# Some Sufficient Conditions for Fixed Points of Multivalued Nonexpansive Mappings

- Zhanfei Zuo
^{1, 2}Email author and - Yunan Cui
^{1, 2}

**2009**:319804

**DOI: **10.1155/2009/319804

© Z. Zuo and Y. Cui. 2009

**Received: **2 July 2009

**Accepted: **3 December 2009

**Published: **13 January 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.

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 :

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

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

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

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 .

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.

Proof.

For every there exists such that

(1) ,

(2)

(3)

(4)

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.

satisfies the DL condition.

Theorem 3.4.

Proof.

For every , there exists such that

(1)

(2)

(3)

(4)

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.

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.

satisfies the DL condition.

Theorem 3.9.

A Banach space has property (D) whenever .

Proof.

- (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.

## Authors’ Affiliations

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