- Research Article
- Open access
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Common Fixed Points for Multimaps in Metric Spaces
Fixed Point Theory and Applications volume 2010, Article number: 204981 (2009)
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.
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F204981/MediaObjects/13663_2009_Article_1225_Equ1_HTML.gif)
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F204981/MediaObjects/13663_2009_Article_1225_Equ2_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F204981/MediaObjects/13663_2009_Article_1225_Equ3_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F204981/MediaObjects/13663_2009_Article_1225_Equ4_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F204981/MediaObjects/13663_2009_Article_1225_Equ5_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F204981/MediaObjects/13663_2009_Article_1225_Equ6_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F204981/MediaObjects/13663_2009_Article_1225_Equ7_HTML.gif)
Define by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F204981/MediaObjects/13663_2009_Article_1225_Equ8_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F204981/MediaObjects/13663_2009_Article_1225_Equ9_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F204981/MediaObjects/13663_2009_Article_1225_Equ10_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F204981/MediaObjects/13663_2009_Article_1225_Equ11_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F204981/MediaObjects/13663_2009_Article_1225_Equ12_HTML.gif)
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 orbitof
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F204981/MediaObjects/13663_2009_Article_1225_Equ13_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F204981/MediaObjects/13663_2009_Article_1225_Equ14_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F204981/MediaObjects/13663_2009_Article_1225_Equ15_HTML.gif)
Then for every
. This implies that
is a
-nonexpansive map. However,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F204981/MediaObjects/13663_2009_Article_1225_Equ16_HTML.gif)
which implies that is not nonexpansive. Let
be any small open neighborhood of
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F204981/MediaObjects/13663_2009_Article_1225_Equ17_HTML.gif)
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
In 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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F204981/MediaObjects/13663_2009_Article_1225_Equ18_HTML.gif)
then there exists an element such that
for all
.
Proof.
By Theorem 4.3, is nonempty closed and convex. Let
. Then for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F204981/MediaObjects/13663_2009_Article_1225_Equ19_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F204981/MediaObjects/13663_2009_Article_1225_Equ20_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F204981/MediaObjects/13663_2009_Article_1225_Equ21_HTML.gif)
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