Fixed Point Theory and Applications

, 2010:204981 | Cite as

Common Fixed Points for Multimaps in Metric Spaces

Open Access
Research Article
Part of the following topical collections:
  1. Takahashi's Legacy in Fixed Point Theory

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, Open image in new window -nonexpansive, or Open image in new window -semicontinuous maps under different conditions of commutativity.

Keywords

Nonexpansive Mapping Multivalued Mapping Common Fixed Point Geodesic Segment Fixed Point Result 

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 Open image in new window (0) spaces where the authors attend to the Bruhat-Tits inequality for Open image in new window (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 Open image in new window (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 Open image in new window -trees for this kind of mappings, as well as for Open image in new window -nonexpansive or Open image in new window -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 Open image in new window be a metric space. A mapping Open image in new window is called nonexpansive if Open image in new window for any Open image in new window . A fixed point of Open image in new window will be a point Open image in new window such that Open image in new window .

At least something else is stated, the set of fixed points of a mapping Open image in new window will be denoted by Fix Open image in new window .

Definition 2.2.

A point Open image in new window is called a center for the mapping Open image in new window if for each Open image in new window , Open image in new window . The set Open image in new window denotes the set of all centers of the mapping Open image in new window .

Definition 2.3.

Let Open image in new window be a metric space. Open image in new window will be said to be an asymptotic pointwise nonexpansive mapping if there exists a sequence of mappings Open image in new window such that

for any Open image in new window .

This notion comes from the notion of asymptotic contraction introduced in [1]. Asymptotic pointwise nonexpansive mappings have been recently studied in [2, 3, 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 Open image in new window is said to be uniformly convex if for any Open image in new window and any Open image in new window there exists Open image in new window such that for all Open image in new window with Open image in new window , Open image in new window and Open image in new window it is the case that

where Open image in new window stands for any midpoint of any geodesic segment Open image in new window . A mapping Open image in new window providing such a Open image in new window for a given Open image in new window and Open image in new window 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 Open image in new window be a metric space, then the metric is said to be convex if for any Open image in new window and Open image in new window in Open image in new window , and Open image in new window a middle point in between Open image in new window and Open image in new window (that is, Open image in new window is such that Open image in new window ), 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 Open image in new window 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 Open image in new window (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 Open image in new window . 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 Open image in new window 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). Open image in new window spaces with small diameters are of type (M) and (L) while their metric needs not to be convex.

Open image in new window -trees are largely studied and their class is a very important within the class of Open image in new window (0)-spaces (and so of uniformly convex metric spaces of type (T)). Open image in new window -trees will be our main object in Section 4.

Definition 2.7.

An Open image in new window -tree is a metric space Open image in new window such that:

(i)there is a unique geodesic segment Open image in new window joining each pair of points Open image in new window

(ii)if Open image in new window , then Open image in new window

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 Open image in new window and Open image in new window in a geodesic space, the (metric) segment joining Open image in new window and Open image in new window is the geodesic joining both points and it is usually denoted by Open image in new window . A subset Open image in new window of a (unique) geodesic space is said to be convex if Open image in new window for any Open image in new window . 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 Open image in new window and Open image in new window , the metric projection Open image in new window from Open image in new window onto Open image in new window is defined by Open image in new window , where Open image in new window .

Theorem 2.8 (see [4, 6]).

Let Open image in new window be a uniformly convex metric space of type (M) or (L), let Open image in new window nonempty complete and convex. Then the metric projection Open image in new window of Open image in new window onto Open image in new window is a singleton for any Open image in new window .

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 Open image in new window (0) spaces in [3] and more recently to uniformly convex metric spaces of type either (M) or (L) in [4].

Theorem 2.9.

Let Open image in new window be a closed bounded convex subset of a complete uniformly convex metric space of type either (M) or (L) and suppose that Open image in new window is a pointwise asymptotically nonexpansive mapping. Then the fixed point set Open image in new window is nonempty closed and convex.

Before introducing more fixed point results, we need to present some notations and definitions. Given a geodesic metric space Open image in new window we will denote by Open image in new window the family of nonempty compact subsets of Open image in new window and by Open image in new window the family of nonempty compact and convex subsets of Open image in new window . If Open image in new window and Open image in new window are bounded subsets of Open image in new window , let Open image in new window denote the Hausdorff metric defined as usual by

where Open image in new window . Let Open image in new window be a subset of a metric space Open image in new window . A mapping Open image in new window with nonempty bounded values is nonexpansive provided that Open image in new window for all Open image in new window .

Theorem 2.10 (see [5]).

Let Open image in new window be a complete uniformly convex metric space of type (T) and Open image in new window nonempty bounded closed and convex. Let Open image in new window be a nonexpansive multivalued mapping, then the set of fixed points of Open image in new window 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 Open image in new window -trees and will play a main role in our last section.

Definition 2.12.

for all Open image in new window

It is shown in [9] that Open image in new window -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 Open image in new window be a nonempty subset of a metric space Open image in new window . Let Open image in new window and Open image in new window with Open image in new window for Open image in new window . Then Open image in new window and Open image in new window are said to be commuting mappings if Open image in new window for all Open image in new window . Open image in new window and Open image in new window are said to commute weakly [12] if Open image in new window for all Open image in new window , where Open image in new window denotes the relative boundary of Open image in new window with respect to Open image in new window . We define a subclass of weakly commuting pair which is different than that of commuting pair as follows.

Definition 2.13.

If Open image in new window and Open image in new window are as what is previously mentioned, then they are said to commute subweakly if Open image in new window for all Open image in new window .

Notice that saying that Open image in new window and Open image in new window commute subweakly is equivalent to saying that Open image in new window and Open image in new window 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 Open image in new window and Open image in new window with Open image in new window for Open image in new window . The ordered pair Open image in new window is a Banach operator pair if Open image in new window for each Open image in new window .

Next examples show that Banach operator pairs need not be neither commuting nor weakly commuting.

Example 2.15.

Let Open image in new window with the usual norm and Open image in new window Let Open image in new window and Open image in new window for all Open image in new window . Then Open image in new window . Note that Open image in new window is a Banach operator pair but Open image in new window and Open image in new window are not commuting.

Example 2.16.

Let Open image in new window with the usual metric. Let Open image in new window be defined by

Then Open image in new window and Open image in new window imply that Open image in new window is a Banach operator pair. Further, Open image in new window and Open image in new window . Thus Open image in new window and Open image in new window 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 Open image in new window be a nonempty closed bounded convex subset of a complete Open image in new window (0) space Open image in new window , Open image in new window a nonexpansive self-mapping of Open image in new window and Open image in new window nonexpansive, where for any Open image in new window , Open image in new window is nonempty compact convex. Assume that for some Open image in new window

is convex for all Open image in new window and Open image in new window . If Open image in new window and Open image in new window commute, then there exists an element Open image in new window such that Open image in new window .

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 Open image in new window (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 Open image in new window 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 Open image in new window -trees (Section 4) for single-valued asymptotic pointwise nonexpansive or nonexpansive mappings and multivalued nonexpansive, Open image in new window -nonexpansive, or Open image in new window -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 Open image in new window be a complete uniformly convex metric space with FPPMM, and, Open image in new window be a bounded closed convex subset of Open image in new window . Assume that Open image in new window is an asymptotic pointwise nonexpansive mapping and Open image in new window a nonexpansive mapping with Open image in new window a nonempty compact convex subset of Open image in new window for each Open image in new window . If the mappings Open image in new window and Open image in new window commute then there is Open image in new window such that Open image in new window .

Proof.

By Theorem 2.9, the fixed point set Open image in new window of Open image in new window of a bounded closed convex subset is a nonempty closed and convex subset of Open image in new window . By the commutativity of Open image in new window and Open image in new window , Open image in new window is Open image in new window -invariant for any Open image in new window and so Open image in new window and convex for any Open image in new window . Therefore, the mapping Open image in new window is well defined.

We will show next that Open image in new window is also nonexpansive as a multivalued mapping. Before that, we claim that Open image in new window for any Open image in new window . In fact, by convexity of Open image in new window and Theorem 2.8, we can take Open image in new window to be the unique point in Open image in new window such that Open image in new window . Now consider the sequence Open image in new window . Since Open image in new window and Open image in new window commute we know that Open image in new window for any Open image in new window . Therefore, by the compactness of Open image in new window , it has a convergent subsequence Open image in new window . Let Open image in new window be the limit of Open image in new window , then we have that

from where, by the uniqueness of Open image in new window , Open image in new window . Consequently, Open image in new window and so Open image in new window . This, in particular, shows that Open image in new window and explains equality (3.1) below. Now, we can argue as follows:

Finally, since Open image in new window has the FPPMM, there exists Open image in new window such that Open image in new window . Therefore, Open image in new window .

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 Open image in new window 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 Open image in new window of a topological space Open image in new window is said to satisfy condition (C) [15, 18] provided Open image in new window for any nonempty Open image in new window -invariant closed set Open image in new window .

Theorem 3.3.

Let Open image in new window be a complete uniformly convex metric space with FPPMM and Open image in new window a bounded closed convex subset of Open image in new window . Assume that Open image in new window is asymptotically pointwise nonexpansive and Open image in new window is nonexpansive with Open image in new window a nonempty compact subset of Open image in new window for each Open image in new window . If the mappings Open image in new window and Open image in new window commute and Open image in new window satisfies condition (C), then there is Open image in new window such that Open image in new window .

Proof.

We know that the fixed point set Open image in new window of Open image in new window is a nonempty closed and convex subset of Open image in new window . Since Open image in new window and Open image in new window commute then Open image in new window is Open image in new window -invariant for Open image in new window , and also, since Open image in new window satisfies condition (C), the mapping Open image in new window is well defined. We prove next that the mapping Open image in new window is nonexpansive.

As in the above proof, we need to show that for any Open image in new window it is the case that Open image in new window . Since Open image in new window and Open image in new window commute, we know that Open image in new window is Open image in new window -invariant. Take Open image in new window such that Open image in new window and consider the sequence Open image in new window . Let Open image in new window be the set of limit points of Open image in new window , then Open image in new window is a nonempty and closed subset of Open image in new window . Consider now Open image in new window , then

and, therefore, Open image in new window . But Open image in new window is also Open image in new window -invariant, so, by condition (C), Open image in new window has a fixed point in Open image in new window and so Open image in new window . 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 Open image in new window of Open image in new window at Open image in new window is proper if Open image in new window or there exists Open image in new window such that Open image in new window is a proper subset of Open image in new window . If Open image in new window is proper for each Open image in new window , we will say that Open image in new window has proper orbits on Open image in new window [19].

Condition (C) in Theorem 3.3 may seem restrictive, however it looks weaker if we recall that the values of Open image in new window 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 Open image in new window having proper orbits then the same conclusion follows.

Proof.

The idea now is that the orbits through Open image in new window of points in Open image in new window are relatively compact, then, by [19, Theorem  3.1], Open image in new window satisfies condition (C).

For any nonempty subset Open image in new window of a metric space Open image in new window , the diameter of Open image in new window is denoted and defined by Open image in new window = sup Open image in new window A mapping Open image in new window has diminishing orbital diameters Open image in new window [19, 20] if for each Open image in new window Open image in new window and whenever Open image in new window there exists Open image in new window such that Open image in new window . Observe that in a metric space Open image in new window if Open image in new window has d.o.d. on Open image in new window , then Open image in new window 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 Open image in new window 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 Open image in new window but, this time, we ask the geodesic space Open image in new window 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 Open image in new window be a geodesic space with no bifurcating geodesics and let Open image in new window be a nonempty subset of Open image in new window . Let Open image in new window , Open image in new window such that Open image in new window , and Open image in new window with Open image in new window . Then the metric projection of Open image in new window onto Open image in new window is the singleton Open image in new window for any Open image in new window .

Now we give another version of Theorem 3.1 without assuming that the values of Open image in new window are convex.

Theorem 3.7.

Let Open image in new window be a complete uniformly convex metric space with FPPMM and with no bifurcating geodesics and Open image in new window a bounded closed convex subset of Open image in new window . Assume that Open image in new window is asymptotically pointwise nonexpansive and Open image in new window nonexpansive with Open image in new window a nonempty compact subset of Open image in new window for each Open image in new window . Assume further that the fixed point set Open image in new window of Open image in new window is such that its topological interior (in Open image in new window ) is dense in Open image in new window . If the mappings Open image in new window and Open image in new window commute, then there exists Open image in new window such that Open image in new window .

Proof.

Just as before, we know that the fixed point set Open image in new window of Open image in new window is a nonempty closed and convex subset of Open image in new window . We are going to see that Open image in new window is well defined. Take Open image in new window and let us see that Open image in new window . Consider Open image in new window such that Open image in new window and let Open image in new window be a limit point of Open image in new window . Fix Open image in new window , then

Therefore, by Lemma 3.6, Open image in new window and so Open image in new window is a convergent sequence to Open image in new window and Open image in new window . Take now Open image in new window , then, by hypothesis, there exists a sequence Open image in new window converging to Open image in new window . Consider the sequence of points Open image in new window given by the above reasoning such that Open image in new window . Define, for each Open image in new window , Open image in new window such that Open image in new window . Since Open image in new window is nonexpansive, Open image in new window . Now, since Open image in new window is compact, take Open image in new window a limit point of Open image in new window . Then Open image in new window because it is also a limit point of Open image in new window and Open image in new window . Therefore our claim that Open image in new window is well defined is correct. Let us see now that Open image in new window is also nonexpansive.

As in the previous theorems, we show that for Open image in new window we have that Open image in new window . Take Open image in new window and consider Open image in new window such that Open image in new window and Open image in new window a limit point of Open image in new window . Take Open image in new window . Then, repeating the same reasoning as above, Open image in new window and so Open image in new window is a fixed point of Open image in new window which proves that Open image in new window for Open image in new window and Open image in new window . For Open image in new window we apply a similar argument as above using that Open image in new window is dense in Open image in new window . Now the result follows as in Theorem 3.1.

Remark 3.8.

The condition about the commutativity of Open image in new window and Open image in new window has been used to guarantee that the orbits Open image in new window for Open image in new window in the fixed point set of Open image in new window remain in a certain compact set and so they are relatively compact. The same conclusion can be reached if we require Open image in new window and Open image in new window 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 Open image in new window be a complete uniformly convex metric space with FPPMM, and, let Open image in new window be a bounded closed convex subset of Open image in new window . Assume that Open image in new window is asymptotically pointwise nonexpansive and Open image in new window nonexpansive with Open image in new window a nonempty compact subset of Open image in new window for each Open image in new window . If the pair Open image in new window is a Banach operator pair, then there is Open image in new window such that Open image in new window .

Proof.

By Theorem 2.9 the fixed point set Open image in new window of Open image in new window is a nonempty closed and convex subset of Open image in new window . Since the pair Open image in new window is a Banach operator pair, Open image in new window for each Open image in new window , and therefore, Open image in new window for Open image in new window . The mapping Open image in new window being the restriction of Open image in new window on Open image in new window 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 Open image in new window is called Open image in new window -nonexpansive [22] if for all Open image in new window and Open image in new window with Open image in new window , there exists Open image in new window with Open image in new window such that Open image in new window . Define Open image in new window by

Husain and Latif [22] introduced the class of Open image in new window -nonexpansive multivalued maps and it has been further studied by Hussain and Khan [23] and many others. The concept of a Open image in new window -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 Open image in new window be the multivalued map defined by
which implies that Open image in new window is not nonexpansive. Let Open image in new window be any small open neighborhood of Open image in new window , then

which is not open. Thus Open image in new window is not continuous. Note also that Open image in new window is a fixed point of Open image in new window .

Theorem 3.12.

Let Open image in new window be a complete uniformly convex metric space with FPPMM and Open image in new window be a bounded closed convex subset of Open image in new window . Assume that Open image in new window is asymptotically pointwise nonexpansive and Open image in new window Open image in new window -nonexpansive with Open image in new window a compact subset of Open image in new window for each Open image in new window . If the pair Open image in new window is a Banach operator pair, then there is Open image in new window such that Open image in new window .

Proof.

As above, the set Open image in new window of fixed points of Open image in new window is nonempty closed convex subset of Open image in new window . Since Open image in new window is compact for each Open image in new window , Open image in new window is well defined and a multivalued nonexpansive selector of Open image in new window [23]. We also have that Open image in new window and Open image in new window for each Open image in new window , so Open image in new window for each Open image in new window . Thus the pair Open image in new window 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 Open image in new window be a complete uniformly convex metric space with FPPMM, and, let Open image in new window be a bounded closed convex subset of Open image in new window . Assumethat Open image in new window is a nonexpansive map and Open image in new window is a Open image in new window -nonexpansive mapping with Open image in new window a compact subset of Open image in new window for each Open image in new window . If the pair Open image in new window is a Banach operator pair, then there is Open image in new window such that Open image in new window .

4. Coincidence Results in Open image in new window-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, 10, 11], existence of fixed points for multivalued mappings happens under very weak conditions if we are working in Open image in new window -tree spaces. This allows us to find much weaker results for Open image in new window -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 Open image in new window -trees.

Theorem 4.1.

Let Open image in new window be a complete Open image in new window -tree, and suppose that Open image in new window is a bounded closed convex subset of Open image in new window . Assume that Open image in new window is asymptotically pointwise nonexpansive and Open image in new window is Open image in new window -semicontinuous mapping with Open image in new window a nonempty closed and convex subset of Open image in new window for each Open image in new window . If the mappings Open image in new window and Open image in new window commute then there is Open image in new window such that Open image in new window .

Proof.

We know that the fixed point set Open image in new window of Open image in new window is a nonempty closed and convex subset of Open image in new window . From the commutativity condition we also have that Open image in new window is Open image in new window -invariant for any Open image in new window and so the mapping defined by Open image in new window is well defined on Open image in new window and takes closed and convex values. By [9, Lemma  2], Open image in new window is a Open image in new window -semicontinuous mapping and so, by [9, Theorem  4] applied to Open image in new window , the conclusion follows.

Remark 4.2.

Actually the only condition we need in the above theorem from Open image in new window is that its set of fixed points is nonempty bounded closed and convex. In the case Open image in new window is nonexpansive then Open image in new window 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 Open image in new window be a complete Open image in new window -tree, and suppose Open image in new window is a bounded closed convex subset of Open image in new window . Then every commuting family Open image in new window of asymptotic pointwise nonexpansive self-mappings of Open image in new window has a nonempty closed and convex common fixed point set.

Proof.

Let Open image in new window . Then by Theorem 2.9, the set Open image in new window of fixed points of Open image in new window is nonempty closed and convex and hence again an Open image in new window -tree. Now suppose Open image in new window . Since Open image in new window and Open image in new window commute it follows Open image in new window , and, by applying the preceding argument to Open image in new window and Open image in new window , we conclude that Open image in new window has a nonempty fixed point set in Open image in new window . In particular the fixed point set of Open image in new window and the fixed point set of Open image in new window 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 Open image in new window be a nonempty bounded closed convex subset of a complete Open image in new window -tree Open image in new window , Open image in new window a commuting family of asymptotic pointwise nonexpansive self-mappings on Open image in new window . Assume that Open image in new window is Open image in new window -semicontinuous mapping on Open image in new window with nonempty closed and convex values and such that Open image in new window and Open image in new window commute weakly for any Open image in new window . If for each Open image in new window

then there exists an element Open image in new window such that Open image in new window for all Open image in new window .

Proof.

By Theorem 4.3, Open image in new window is nonempty closed and convex. Let Open image in new window . Then for any Open image in new window ,
Let Open image in new window the unique closest point to Open image in new window from Open image in new window . Now (4.2) implies that Open image in new window is in Open image in new window . Further, (4.1) implies that Open image in new window and so, by the uniqueness of Open image in new window , Open image in new window . Thus Open image in new window is a common fixed point of Open image in new window , which implies

for each Open image in new window . Let Open image in new window . Now the proof follows as the proof of Theorem 4.1.

Remark 4.5.

Note that condition (4.1) is satisfied if each Open image in new window is nonexpansive with respect to Open image in new window .

Corollary 4.6.

Let Open image in new window be a nonempty bounded closed convex subset of a complete Open image in new window -tree Open image in new window and Open image in new window a commuting family of asymptotic pointwise nonexpansive self-mappings of Open image in new window . Assume that Open image in new window is Open image in new window -semicontinuous, where for any Open image in new window , Open image in new window is nonempty closed and convex and Open image in new window and Open image in new window commute weakly. If

then there exists an element Open image in new window such that Open image in new window for all Open image in new window .

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 Open image in new window -trees are gated; that is, if Open image in new window is a closed and convex subset of the Open image in new window -tree Open image in new window , Open image in new window and Open image in new window is the metric projection of Open image in new window onto Open image in new window then Open image in new window is in the metric segment joining Open image in new window and Open image in new window for any Open image in new window . Notice that condition (4.1) is dropped in the next theorem.

Theorem 4.7.

Let Open image in new window be a nonempty bounded closed convex subset of a complete Open image in new window -tree Open image in new window , Open image in new window a commuting family of asymptotic pointwise nonexpansive self-mappings on Open image in new window . Assume that Open image in new window is Open image in new window -semicontinuous mapping on Open image in new window with nonempty closed and convex values and such that Open image in new window and Open image in new window commute for any Open image in new window , then there exists an element Open image in new window such that Open image in new window for all Open image in new window .

Proof.

As in the proof of Theorem 4.4, the only thing that really needs to be proved is that Open image in new window for each Open image in new window . From the commutativity condition we know that Open image in new window is Open image in new window -invariant for any Open image in new window . Therefore each Open image in new window has a fixed point Open image in new window . But, since the fixed point set of Open image in new window is convex and Open image in new window , then the metric segment joining Open image in new window and Open image in new window is contained in Open image in new window . From the gated property, we know that the closest point Open image in new window to Open image in new window from Open image in new window is in such segment for any Open image in new window . In consequence, Open image in new window is a fixed point for any Open image in new window and, therefore, Open image in new window .

The next theorem follows as a consequence of Theorem 4.7.

Theorem 4.8.

If in the previous theorem Open image in new window is supposed to be either upper semicontinuous or almost lower semicontinuous then the same conclusion follows.

Notes

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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Copyright information

© Esp´ınola and N. Hussain. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Departamento de Análisis MatemáticoUniversidad de SevillaSevillaSpain
  2. 2.Department of MathematicsKing Abdul Aziz UniversityJeddahSaudi Arabia

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