Abstract
We discuss the existence of common fixed points in uniformly convex metric spaces for single-valued pointwise asymptotically nonexpansive or nonexpansive mappings and multivalued nonexpansive, 
-nonexpansive, or 
-semicontinuous maps under different conditions of commutativity.
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1. Introduction
Fixed point theory for nonexpansive and related mappings has played a fundamental role in many aspects of nonlinear functional analysis for many years. The notion of asymptotic pointwise nonexpansive mapping was introduced and studied in [1, 2]. Very recently, in [3], techniques developed in [1, 2] were applied in metric spaces and 
(0) spaces where the authors attend to the Bruhat-Tits inequality for 
(0) spaces in order to obtain such results. In [4] it has been shown that these results hold even for a more general class of uniformly convex metric spaces than 
(0) spaces. Here, we take advantage of this recent progress on asymptotic pointwise nonexpansive mappings and existence of fixed points for multivalued nonexpansive mappings in metric spaces to discuss the existence of common fixed points in either uniformly convex metric spaces or 
-trees for this kind of mappings, as well as for 
-nonexpansive or 
-semicontinuous multivalued mappings under different kinds of commutativity conditions.
2. Basic Definitions and Results
First let us start by making some basic definitions.
Definition 2.1.
Let 
be a metric space. A mapping 
is called nonexpansive if 
for any 
. A fixed point of 
will be a point 
such that 
.
At least something else is stated, the set of fixed points of a mapping 
will be denoted by Fix
.
Definition 2.2.
A point 
is called a center for the mapping 
if for each 
, 
. The set 
denotes the set of all centers of the mapping 
.
Definition 2.3.
Let 
be a metric space. 
will be said to be an asymptotic pointwise nonexpansive mapping if there exists a sequence of mappings 
such that
and
for any 
.
This notion comes from the notion of asymptotic contraction introduced in [1]. Asymptotic pointwise nonexpansive mappings have been recently studied in [2–4].
In this paper we will mainly work with uniformly convex geodesic metric space. Since the definition of convexity requires the existence of midpoint, the word geodesic is redundant and so, for simplicity, we will omit it.
Definition 2.4.
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.
A particular case of this kind of spaces was studied by Takahashi and others in the 90s [5]. To define them we first need to introduce the notion of convex metric.
Definition 2.5.
Let 
be a metric space, then the metric is said to be convex if for any 
and 
in 
, and 
a middle point in between 
and 
(that is, 
is such that 
), it is the case that
Definition 2.6.
A uniformly convex metric space will be said to be of type (T) if it has a modulus of convexity which does not depend on 
and its metric is convex.
Notice that some of the most relevant examples of uniformly convex metric spaces, as it is the case of uniformly convex Banach spaces or 
(0) spaces, are of type (T).
Another situation where the geometry of uniformly convex metric spaces has been shown to be specially rich is when certain conditions are found in at least one of their modulus of convexity even though it may depend on 
. These cases have been recently studied in [4, 6, 7]. After these works we will say that given a uniformly convex metric space, this space will be of type (M) (or [L]) if it has an adequate monotone (lower semicontinuous from the right) with respect to 
modulus of convexity (see [4, 6, 7] for proper definitions). It is immediate to see that any space of type (T) is also of type (M) and (L). 
spaces with small diameters are of type (M) and (L) while their metric needs not to be convex.
-trees are largely studied and their class is a very important within the class of 
(0)-spaces (and so of uniformly convex metric spaces of type (T)). 
-trees will be our main object in Section 4.
Definition 2.7.
An 
-tree is a metric space 
such that:
(i)there is a unique geodesic segment 
joining each pair of points 
(ii)if 
, then 
It is easy to see that uniform convex metric spaces are unique geodesic; that is, for each two points there is just one geodesic joining them. Therefore midpoints and geodesic segments are unique. In this case there is a natural way to define convexity. Given two points 
and 
in a geodesic space, the (metric) segment joining 
and 
is the geodesic joining both points and it is usually denoted by 
. A subset 
of a (unique) geodesic space is said to be convex if 
for any 
. For more about geodesic spaces the reader may check [8].
The following theorem is relevant to our results. Recall first that given a metric space 
and 
, the metric projection 
from 
onto 
is defined by 
, where 
.
Let 
be a uniformly convex metric space of type (M) or (L), let 
nonempty complete and convex. Then the metric projection 
of 
onto 
is a singleton for any 
.
These spaces have also been proved to enjoy very good properties regarding the existence of fixed points [4, 5] for both single and multivalued mappings. In [2] we can find the central fixed point result for asymptotic pointwise nonexpansive mappings in uniformly convex Banach spaces. This result was later extended to 
(0) spaces in [3] and more recently to uniformly convex metric spaces of type either (M) or (L) in [4].
Theorem 2.9.
Let 
be a closed bounded convex subset of a complete uniformly convex metric space of type either (M) or (L) and suppose that 
is a pointwise asymptotically nonexpansive mapping. Then the fixed point set 
is nonempty closed and convex.
Before introducing more fixed point results, we need to present some notations and definitions. Given a geodesic metric space 
we will denote by 
the family of nonempty compact subsets of 
and by 
the family of nonempty compact and convex subsets of 
. If 
and 
are bounded subsets of 
, let 
denote the Hausdorff metric defined as usual by
where 
. Let 
be a subset of a metric space 
. A mapping 
with nonempty bounded values is nonexpansive provided that 
for all 
.
Theorem 2.10 (see [5]).
Let 
be a complete uniformly convex metric space of type (T) and 
nonempty bounded closed and convex. Let 
be a nonexpansive multivalued mapping, then the set of fixed points of 
is nonempty.
We next give the definition of those uniformly convex metric spaces for which most of the results in the present work will apply.
Definition 2.11.
A uniformly convex metric space with the fixed point property for nonexpansive multivalued mappings (FPPMM) will be any such space of type either (M) or (L) or both verifying the above theorem.
The problem of studying whether more general uniformly convex metric spaces than those of type (T) enjoy that the FPPMM has been recently taken up in [4], where it has been shown that under additional geometrical conditions certain spaces of type (M) and (L) also enjoy the FPPMM.
The following notion of semicontinuity for multivalued mappings has been considered in [9] to obtain different results on coincidence fixed points in 
-trees and will play a main role in our last section.
Definition 2.12.
For a subset 
of 
, a set-valued mapping 
is said to be 
-semicontinuous at 
if for each 
there exists an open neighborhood 
of 
in 
such that
for all 
It is shown in [9] that 
-semicontinuity of multivalued mappings is a strictly weaker notion than upper semicontinuity and almost lower semicontinuity. Similar results to those presented in [9] had been previously obtained under these other semicontinuity conditions in [10, 11].
Let 
be a nonempty subset of a metric space 
. Let 
and 
with 
for 
. Then 
and 
are said to be commuting mappings if 
for all 
. 
and 
are said to commute weakly [12] if 
for all 
, where 
denotes the relative boundary of 
with respect to 
. We define a subclass of weakly commuting pair which is different than that of commuting pair as follows.
Definition 2.13.
If 
and 
are as what is previously mentioned, then they are said to commute subweakly if 
for all 
.
Notice that saying that 
and 
commute subweakly is equivalent to saying that 
and 
commute.
Recently, Chen and Li [13] introduced the class of Banach operator pairs as a new class of noncommuting maps which has been further studied by Hussain [14] and Pathak and Hussain [15]. Here we extend this concept to multivalued mappings.
Definition 2.14.
Let 
and 
with 
for 
. The ordered pair 
is a Banach operator pair if 
for each 
.
Next examples show that Banach operator pairs need not be neither commuting nor weakly commuting.
Example 2.15.
Let 
with the usual norm and 
Let 
and 
for all 
. Then 
. Note that 
is a Banach operator pair but 
and 
are not commuting.
Example 2.16.
Let 
with the usual metric. Let 
be defined by
Define 
by
Then 
and 
imply that 
is a Banach operator pair. Further, 
and 
. Thus 
and 
are neither commuting nor weakly commuting.
In 2005, Dhompongsa et al. [16] proved the following fixed point result for commuting mappings.
Theorem 2 DKP.
Let 
be a nonempty closed bounded convex subset of a complete 
(0) space 
, 
a nonexpansive self-mapping of 
and 
nonexpansive, where for any 
, 
is nonempty compact convex. Assume that for some 
is convex for all 
and 
. If 
and 
commute, then there exists an element 
such that 
.
This result has been recently improved by Shahzad in [17, Theorem 3.3]. More specifically, the same coincidence result was achieved in [17] for quasi-nonexpansive mappings (i.e., mappings for which its fixed points are centers) with nonempty fixed point sets in 
(0) spaces and dropping the condition given by (2.9) at the time that the commutativity condition was weakened to weakly commutativity. Our main results provide further extensions of this result for asymptotic pointwise nonexpansive mappings and for nonexpansive multivalued mappings 
with convex and nonconvex values. Earlier versions of such results for asymptotically nonexpansive mappings can already be found in [3, 4].
Summarizing, in this paper we prove some common fixed point results either in uniformly convex metric space with the FPPMM (Section 3) or 
-trees (Section 4) for single-valued asymptotic pointwise nonexpansive or nonexpansive mappings and multivalued nonexpansive, 
-nonexpansive, or 
-semicontinuous maps which improve and/or complement Theorem DKP, [17, Theorem 3.3], and many others.
3. Main Results
Our first result gives the counterpart of [17, Theorem 3.3] to asymptotic pointwise nonexpansive mappings.
Theorem 3.1.
Let 
be a complete uniformly convex metric space with FPPMM, and, 
be a bounded closed convex subset of 
. Assume that 
is an asymptotic pointwise nonexpansive mapping and 
a nonexpansive mapping with 
a nonempty compact convex subset of 
for each 
. If the mappings 
and 
commute then there is 
such that 
.
Proof.
By Theorem 2.9, the fixed point set 
of 
of a bounded closed convex subset is a nonempty closed and convex subset of 
. By the commutativity of 
and 
, 
is 
-invariant for any 
and so 
and convex for any 
. Therefore, the mapping 
is well defined.
We will show next that 
is also nonexpansive as a multivalued mapping. Before that, we claim that 
for any 
. In fact, by convexity of 
and Theorem 2.8, we can take 
to be the unique point in 
such that 
. Now consider the sequence 
. Since 
and 
commute we know that 
for any 
. Therefore, by the compactness of 
, it has a convergent subsequence 
. Let 
be the limit of 
, then we have that
from where, by the uniqueness of 
, 
. Consequently, 
and so 
. This, in particular, shows that 
and explains equality (3.1) below. Now, we can argue as follows:
Finally, since 
has the FPPMM, there exists 
such that 
. Therefore, 
.
Remark 3.2.
The proof of our result is inspired on that one [17, Theorem 3.3]. Notice, however, that equality (3.1) is given as trivial in [17] while this is not the case. Notice also that there is no direct relation between the families of quasi-nonexpansive mappings and asymptotically pointwise nonexpansive mappings which make both results independent and complementary to each other.
The condition that 
is a mapping with convex values is crucial to get the desired conclusion in the previous theorem, Theorem DKP and all the results in [17]. Next we give conditions under which this hypothesis can be dropped. A self-map 
of a topological space 
is said to satisfy condition (C) [15, 18] provided 
for any nonempty 
-invariant closed set 
.
Theorem 3.3.
Let 
be a complete uniformly convex metric space with FPPMM and 
a bounded closed convex subset of 
. Assume that 
is asymptotically pointwise nonexpansive and 
is nonexpansive with 
a nonempty compact subset of 
for each 
. If the mappings 
and 
commute and 
satisfies condition (C), then there is 
such that 
.
Proof.
We know that the fixed point set 
of 
is a nonempty closed and convex subset of 
. Since 
and 
commute then 
is 
-invariant for 
, and also, since 
satisfies condition (C), the mapping 
is well defined. We prove next that the mapping 
is nonexpansive.
As in the above proof, we need to show that for any 
it is the case that 
. Since 
and 
commute, we know that 
is 
-invariant. Take 
such that 
and consider the sequence 
. Let 
be the set of limit points of 
, then 
is a nonempty and closed subset of 
. Consider now 
, then
and, therefore, 
. But 
is also 
-invariant, so, by condition (C), 
has a fixed point in 
and so 
. The rest of the proof follows as in Theorem 3.1.
For the next corollary we need to recall some definitions about orbits. The orbit
of
at
is proper if 
or there exists 
such that 
is a proper subset of 
. If 
is proper for each 
, we will say that 
has proper orbits on 
[19].
Condition (C) in Theorem 3.3 may seem restrictive, however it looks weaker if we recall that the values of 
are compact. This is shown in the next corollary.
Corollary 3.4.
Under the same conditions of the previous theorem, if condition (C) is replaced with 
having proper orbits then the same conclusion follows.
Proof.
The idea now is that the orbits through 
of points in 
are relatively compact, then, by [19, Theorem 3.1], 
satisfies condition (C).
For any nonempty subset 
of a metric space 
, the diameter of
is denoted and defined by 
= sup
A mapping 
has diminishing orbital diameters
[19, 20] if for each 
and whenever 
there exists 
such that 
. Observe that in a metric space 
if 
has d.o.d. on 
, then 
has proper orbits [15, 19]; consequently, we obtain the following generalization of the corresponding result of Kirk [20].
Corollary 3.5.
Under the same conditions of the previous theorem, if condition (C) is replaced with 
having d.o.d. then the same conclusion follows.
In our next result we also drop the condition on the convexity of the values of 
but, this time, we ask the geodesic space 
not to have bifurcating geodesics. That is, for any two segments starting at the same point and having another common point, this second point is a common endpoint of both or one segment that includes the other. This condition has been studied by Zamfirescu in [21] in order to obtain stronger versions of the next lemma which is the one we need and which proof is immediate.
Lemma 3.6.
Let 
be a geodesic space with no bifurcating geodesics and let 
be a nonempty subset of 
. Let 
, 
such that 
, and 
with 
. Then the metric projection of 
onto 
is the singleton 
for any 
.
Now we give another version of Theorem 3.1 without assuming that the values of 
are convex.
Theorem 3.7.
Let 
be a complete uniformly convex metric space with FPPMM and with no bifurcating geodesics and 
a bounded closed convex subset of 
. Assume that 
is asymptotically pointwise nonexpansive and 
nonexpansive with 
a nonempty compact subset of 
for each 
. Assume further that the fixed point set 
of 
is such that its topological interior (in 
) is dense in 
. If the mappings 
and 
commute, then there exists 
such that 
.
Proof.
Just as before, we know that the fixed point set 
of 
is a nonempty closed and convex subset of 
. We are going to see that 
is well defined. Take 
and let us see that 
. Consider 
such that 
and let 
be a limit point of 
. Fix 
, then
Therefore, by Lemma 3.6, 
and so 
is a convergent sequence to 
and 
. Take now 
, then, by hypothesis, there exists a sequence 
converging to 
. Consider the sequence of points 
given by the above reasoning such that 
. Define, for each 
, 
such that 
. Since 
is nonexpansive, 
. Now, since 
is compact, take 
a limit point of 
. Then 
because it is also a limit point of 
and 
. Therefore our claim that 
is well defined is correct. Let us see now that 
is also nonexpansive.
As in the previous theorems, we show that for 
we have that 
. Take 
and consider 
such that 
and 
a limit point of 
. Take 
. Then, repeating the same reasoning as above, 
and so 
is a fixed point of 
which proves that 
for 
and 
. For 
we apply a similar argument as above using that 
is dense in 
. Now the result follows as in Theorem 3.1.
Remark 3.8.
The condition about the commutativity of 
and 
has been used to guarantee that the orbits 
for 
in the fixed point set of 
remain in a certain compact set and so they are relatively compact. The same conclusion can be reached if we require 
and 
to commute subweakly. Therefore, Theorems 3.1, 3.3 and 3.7, and stated corollaries remain true under this other condition.
In the next result the convexity condition on the multivalued mappings is also removed.
Theorem 3.9.
Let 
be a complete uniformly convex metric space with FPPMM, and, let 
be a bounded closed convex subset of 
. Assume that 
is asymptotically pointwise nonexpansive and 
nonexpansive with 
a nonempty compact subset of 
for each 
. If the pair 
is a Banach operator pair, then there is 
such that 
.
Proof.
By Theorem 2.9 the fixed point set 
of 
is a nonempty closed and convex subset of 
. Since the pair 
is a Banach operator pair, 
for each 
, and therefore, 
for 
. The mapping 
being the restriction of 
on 
is nonexpansive. Now the proof follows as in Theorem 3.1.
Remark 3.10.
Since asymptotically nonexpansive and nonexpansive maps are asymptotically pointwise nonexpansive maps, all the so far obtained results also apply for any of these mappings.
A set-valued map 
is called 
-nonexpansive [22] if for all 
and 
with 
, there exists 
with 
such that 
. Define 
by
Husain and Latif [22] introduced the class of 
-nonexpansive multivalued maps and it has been further studied by Hussain and Khan [23] and many others. The concept of a 
-nonexpansive multivalued mapping is different from that one of continuity and nonexpansivity, as it is clear from the following example [23].
Example 3.11.
Let 
be the multivalued map defined by
Then 
for every 
. This implies that 
is a 
-nonexpansive map. However,
which implies that 
is not nonexpansive. Let 
be any small open neighborhood of 
, then
which is not open. Thus 
is not continuous. Note also that 
is a fixed point of 
.
Theorem 3.12.
Let 
be a complete uniformly convex metric space with FPPMM and 
be a bounded closed convex subset of 
. Assume that 
is asymptotically pointwise nonexpansive and 
-nonexpansive with 
a compact subset of 
for each 
. If the pair 
is a Banach operator pair, then there is 
such that 
.
Proof.
As above, the set 
of fixed points of 
is nonempty closed convex subset of 
. Since 
is compact for each 
, 
is well defined and a multivalued nonexpansive selector of 
[23]. We also have that 
and 
for each 
, so 
for each 
. Thus the pair 
is a Banach operator pair. By Theorem 3.9, the desired conclusion follows.
The following corollary is a particular case of Theorem 3.12.
Corollary 3.13.
Let 
be a complete uniformly convex metric space with FPPMM, and, let 
be a bounded closed convex subset of 
. Assumethat 
is a nonexpansive map and 
is a 
-nonexpansive mapping with 
a compact subset of 
for each 
. If the pair 
is a Banach operator pair, then there is 
such that 
.
4. Coincidence Results in 
-Trees
-TreesIn this section we present different results on common fixed points for a family of commuting asymptotic pointwise nonexpansive mappings. As it can be seen in [9–11], existence of fixed points for multivalued mappings happens under very weak conditions if we are working in 
-tree spaces. This allows us to find much weaker results for 
-trees than those in the previos section. Close results to those presented in this section can be found in [17]. We begin with the adaptation of Theorem 3.1 to 
-trees.
Theorem 4.1.
Let 
be a complete 
-tree, and suppose that 
is a bounded closed convex subset of 
. Assume that 
is asymptotically pointwise nonexpansive and 
is 
-semicontinuous mapping with 
a nonempty closed and convex subset of 
for each 
. If the mappings 
and 
commute then there is 
such that 
.
Proof.
We know that the fixed point set 
of 
is a nonempty closed and convex subset of 
. From the commutativity condition we also have that 
is 
-invariant for any 
and so the mapping defined by 
is well defined on 
and takes closed and convex values. By [9, Lemma 2], 
is a 
-semicontinuous mapping and so, by [9, Theorem 4] applied to 
, the conclusion follows.
Remark 4.2.
Actually the only condition we need in the above theorem from 
is that its set of fixed points is nonempty bounded closed and convex. In the case 
is nonexpansive then 
may be supposed to be geodesically bounded instead of bounded, as shown in [24].
The next theorem is the counterpart of Espínola and Kirk [24, Theorem 4.3] to asymptotic pointwise nonexpansive mappings.
Theorem 4.3.
Let 
be a complete 
-tree, and suppose 
is a bounded closed convex subset of 
. Then every commuting family 
of asymptotic pointwise nonexpansive self-mappings of 
has a nonempty closed and convex common fixed point set.
Proof.
Let 
. Then by Theorem 2.9, the set 
of fixed points of 
is nonempty closed and convex and hence again an 
-tree. Now suppose 
. Since 
and 
commute it follows 
, and, by applying the preceding argument to 
and 
, we conclude that 
has a nonempty fixed point set in 
. In particular the fixed point set of 
and the fixed point set of 
intersect. The rest of the proof is similar to that of Espínola and Kirk [24, Theorem 4.3] and so is omitted.
In the next result, we combine a family of commuting asymptotic pointwise nonexpansive mappings with a multivalued mapping.
Theorem 4.4.
Let 
be a nonempty bounded closed convex subset of a complete 
-tree 
, 
a commuting family of asymptotic pointwise nonexpansive self-mappings on 
. Assume that 
is 
-semicontinuous mapping on 
with nonempty closed and convex values and such that 
and 
commute weakly for any 
. If for each 
then there exists an element 
such that 
for all 
.
Proof.
By Theorem 4.3, 
is nonempty closed and convex. Let 
. Then for any 
,
Let 
the unique closest point to 
from 
. Now (4.2) implies that 
is in 
. Further, (4.1) implies that 
and so, by the uniqueness of 
, 
. Thus 
is a common fixed point of 
, which implies
for each 
. Let 
. Now the proof follows as the proof of Theorem 4.1.
Remark 4.5.
Note that condition (4.1) is satisfied if each 
is nonexpansive with respect to 
.
Corollary 4.6.
Let 
be a nonempty bounded closed convex subset of a complete 
-tree 
and 
a commuting family of asymptotic pointwise nonexpansive self-mappings of 
. Assume that 
is 
-semicontinuous, where for any 
, 
is nonempty closed and convex and 
and 
commute weakly. If
then there exists an element 
such that 
for all 
.
Proof.
Condition (4.4) implies (4.1). The desired conclusion now follows from the previous theorem.
In our next result we make use of the fact that convex subsets of 
-trees are gated; that is, if 
is a closed and convex subset of the 
-tree 
, 
and 
is the metric projection of 
onto 
then 
is in the metric segment joining 
and 
for any 
. Notice that condition (4.1) is dropped in the next theorem.
Theorem 4.7.
Let 
be a nonempty bounded closed convex subset of a complete 
-tree 
, 
a commuting family of asymptotic pointwise nonexpansive self-mappings on 
. Assume that 
is 
-semicontinuous mapping on 
with nonempty closed and convex values and such that 
and 
commute for any 
, then there exists an element 
such that 
for all 
.
Proof.
As in the proof of Theorem 4.4, the only thing that really needs to be proved is that 
for each 
. From the commutativity condition we know that 
is 
-invariant for any 
. Therefore each 
has a fixed point 
. But, since the fixed point set of 
is convex and 
, then the metric segment joining 
and 
is contained in 
. From the gated property, we know that the closest point 
to 
from 
is in such segment for any 
. In consequence, 
is a fixed point for any 
and, therefore, 
.
The next theorem follows as a consequence of Theorem 4.7.
Theorem 4.8.
If in the previous theorem 
is supposed to be either upper semicontinuous or almost lower semicontinuous then the same conclusion follows.
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Acknowledgments
The first author was 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 is dedicated to Professor. W. Takahashi on the occasion of his retirement.
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Espínola, R., Hussain, N. Common Fixed Points for Multimaps in Metric Spaces. Fixed Point Theory Appl 2010, 204981 (2009). https://doi.org/10.1155/2010/204981
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DOI: https://doi.org/10.1155/2010/204981