Common Fixed Points for Multimaps in Metric Spaces
© Esp´ınola and N. Hussain. 2010
Received: 10 June 2009
Accepted: 15 September 2009
Published: 1 November 2009
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.
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 , 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  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.
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 .
A point is called a center for the mapping if for each , . The set denotes the set of all centers of the mapping .
for any .
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.
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 . To define them we first need to introduce the notion of convex metric.
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.
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 .
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  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  and more recently to uniformly convex metric spaces of type either (M) or (L) in .
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 ).
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.
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 , 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  to obtain different results on coincidence fixed points in -trees and will play a main role in our last section.
It is shown in  that -semicontinuity of multivalued mappings is a strictly weaker notion than upper semicontinuity and almost lower semicontinuity. Similar results to those presented in  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  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.
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  introduced the class of Banach operator pairs as a new class of noncommuting maps which has been further studied by Hussain  and Pathak and Hussain . Here we extend this concept to multivalued mappings.
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.
Let with the usual norm and Let and for all . Then . Note that is a Banach operator pair but and are not commuting.
Then and imply that is a Banach operator pair. Further, and . Thus and are neither commuting nor weakly commuting.
In 2005, Dhompongsa et al.  proved the following fixed point result for commuting mappings.
Theorem 2 DKP.
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  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.
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 .
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
Finally, since has the FPPMM, there exists such that . Therefore, .
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  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 . 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 .
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 .
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 .
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.
Under the same conditions of the previous theorem, if condition (C) is replaced with having proper orbits then the same conclusion follows.
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 .
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  in order to obtain stronger versions of the next lemma which is the one we need and which proof is immediate.
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.
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 .
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.
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.
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 .
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.
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  if for all and with , there exists with such that . Define by
Husain and Latif  introduced the class of -nonexpansive multivalued maps and it has been further studied by Hussain and Khan  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 .
which is not open. Thus is not continuous. Note also that is a fixed point of .
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 .
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 . 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.
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 . We begin with the adaptation of Theorem 3.1 to -trees.
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 .
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.
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 .
The next theorem is the counterpart of Espínola and Kirk [24, Theorem 4.3] to asymptotic pointwise nonexpansive mappings.
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.
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.
then there exists an element such that for all .
for each . Let . Now the proof follows as the proof of Theorem 4.1.
Note that condition (4.1) is satisfied if each is nonexpansive with respect to .
then there exists an element such that for all .
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.
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 .
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.
If in the previous theorem is supposed to be either upper semicontinuous or almost lower semicontinuous then the same conclusion follows.
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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