Abstract
The purpose of this paper is to ensure the existence of fixed points for multivalued nonexpansive weakly inward nonself-mappings in uniformly convex metric spaces. This extends a result of Lim (1980) in Banach spaces. All results of Dhompongsa et al. (2005) and Chaoha and Phon-on (2006) are also extended.
1. Introduction
In 1974, Lim [1] developed a result concerning the existence of fixed points for multivalued nonexpansive self-mappings in uniformly convex Banach spaces. This result was extended to nonself-mappings satisfying the inwardness condition independently by Downing and Kirk [2] and Reich [3]. This result was extended to weak inward mappings independently by Lim [4] and Xu [5]. Recently, Dhompongsa et al. [6] presented an analog of Lim-Xu's result in CAT(0) spaces. In this note, we extend the result to uniformly convex metric spaces which improve results of both Lim-Xu and Dhompongsa et al. In addition, we also give a new proof of a result of Lim [7] by using Caristi's theorem [8]. Finally, we give some basic properties of fixed point sets for quasi-nonexpansive mappings for these spaces.
2. Preliminaries
A concept of convexity in metric spaces was introduced by Takahashi [9]. Definition 2.1. Let be a metric space and . A mapping is said to be a convex structure on if for each and A metric space together with a convex structure is called a convex metric space which will be denoted by .
Definition 2.2. A convex metric space is said to be uniformly convex [10] if for any , there exists such that for all and with and Obviously, uniformly convex Banach spaces are uniformly convex metric spaces. By using the (CN) inequality [11], it is easy to see that CAT(0) spaces are also uniformly convex.
For , , and , we denote , , , and So we can define the inward set of as follows: Let be a nonempty subset of a metric space . Then is called convex if for We will denote by the family of nonempty closed subsets of , by the family of nonempty compact subsets of and by the family of nonempty compact convex subsets of Let be the Hausdorff distance on That is,
Definition 2.3. A multivalued mapping is said to be inward on if for some and weakly inward on if for some where denotes the closure of a subset of . In a Banach space setting, if is convex, then so is . Therefore, the conditions above can be replaced by and , respectively.
Definition 2.4. A multivalued mapping satisfyingis called a contraction if and nonexpansive if A point is a fixed point of if
Given a metric space one way to describe a metric space ultrapower of is to first embed as a closed subset of a Banach space (see, e.g., [12, page 129]). Let denote a Banach space ultrapower of relative to some nontrivial ultrafilter (see, e.g., [13]). Then takeOne can then let denote the metric on inherited from the ultrapower norm in . If is complete, then so is since is a closed subset of the Banach space In particular, the metric on is given bywith if and only if
If is a convex metric space, we consider a metric space ultrapower of and define a function byIn order to show that the function is well defined, we need the following condition. For each and which is equivalent tofor all and
By using condition (2.12), it is easy to see that is a convex structure on This implies that is a convex metric space.
Example 2.5. Every Banach space satisfies condition (2.11).
Example 2.6. Condition (2.11) is satisfied for spaces of hyperbolic type (for more details of these spaces see [14]). This is also true for CAT(0) spaces and -trees.
Example 2.7. Let be a hyperconvex metric space. Then there exists a nonexpansive retract (see, e.g., [15] for more on this). For any and we letSince is a Banach space, also satisfies condition (2.11).
Let be a nontrivial ultrafilter on the natural number . If is a uniformly convex metric space satisfying condition (2.11), then the metric space ultrapower relative to is also uniformly convex. Indeed, let any and let be a positive number corresponding to the uniform convexity of Let and be such thatThen there are some representatives and of and and a set such thatFor such This implies
Recall that a subset of a metric space is said to be (uniquely) proximinal if each point has a (unique) nearest point in A convex metric space is said to have property (C) if every decreasing sequence of nonempty bounded closed convex subsets of has nonempty intersection. In [10], Shimizu and Takahashi proved that property (C) holds in complete uniformly convex metric spaces. This implies that every nonempty closed convex subset of a complete uniformly convex metric space is uniquely proximinal. Indeed, let be a nonempty closed convex subset of a complete uniformly convex metric space , and
Let For each we defineThen is a decreasing sequence of nonempty bounded closed convex subsets of Moreover,which is nonempty by the above observation. The uniqueness follows from the uniform convexity of
3. Main Results
We first establish the following lemma.
Lemma 3.1. Let be a complete uniformly convex metric space satisfying condition (2.11), a nonempty closed convex subset of , and the unique nearest point of in Then
Proof. Let . Then there is a sequence in and Choose such that for all . For such , let and write ThenThis impliesIf we let By the uniform
convexity of we have On the other hand,
for each , let We will show
that
Case 1. We are done.
Case 2. Let This impliesThereforeWe claim that If not, let
From (3.4), (3.6), and the uniform convexity of , we haveThis implieswhich is a contradiction. Hence and so by the
convexity of
Case 3. let By the same
arguments in the proof of Case 2, we can show that This means
By condition (2.11), which implies By the same arguments
in the first part of the proof, which is a
contradiction. Hence as
desired.
From [6, Lemma 3.4], we observe that the space is not necessarity assumed to be a CAT(0) space since the proof is only involved with condition (2.11) which is weaker than the (CN) inequality (see [11, Lemma 3]). Therefore, we can obtain the following lemma.
Lemma 3.2. Let be a complete convex metric space satisfying condition (2.11), a nonempty closed subset of and a contraction mapping satisfying, for all Then has a fixed point.
This lemma was first proved in Banach spaces by Lim [7], using transfinite induction, while we apply directly Caristi's theorem. For completeness, we include the details.
Proof of Lemma 3.2.. Let be the
contraction constant of and let be such that Let and define a
metric on by It is easy to
see that is a complete
metric space.
Now define by . Then is continuous
on Suppose that has no fixed
points, that is, for all Let By (3.9), we can find satisfying Now choose and write for some For such , we haveSince , we can find satisfying
Now we define a mapping by for all We claim that satisfiesCaristi's
theorem [8] then
implies that has a fixed
point, which contradicts to the strict inequality (3.12) and the proof is
complete. So it remains to prove (3.12). In fact, it is enough to show thatBut it only needs
to prove that
Now,ThereforeIt follows thatWe now let then by the
condition (2.11),ThusInequalities (3.15), (3.18), and (3.10) imply thatTherefore as
desired.
By Lemmas 3.1 and 3.2 with the same arguments in the proof of Theorem 3.3 of [6], we can obtain the following theorem which extends [4, Theorem 8] by Lim and [6, Theorem 3.3] by Dhompongsa et al.
Theorem 3.3. Let be a complete uniformly convex metric space satisfying condition (2.11), a nonempty bounded closed convex subset of and a nonexpansive weakly inward mapping. Then has a fixed point.
As an immediate consequence of Theorem 3.3, we obtain the following corollary.
Corollary 3.4. Let be a complete uniformly convex metric space satisfying condition (2.11), a nonempty bounded closed convex subset of , and a nonexpansive mapping. Then has a fixed point.
In fact, this corollary is a special case of [10, Theorem 2] in which condition (2.11) was not assumed. An interesting question is whether condition (2.11) in Theorem 3.3 can be dropped.
Let be a nonempty subset of a metric space Recall that a single-valued mapping and a multivalued mapping are said to be commuting if for all and If is nonexpansive with being bounded closed convex and complete uniformly convex satisfying condition (2.11), then is nonempty by the above corollary. Moreover, by a standard argument, we can show that it is also closed and convex. So we can obtain a common fixed point theorem in uniformly convex metric spaces as [6, Theorem 4.1] (see also [16, Theorem 4.2] for a related result in Banach spaces).
Theorem 3.5. Let be a complete uniformly convex metric space satisfying condition (2.11), let be a nonempty bounded closed convex subset of and let and be nonexpansive. Assume that for some If and are commuting, then there exists a point such that
4. Fixed Point Sets of Quasi-Nonexpansive Mappings
Let be a metric space. Recall that a mapping is said to be quasi-nonexpansive if for all and In this case, we will assume that In [17], Chaoha and Phon-on showed that if is a CAT(0) space, then is closed convex. Furthermore, they gave an explicit construction of a continuous function defined on whose fixed point set is any prescribed closed subset of In this section, we extend these results to uniformly convex metric spaces.
We begin by proving the following lemma.
Lemma 4.1. Let be a uniformly convex metric space, and let for which Then
Proof. Let be such that Then and also by (4.1). We will show that Suppose not, we let and Since choose so that By the uniform convexity of there exists such thatBy using the same arguments, we can show that Thereforewhich is a contradiction.
By using the above lemma with the proof of Theorem 1.3 of [17], we obtain the following result.
Theorem 4.2. Let be a convex subset of a uniformly convex metric space and a quasi-nonexpansive mapping whose fixed point set is nonempty. Then is closed convex.
In [17], the authors constructed a continuous function defined on a CAT(0) space whose fixed point set is any prescribed closed subset of by using the following two implications of the (CN) inequality: In fact, condition (4.4) holds in uniformly convex metric spaces as the following lemma shows.
Lemma 4.3. Condition (4.4) holds in uniformly convex metric spaces.
Proof. We first note that the conclusion holds if or We now let be a uniformly convex metric space, , and Let and . Without loss of generality, we can assume that . Let thenIf we let Then by the uniform convexity of we can show that and This implieswhich is a contradiction, hence Therefore
It is unclear that condition (4.5) holds for uniformly convex metric spaces. However, the following theorem is a generalization of [17, Theorem 2.1], we omit the proof because it is similar to the one given in [17].
Theorem 4.4. Let be a nonempty subset of a uniformly convex metric space satisfying condition (4.5). Then there exists a continuous function such that
Acknowledgment
This research was supported by the Commission on Higher Education and Thailand Research Fund under Grant MRG5080188.