#### Abstract

We use an approach on ultra-asymptotic centers to obtain fixed point theorems for two classes of nonself multivalued mappings. The results extend and improve several known ones.

#### 1. Introduction

Domínguez Benavides and Lorenzo Ramírez  introduced a new method to prove the existence of fixed points using asymptotic centers as main tools by comparing their asymptotic radii with various geometric moduli of Banach spaces. The method led Dhompongsa et al.  to define the (DL) condition and obtained a fixed point theorem by following the proof in .

Definition 1.1 (see [4, Definition  3.1]). A Banach space is said to satisfy the Domínguez-Lorenzo condition ((DL) condition, in short), if there exists such that for every weakly compact convex subset of and for every bounded sequence in which is regular relative to ,

Theorem 1.2 (see [4, Theorem  3.3]). Let be a reflexive Banach space satisfying the condition, and let be a bounded closed and convex separable subset of . If is a nonexpansive and -contractive mapping such that is a bounded set and which satisfies the inwardness condition:, then has a fixed point.

Wiśnicki and Wośko  introduced an ultrafilter coefficient for a Banach space and presented a fixed point result under a stronger condition than the condition.

Definition 1.3 (see [5, Definition  5.2]). Let be a free ultrafilter defined on the set of natural numbers . The coefficient of a Banach space is defined as where the supremum is taken over all nonempty weakly compact convex subsets of and all weakly, not norm-convergent sequences in which are regular relative to .

Theorem 1.4 (see [5, Theorem  5.3]). Let be a nonempty weakly compact convex subset of a Banach space with . Assume that is a nonexpansive and -contractive mapping such that for all . Then has a fixed point.

Observe that, unlike Theorem 1.2, it does not assume the “separability” condition on in Theorem 1.4. However, it should be mentioned that the idea of the proof came from the original one of Domínguez Benavides and Lorenzo Ramírez . At the same time, with the same purpose, Garvira  introduced independently its counter part in terms of ultranets.

Definition 1.5 (see [6, Definition  3.1.2]). A Banach space is said to have the condition with respect to a topology condition) if there exists such that for every -compact convex subset of and for every bounded ultranet in When is the weak topology we write condition instead of condition.

It follows from [6, Proposition  3.1.1] that the condition is stronger than the condition. The ultranet counterpart of Theorem 1.4 becomes:

Theorem 1.6 (special case of [6, Theorem  3.5.2]). Let be a Banach space satisfying the condition. Let be a weakly compact convex subset of . If is a nonexpansive and -contractive mapping such that for , then has a fixed point.

In 2006, Dhompongsa et al.  introduced a property for a Banach space .

Definition 1.7 (see [7, Definition  3.1]). A Banach space is said to have property if there exists such that for any nonempty weakly compact convex subset of , any sequence which is regular and asymptotically uniform relative to , and any sequence which is regular and asymptotically uniform relative to one has

Theorem 1.8 (see [7, Theorem  3.6]). Let be a nonempty weakly compact convex subset of a Banach space which has property . Assume that is a nonexpansive mapping. Then has a fixed point.

The following definition is due to Butsan et al. .

Definition 1.9 (see [8, Definition  3.1]). Let be a mapping on a subset of a Banach space . Then is said to satisfy condition if (1)for each -invariant subset of , has an afps in , (2)for each pair of -invariant subsets and of , is -invariant for each afps in .

The main fixed point theorem concerning condition in  is Theorem 3.5 of which the correct statement should be stated as follows.

Theorem 1.10 (see [8, Theorem  3.5]). Let be a Banach space having property , and let be a weakly compact convex subset of . Let satisfy conditon . If is continuous, then has a fixed point.

In fact, we can replace “continuity” by a weaker condition, namely “ is strongly demiclosed at 0”: for every sequence in strongly converges to and such that we have (cf. ).

Following the concept of , Dhompongsa and Inthakon  introduced the following coefficient.

Definition 1.11 (see [10, Definition  3.2]). Let be a free ultrafilter defined on . The coefficient of a Banach space is defined as where the supremum is taken over all nonempty weakly compact convex subsets of , all sequences in which are weakly, not norm-convergent and are regular relative to and all weakly, not norm-convergent sequences which are regular relative to .

A concept corresponding to the coefficient is the following property.

Definition 1.12 (see [10, Definition  3.1]). A Banach space is said to have property if there exists such that for any nonempty weakly compact convex subset of , any sequence which is regular relative to , and any sequence which is regular relative to one has

Obviously, . Several well-known spaces have been proved to satisfy the condition and to have property (see, e.g., [1, 4, 7, 1116]). The class of spaces having property includes both spaces satisfying the condition as well as spaces satisfying the Kirk-Massa condition (see Section 3.2 for the definition of the condition). One of the main results in  is the following theorem.

Theorem 1.13 (see [10, Theorem  1.9]). Let be a nonempty weakly compact convex subset of a Banach space , and let have property . Assume that is a nonexpansive and -contractive mapping such that for every . Then has a fixed point.

Due to Kuczumow and Prus , we can assume without loss of generality that in Theorem 1.8 is separable. Theorem 1.13 does not only extend Theorem 1.8 to nonself-mappings, but it can remove “separability” condition on the domains of the mappings in Theorem 1.2 without refering to ultrafilters or ultranets in its statement. The key to the proof of Theorem 1.13 is the following.

Theorem 1.14 (see [10, Theorem  3.4]). Let be a weakly compact convex subset of a Banach space and a free ultrafilter defined on . Then if and only if has property .

Thus, unlike the coefficient , we have, for any two free ultrafilters and , if and only if .

In Section 3.1, we extend Theorem 1.10 to multivalued nonself-mappings for spaces having property . Examples of such mappings are given. To obtain a fixed point result for more mappings, a new class of mappings is introduced in Section 3.2. Some examples of those mappings are also given. Our results extend and improve several known results in [810, 1825] and many others (see Remark 3.4).

#### 2. Preliminaries

Let be a nonempty closed and convex subset of a Banach space . We will denote by the family of all subsets of , the family of all nonempty bounded and closed subsets of and denote by the family of all nonempty compact convex subsets of . For a given mapping the set of all fixed points of will be denoted by , that is, . Let be the Hausdorff distance defined on , that is,

where is the distance from a point to a subset . A multivalued mapping is said to be nonexpansive if

and is said to be a contraction if there exists a constant such that A multivalued mapping is called -condensing (resp., -contractive), where is a measure of noncompactness, if for each bounded subset of with , there holds the inequality

where .

Recall that the inward set of at is defined by A sequence in for which for a mapping is called an approximate fixed point sequence (afps for short) for . Analogously for a multivalued mapping , a sequence in of a Banach space for which is called an approximate fixed point sequence (afps for short) for .

We denote by to indicate that the sequence in converges to .

Let be a nonempty closed and convex subset of a Banach space and a bounded sequence in . For , define the asymptotic radius of at as the number Let The number and the set are, respectively, called the asymptotic radius and asymptotic center of relative to . The sequence is called regular relative to if for each subsequence of . It was noted in  that if is nonempty and weakly compact, then is nonempty and weakly compact, and if is convex, then is convex.

Proposition 2.1 (see [27, Theorem  1]). Let and be as above. Then there exists a subsequence of which is regular relative to .

We now present the formulation of an ultrapower of Banach spaces. Let be a free ultrafilter on . Recall ([26, 2830]) that the ultrapower of a Banach space is the quotient space of by One can proof that is a Banach space with the quotient norm given by , where is the equivalence class of . It is also clear that is isometric to a subspace of by the canonical embedding . If , we will use the symbols and to denote the image of and in under this isometry, respectively, and denote

Thus and .

If is a multivalued mapping, we define a corresponding multivalued mapping by

where . Moreover the set is bounded and closed (see [28, 29]). The Hausdorff metric on will be denoted by .

Proposition 2.2 (see [5, Proposition  3.1]). For every and in ,

Proposition 2.3 (see [31, Page 37], [5, Proposition  3.2]). Let be a nonempty subset of a Banach space and . (i)If is convex-valued, then is convex-valued. (ii)If is compact-valued, then is compact-valued and for every . (iii)If is nonexpansive, then is nonexpansive.

Let denote a free ultrafilter defined on . Wiśnicki and Wośko  defined the ultra-asymptotic radius and the ultra-asymptotic center of relative to by It is not difficult to see that is a nonempty weakly compact convex set if is. Notice that the above notions have a natural interpretation in the ultrapower :

is the relative Chebyshev radius of , and

is the relative Chebyshev center of relative to in the ultrapower . (Here denotes the ball in centered at and of radius .) It should be noted that, in general, and may be different. The notion of the asymptotic radius is closely related to the notion of the relative Hausdorff measure of noncompactness defined by Domínguez Benavides and Lorenzo Ramírez  as

Proposition 2.4 (see [5, Proposition  4.5]). If is a bounded sequence which is regular relative to , then

From Proposition 2.4, we have, for ,

Therefore, .

The following result plays an important role in our proofs.

Lemma 2.5 (see [10, Lemma  3.3]). Let be a nonempty closed and convex subset of a Banach space and a bounded sequence in which is regular relative to . For each , there exists a subsequence of such that .

A direct consequence of Lemma 2.5 is as follows. If every center is compact for every bounded sequence in which is regular relative to , then is also compact for every bounded sequence in which is regular relative to .

#### 3. Main Results

##### 3.1. Property (𝐷′)

Lemma 3.1. Let be a nonempty subset of a Banach space and . Then (1)if is uniformly continuous, then is uniformly continuous; (2)if is continuous at , then is continuous at .

Proof. (1) Let . Since is uniformly continuous, there exists such that for each with . Suppose , and . Let and . Since and , . Thus, by Proposition 2.2.
(2) Let . Since is continuous at , there exists such that for each with . If such that , then, letting and , we see that and . Thus, by Proposition 2.2.

We now introduce condition for multivalued mappings.

Definition 3.2. Let be a nonempty subset of a Banach space . A mapping is said to satisfy condition if (1) has an afps in , (2) has an afps in for some subsequence of any given afps for in .

Theorem 3.3. Let be a Banach space having property , and let be a weakly compact convex subset of . Assume that is a multivalued mapping satisfying condition . If is continuous, then has a fixed point.

Proof. The proof follows by adapting the proof of [10, Theorem  1.9]. By (1) of Definition 3.2, let be an afps for in . We can assume by Proposition 2.1 that is regular relative to . Condition (2) of Definition 3.2 gives us a subsequence of so that the center contains an afps for . Denote , and let be an afps in . Assume that is regular relative to . As before, has an afps in for some subsequence of . Since has property , put . Then, by Proposition 2.4 and Definition 1.11, Continue the procedure to obtain, for each , a regular sequence relative to in such that and for all , Consequently, We show that is a Cauchy sequence in . Indeed, for each , take an element . Then for all , and hence Thus implying that is a Cauchy sequence and hence converges to some in as . Next, we will show that . For each , Taking we see that Thus, it follows that there exists such that . By Lemma 3.1, is continuous at , and thus as . For every , Taking we then obtain . By Proposition 2.3, , and therefore, .

Remark 3.4. The proof presented here based on a standard proof appeared in a series of papers [1, 3, 5, 10]. However, we cannot follow its proof directly to be able to obtain a result for larger classes of spaces and mappings. We choose an ultralimit approach by using an ultra-asymptotic center as our main tool. As mentioned earlier, this powerful tool was introduced in  by Wiśnicki and Wośko. Thus our proof may not be totally new, but it significantly improves, generalizes, or extends many known results. (i)Theorem 3.3 (as well as Theorem 3.16) unifies many known theorems in one. Examples of mappings in both theorems are given throughout the rest of the paper. (ii)Theorem 3.3 improves condition in [8, Definition  3.1] in which the mappings under consideration only are single valued and are self-mappings. Consequently [8, Theorem  3.5] is improved significantly. Obviously, [10, Theorem  1.9] is a special case of Theorem 3.3. (iii)In Remark 3.15(ii) below, we show the following implication:Thus results in [18, Corollaries  3.5 and  3.6], [19, Theorems  1 and 2], [20, Theorem  3.3], [9, Theorem  5], [21, Theorem  2.4], [22, Theorem  2], [23, Theorem  4.2, Corollary   4.3, Theorem  4.4], [24, Theorem  2.6], and [25, Theorem  2.3.1] are either improved, generalized, or extended. See Remark 3.17, Corollaries 3.18 and 3.19.See also Remark 3.24(iii) and (iv).

We now give some examples of mappings satisfying condition . We will see that the ultracenter plays a significant role in verifying condition (2) of condition for a given mapping.

Nonexpansive Mappings
We will show by following the proof of Theorem  5.3 in  that if is nonexpansive and -contractive such that for every . Then satisfies condition . The main tools are Lemma 2.5 and the following result.
Theorem 3.5 (see [32, Theorem  11.5]). Let be a nonempty bounded closed and convex subset of a Banach space and an upper semicontinuous and -condensing mapping. If for all , then has a fixed point.
Proposition 3.6. Let be a nonempty weakly compact convex subset of a Banach space . Assume that is nonexpansive and -contractive such that for every . Then satisfies condition .
Proof. First, we will show that has an afps in . Let , and consider, for each , the contraction defined by It is not difficult to see that for every . Since is -contractive, is -contractive, and by Theorem 3.5, there exists a fixed point of . Clearly, is an afps for in .
Next, let us see that has an afps in for some subsequence of an afps for in . Let be an afps in . By Proposition 2.1, we can assume that is weakly convergent and regular relative to . Let . We show that Let . Observe first that . By Proposition 2.3, is compact, and hence there exists such that Since , there exists and such that . If then , and it follows from (3.14) that . If then , and therefore, we have Hence and consequently . Thus (3.13) is justified.
Fixed , and consider for each , the contraction defined by As before, is -contractive, and by Theorem 3.5, there exists a fixed point of . Again, as above, is an afps for in . By Lemma 2.5, there exists a subsequence of such that .

Diametrically Contractive Mappings
In  Istratescu introduced a new class of mappings.
Definition 3.7 (see ). A mapping defined on a complete metric space is said to be diametrically contractive if for all closed subsets with . (Here denotes the diameter of .)
Xu  proved the fixed point theorem for a diametrically contractive mapping in the framework of Banach spaces.
Theorem 3.8 (see [34, Theorem  2.3]). Let be a weakly compact subset of a Banach space , and let be a diametrically contractive mapping. Then has a fixed point.
Dhompongsa and Yingtaweesittikul  defined a multivalued version of mappings in Theorem 3.8 which is weaker than the condition required in Definition 3.7. Recall that and is said to be -invariant if for all .
Theorem 3.9 (see [35, Theorem  2.2]). Let be a weakly compact subset of a Banach space , and let be a multivalued mapping such that for all closed sets with and is invariant under . Then has a unique fixed point.
The following result extends Theorem 3.9 partially.
Proposition 3.10. Let be a nonempty weakly compact convex subset of a Banach space , and let be a multivalued mapping such that for all closed sets with and is invariant under . Then satisfies condition .
Proof. First, we will see that has an afps in . Let , and consider, for each , the contraction defined by For , let . Thus , and therefore, for every . We show that . Let be a closed subset of with . For , there exist such that and this entails . Hence . By Theorem 3.9, there exists a fixed point of , and thus the sequence forms an afps for in .
Next, let us see that has an afps in for some subsequence of an afps for in . Let be an afps in . We can assume that is weakly convergent and regular relative to . Let . First, we show that Let , and for each , we see that . Take so that and select for each such that Let . Note that We obtain proving that . Thus (3.19) is satisfied. Fix , and consider, for each , the contraction defined by For , let . Thus . Therefore, for every . Let be a closed subset of with . As before, . By Theorem 3.9 (or we can apply Theorem 3.5), there exists a fixed point of . Again, as above, is an afps for in . Finally, by Lemma 2.5, there exists a subsequence of such that .

##### 3.2. Kirk-Massa Condition

In 1990, Kirk and Massa  generalized Lim's Theorem  using asymptotic centers of sequences and nets and obtained the following result.

Theorem 3.11 (Kirk and Massa theorem). Let be a nonempty bounded closed and convex subset of a Banach space and a nonexpansive mapping. Suppose that the asymptotic center in of each bounded sequence of is nonempty and compact. Then has a fixed point.

We call the assumption in Kirk and Massa theorem the Kirk-Massa condition. Xu  extended Kirk and Massa theorem to nonexpansive nonself-mappings.

Theorem 3.12 ([25, Theorem  2.3.1]). Let be a Banach space satisfying the Kirk-Massa condition and let be a nonempty bounded closed and convex subset of . Let be a nonexpansive mapping which satisfies the inwardness condition. Then has a fixed point.

Remark 3.13. Obviously, every space that satisfies the Kirk-Massa condition always has property . Thus, particularly, the fixed point result in Section 3.1 holds for uniform convex Banach spaces, uniformly convex in every direction (UCED) and spaces satisfying the Opial condition.

We aim to extend Xu's result to a wider class of mappings. Thus, the domains of mappings are more general than the ones in Section 3.1.

Definition 3.14. Let be a free ultrafilter defined on . Let be a bounded closed and convex subset of a Banach space . A mapping is said to satisfy condition if it fulfills the following conditions. (1) has an afps in ; (2)if is an apfs for in and , then .

Remark 3.15. (i)  Let be a bounded closed and convex subset of a Banach space , and let a mapping satisfy condition . If in addition, satisfies the following:
(A)  every -invariant, closed, and convex subset possesses an afps,
then satisfies condition .

Proof. By (1) of condition , let be an apfs for in . From Proposition 2.1 by passing through a subsequence, we may assume that is regular relative to . Let and . The compactness of implies that for each we can take so that Since is compact, select for each such that Let . Note that We obtain proving that and hence for all , that is, is -invariant. By assumption, there exists an afps in . By Lemma 2.5, there exists a subsequence of such that . Thus, satisfies condition .

We wonder if we can drop condition in proving the implication: . An example of a mapping satisfies condition but not condition is given in Remark 3.24(i).

(ii)  In [23, Definition  3.1] the following concept of mappings is defined: a mapping satisfies condition on provided that it fulfills the following two conditions. (1)If a set is nonempty, closed, convex, and -invariant, then there exists an afps for in . (2)For any afps of in and each ,

Therefore, (i) shows that the class of mappings satisfying condition contains and extends mappings satisfying condition as a multivalued nonself version.

The main idea of the proof of the following theorem is originated from Kirk and Massa .

Theorem 3.16. Let be a Banach space satisfying the Kirk-Massa condition, and let be a nonempty bounded closed and convex subset of . Let be a multivalued mapping satisfying condition . If is an upper semicontinuous mapping, then has a fixed point.

Proof. Let be an afps for in . From Proposition 2.1 by passing through a subsequence, we may assume that is regular relative to . Let . The compactness of implies that for each we can take such that If , since is compact, select for each such that Let . Note that Thus proving that and hence for all . By assumption, is nonempty and compact which implies that is also nonempty and compact. Now define a mapping by for all . Thus is upper semicontinuous. Indeed, let be such that , and let be such that . Since is upper semicontinuous and is compact, we have and , that is . By the Bohnenblust-Karlin fixed point theorem , and hence , have a fixed point in .

Remark 3.17. If, in addition, mappings in Theorem 3.16 also satisfy condition , then the condition on “upper semicontinuity” can be dropped. This is because an afps in a compact set can be chosen so that its asymptotic center is only a singleton, and a fixed point can be easily derived. Consequently, Theorem 3.3 can be extended to a bigger class of domains, namely, the bounded, closed, and convex ones. And the following results are immediate.

Corollary 3.18 (see [23, Theorem  4.2]). Let be a nonempty compact convex subset of a Banach space and a mapping satisfying condition . Then, has a fixed point.

Corollary 3.19 (see [23, Corollary  4.3]). Let be a nonempty compact convex subset of a Banach space and a mapping satisfying condition . Suppose that the asymptotic center in of each sequence in is nonempty and compact. Then, has a fixed point.

We give some examples of mappings satisfying condition . The first example is of course the mapping described in Theorem 3.12.

Condition 3.2 (). García-Falset et al.  introduced the following mappings.
Let E be a nonempty subset of a Banach space . For , we say that a mapping satisfies Condition on if, for each , It is natural to define a multivalued version of Condition (see ).
Let be a nonempty subset of a Banach space , and let be a multivalued mapping. Then is said to satisfy condition for some if, for each , Clearly, satisfies (2) of condition .

Proposition 3.20. Let be a nonempty bounded closed and convex subset of a Banach space . If satisfies condition for some , then satisfies condition .

Proof. We only show that contains an afps for . But this follows from [37, Lemma  2.8].

Generalized Nonexpansive Mappings
Let be a nonempty subset of a Banach space . Following , a mapping is a generalized nonexpansive mapping if for some nonnegative constants with , for each .

We will use the following equivalent condition.

For some nonnegative constants with ,

for all .

We introduce a multivalued version of these mappings.

Let be a multivalued mapping. is called a generalized nonexpansive mapping if there exist nonnegative constants with such that, for each , there holds

Proposition 3.21. Let be a nonempty subset of a Banach space . If is a generalized nonexpansive mapping, then satisfies (2) of condition .

Proof. Let be an afps for in and . By assumption we obtain Since ,, , By (3.39), Thus and therefore,

Takahashi Generalized Nonexpansive Mappings
Definition 3.22. Let be a nonempty subset of a Banach space . A mapping is said to be a Takahashi generalized nonexpansive mapping if, for some with , there holds
The following are examples of Takahashi generalized nonexpansive mappings: (i)nonexpansive mappings ; (ii)nonspreading mappings : ; (iii)hybrid mappings : ; (iv)mappings : ; (v)mappings : .
We define a multivalued version of Takahashi generalized nonexpansive mappings and prove that these mappings satisfy (2) of condition .
Proposition 3.23. Let be a nonempty subset of a Banach space . For nonnegative constants with , if is a multivalued mapping such that then satisfies (2) of condition .
Proof. Let be an afps for in and . By (3.41) and (3.42), Thus Therefore,
Remark 3.24. (i)  A mapping that satisfies condition need not satisfy condition . Consider a mapping defined by . Since 0 is a fixed point of , the sequence given by for all forms an afps for . Thus, fulfills condition (1) of Definition 3.2. If is an apfs for , then converges to 0 and . This implies that has an apfs for , and satisfies condition . On the other hand, for the afps given by , if , then Thus, fails to satisfy condition .
As mentioned earlier, it is unclear if a mapping, satisfies condition also satisfies condition .
(ii)  We do not know if Theorem 3.16 is still valid when “” in Definition 3.14 is replaced by “.” It is possible that the theorem holds true when the domain is separable. Indeed, by [27, 40] and Kirk , we assume the afps for to be regular and asymptotically uniform relative to . Thus, for all subsequences of . Therefore, by Lemma 2.5, it is easy to see that . If , then (3.33) becomes Hence , and the rest of the proof follows.
It is observed that if and is a singleton, then is automatically a fixed point of . Thus our method provides another proof of Lim [42, Theorem  8], where is a uniformly convex Banach space, assumes only compact values and need not be separable.
(iii)  If is a generalized nonexpansive mapping with any of the following conditions holding, then satisfies condition :(1) (see [43, Theorem  4]);(2) and (see [19, Theorem  1]);(3) and (see [44, Theorem  1.1]);(4) and (see [45, Lemma  2.1]).
(iv)  Regarding the proof of Theorem 3.16, the fixed point result also holds for weak*-nonexpansive mappings (see [46, Definition  1.3]). Thereby [46, Theorem  1.7] is extended to another circumstance.

#### Acknowledgment

The authors wish to thank the National Research University Project under Thailand's Office of the Higher Education Commission for financial support.