Fixed point sets of setvalued mappings
 Parunyou Chanthorn^{1} and
 Phichet Chaoha^{1, 2}Email author
https://doi.org/10.1186/s1366301503056
© Chanthorn and Chaoha; licensee Springer. 2015
Received: 23 December 2014
Accepted: 1 April 2015
Published: 22 April 2015
Abstract
We present new results regarding fixed point sets of various setvalued mappings using the concept of fixed point iteration schemes and the newly defined concept of fixed point resolutions. In particular, we prove that the fixed point sets of certain nonexpansive setvalued mappings are contractible.
Keywords
MSC
1 Introduction
In 1973, Bruck [1] gave an intriguing result on the structure of fixed point sets by proving that the fixed point set of a certain nonexpansive mapping is always a nonexpansive retract of its domain. It took almost thirty years before the result was extended to asymptotically nonexpansive mappings in [2]. From the point of view of topological theory, such retraction results enable us to pass various topological properties (for example, connectedness and contractibility) from the domain of the mapping onto its fixed point set. Recently in [3], Chaoha introduced the notion of virtually nonexpansive mappings (which includes various nonexpansivetype mappings) on metric spaces and proved that the fixed point set of a virtually nonexpansive mapping is always a retract of a certain subset called the convergence set. Consequently, Chaoha and Atiponrat [4] extended the notion of virtually nonexpansive mappings on metric spaces to virtually stable mappings on Hausdorff spaces and presented a retraction result similar to [3] for regular spaces. Recently, Chaoha and Chanthorn [5] presented the concept of fixed point iteration schemes that unifies wellknown iteration processes (for example, Picard, Mann and Ishikawa iteration processes [6]) and showed that in regular spaces the fixed point set of a certain virtually stable scheme is a retract of its convergence set. Combined with numerous convergence results of iteration processes in the literature, the authors were able to derived some contractibility criteria for fixed point sets of mappings in various situations. For setvalued mappings, fewer results on the structure of fixed point sets have been explored. It was recently proved in [7] that the fixed point set of a certain quasinonexpansive setvalued mapping on a CAT(0) space is always convex and hence contractible. In this work, we use the concept of virtually stable schemes to acquire retraction results for the fixed point sets of setvalued mappings in appropriate settings. Especially, combined with Nadler’s result in [8], we obtain a new contractibility criterion for the fixed point set of a certain setvalued αcontraction. Then we will construct a sequence of mappings that is not naturally a scheme, but surprisingly yields similar retraction and contractibility results for the fixed point set of a certain nonexpansive setvalued mapping. This immediately calls for the introduction of fixed point resolutions generalizing fixed point iteration schemes at the end of this work.
The paper is organized as follows. In Section 2, we recall the backgrounds used throughout this work. In particular, we present some useful conditions involving setvalued mappings as well as the new concept of αcontractive schemes. In Section 3, we first construct, with the aid of Michael’s selection theorem, virtually stable schemes for certain setvalued mappings to obtain retraction results for fixed point sets. Then we construct an αcontractive scheme for a setvalued αcontraction to obtain a new contractibility criterion for its fixed point set. Finally, we construct a sequence of mappings that is not a fixed point iteration scheme, but still induces retraction and contractibility results for the fixed point set of a certain nonexpansive setvalued mapping.
2 Preliminaries
For a nonempty set X, we let \(\mathcal{P}(X)\) be the power set of X and \(2^{X} =\mathcal{P}(X)  \{\emptyset\}\). Also, we denote the fixed point sets of a singlevalued mapping \(f:X\rightarrow X\) and a setvalued mapping \(F:X\rightarrow2^{X}\) by \(\operatorname{Fix}(f) = \{x \in X: x=f(x)\}\) and \(\operatorname{Fix}(F) = \{x \in X: x\in F(x)\}\), respectively. Throughout this paper, we always assume that every mapping has nonempty fixed point sets.

\(B(a;\varepsilon) =\{ y\in X: d(a,y)< \varepsilon \}\),

\(D(a;\varepsilon) =\{ y\in X: d(a,y)\leq\varepsilon \}\),

\(S(a;\varepsilon) =\{ y\in X: d(a,y)= \varepsilon \}\),

\(\eta(A; \varepsilon) =\bigcup_{a\in A} B(a;\varepsilon)\),

\(d(A,B)=\inf\{ d(a,b): a \in A \mbox{ and } b\in B \}\),

\(d(a,B)=d(\{a\},B)=\inf_{b\in B} d(a,b) \),

\(h(A,B)=\sup_{a\in A} d(a,B) \), and

\(\operatorname{CB}(X)=\{A\in2^{X}: A \mbox{ is closed and bounded}\}\).
The following facts can be found in [8, 10, 11].
Lemma 2.1
 (1)
\(d(x,A)\leq d(x,y)+d(y,A) \).
 (2)
\(d(y,A)\leq\sup_{b\in B} d(b,A)= h(B,A)\) for each \(y\in B \).
 (3)
\(H(A,B)= \max \{ h(A,B), h(B,A) \} \) for each \(A,B\in \operatorname{CB}(X)\).
 (4)
For each \(a\in A\) and \(\varepsilon>0\), there is \(b\in B\) such that \(d(a,b)\leq H(A,B)+\varepsilon\).

the endpoint condition [12] if \(F(p)=\{p\}\) for each \(p\in \operatorname{Fix}(F) \),

the proximal condition if \(P_{F}(x)\neq\emptyset\) for each \(x\in X\),

the Chebyshev condition if \(P_{F}(x)\) is a singleton for each \(x\in X\).
Remark 2.2
The definitions of the proximal condition and the Chebyshev condition are motivated by the proximal set and the Chebyshev set, respectively, in [13]. When a mapping \(F: X\rightarrow2^{X}\) satisfies the Chebyshev condition, we will identify \(P_{F}(x)\) with its element, i.e., \(P_{F}\) can be considered as a mapping on X in this case. Note that every setvalued mapping with compact values always satisfies the proximal condition.
Recall that \(f:X\rightarrow Y\) is called a selection of the setvalued mapping \(F:X\rightarrow2^{Y}\) if \(f(x)\in F(x)\) for each \(x\in X\).
Proposition 2.3
If \(F:X\rightarrow2^{X}\) satisfies the endpoint condition and \(f: X\rightarrow X\) is a selection of F, then \(\operatorname{Fix}(F)=\operatorname{Fix}(f) \).
Lemma 2.4
 (1)
\(\operatorname{Fix}(F)= \operatorname{Fix}(P_{F})\).
 (2)
\(P_{F}\) satisfies the endpoint condition.
 (3)
\(P_{F}(x)= F(x) \cap S(x; d(x,F(x))) \) for each \(x\in X\).
 (4)
If F has closed values, then \(P_{F}(x) \) is closed and bounded for each \(x\in X\).
Proof
(1) and (2) are obvious. (3) is straightforward from the definition of \(P_{F}\). (4) follows directly from (3) and the fact that \(F(x)\) is closed for each \(x\in X\). □

upper semicontinuous at x if \((x_{n})\) is a sequence in X converging to x and U is an open subset of Y such that \(F(x) \subseteq U\), then there exists \(N\in \mathbb{N}\) such that \(F(x_{n})\subseteq U\) for each \(n\geq N\),

lower semicontinuous at x if \((x_{n})\) is a sequence in X converging to x and \(y\in F(x) \), then there exists a sequence \((y_{n})\) in Y such that \(y_{n}\in F(x_{n})\) and \((y_{n})\) converges to y,

Hupper semicontinuous at x if for each \(\varepsilon >0\), there is \(\delta>0\) such that \(h(F(y),F(x))<\varepsilon\) for each \(y\in B(x,\delta)\),

Hlower semicontinuous at x if for each \(\varepsilon >0\), there is \(\delta>0\) such that \(h(F(x),F(y))<\varepsilon\) for each \(y\in B(x,\delta)\),

(H) continuous at x if F is (H) upper and (H) lower semicontinuous at x.

an αcontraction if \(H(F(x),F(y))\leq\alpha d(x,y)\) for each \(x,y\in X\),

nonexpansive if \(H(F(x),F(y))\leq d(x,y)\) for each \(x,y\in X\),

quasinonexpansive if \(H(F(x),F(p))\leq d(x,p)\) for each \(x \in X\) and \(p\in \operatorname{Fix}(F)\),

∗nonexpansive [14] if for each \(x,y\in X\) and \(u_{x}\in P_{F}(x)\), there is \(u_{y}\in P_{F}(y)\) such that \(d(u_{x},u_{y})\leq d(x,y)\).
It is not difficult to see that every setvalued αcontraction is nonexpansive and Hcontinuous, while every nonexpansive setvalued mapping is quasinonexpansive and continuous. Moreover, when F is singlevalued, the above definitions of αcontraction, nonexpansive mapping and quasinonexpansive setvalued mapping coincide with the usual definitions for singlevalued mapping.
Lemma 2.5
([10], Proposition 5.3.42 and Proposition 5.3.43)
 (1)
If F is Hupper semicontinuous and has compact values, then it is upper semicontinuous.
 (2)
If F is Hlower semicontinuous, then it is lower semicontinuous.
Lemma 2.6
([15], Proposition 2.3)
Let X and Y be metric spaces and \(F:X\rightarrow2^{Y} \) be lower semicontinuous. If a mapping \(G:X\rightarrow2^{Y} \) satisfies \(\overline{G(x)}= \overline{F(x)}\) for each \(x\in X\), then G is lower semicontinuous.
Lemma 2.7
([10], Proposition 5.3.20)
Lemma 2.8
Every selection of a quasinonexpansive setvalued mapping on a metric space satisfying the endpoint condition is quasinonexpansive.
Proof
It is straightforward from the definition. □
Lemma 2.9
Let X be a metric space. If \(F:X\rightarrow2^{X}\) is ∗nonexpansive with closed values and satisfies the proximal condition, then the mapping \(P_{F}: X\rightarrow \operatorname{CB}(X)\) is nonexpansive.
Proof
We will see later that the continuity of the mapping \(P_{F}\) plays a crucial role in proving the main result when the setvalued mapping F satisfies the Chebyshev condition. Also in Theorem 2.1.1 [16], it is shown, by using the compactness of the unit disc of the Euclidean space, that every continuous mapping F from the Euclidean space \(\mathbb{R}^{n}\) into \(2^{\mathbb{R}^{n}} \) satisfying the Chebyshev condition induces the continuity of \(P_{F}\). Therefore, to generalize such a result, we will consider a continuous setvalued mapping on a compact metric space as follows.
Lemma 2.10
Let X be a compact metric space, \(p\in X\) and \((x_{n})\) be a sequence in X. If every convergent subsequence of \((x_{n})\) converges to p, then so does \((x_{n})\).
Proof
Suppose that \((x_{n})\) does not converge to p. Then there exist a neighborhood U of p and a subsequence \(( x_{n_{k}})\) of \((x_{n})\) such that \(x_{n_{k}}\notin U\) for each \(k\in \mathbb{N}\). By the compactness of X, the sequence \((x_{n_{k}}) \) has a convergent subsequence which is also a subsequence of \((x_{n})\), say \((x_{m_{k}}) \). Hence, \((x_{m_{k}})\) converges to a point p in X by the assumption, and we get a contradiction. □
Theorem 2.11
Let X be a compact metric space. If \(F :X\rightarrow2^{X}\) is continuous, satisfies the Chebyshev condition and has closed values, then \(P_{F}: X\rightarrow X\) is continuous.
Proof
Recall that a metric space X is said to be metrically convex [17] if for each \(x,y\in X \) with \(x\neq y\), there exists an element \(z\in X\) such that \(x\neq z\), \(y\neq z\), and \(d(x,y)=d(x,z)+ d(z,y)\). Notice that every linear space is metrically convex.
Lemma 2.12
([17])
Lemma 2.13
Let X be a complete and metrically convex metric space, \(x,y\in X \) and \(s,t\in[0,\infty)\). Then we have \(H(D(x;s), D(x;t))= st \). Moreover, if X is a normed space (over \(\mathbb{R}\)), then \(H(D(x; t), D(y;t))= \Vert x y\Vert \).
Proof
The assumption that X is a normed space in the previous lemma is necessary as shown in the following example.
Example 2.14
Consider \([0,1]\) with the standard metric. Let \(x=0\), \(y=1\), and \(t=1\). Then \(H(D(x;t), D(y;t))=0\) but \(\vert xy\vert =1\).
The following theorem follows directly from the classical Michael’s selection theorem ([15], Theorem 3.2^{′′}).
Theorem 2.15
Every lower semicontinuous setvalued mapping from a metric space into a Banach space with closed and convex values admits a continuous selection.
Lemma 2.16
 (1)
The mapping \(\overline{\Phi}: E\rightarrow2^{E} \) defined by \(\overline{\Phi} ( x)=D(f(x);\varphi(x))\) for each \(x\in E\) is lower semicontinuous.
 (2)
The mapping \(\Phi: E\rightarrow2^{E} \) defined by \(\Phi(x)= B(f(x);\varphi(x))\) for each \(x\in E\) is lower semicontinuous.
 (3)
If \(F:X\rightarrow \operatorname{CC}(X) \) is a lower semicontinuous mapping satisfying \(\Phi(x)\cap F(x) \neq\emptyset\) for each \(x\in X\), then there is a continuous mapping \(g:X\rightarrow X\) such that \(g(x)\in\overline{\Phi} (x) \cap F(x)\) for each \(x\in X\).
Proof
(2) It follows directly from (1) and Lemma 2.6.
Definition 2.17
A scheme \(\mathcal{S}= (\prod_{i=1}^{n} f_{i})\) on X is said to be a virtually stable scheme if for each common fixed point p of \(\mathcal{S}\) and each neighborhood U of p, there exist a neighborhood V of p and a subsequence \(( {n_{k}})\) of \((n)\) such that \(\prod_{i=j}^{n_{k}} f_{i} (V)\subseteq U \) for each \(k\in \mathbb{N}\) and \(j\leq n_{k}\).
Example 2.18
From [5] it follows that the Picard iteration scheme \((f^{n}) \) for a continuous quasinonexpansive mapping f on a metric space is virtually stable.
Recall that a subset A of the space X is said to be a retract of X if there is a continuous \(r: X\rightarrow A\) such that \(r(a)=a\) for each \(a\in A\). Moreover, r is called a retraction. In addition, a (topological) space X is said to be contractible if there are a point \(x_{0}\in X\) and a continuous mapping \(f: X\times[0,1]\rightarrow X \) such that \(f(x,1)=x\) and \(f(x,0)=x_{0} \) for all \(x\in X\). Note that a convex subset of a Banach space and a retract of a contractible space are always contractible.
Theorem 2.19
([5], Theorem 2.5)
Let X be a regular space. If \(\mathcal{S}\) is a virtually stable scheme having a continuous subsequence, i.e., there is a subsequence \((s_{n_{k}} )\) of \(\mathcal{S}\) such that each \(s_{n_{k}}\) is continuous, then \(r: \mathrm{C}(\mathcal{S})\rightarrow \operatorname{Fix}(\mathcal{S})\) is a retraction.
Next we will introduce the concept of an αcontractive scheme on a metric space X which resembles an αcontraction in the sense that it is always virtually stable, and when X is complete, its convergence set is the whole space X.
Definition 2.20
 (1)For each sequence \((t_{n})\in\{ (\prod_{i=0}^{n} f_{k+i}) : k\in \mathbb{N}\} \),for each \(n\in \mathbb{N}\), where \(t_{0}(x)=x\) for each \(x\in X\).$$d\bigl(t_{n+1}(x),t_{n}(x)\bigr)\leq\alpha d \bigl(t_{n}(x), t_{n1}(x)\bigr) $$
 (2)
The set \(\mathcal{F}=\{ f_{n}:n\in \mathbb{N}\}\) is equicontinuous on \(\operatorname{Fix}(\mathcal{F})\).
Example 2.21
If f is an αcontraction on a metric space, then the Picard iteration scheme \((f^{n})\) is αcontractive.
Theorem 2.22
Every αcontractive scheme \(\mathcal{S}\) on a metric space X is virtually stable. Moreover, if X is complete, then \(\mathrm{C}(\mathcal{S})=X\).
Proof
Notice that if a sequence \(( x_{n})\) satisfies \(d( x_{n+1} ,x_{n} ) \leq \alpha d( x_{n } , x_{n1} )\) for some \(\alpha\in[0,1)\) and all \(n\in \mathbb{N}\), then \((x_{n} )\) is a Cauchy sequence. This fact immediately implies that \(\mathrm{C}(\mathcal{S})=X\) when X is complete. □
Corollary 2.23
Proof
Observe that \(\mathcal{F}=\{f\}\) is equicontinuous on \(\operatorname{Fix}(\mathcal{F})\) by the continuity of f. Also, since \((\prod_{i=0}^{n} f_{k+i})= (f^{n})=\mathcal{S}\) for each \(k\in \mathbb{N}\), then \(\mathcal{S}\) is αcontractive and hence virtually stable by the previous theorem. □
3 Main results
We now present some new retraction and contractibility results for fixed point sets of certain setvalued mappings.
Theorem 3.1
Let X be a closed subset of a Banach space E. If \(F : X \to \operatorname{CCB}(X)\) is a lower semicontinuous quasinonexpansive mapping satisfying the endpoint condition, then there is a virtually stable scheme \(\mathcal{S}\) such that \(\operatorname{Fix}(\mathcal{S}) = \operatorname{Fix}(F)\), and hence \(\operatorname{Fix}(F)\) is a retract of \(\mathrm{C}(\mathcal{S})\).
Proof
Note that \(\operatorname{CC}(X)\subseteq \operatorname{CC}(E)\) because X is closed in E. Since F is lower semicontinuous, by Theorem 2.15, F admits a continuous selection, say \(f:X\rightarrow X\). By Lemma 2.8, f is quasinonexpansive, and the scheme \(\mathcal{S}=(f^{n})\) is virtually stable by Example 2.18. Also, by Proposition 2.3, \(\operatorname{Fix}(\mathcal{S})=\operatorname{Fix}(f)=\operatorname{Fix}(F)\). □
Corollary 3.2
Let X be a closed subset of a Banach space. If \(F : X \to \operatorname{CC}(X)\) is a ∗nonexpansive mapping satisfying the proximal condition, then there is a virtually stable scheme \(\mathcal{S}\) such that \(\operatorname{Fix}(\mathcal{S}) = \operatorname{Fix}(F)\), and hence \(\operatorname{Fix}(F)\) is a retract of \(\mathrm{C}(\mathcal{S})\).
Proof
From Lemma 2.4(1), (2) and Lemma 2.9, the mapping \(P_{F}: X \rightarrow \operatorname{CCB}(X)\) is nonexpansive satisfying the endpoint condition and \(\operatorname{Fix}(F)=\operatorname{Fix}(P_{F})\). Moreover, \(P_{F}\) is lower semicontinuous by Lemma 2.5(2). Following Theorem 3.1, the sequence \(\mathcal{S}= (f^{n})\) is a virtually stable scheme with \(\operatorname{Fix}(\mathcal{S}) = \operatorname{Fix}(f)= \operatorname{Fix}(F)\), where f is a continuous selection of \(P_{F}\). □
Theorem 3.3
Proof
Remark 3.4
Corollary 3.5
Let X be a compact contractible metric space. If \(F : X \to \operatorname{CB}(X)\) is an αcontraction satisfying the Chebyshev condition, then \(\operatorname{Fix}(F)\) is contractible.
Proof
It follows directly from Theorem 3.3 and the contractibility of X. □
The Chebyshev condition in Corollary 3.5 is necessary. For example, in Example 1 [11], the mapping F is a contraction that does not satisfy the Chebyshev condition (consider \(x = \frac{1}{3}\)), and \(\operatorname{Fix}(F)\) is clearly not contractible. Moreover, the following example shows that the fixed point set of an αcontraction satisfying the Chebyshev condition may not be convex.
Example 3.6
The following construction, motivated by the iteration process (1.3) in [12], indicates that there is a sequence of mappings that is not naturally a scheme, but still gives rise to a retraction result as well as a contractibility criterion for the fixed point set of a certain nonexpansive setvalued mapping.
Let X be a closed convex subset of a Banach space, \(F: X \rightarrow \operatorname{CCB}(X)\) be an Hcontinuous mapping, \((\alpha _{n})_{n=0}^{\infty}\) be a sequence in \([a,b]\subseteq (0,1)\), and \((\gamma_{n})_{n=0}^{\infty}\) be a sequence in \((0,+\infty) \) satisfying \(\lim_{n\rightarrow\infty} \gamma_{n}=0 \).
Lemma 3.7
Suppose that \(F:X\rightarrow \operatorname{CCB}(X)\) satisfies the endpoint condition. Then \(\operatorname{Fix}(F)=\operatorname{Fix}(\mathcal{S})= r(\mathrm{C}(\mathcal{S}))\).
Proof
It suffices to show that \(\operatorname{Fix}(F)\subseteq \operatorname{Fix}(\mathcal{S})\) and \(r(\mathrm{C}(\mathcal{S}))\subseteq \operatorname{Fix}(F) \).
Now, suppose \(p\in r(\mathrm{C}(\mathcal{S})) \). Then \(p = r(x)\) for some \(x\in \mathrm{C}(\mathcal{S})\).
Lemma 3.8
Proof
By Lemma 3.7 and Lemma 3.8, we immediately obtain the following.
Theorem 3.9
Let X be a closed convex subset of a Banach space. If \(F: X\rightarrow \operatorname{CCB}(X)\) is Hcontinuous, quasinonexpansive and satisfies the endpoint condition, then \(\operatorname{Fix}(F) \) is a retract of \(\mathrm{C}(\mathcal{S})\), where the sequence \(\mathcal{S}\) is defined as in (1).
Corollary 3.10
Let X be a compact convex subset of a Banach space and \(F : X \to \operatorname{CCB}(X)\) be a nonexpansive setvalued mapping satisfying the endpoint condition. Then the fixed point set of F is contractible.
Proof
By Theorem 2.3 [12], we have \(X=\mathrm{C}(\mathcal{S})\), and hence the result follows immediately from the previous theorem. □
The example below shows that, under the assumption of the above corollary, the fixed point set may not be convex.
Example 3.11
Motivated by the last construction, we are ready to extend the notion of fixed point iteration schemes to the notion of fixed point resolutions as follows.
Definition 3.12
Hence, the fixed point set of a resolution is always a retract of its convergence set, and when X is a regular space, every virtually stable scheme on X having a continuous subsequence is always a resolution. The last theorem below summarizes all retraction results in this section in terms of resolutions.
Theorem 3.13
 (1)
X is closed, \(Y= \operatorname{CCB}(X)\) and F is a quasinonexpansive and lower semicontinuous mapping satisfying the endpoint condition.
 (2)
X is closed, \(Y=\operatorname{CC}(X)\) and F is a ∗nonexpansive mapping satisfying the proximal condition.
 (3)
X is compact, \(Y=\operatorname{CB}(X)\) and F is a setvalued αcontraction satisfying the Chebyshev condition.
 (4)
X is closed and convex, \(Y= \operatorname{CCB}(X)\) and F is a nonexpansive mapping satisfying the endpoint condition.
Declarations
Acknowledgements
The authors would like to thank A Thamrongthanyalak for useful discussion during the manuscript preparation. The second author is (partially) supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.
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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