new window has a fixed point.

Proof.

When Open image in new window , then Open image in new window satisfies the DL condition by Theorem 3.1. So Open image in new window has a fixed point by Theorem 2.3.

Remark 3.3.

In particular, when Open image in new window , we get the result of Kaewkhao; a Banach space Open image in new window with

satisfies the DL condition.

Theorem 3.4.

Let Open image in new window be a weakly compact convex subset of a Banach space Open image in new window and let Open image in new window be a bounded sequence in Open image in new window regular with respect to Open image in new window . Then for every Open image in new window ,

Proof.

Let Open image in new window , and Open image in new window be as in the proof of the previous theorem. Thus,
Since Open image in new window and by the definition of Open image in new window , we obtain
On the other hand, by the weak lower semicontinuity of the norm, we have that

For every Open image in new window , there exists Open image in new window such that

(1) Open image in new window

(2) Open image in new window

(3) Open image in new window

(4) Open image in new window

By the definition of Open image in new window , we get
This holds for arbitrary Open image in new window ; hence, we have that

Corollary 3.5.

Let Open image in new window be a nonempty bounded closed convex subset of a Banach space Open image in new window such that Open image in new window and let Open image in new window be a nonexpansive mapping. Then Open image in new window has a fixed point.

Proof.

When Open image in new window , then Open image in new window satisfies the DL condition by Theorem 3.4. So Open image in new window has a fixed point by Theorem 2.3.

Remark 3.6.

In particular, when Open image in new window , we get the result of Kaewkhao; a Banach space Open image in new window 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 Open image in new window be a nonempty bounded closed convex subset of a Banach space Open image in new window such that Open image in new window and let Open image in new window be a nonexpansive mapping. Then Open image in new window has a fixed point.

Remark 3.8.

In particular, when Open image in new window , we get that
satisfies the DL condition which improves the result of Kaewkhao; a Banach space Open image in new window with

satisfies the DL condition.

Theorem 3.9.

A Banach space Open image in new window has property (D) whenever Open image in new window .

Proof.

Let Open image in new window be a nonempty weakly compact convex subset of Open image in new window . Suppose that Open image in new window and Open image in new window are regular asymptotically uniforms relative to Open image in new window . Passing to a subsequence, we may assume that Open image in new window is weakly convergent to a point Open image in new window and Open image in new window exists. Let Open image in new window . Again, passing to a subsequence of Open image in new window , still denoted by Open image in new window , we assume in addition that
Remark 3.10.
  1. (1)

    Theorem 3.9 strengthens the result of Saejung [7] and Open image in new window has property (D) whenever Open image in new window .

     
  2. (2)

    Theorem 3.9 also improves the result Open image in new window implying that the Banach space Open image in new window has normal structure from Theorem 2.2.

     

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Copyright information

© Z. Zuo and Y. Cui. 2009

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceChongqing Three Gorges UniversityWanzhouChina
  2. 2.Department of MathematicsHarbin University of Science and TechnologyHarbinChina

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