1. Introduction

This paper is motivated by the recent paper [1]. In [1] the authors study different questions related to fixed points of asymptotic pointwise contractive/nonexpansive mappings in CAT(0) spaces. CAT(0) spaces are studied in [1] as a very significant example within the class of uniformly convex metric spaces (the reader can consult [2] for details on CAT(0) spaces). In our present paper we propose to consider similar questions on uniformly convex metric spaces under the mildest additional conditions we may impose. More precisely, we will work with uniformly convex metric spaces with either a monotone modulus of convexity in the sense first given in [3] or a lower semicontinuous from the right modulus of convexity (see Section 2 for proper definitions). For a recent survey on the existence of fixed points in geodesic spaces, the reader may check [4], for recent achievements on related topics the reader may also check [5].

The notion of asymptotic pointwise contractions was introduced in [6]. Then it was also studied in [7] where, by means of ultrapower techniques, different results about the existence of fixed points and convergence of iterates were proved. In [8] new proofs were presented but this time after applying only elementary techniques. Very recently, in [1], these techniques were applied in CAT(0), where the authors attend to the Bruhat-Tits inequality for CAT(0) spaces in order to obtain such results. In the present paper we show that actually most of those results still hold for general uniformly convex metric spaces under mild conditions on the modulus of convexity. In Section 3 we focus on single-valued mappings and, in particular, on mappings which are asymptotically pointwise contractive/nonexpansive to study the existence of fixed points, convergence of Picard's iterates, and the structure of their sets of fixed points. As a technical result we need to show that bounded sequences in these spaces have a unique asymptotic center which, as a by-product, leads to Kirk's Fixed Point Theorem. In Section 4 we study different problems regarding set-valued mappings in these spaces. The main technical difficulty to achieve similar results to those shown in [1] is that now we cannot count on the existence of fixed points for nonexpansive set-valued mappings for the kind of spaces we deal with. Finding fixed point for set-valued nonexpansive mappings in uniformly convex metric spaces was first studied by Shimizu and Takahashi [9], where the existence of fixed points was guaranteed under stronger conditions on the modulus of convexity and the additional condition of metric convexity of the space. The fact that we do not have that the metric are convex will make the problem more complicated and this will take us to impose new conditions on the modulus of convexity which we will relate with the geometry of the space.

2. Basic Definitions and Results

We introduce next some basic definitions.

Definition 2.1. Let be a metric space. A mapping is called a pointwise contraction if there exists a mapping such that for any .

It is proved in [8] (see also [6]) that if is a weakly compact convex subset of a Banach space and is a pointwise contraction, then has a unique fixed point and the sequence of the iterates of converges to the fixed point for any . As it is pointed out in [1], the uniqueness of fixed points and convergence of iterates for these mappings directly follow if existence is guaranteed.

Definition 2.2. Let be a metric space. Let be a mapping, and let for each be such that Then(i) is called an asymptotic pointwise contraction if converges pointwise to ;(ii) is called an asymptotic pointwise nonexpansive mapping if for any ;(iii) is called a strongly asymptotic pointwise contraction if , with , for any .

In this paper we will mainly work with uniformly convex geodesic metric space. Since the definition of uniform convexity requires the existence of midpoints, the word geodesic is redundant and so, for simplicity, we will usually omit it.

Definition 2.3. A geodesic metric space is said to be uniformly convex if for any and any there exists such that for all with , and it is the case that where stands for any midpoint of any geodesic segment . A mapping providing such a for a given and is called a modulus of uniform convexity.

Notice that this definition of uniform convex metric spaces is weaker than the one used in [9] in two ways. First, we do not impose that the metric is convex and, second, our modulus of convexity does depend on the two variables and while it is assumed to depend only on in [9].

Definition 2.4. Let be a metric space, then the metric is said to be convex if for any and in , and a midpoint in between and ,

It is easy to see that uniformly convex metric spaces are uniquely geodesic, that is, for each two points there is just one geodesic joining them. Therefore midpoints and geodesic segments joining two points are unique. In this case there is a natural way to define convexity. A subset of a (uniquely) geodesic space is said to be convex if for any . For more about geodesic spaces the reader may check [2].

To obtain our results we will need to impose additional conditions on the modulus of convexity. Following [3, 10] we consider the notion of monotone modulus of convexity as follows.

Definition 2.5. If a uniformly convex metric space admits a modulus of convexity such that it decreases with (for each fixed ) then we say that is a monotone modulus of convexity for .

In the same way we define a lower semicontinuous from the right modulus of convexity as follows.

Definition 2.6. If a uniformly convex metric space admits a modulus of convexity such that it is lower semicontinuous from the right with respect to (for each fixed ) then we say is a lower semicontinuous from the right modulus of convexity for .

Let be a metric space and a family of subsets of . Then, following [1], we say that defines a convexity structure on if it contains the closed balls and is stable by intersection.

Let be a metric space and a convexity structure on . Given , we say that is -convex if for any .

If we consider a bounded sequence in , we are able to define a function , called type, such that for each The asymptotic center of a bounded sequence with respect to a subset of is then defined as If the asymptotic center is taken with respect to then it is simply denoted by .

Definition 2.7. We say that a convexity structure is -stable if types are -convex.

In [1] the following definition of compactness for convexity structure was considered.

Definition 2.8. Given a convexity structure, we will say that is compact if any family of elements of has nonempty intersection provided for any finite subset .

In our paper we will rather use the idea of compactness given in [11]. Notice that this second notion of compactness is weaker than the previous one.

Definition 2.9. Given a convexity structure, we will say that is nested compact if any decreasing chain of nonempty bounded elements of has nonempty intersection.

A very important property given in [3] about complete uniformly convex metric spaces with monotone modulus of convexity is that decreasing sequences of nonempty bounded closed and convex subsets of these spaces have nonempty intersection. As a consequence, we have that if stands for the collection of nonempty closed and convex subsets of a complete uniformly convex metric space with monotone modulus of convexity, then is a nested compact convexity structure.

Remark 2.10. It is not hard to see that the same remains true if the monotone condition on the modulus is replaced by lower semicontinuity from the right.

3. Asymptotic Pointwise Contractions in Uniformly Convex Metric Spaces

In this section we give different results for the above defined mappings in uniformly convex metric spaces. Although, for expository reasons, our results will be usually proved only for uniformly convex metric spaces with a monotone modulus of convexity, they also hold when there is a lower semicontinuous modulus of convexity. Some indications about differences in both cases will be given. We begin with a technical result.

Proposition 3.1. Let be a complete uniformly convex metric space with a monotone (or lower semicontinuous from the right) modulus of convexity . Consider the family of all nonempty closed and convex subsets of . Then defines a nested compact and -stable convexity structure on .

Proof. It only remains to be proved that is -stable. Let be a bounded sequence in and consider the type defined by . We need to show that for any positive . It is immediate to see that is closed and nonempty. To see that is also convex, consider and to be two different points in . There is no restriction if we assume that . Let be the midpoint of the segment and take , then, by uniform convexity, we have that and so, Hence, .

The following theorems were proved in [1] under the hypothesis of compactness on the convexity structure. We state it, however, under the hypothesis of nested compactness since this is all it is actually required in the proofs given in [1].

Theorem 3.2. Let be a bounded metric space. Assume that the convexity structure is nested compact. Let be a pointwise contraction. Then has a unique fixed point . Moreover the orbit converges to , for each .

Theorem 3.3. Let be a bounded metric space. Assume that the convexity structure is nested compact. Let be a strongly asymptotic pointwise contraction. Then has a unique fixed point . Moreover the orbit converges to , for each .

Now the next corollary follows.

Corollary 3.4. The above theorems hold for complete bounded uniformly convex metric spaces with either monotone or lower semicontinuous from the right modulus of convexity.

The following lemma is immediate.

Lemma 3.5. Let be a metric space and a nested compact convexity structure on which is -stable. Then for any type , there exists such that

As a direct consequence of Proposition 3.1 and the previous lemma we get the following result for asymptotic pointwise contractions. We omit the details of its proof as it follows similar patterns as in [1, Theorem  4.2].

Theorem 3.6. Let be a complete uniformly convex metric space with a monotone (or lower semicontinuous from the right) modulus of convexity . Suppose is bounded. Then every asymptotic pointwise contraction has a unique fixed point . Moreover, the orbit converges to for each .

Next we show some consequences of Proposition 3.1 and Lemma 3.5. The cases for monotone and lower semicontinuous from the right modulus of convexity are shown separately as they require different proofs.

Corollary 3.7. Let be a complete uniformly convex metric space with a monotone modulus of convexity and a bounded sequence in . Then the set of asymptotic centers of is a singleton.

Proof. Let and be two different points in and let be the midpoint of . Let , and . By the uniform convexity, there exists such that for every ,
If we let go to infinite, we obtain that which is clearly a contradiction.

Remark 3.8. This corollary has been first proved in [12, Proposition  3.3] for a certain class of uniformly convex hyperbolic spaces with monotone modulus of convexity.

Now we show the lower semicontinuous case.

Corollary 3.9. Let be a complete uniformly convex metric space with a lower semicontinuous from the right modulus of convexity and a bounded sequence in . Then the set of asymptotic centers of is a singleton.

Proof. Let and be two different points in and let be the midpoint of . Let , and let us fix . Then for each large enough. By the uniform convexity, for the same as above and finally Now it suffices to observe that for large enough. Combining it with (3.6) and taking we obtain as in the former corollary, and thus the contradiction.

Another consequence is Kirk Fixed Point Theorem in uniformly convex metric spaces.

Corollary 3.10. Let be a complete uniformly convex geodesic metric space with a monotone (or lower semicontinuous from the right) modulus of convexity. Suppose is bounded, then any nonexpansive mapping has a fixed point.

Proof. Consider and the sequence of its iterates. Let be the only asymptotic center of in . Then, by the nonexpansiveness of , it follows that and so, .

Now we present a counterpart for [1, Theorem  5.1].

Theorem 3.11. Let be a complete uniformly convex metric space with a monotone (or lower semicontinuous from the right) modulus of convexity . Let be a bounded closed convex nonempty subset of . Then any asymptotic pointwise nonexpansive mapping has a fixed point, and the set of fixed points of , , is closed and convex.

Proof. Let and consider . From Corollary 3.7, we know that is a singleton. Let be the only point in that set, that is, is such that . We want to show that is a Cauchy sequence. Suppose this is not the case. Then there exists a separated subsequence of , that is, there exists such that for every in .
Let be the midpoint of the segment , and . The uniform convexity of the space, together with its monotone character, implies that for every and in Notice that, by definition of , Then, if we let go to infinity, Since is pointwise asymptotic nonexpansive, then and so , which is a contradiction since, in virtue of (3.9), this implies that converges to . Therefore, is a Cauchy sequence and its limit, again by (3.9), is . Then, from the continuity of , .
In consequence, is nonempty. Now, since is continuous, is closed. We show next that is also convex. Let be two different points in and the midpoint of the segment . We need to show that . Now, since is pointwise asymptotic nonexpansive, and, equally, Therefore, for there exists such that if then but, from the proof of Proposition  2.2 in [3], the diameters of the sets tend to as tends to and so , which proves is a fixed point of .

Remark 3.12. The proof for the lower semicontinuous case follows in a similar way but following the reasoning of Corollary 3.9.

In [1] a demiclosed principle is also given for asymptotic pointwise nonexpansive mappings in CAT(0) spaces. Next we show that an equivalent result is also possible for uniformly convex metric spaces. Following [1] we define where is a closed and convex subset of a uniformly convex metric space containing the bounded sequence . Notice that this definition does not depend on the set when the space is a complete CAT(0) space. This is due to the fact that the asymptotic center of a bounded sequence of a complete CAT(0) space belongs to the closed convex hull of the sequence, which easily follows from the very well-known fact that the metric projection onto closed convex subsets of a complete CAT(0) space is nonexpansive (see [2] for details). Recall that the existence and uniqueness of such a in a complete uniformly convex metric spaces with monotone modulus of convexity is guaranteed by Corollary 3.7.

Proposition 3.13. Let be a complete uniformly convex metric space with a monotone modulus of convexity . Let be a bounded closed convex nonempty subset of . Let an asymptotic pointwise nonexpansive mapping. Let be an approximate fixed point sequence, that is, , and such that for a certain . Then .

Proof. Since is an approximate fixed point sequence, then we have that for any (see Note Added in Proof at the end of the paper). In consequence, since for , (3.9) holds for any .
Therefore, particularizing for , we have that . Now we claim that as . Suppose on the contrary that there exist an and a subsequence of such that for every . Let be the midpoint of the geodesic segment , and . By uniform convexity, for every , we have that If we consider the upper limit of the above inequality when , we get If we do the same when , we finally obtain that . Therefore and the existence of fixed point follows the same as in Theorem 3.11.

Remark 3.14. The proof for the lower semicontinuous case follows in a similar way but following the reasoning of Corollary 3.9.

4. Fixed Points of Set-Valued Mappings

In this section we present fixed points theorems for set-valued mappings defined on uniformly convex metric spaces. Results stated for uniformly convex metric space with a monotone modulus of convexity also hold if there is a lower semicontinuous from the right modulus of convexity. Proofs of this second case will be omitted as they are based on technical results already proved for both kinds of modulus in Section 3. The Hausdorff metric on the closed and bounded parts of a metric space is defined as follows. If and are bounded and closed subsets of a metric space , then where . Let be a subset of a metric space . A mapping with nonempty bounded closed values is nonexpansive if for all . Our main goal in this section is to study if given is a bounded uniformly convex metric space with monotone modulus of convexity, then every nonexpansive mapping with nonempty and compact values has a fixed point, that is, a point such that . This problem was first solved in the affirmative by Shimizu and Takahashi in [9] under the assumption that the metric is convex. If we lack this condition the problem is much more complicated and we can offer only partial answers. Our first answer will be achieved after imposing condition (i):

Remark 4.1. Condition (i) can be seen as a kind of very weak hyperbolicity condition. In fact, it is immediate to see that hyperbolic uniformly convex spaces studied in [3, 10, 12] satisfy condition (i) as well as for any uniformly convex CAT space with . Notice here that CAT spaces with are particular examples of hyperbolic uniformly convex spaces.

Theorem 4.2. Let be a complete uniformly convex metric space with a monotone modulus of convexity. Suppose that is bounded and the condition (i) holds true. Then each nonexpansive set-valued mapping with compact values has a fixed point.

Proof. Let us fix satisfying the condition (i). Now, from (i) and the fact that is compact for any , it follows that set-valued mappings , defined by are compact-valued contractions with constants . Nadler's Fixed Point Theorem for set-valued contractions implies that has a fixed point for each and therefore has an approximate fixed point sequence, that is, a sequence such that . According to Corollary 3.7 there is a unique asymptotic center of in . Now the rest of the proof follows the same patterns of the standard one for uniformly convex Banach spaces (see [13, Theorem  15.3, page 165]).

It is wellknown that a CAT(1) space needs not to be uniformly convex if its diameter is not smaller than . Next we show, however, that the same above idea can be applied to CAT(1) spaces of radius smaller than . Remember that for the number is defined as and that the radius of a bounded subset of is given by

Theorem 4.3. Let and let be a complete space with . Then each nonexpansive mapping with compact values has at least one fixed point.

Proof. Take in such a way that . As it is shown in [14, Lemma  3], it is enough to take for to verify condition (i) with in . In a similar manner as above we obtain an approximate fixed point sequence . On account of and [15, Proposition  4.1] the asymptotic center of each subsequence of is unique, and the rest of the proof is not different from the case of uniformly convex Banach spaces.

Next we consider two further conditions to guarantee the existence of fixed points for nonexpansive set-valued mappings with compact values in uniformly convex metric spaces.

Remark 4.4. Notice that, roughly speaking, conditions (i) and (ii) give opposite information about the geometry of the space. While condition (i) implies that geodesic emanating from a same point must separate and no matters how fast, condition (ii) imposes a superior bound about how much two geodesics emanating from a same point are allowed to separate. It is easy to see that any geodesic space admitting bifurcating geodesics cannot verify condition (ii). In particular, condition (ii) does not hold in -trees. It easily follows from the definitions that geodesic spaces with curvature bounded below by a real number (check [16] for a detailed exposition about these spaces) satisfy condition (ii).

Before stating our next result we need to introduce a definition.

Definition 4.5. A geodesic space is said to have the geodesic extension property if any geodesic segment in is actually contained in a geodesic line, that is, in a geodesic .

Theorem 4.6. Let be a complete uniformly convex metric space with a monotone modulus of convexity and the geodesic extension property. Moreover, suppose that satisfies conditions (ii) and (iii) and is a nonempty bounded closed and convex subset of . If is a nonexpansive mappings with compact values then has at least one fixed point.

Proof. From the proof of Theorem 4.2, we know that if then the conclusion follows. So let us suppose that and fix small enough. Let be chosen in such a way that and let us denote this distance by . Clearly, from the compactness of it follows the existence of for which . Now assume that is a midpoint of the metric segment . The conditions imposed on imply that there is such that and .
To estimate the distance between and let us consider the midpoint of . Using the uniformly convexity of we obtain that If we denote by the point of the geodesic ray containing such that and , then and
Repeating our reasoning for , and , we obtain points and satisfying where If we denote by the point in for which and , then it is easy to see that . Hence