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.