Endpoints of multivalued nonexpansive mappings in geodesic spaces
- Bancha Panyanak^{1}Email author
https://doi.org/10.1186/s13663-015-0398-y
© Panyanak 2015
Received: 14 May 2015
Accepted: 2 August 2015
Published: 19 August 2015
Abstract
Let X be either a uniformly convex Banach space or a reflexive Banach space having the Opial property. It is shown that a multivalued nonexpansive mapping on a bounded closed convex subset of X has an endpoint if and only if it has the approximate endpoint property. This is the first result regarding the existence of endpoints for such kind of mappings even in Hilbert spaces. The related result in a complete CAT(0) space is also given.
Keywords
1 Introduction
A point \(x\in E\) is called a fixed point of T if \(x\in T(x)\). A point \(x\in E\) is called an endpoint (or stationary point) of T if x is a fixed point of T and \(T(x)=\{x\}\). We shall denote by \(\operatorname{Fix}(T)\) the set of all fixed points of T and by \(\operatorname{End}(T)\) the set of all endpoints of T. We see that for each mapping T, \(\operatorname{End}(T)\subseteq \operatorname{Fix}(T)\). Thus, the concept of endpoints seems to be more difficult (but more important) than the concept of fixed points. However, both concepts are equivalent when T is a single-valued mapping since, in this case, \(\operatorname{End}(T)= \operatorname{Fix}(T)\).
The existence of endpoints for a special kind of contractive mappings was first studied by Aubin and Siegel [1]. They proved that every multivalued dissipative mapping on a complete metric space always has an endpoint. Since then the endpoint results for several kinds of contractive mappings have been rapidly developed and many papers have appeared (see, e.g., [2–20]).
The first result regarding the existence of endpoints for non-contractive type mappings was discovered by Garcia-Falset et al. [21]. They proved that every J-type mapping on a weakly compact convex subset of a Banach space with compact faces always has an endpoint. Later on, Garcia-Falset et al. [22] introduced the class of (SL)-type mappings and proved that every (SL)-type mapping on a weakly compact convex subset of a Banach space with normal structure always has an endpoint. But, both classes of J-type and (SL)-type mappings are different from the class of nonexpansive mappings (see Remark 2.5, [22], Example 4, [21], Example 27 and [21], p.1260). Summary: there is no result in metric or Banach spaces regarding the existence of endpoints for nonexpansive mappings.
In this article, we give a necessary and sufficient condition for the existence of endpoints for multivalued nonexpansive mappings in uniformly convex Banach spaces and reflexive Banach spaces having the Opial property. We also obtain the related result in a special kind of metric spaces, namely, CAT(0) spaces. Our main discoveries are Theorems 3.1, 3.4 and 4.7.
2 Preliminaries
In this section we collect some geometric properties of Banach spaces. For more details the reader is referred to [23, 24].
The sequence \(\{x_{n}\}\) is called regular relative to E if \(r(E,\{x_{n}\})=r(E,\{x_{n_{k}}\})\) for all subsequences \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\). It is known that there always exists a subsequence of \(\{x_{n}\}\) which is regular relative to E (see, e.g., [25, 26]).
It is well known that if E is a bounded closed convex subset of a uniformly convex Banach space, then \(A(E,\{x_{n}\})\) consists of exactly one point (see, e.g., [24], p.223).
From now on, we will use the notation ‘\(x_{n}\rightharpoonup x\)’ for a sequence \(\{x_{n}\}\) converging weakly to a point x.
Proposition 2.1
- (1)
Every Hilbert space is a uniformly convex Banach space.
- (2)
Every Hilbert space is a reflexive Banach space having the Opial property.
- (3)
Every uniformly convex Banach space has normal structure.
- (4)
Every reflexive Banach space with the Opial property has normal structure.
Let E be a nonempty subset of a metric space X and \(T:E\to\mathcal{CB}(X)\) be a mapping. A sequence \(\{x_{n}\}\) in E is called an approximate fixed point sequence for T (a.f.p.s. in short) if \(\lim_{n\to\infty}\operatorname{dist}(x_{n},T(x_{n}))=0\). The mapping T is said to have the approximate fixed point property if it has an a.f.p.s. in E (or, equivalently, \(\inf_{x\in E} \operatorname{dist}(x,T(x))=0\)). The mapping T is said to have the approximate endpoint property [11] if \(\inf_{x\in E} r_{x}(T(x))=0\).
Proposition 2.2
- (1)
If T has the approximate endpoint property, then T has the approximate fixed point property.
- (2)
If T is a single-valued mapping, then T has the approximate endpoint property if and only if T has the approximate fixed point property.
The following example shows that the converse of (1) in Proposition 2.2 may not be true if T is a multivalued mapping.
Example 2.3
Case 1. \(0\leq x<1/4\). We have \(r_{x}(T(x))=\sup\{|x-y|: y\in[0,1-x]\}\geq|x-(1-x)|=1-2x>1/2\).
Case 2. \(x\geq1/4\). We have \(r_{x}(T(x))=\sup\{|x-y|: y\in[0,1-x]\}\geq|x-0|= x\geq1/4\).
Thus, \(\inf_{x\in E}r_{x}(T(x)) \geq1/4\). Therefore, T does not have the approximate endpoint property.
However, the converse of (1) in Proposition 2.2 is true under some additional conditions.
Proposition 2.4
Let E be a nonempty subset of a metric space \((X,d)\), \(\{x_{n}\}\) be a sequence in E, and \(T : E\to\mathcal{K}(X)\) be a mapping. Then \(r_{x_{n}}(T(x_{n}))\to0\) if and only if \(\operatorname{dist}(x_{n}, T(x_{n}))\to 0\) and \(\operatorname{diam}(T(x_{n}))\to0\).
Proof
- (1)
There exists an a.f.p.s. for T in each nonempty closed convex and T-invariant subset D of E. Here, T-invariant means \(T(x)\subseteq D\) for all \(x\in D\).
- (2)For any a.f.p.s. \(\{x_{n}\}\) of T in E and each \(x\in E\), one has$$\limsup_{n\to\infty}H\bigl(\{x_{n}\}, T(x)\bigr)\leq\limsup _{n\to\infty} \|x_{n}-x\|. $$
Remark 2.5
The mapping T in Example 2.3 is nonexpansive but is not (SL)-type.
Proof
3 Main results
We begin this section by proving a result in uniformly convex Banach spaces.
Theorem 3.1
Let \((X,\|\cdot\|)\) be a uniformly convex Banach space, E be a nonempty bounded closed convex subset of X, and \(T : E\to \mathcal{K}(E)\) be a nonexpansive mapping. Then T has an endpoint if and only if T has the approximate endpoint property.
Proof
We observe that if X has the Opial property, then the assumption that \(T:E\to\mathcal{K}(E)\) in Theorem 3.1 can be weakened to \(T:E\to\mathcal{K}(X)\). For this, we need the following fact which is known as the demiclosed principle.
Proposition 3.2
Let \((X,\|\cdot\|)\) be a Banach space having the Opial property, E be a nonempty closed convex subset of X, and \(T : E\to \mathcal{K}(X)\) be a nonexpansive mapping. If \(\{x_{n}\}\) is a sequence in E and \(x\in E\), then the conditions \(x_{n}\rightharpoonup x\), \(\operatorname{dist}(x_{n}, T(x_{n}))\to0\), and \(\operatorname{diam}(T(x_{n}))\to0\) imply \(x\in \operatorname{End}(T)\).
Proof
The following fact is an immediate consequence of Propositions 2.4 and 3.2.
Proposition 3.3
Let \((X,\|\cdot\|)\) be a Banach space having the Opial property, E be a nonempty closed convex subset of X, and \(T : E\to \mathcal{K}(X)\) be a nonexpansive mapping. If \(\{x_{n}\}\) is a sequence in E such that \(x_{n}\rightharpoonup x\in E\) and \(r_{x_{n}}(T(x_{n}))\to0\), then \(x\in \operatorname{End}(T)\).
Theorem 3.4
Let \((X,\|\cdot\|)\) be a reflexive Banach space having the Opial property, E be a nonempty bounded closed convex subset of X, and \(T : E\to\mathcal{K}(X)\) be a nonexpansive mapping. Then T has an endpoint if and only if T has the approximate endpoint property.
Proof
The necessity is clear. For the sufficiency, we suppose that T has the approximate endpoint property. Then there exists a sequence \(\{x_{n}\}\) in E such that \(r_{x_{n}}(T(x_{n}))\to0\). Since \(\{x_{n}\}\) is bounded, by the reflexivity of X, there exists a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) such that \(x_{n_{k}}\rightharpoonup x\in E\). The conclusion follows from Proposition 3.3. □
As an immediate consequence of Theorem 3.4 and Proposition 2.2, we can obtain the following.
Corollary 3.5
Let \((X,\|\cdot\|)\) be a reflexive Banach space having the Opial property, E be a nonempty bounded closed convex subset of X, and \(f : E\to X\) be a single-valued nonexpansive mapping. Then f has a fixed point if and only if f has the approximate fixed point property.
4 CAT(0) spaces
Let \([0,l]\) be a closed interval in \(\mathbb{R}\) and x, y be two points in a metric space \((X,d)\). A geodesic joining x to y is a map \(\xi:[0,l]\to X\) such that \(\xi(0)=x\), \(\xi(l)=y\), and \(d(\xi(s),\xi(t))=|s-t|\) for all \(s, t\in[0,l]\). The image of ξ is called a geodesic segment joining x and y which when unique is denoted by \([x,y]\). The space \((X,d)\) is said to be a geodesic space if every two points in X are joined by a geodesic, and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each \(x,y\in X\). A subset E of X is said to be convex if every pair of points \(x,y\in E\) can be joined by a geodesic in X and the image of every such geodesic is contained in E.
A geodesic triangle \(\triangle(p, q, r)\) in a geodesic space \((X,d)\) consists of three points p, q, r in X and a choice of three geodesic segments \([p, q]\), \([q, r]\), \([r, p]\) joining them. A comparison triangle for geodesic triangle \(\triangle(p, q, r)\) in X is a triangle \(\overline{\triangle}(\bar{p}, \bar{q}, \bar{r})\) in the Euclidean plane \(\mathbb{R}^{2}\) such that \(d_{\mathbb{R}^{2}} ( \bar{p},\bar{q} ) = d(p, q)\), \(d_{\mathbb{R}^{2}} ( \bar{q},\bar{r} ) = d(q, r)\), and \(d_{\mathbb{R}^{2}} ( \bar{r},\bar{p} ) = d(r, p)\). A point \(\bar{u}\in[\bar{p}, \bar{q}]\) is called a comparison point for \(u\in[p, q]\) if \(d(p, u)=d_{\mathbb{R}^{2}}(\bar{p},\bar{u})\). Comparison points on \([\bar{q}, \bar{r}]\) and \([\bar{r}, \bar{p}]\) are defined in the same way.
Definition 4.1
A geodesic space X is said to be a CAT(0) space if all of its geodesic triangles satisfy the CAT(0) inequality. For other equivalent definitions and basic properties of CAT(0) spaces, we refer the reader to standard texts such as [27, 28]. It is well known that every CAT(0) space is uniquely geodesic. Notice also that pre-Hilbert spaces, \(\mathbb{R}\)-trees, and Euclidean buildings are examples of CAT(0) spaces (see [27, 29]).
It is known from Proposition 7 of [30] that if \(\{x_{n}\} \) is a bounded sequence in a complete CAT(0) space X, then its asymptotic center \(A(\{x_{n}\})\) consists of exactly one point.
We now give the concept of Δ-convergence and collect some of its basic properties.
Definition 4.2
([31])
A sequence \(\{x_{n}\}\) in a CAT(0) space X is said to Δ-converge to \(x\in X\) if x is the unique asymptotic center of \(\{x_{n_{k}}\}\) for every subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\). In this case we write \(x_{n}\xrightarrow{\Delta}x\) and call x the Δ-limit of \(\{x_{n}\} \).
Lemma 4.3
([31])
Every bounded sequence in a complete CAT(0) space always has a Δ-convergent subsequence.
Lemma 4.4
([32])
If E is a closed convex subset of a complete CAT(0) space and if \(\{x_{n}\}\) is a bounded sequence in E, then the asymptotic center of \(\{x_{n}\}\) is in E.
We use the notation \(t x\oplus(1-t)y\) for the unique point z satisfying (2).
Lemma 4.5
([33])
Inequality (3) is known as the (CN) inequality of Bruhat and Tits [34]. The following lemma is an analog of Proposition 3.2. It can be viewed as an extension of Proposition 3.7 in [31].
Lemma 4.6
Let E be a nonempty closed convex subset of a complete CAT(0) space \((X,d)\) and \(T : E\to\mathcal{K}(X)\) be a nonexpansive mapping. If \(\{x_{n}\}\) is a sequence in E and \(x\in E\), then the conditions \(x_{n}\xrightarrow{\Delta}x\), \(\operatorname{dist}(x_{n}, T(x_{n}))\to0\), and \(\operatorname{diam}(T(x_{n}))\to0\) imply \(x\in \operatorname{End}(T)\).
Proof
Theorem 4.7
Let E be a nonempty bounded closed convex subset of a complete CAT(0) space \((X,d)\) and \(T : E\to\mathcal{K}(X)\) be a nonexpansive mapping. Then T has an endpoint if and only if T has the approximate endpoint property.
Proof
The necessity is clear. For the sufficiency, we suppose that T has the approximate endpoint property. Then there exists a sequence \(\{x_{n}\}\) in E such that \(r_{x_{n}}(T(x_{n}))\to0\) and hence \(\operatorname{dist}(x_{n}, T(x_{n}))\to0\) and \(\operatorname{diam}(T(x_{n}))\to0\) by Proposition 2.4. Since \(\{x_{n}\}\) is bounded, by Lemmas 4.3 and 4.4, there exists a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) such that \(x_{n_{k}}\xrightarrow{\Delta}x\in E\). By Lemma 4.6, x is an endpoint of T. □
As a consequence of Theorem 4.7, we can obtain the following.
Corollary 4.8
([35], Theorem 21)
Let E be a nonempty bounded closed convex subset of a complete CAT(0) space \((X,d)\) and \(f : E\to X\) be a single-valued nonexpansive mapping. Then f has a fixed point if and only if \(\inf\{d(x,f(x)): x\in E\}=0\).
If T is a single-valued nonexpansive mapping on a closed convex subset of a complete CAT(0) space, then \(\operatorname{Fix}(T)\) is closed and convex (see, e.g., Kirk [36]). The closedness of \(\operatorname{Fix}(T)\) can be easily extended to the multivalued case. But the convexity of \(\operatorname{Fix}(T)\) cannot be extended (see, e.g., [37, 38]). However, if T is a multivalued nonexpansive mapping, then \(\operatorname{End}(T)\) is always closed and convex as the following result.
Theorem 4.9
Let E be a nonempty closed convex subset of a complete CAT(0) space \((X,d)\) and \(T:E\to \mathcal{CB}(X)\) be a nonexpansive mapping with \(\operatorname{End}(T)\neq\emptyset\). Then \(\operatorname{End}(T)\) is closed and convex.
Proof
5 Concluding remarks and open questions
Remark 5.1
As we have observed from Proposition 2.1, every Hilbert space is a uniformly convex Banach space and is a reflexive Banach space having the Opial property and is even a CAT(0) space. Thus, all results in this article also hold in Hilbert spaces.
In view of Theorems 3.4 and 4.7, we do not know if Theorem 3.1 can be extended to nonself-mappings. Therefore, the following question remains open.
Question 5.2
Let X be a uniformly convex Banach space, E be a nonempty bounded closed convex subset of X, and \(T : E\to\mathcal{K}(X)\) be a nonexpansive mapping. If T has the approximate endpoint property, then does T have an endpoint?
In view of Theorems 3.1 and 3.4, along with Proposition 2.1, the following question should be of interest.
Question 5.3
Let X be a reflexive Banach space with normal structure, E be a nonempty bounded closed convex subset of X, and \(T : E\to\mathcal{K}(E)\) be a nonexpansive mapping. If T has the approximate endpoint property, then does T have an endpoint?
One may observe that the (CN) inequality is a key tool in the proof of Theorem 4.9 and there is an inequality in uniformly convex Banach spaces similar to it (see [39], p.1133). However, Theorem 4.9 for uniformly convex Banach spaces is unknown. Therefore, the following question remains open.
Question 5.4
Let X be a uniformly convex Banach space, E be a nonempty closed convex subset of X, and \(T : E\to\mathcal{CB}(X)\) be a nonexpansive mapping. Is \(\operatorname{End}(T)\) convex?
Declarations
Acknowledgements
The author thanks Chiang Mai University for financial support.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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