Fixed Points of Single- and Set-Valued Mappings in Uniformly Convex Metric Spaces with No Metric Convexity
© Rafa Espínola et al. 2010
Received: 20 April 2009
Accepted: 28 May 2009
Published: 29 June 2009
We study the existence of fixed points and convergence of iterates for asymptotic pointwise contractions in uniformly convex metric spaces. We also study the existence of fixed points for set-valued nonexpansive mappings in the same class of spaces. Our results do not assume convexity of the metric which makes a big difference when studying the existence of fixed points for set-valued mappings.
This paper is motivated by the recent paper . In  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  as a very significant example within the class of uniformly convex metric spaces (the reader can consult  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  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 , for recent achievements on related topics the reader may also check .
The notion of asymptotic pointwise contractions was introduced in . Then it was also studied in  where, by means of ultrapower techniques, different results about the existence of fixed points and convergence of iterates were proved. In  new proofs were presented but this time after applying only elementary techniques. Very recently, in , 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  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 , 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.
It is proved in  (see also ) 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 , the uniqueness of fixed points and convergence of iterates for these mappings directly follow if existence is guaranteed.
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.
Notice that this definition of uniform convex metric spaces is weaker than the one used in  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 .
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 .
In the same way we define a lower semicontinuous from the right modulus of convexity as follows.
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 , we say that defines a convexity structure on if it contains the closed balls and is stable by intersection.
In  the following definition of compactness for convexity structure was considered.
In our paper we will rather use the idea of compactness given in . Notice that this second notion of compactness is weaker than the previous one.
A very important property given in  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.
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.
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 .
The following theorems were proved in  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 .
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.
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.
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].
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.
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.
Another consequence is Kirk Fixed Point Theorem in uniformly convex metric spaces.
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.
Now we present a counterpart for [1, Theorem 5.1].
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.
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 .
but, from the proof of Proposition 2.2 in , the diameters of the sets tend to as tends to and so , which proves is a fixed point of .
The proof for the lower semicontinuous case follows in a similar way but following the reasoning of Corollary 3.9.
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  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.
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 .
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
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  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):
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.
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.
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]).
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.
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  for a detailed exposition about these spaces) satisfy condition (ii).
Before stating our next result we need to introduce a definition.
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.
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 .
From Theorems 3.11 and 4.2 one may get the following generalization of [1, Theorem 5.2]. We omit the proof as it is analog to the one given for CAT(0) spaces in . We first need some notations and definitions.
Let be a uniformly convex metric space. Consider the mappings and , then and are said to be commuting mappings if for all and for all . A point is called a center for the mapping if for each , . The set denotes the set of all centers of the mappings .
The same result remains true if the uniformly convex metric space is supposed to have the geodesic extension property, and conditions (ii)-(iii) hold instead of condition (i).
The Stečkin property was introduced in  to obtain different results regarding the existence of unique nearest and farthest points to closed subsets of normed linear spaces. This property has been studied by many authors since then being of special relevance in the study of uniformly convex Banach spaces; see for instance [18, 19]. It is known that the Stečkin property does not happen in geodesic spaces with bifurcating geodesics [19, 20] and it has been shown to be related to the property of having curvature bounded below (see  for details about spaces with curvature bounded below); see [19, 20]. We next show that this property, in addition to a similar condition to (ii), leads to the existence of fixed points for set-valued nonexpansive mappings. We first introduce condition :
Let be a complete uniformly convex metric space with a monotone modulus of convexity and the geodesic extension property. Moreover, suppose that satisfies conditions and the Stečkin property. Let be a nonempty bounded closed and convex subset of . If is a nonexpansive mappings with compact values then has at least one fixed point.
We will omit details for this proof as it follows the same patterns of the proof of Theorem 4.6. We will just point out that the main difference happens when the modulus of convexity is used to estimate first (see (4.10), and later (see (4.15), now we use property S to achieve similar estimations.
Note Added in Proof
The two first authors were partially supported by the Ministery of Science and Technology of Spain, Grant BFM 2000-0344-CO2-01 and La Junta de Antalucía Project FQM-127. This work was carried out while the third author was visiting the University of Seville. She acknowledges the kind hospitality of the Departamento de Análisis Matemático. This work is dedicated to W. A. Kirk.
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