On Some Geometric Constants and the Fixed Point Property for Multivalued Nonexpansive Mappings

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.

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

implies property (D).

(3)Satit Saejung [5] proved that the condition implies property (D).

(4)Gavira [6] showed that the condition

implies (DL) condition.

In 2007, Domínguez 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, Domínguez 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.

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

holds for every , where is the Hausdorff distance on , that is,

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,

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

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 ,

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 ,

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 Domínguez Benavides [12] as

where the supremum is taken over all with and all weakly null sequence in such that

Obviously, .

The weakly convergent sequence coefficient is equivalently defined by (see [13])

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 towith 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

Let be an ultrafilter on and let

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

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

We first give some sufficient conditions which imply (DL) condition. The Jordan-von Neumann constant was defined in 1937 by Clarkson [15] as

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

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

Let ; taking a subsequence if necessary, we can assume that for all .

Let . Then we have and . Denote ; by definition, we have

On the other hand, observe that the convexity of implies ; since the norm is weak lower semicontinuity, we have

In the ultrapower of , we consider

Using the above estimates, we obtain

Therefore, we have

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

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 Zbganu constant: (see [16])

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

the set belongs to . In particular, noticing that for , there exists such that

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

Proof.

Let be as in Theorem 3.1. Then

Therefore, by the definition of Zbganu constant, we have

Since Zbganu constant of equals to of , we obtain

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

Let us consider an ultrapower of . Put

then we know that . We see that

Thus, By the definition of Zbganu constant, we have

Since the Zbganu constants of and of are the same, we obtain . Now we estimate as follows:

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

for any bounded subset of a Banach space , where

The modulus of noncompact convexity associated to is defined in the following way:

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

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

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

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

By the definition of , we have

Since the last inequality is true for any , we obtain ; thus . Now we estimate as follows:

Hence,

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.

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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