Fixed Point Theorems for Suzuki Generalized Nonexpansive Multivalued Mappings in Banach Spaces

Abstract

In the first part of this paper, we prove the existence of common fixed points for a commuting pair consisting of a single-valued and a multivalued mapping both satisfying the Suzuki condition in a uniformly convex Banach space. In this way, we generalize the result of Dhompongsa et al. (2006). In the second part of this paper, we prove a fixed point theorem for upper semicontinuous mappings satisfying the Suzuki condition in strictly spaces; our result generalizes a recent result of Domínguez-Benavides et al. (2009).

1. Introduction

A mapping on a subset of a Banach space is said to be nonexpansive if

(1.1)

In 2008, Suzuki [1] introduced a condition which is weaker than nonexpansiveness. Suzuki's condition which was named by him the condition reads as follows: a mapping is said to satisfy the condition on if

(1.2)

He then proved some fixed point and convergence theorems for such mappings. We shall at times refer to this concept by saying that is a generalized nonexpansive mapping in the sense of Suzuki. Very recently, the current authors used a modified Suzuki condition for multivalued mappings and proved a fixed point theorem for multivalued mappings satisfying this condition in uniformly convex Banach spaces (see [2]).

In this paper, we first present a common fixed point theorem for commuting pairs consisting of a single-valued and a multivalued mapping both satisfying the Suzuki condition. This result extends a result of Dhompongsa et al. [3].

In the next part, we shall consider a recent result of Domínguez-Benavides et al. [4] on the existence of fixed points in an important class of spaces which are usually called strictly spaces. These spaces contain all Lebesgue function spaces for . In this paper, we also generalize results of Domínguez-Benavides et al. [4] to upper semicontinuous mappings satisfying the Suzuki condition.

2. Preliminaries

Given a mapping on a subset of a Banach space , the set of its fixed points will be denoted by

(2.1)

We start by the following definition due to Suzuki.

Definition 2.1 (see [1]).

Let be a mapping on a subset of a Banach space . The mapping is said to satisfy the Suzuki condition if

(2.2)

As the following example shows, the Suzuki condition is weaker than nonexpansiveness. Therefore, it is natural to call these mappings as "generalized nonexpansive mappings". However, we shall at times refer to these mappings as those satisfying the condition .

Example 2.2.

Let be equipped with the usual metric , and let . We put

(2.3)

The mapping is continuous and satisfies the condition . However, is not nonexpansive.

Lemma 2.3 (see [1, Lemma ]).

Let be a mapping defined on a closed subset of a Banach space . Assume that satisfies the condition . Then is closed. Moreover, if is strictly convex and is convex, then is also convex.

Theorem 2.4 (see [5, Theorem ]).

Let be a nonempty bounded closed convex subset of a uniformly convex Banach space . Let be a mapping satisfying the condition (). Then has a fixed point.

Let be a metric space. We denote by the collection of all nonempty closed bounded subsets of ; we also write to denote the collection of all nonempty compact convex subsets of . Let be the Hausdorff metric with respect to , that is,

(2.4)

for all where

Let be a multivalued mapping. An element is said to be a fixed point of provided that

Definition 2.5.

A multivalued mapping is said to be nonexpansive provided that

(2.5)

Suzuki's condition can be modified to incorporate multivalued mappings. This was done by the current authors in [2]. We call these mappings generalized multivalued nonexpansive mappings in the sense of Suzuki or multivalued mappings satisfying the condition . We now state Suzuki's condition for multivalued mappings as follows.

Definition 2.6.

A multivalued mapping is said to satisfy the condition provided that

(2.6)

Example 2.7.

Define a mapping on by

(2.7)

It is not difficult to see that satisfies the Suzuki condition; however, is not nonexpansive.

The following lemma, proved by Goebel and Kirk [6], plays an important role in the coming discussions.

Lemma 2.8.

Let and be two bounded sequences in a Banach space , and let . If for every natural number we have and , then .

Definition 2.9.

A multivalued mapping is said to be upper semicontinuous on if is open in whenever is open.

We recall that if is single valued, then reduces to a continuous function.

3. Fixed Points in Uniformly Convex Banach Spaces

Let be a nonempty closed convex subset of a Banach space . Assume that is a bounded sequence in . For each , the asymptotic radius of at is defined by

(1)

Let

(3.2)

The number is known as the asymptotic radius of relative to . Similarly, the set is called the asymptotic center of relative to . In the case that is a reflexive Banach space and is a nonempty closed convex bounded subset of , the set is always a nonempty closed convex subset of . To see this, observe that by the definition of , for each , the set

(3.3)

is nonempty. It is not difficult to see that each is closed and convex; hence

(3.4)

is closed and convex. Moreover, it follows from the weak compactness of that is nonempty. It is easy to see that if is uniformly convex and if is a closed convex subset of , then consists of exactly one point.

A bounded sequence is said to be regular with respect to if for every subsequence we have

(3.5)

It is also known that if is uniformly convex and if is a nonempty closed convex subset of , then for any , there exists a unique point such that .

The following lemma was proved by Goebel and Lim.

Lemma 3.1 (see [7, 8]).

Let be a bounded sequence in and let be a nonempty closed convex subset of . Then has a subsequence which is regular relative to .

Definition 3.2.

Let be a nonempty closed convex bounded subset of a Banach space , and let and be two mappings. Then and are said to be commuting mappings if for every such that and , we have .

Now the time is ripe to state and prove the main result of this section.

Theorem 3.3.

Let be a nonempty closed convex bounded subset of a uniformly convex Banach space . Let be a single-valued mapping, and let be a multivalued mapping. If both and satisfy the condition () and if and are commuting, then they have a common fixed point, that is, there exists a point such that .

Proof.

By Theorem 2.4, the mapping has a nonempty fixed point set which is a closed convex subset of (by Lemma 2.3). We show that for , . To see this, let ; since and are commuting, we have for each . Therefore, is invariant under for each . Since is a bounded closed convex subset of the uniformly convex Banach space , we conclude that has a fixed point in and therefore, for

Now we find an approximate fixed point sequence for in . Take , since , therefore, we can choose . Define

(3.6)

Since is a convex set, we have . Let be chosen in such a way that

(3.7)

We see that Indeed, we have

(3.8)

Since satisfies the condition , we get

(3.9)

which contradicts the uniqueness of as the unique nearest point of (note that ). Similarly, put ; again we choose in such a way that

(3.10)

By the same argument, we get In this way, we will find a sequence in such that where and

(3.11)

Therefore, for every natural number we have

(3.12)

from which it follows that

(3.13)

Our assumption now gives

(3.14)

hence

(3.15)

We now apply Lemma 2.8 to conclude that where . Moreover, by passing to a subsequence we may assume that is regular (see Lemma 3.1). Since is uniformly convex, is singleton, say (note that . Let . For each , we choose such that

(3.16)

On the other hand, there is a natural number such that for every we have . This implies that

(3.17)

and hence from our assumption we obtain

(3.18)

Therefore,

(3.19)

Moreover, for all natural numbers Indeed, since

(3.20)

we have

(3.21)

Since and , by the fact that the mappings and are commuting, we obtain . Now, by the uniqueness of as the nearest point to , we get

Since is compact, the sequence has a convergent subsequence with . Because for all , and is closed, we obtain . Note that

(3.22)

and for we have . This entails

(3.23)

Since is regular, this shows that . And hence .

As a consequence, we obtain the theorem already proved by Dhompongsa et al. (see [3, Theorem ]).

Corollary 3.4.

Let be a nonempty closed convex bounded subset of a uniformly convex Banach space , , and a single-valued and a multivalued nonexpansive mapping, respectively. Assume that and are commuting mappings. Then there exists a point such that .

Corollary 3.5.

Let be a nonempty bounded closed convex subset of a uniformly convex Banach space , and let be a multivalued mapping satisfying the Suzuki condition (C). Then has a fixed point.

Corollary 3.6.

Let be a nonempty closed convex bounded subset of a uniformly convex Banach space , and let be a nonexpansive multivalued mapping. Then has a fixed point.

4. Strictly Spaces

Definition 4.1.

Let be a Banach space and let be a linear topology on . We say that is a strictly space if there exists a function satisfying the following

(i) is continuous;

(ii) is strictly increasing;

(iii), for every

(iv) is strictly increasing;

(v), for every and for every bounded and -null sequence , where is defined by

(4.1)

In this case we also say that satisfies the strict property with respect to

Example 4.2 (see [9]).

Let be a positive -finite measure space. For every , consider the Banach space with the usual norm. Let be the topology of convergence locally in measure (clm). Then endowed with the clm-topology satisfies the property with

Definition 4.3.

Let be a Banach space and let be a linear topology on which is weaker than the norm topology. Let be a closed convex bounded subset of ; then for we write We say that has property () if for every the set is a nonempty and norm-compact subset of .

Theorem 4.4.

Let be a strictly Banach space and let be a nonempty closed convex bounded subset of satisfying the property . Suppose, in addition, that is -sequentially compact. If satisfies the condition , and if is an upper semicontinuous mapping, then has a fixed point.

Proof.

First, we find an approximate fixed point sequence. Choose and . Define

(4.2)

Let be chosen in such a way that

(4.3)

Similarly, put ; again we choose in such a way that

(4.4)

In this way, we will find a sequence in such that , where and

(4.5)

Therefore, for every natural number we have

(4.6)

from which it follows that

(4.7)

Our assumption now gives

(4.8)

Hence

(4.9)

We now apply Lemma 2.8 to conclude that where . Since is -sequentially compact, by passing to a subsequence, we may assume that is -convergent to Now we are going to show that

(4.10)

Taking any , by the compactness of , we can find such that . On the other hand, there is a natural number such that for every we have . This implies that

(4.11)

and hence from the assumption we obtain

(4.12)

Therefore,

(4.13)

Since is compact, the sequence has a convergent subsequence with . It follows that

(4.14)

On the other hand, we have that Since is strictly increasing, it follows that Hence and so . Now we define the mapping

(4.15)

by . From [10, Proposition ], we know that the mapping is upper semicontinuous. Since is a compact convex set, we can apply the Kakutani-Bohnenblust-Karlin Theorem (see [11]) to obtain a fixed point for and hence for .

Corollary 4.5.

Let be a strictly Banach space and let be a nonempty closed convex bounded subset of satisfying the property . Suppose, in addition, that is -sequentially compact. If is a nonexpansive mapping, then has a fixed point.

Corollary 4.6.

Let be a strictly Banach space and let be a nonempty closed convex and bounded subset of satisfying the property . Suppose, in addition, that is -sequentially compact. If is a continuous mapping satisfying the condition (C), then has a fixed point.

Finally we mention that by Example 2.2, this corollary generalizes the recent result of Domínguez-Benavides et al. [4].