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Fixed point sets of setvalued mappings
Fixed Point Theory and Applications volume 2015, Article number: 56 (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.
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.
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.
For a metric space \((X,d)\), \(a\in X\), and \(A,B\in2^{X}\), let

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

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

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

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

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

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

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

\(\operatorname{CB}(X)=\{A\in2^{X}: A \mbox{ is closed and bounded}\}\).
The Hausdorff metric [9] is the mapping \(H : \operatorname{CB}(X)\times \operatorname{CB}(X) \rightarrow \mathbb{R}\) defined by
for each \(A,B\in \operatorname{CB}(X)\).
The following facts can be found in [8, 10, 11].
Lemma 2.1
Let \(x,y\in X\) and \(A,B\subseteq X\). Then we obtain the following:

(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 \(d(a,b)\leq H(A,B)+\varepsilon\).
For a setvalued mapping \(F :X\rightarrow2^{X}\), let \(P_{F}: X\rightarrow \mathcal{P}(X)\) be defined by
for each \(x\in X\), and we say that F satisfies

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 \(x\in X\).
Notice that every singlevalued mapping (considered as a setvalued mapping in the trivial way) satisfies all above conditions.
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
For a metric space \((X,d)\), if \(F:X\rightarrow2^{X}\) satisfies the proximal condition, then we have the following:

(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 \(x\in X\). □
We now recall some definitions of continuity for setvalued mappings (see [10] for more details). For our purpose, let X and Y be metric spaces (with no ambiguity, their metrics will be denoted by the same symbol ‘d’). A setvalued mapping \(F: X\rightarrow2^{Y}\) is said to be

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.
We say that the setvalued mapping \(F: X\rightarrow2^{Y}\) is (upper, Hupper, lower, Hlower semi, H) continuous if it is (upper, Hupper, lower, Hlower semi, H) continuous at each point in X. Moreover, for \(\alpha\in[0,1)\), we say that the mapping \(F:X\rightarrow \operatorname{CB}(X)\) is

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)
Let X and Y be metric spaces and \(F:X\rightarrow2^{Y}\) be a mapping.

(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)
Let X be a metric space and Y be a Banach space. Assume that \(F:X\rightarrow2^{Y} \) and \(G:X\rightarrow2^{Y} \) are lower semicontinuous. If G has open convex values and \(F(x)\cap G(x) \neq\emptyset\) for each \(x\in X\), then the mapping \(\Phi:X\rightarrow 2^{Y} \) defined by
for each \(x\in X\) is lower semicontinuous.
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
Let \(x,y\in X\) and \(\varepsilon>0\). For each \(z\in P_{F}(x)\), there is \(u_{y}\in P_{F}(y)\) such that
Thus, \(P_{F}(x)\subseteq\eta(P_{F}(y); d(x,y)+\varepsilon)\). Similarly, we have \(P_{F}(y)\subseteq\eta(P_{F}(x); d(x,y)+\varepsilon) \). Therefore, since ε is arbitrary, \(H(P_{F}(x), P_{F}(y))\leq d(x,y) \). □
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
Let \((x_{n})\) be a sequence in X converging to \(x_{0} \in X \). Suppose that \((P_{F}(x_{n_{k}})) \) is a subsequence of \((P_{F}(x_{n}))\) converging to a point \(p\in X\). By the upper semicontinuity of F, for each \(\varepsilon>0\), there is \(N\in \mathbb{N}\) such that
for each \(k\geq N\). Hence, \(\lim_{k\rightarrow\infty} d(P_{F}(x_{n_{k} }), F(x_{0}))=0\) and
Thus, \(p\in F(x_{0}) \) since \(F(x_{0})\) is closed. Also, for each \(a\in F(x_{n_{k}})\) and \(k\in \mathbb{N}\),
Since F is lower semicontinuous and \(P_{F}(x_{0})\in F(x_{0})\), there is a sequence \((y_{n_{k}}) \) in X such that \(y_{n_{k}}\in F(x_{n_{k}})\) and \((y_{n_{k}})\) converges to \(P_{F}(x_{0})\). Consequently,
It follows that \(p \in F(x_{0}) \cap S( x_{0};d(x_{0},F(x_{0})))= P_{F}(x_{0}) \) by Lemma 2.4(3), and hence \(p= P_{F}(x_{0})\) by the Chebyshev condition on F. This proves that each convergent subsequence of \((P_{F}(x_{n}))\) converges to the same point \(P_{F}(x_{0})\). Then, by Lemma 2.10, the sequence \((P_{F}(x_{n}))\) converges to \(P_{F}(x_{0})\). Therefore, the mapping \(P_{F}\) is continuous. □
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])
Let X be a complete and metrically convex metric space. If A is a nonempty closed subset of X, then for each \(x\in A\) and \(y\notin A\), there exists \(z\in \partial A\) (the boundary of A) such that
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
WLOG, we assume that \(s\leq t\). Since \(D(x;s)\subseteq D(x;t)\), we have
and hence \(H(D(x;t), D(x;s))=h(D(x;t) , D(x;s) )\) by Lemma 2.1(3). It is easy to see that \(ts\leq d(y,z) \) for each \(y\in S(x;t)\) and \(z\in D(x;s)\). Also,
On the other hand, for each \(y\in S(x;t)\), there exists \(z\in S(x;s)\) such that \(d(x,z)+d(z,y)=d(x,y)\) by Lemma 2.12. Consequently,
for each \(y\in S(x;t)\). Then \(h(D(x;t) , D(x;s) )\leq ts\).
Now if X is a normed space and \(x\neq y\), let \(a=x+ \frac{ t}{\Vert xy\Vert } (xy) \in S(x;t)\). Then
for each \(z\in D(y;t)\). Therefore, \(\Vert xy\Vert \leq \Vert az\Vert \) for each \(z\in D(y;t)\), and hence
On the other hand, let \(a\in D(x;t)\) and \(\varepsilon>0\). Consider the point \(b = a+ yx\). Then \(\Vert ab\Vert =\Vert xy\Vert \) and \(\Vert by\Vert = \Vert ax\Vert \leq t\). Hence, we have
Consequently, \(D(x; t) \subseteq \eta(D(y; t) ; \Vert xy\Vert +\varepsilon)\). By a similar argument, we can show that \(D(y; t) \subseteq \eta(D(x; t) ; \Vert xy\Vert +\varepsilon)\). It follows that \(H(D(x; t), D(y;t))\leq \Vert xy\Vert +\varepsilon \), and since ε is arbitrary, \(H(D(x; t), D(y;t))\leq \Vert xy\Vert \). □
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.
In a (real) Banach space \(( X,\Vert \cdot \Vert ) \), we let
where \(\operatorname{CCB}(X)\) is considered as a subspace of the metric space \((\operatorname{CB}(X) , H)\).
Lemma 2.16
Let X be a closed subset of a Banach space E. Assume that \(f:E\rightarrow E \) and \(\varphi: E\rightarrow(0,\infty)\) are continuous mappings.

(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
(1) Let \(x\in E\) and \(\varepsilon>0\). Since f and φ are continuous, there is \(\delta>0\) such that
for each \(y\in B(x;\delta) \). Thus, by Lemma 2.13,
for each \(y\in B(x;\delta)\). Therefore, \(\overline{\Phi} \) is Hcontinuous and hence lower semicontinuous by Lemma 2.5(2).
(2) It follows directly from (1) and Lemma 2.6.
(3) By (2), the restriction \(\Phi_{X}: X\rightarrow2^{E}\) is lower semicontinuous. For each \(x\in X\), since \(\Phi(x)\) and \(F(x)\) are convex, so is \(\overline{\Phi(x) \cap F(x)} \). Following Lemma 2.6 and Lemma 2.7, the mapping \(\Psi:X\rightarrow \operatorname{CC}(X) \subseteq \operatorname{CC}(E) \) defined by
for each \(x\in X \) is lower semicontinuous. By Theorem 2.15, there is a continuous mapping \(g:X\rightarrow E\) such that
for each \(x\in X\). □
We now recall the notion of fixed point iteration schemes and virtually stable schemes defined in [5]. For a given (nonempty) Hausdorff space X and a sequence \(\mathcal{S}=(s_{n})\) of mappings on X, let \(\operatorname{Fix}(\mathcal{S}) = \bigcap_{n=1}^{\infty} \operatorname{Fix}(s_{n})\) and \(\mathrm{C}(\mathcal{S}) = \{x : \lim_{n \to\infty} s_{n}(x) \mbox{ exists}\}\) denote the fixed point set and the convergence set of \(\mathcal{S}\), respectively. We always have \(\operatorname{Fix}(\mathcal{S})\subseteq \mathrm{C}(\mathcal{S})\), and there is a natural mapping \(r : \mathrm{C}(\mathcal{S}) \to X\) defined by
for each \(x\in X\). Notice that r may not be continuous in general, and \(\operatorname{Fix}(\mathcal{S}) \subseteq r(\mathrm{C}(\mathcal{S}))\). For a sequence \((f_{n})\) of mappings on X, we write \(\prod_{i=j}^{n} f_{i}\) for the composition \(f_{n} \circ f_{n1}\circ\cdots\circ f_{j}\). Notice that if \(\mathcal{F}=(f_{n} )\) is a sequence of mappings and \(\mathcal{S}= ( \prod_{i=1}^{n} f_{i} )\), then \(\operatorname{Fix}(\mathcal{S})=\operatorname{Fix}(\mathcal{F}) \). The sequence \(\mathcal{S}=(\prod_{i=1}^{n} f_{i}) \) is said to be a fixed point iteration scheme, or a scheme in short, if \(\operatorname{Fix}(\mathcal{S}) = r(\mathrm{C}(\mathcal{S}))\).
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
For \(\alpha\in[0,1)\), the scheme \(\mathcal{S}= (\prod_{i=1}^{n} f_{i})\) on a metric space X is said to be αcontractive if it satisfies the following conditions:

(1)
For each sequence \((t_{n})\in\{ (\prod_{i=0}^{n} f_{k+i}) : k\in \mathbb{N}\} \),
$$d\bigl(t_{n+1}(x),t_{n}(x)\bigr)\leq\alpha d \bigl(t_{n}(x), t_{n1}(x)\bigr) $$for each \(n\in \mathbb{N}\), where \(t_{0}(x)=x\) for each \(x\in X\).

(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
Let \(\mathcal{S}= ( \prod_{i=1}^{n} f_{i} )\) be an αcontractive scheme. To show that \(\mathcal{S}\) is virtually stable, let \(p\in \operatorname{Fix}(\mathcal{S})=\operatorname{Fix}(\mathcal{F})\), \(\varepsilon>0\), and \(\varepsilon_{0} = \min \{ \frac{\varepsilon(1\alpha)}{4 },\frac {\varepsilon}{4} \}\). Since \(\mathcal{F}\) is equicontinuous at \(p\in \operatorname{Fix}(\mathcal{F})\), there exists \(\delta >0 \) such that
for each \(f_{i}\in \mathcal{F}\). WLOG, we may assume that \(\delta\leq\varepsilon_{0}\).
Let \(n\in \mathbb{N}\), \(j\leq n\), \(\mathcal{T}=(t_{n})= (\prod_{i=0}^{n} f_{j+i} ) \), and \(y\in B(p;\delta)\). Since \(p\in \operatorname{Fix}(\mathcal{S})\subseteq \operatorname{Fix}(\mathcal{T})\), we have \(t_{1}(y)=f_{j}(y)\in B(f_{j}(p); \varepsilon_{0} )=B(p; \varepsilon_{0} )\), and
Consequently,
So,
Hence, for each \(p\in \operatorname{Fix}(\mathcal{S})\) and \(\varepsilon>0\), there is \(\delta >0\) such that
for all \(n\in \mathbb{N}\) and \(j\leq n\). Therefore, \(\mathcal{S}\) is virtually stable.
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
Let X be a metric space and \(f:X\rightarrow X\) continuous on \(\operatorname{Fix}(f)\). If the (Picard) scheme \(\mathcal{S}=( f^{n})\) satisfies
for some \(\alpha\in[0,1) \) and for all \(x\in X\), then \(\mathcal{S}\) is αcontractive and hence virtually stable.
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. □
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
Let X be a compact metric space, and \(F : X \rightarrow \operatorname{CB}(X)\) be a setvalued αcontraction satisfying the Chebyshev condition. Then the sequence \(\mathcal{S}=(s_{n})\) of mappings on X, defined by
for each \(n \in \mathbb{N}\) and \(x\in X \), is an αcontractive scheme with \(\operatorname{Fix}(\mathcal{S}) = \operatorname{Fix}(F)\), and hence \(\operatorname{Fix}(F)\) is a retract of X.
Proof
Let \(x\in X\) and \(n\geq1\). By Lemma 2.1(2) and (3),
It follows that \((s_{n}(x)) \) is a Cauchy sequence, and hence it converges to a point \(p\in X\). Consider the following inequality:
Since \((s_{n}(x)) \) converges to p and F is Hcontinuous, \((F \circ s_{n}(x))\) converges to \(F(p) \). Consequently, \(d(p, F(p))= 0\), and since \(F(p)\) is closed, \(p\in F(p)\). This implies \(r(\mathrm{C}(\mathcal{S}))\subseteq \operatorname{Fix}(F)\). By Lemma 2.4(1),
It follows that the sequence \(\mathcal{S}\) is a scheme satisfying the condition in Corollary 2.23 with \(\operatorname{Fix}(\mathcal{S})=\operatorname{Fix}(F)\). Therefore, the scheme \(\mathcal{S}\) is αcontractive with \(\operatorname{Fix}(\mathcal{S})=\operatorname{Fix}(F)\), and hence \(\operatorname{Fix}(F)\) is a retract of X by Lemma 2.22 and Theorem 2.19. □
Remark 3.4
The sequence \(((P_{F})^{n})\) in the proof of Theorem 3.3 is motivated by the iteration process defined in Theorem 5 [11]. Moreover, in [18], Yanagi studied the convergence of an iteration sequence for a sequence \(\mathcal{F}=(F_{n})\) of setvalued mappings from a complete metric space X into \(\operatorname{CB}(X) \) satisfying the relation
for all \(x,y\in X\) and \(i,j\in \mathbb{N}\), where \(a,b,c\geq0\) and \(a+(a+3)(b+c)<1\). In this case, we observe that the sequence \(\mathcal{S}=( \prod_{i=1}^{n}P_{F_{i}})\) is an αcontractive scheme for some \(\alpha\in[0,1)\) if each \(F_{i} \) satisfies the Chebyshev condition, and hence, by a similar argument in Theorem 3.3, the common fixed point set \(\operatorname{Fix}(\mathcal{F})\) is a retract of X when X is compact.
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
Consider the subset \(X=[1,1]\times[0,1]\) of \(\mathbb{R}^{2}\). Define \(F:X\rightarrow \operatorname{CB}(X) \) by
Then, for each \((x,y), (x_{1},y_{1}), (x_{2},y_{2})\in X\),
and \(P_{F}(x,y)=( x,\vert \frac{x}{2}\vert )\). Therefore, the mapping F is a \(\frac{1}{2}\)contraction satisfying the Chebyshev condition, and \(\operatorname{Fix}(F)= \{(x,y): y= \vert \frac {x}{2}\vert \}\) which is not convex.
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 \(\lim_{n\rightarrow\infty} \gamma_{n}=0 \).
By Lemma 2.5(2) and Theorem 2.15, F admits a continuous selection, say \(g_{0}:X\rightarrow X\). We define mappings \(s_{0}:X\rightarrow X\) and \(s_{1}:X\rightarrow X\) by
for each \(x\in X\), respectively.
To construct \(s_{2}\), consider the mapping \(\varphi: X\rightarrow (0,\infty)\) defined by
for each \(x\in X\). Let \(x\in X\) and \(\varepsilon>0\). Since \(s_{1}\) and \(s_{0}\) are continuous and F is Hcontinuous, \(F\circ s_{1}\) and \(F\circ s_{0} \) are Hcontinuous. Then there is \(\delta>0 \) such that
for each \(y\in B(x,\delta) \). Let \(y\in B(x;\delta) \), we may assume that \(\varphi(x)\varphi(y)\geq0 \), and hence
Therefore, φ is continuous. Also, since \(g_{0}(x)\in F(x)=F\circ s_{0}(x)\), then by Lemma 2.1(4), there exists \(a\in F\circ s_{1}(x)\) such that \(\Vert ag_{0}(x) \Vert < \varphi(x)\). Thus,
Following Lemma 2.16, there exists a continuous mapping \(g_{1}:X\rightarrow X\) such that
for each \(x\in X\). That is, \(g_{1}\) is a continuous selection of \(F\circ s_{1}\) satisfying
for each \(x \in X\). We now define \(s_{2}:X\rightarrow X\) by
for each \(x\in X\).
Inductively, we obtain a sequence \(\mathcal{S}=(s_{n})\) of continuous mappings such that
for each \(x\in X \mbox{ and } n\geq2\), where \(g_{n1}:X\rightarrow X\) is a continuous selection of \(F\circ s_{n1}\) satisfying
for each \(x\in X\).
Recall that \(r: \mathrm{C}(\mathcal{S})\rightarrow X\) is given by
for each \(x\in \mathrm{C}(\mathcal{S})\).
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 \(r(\mathrm{C}(\mathcal{S}))\subseteq \operatorname{Fix}(F) \).
Suppose \(p\in \operatorname{Fix}(F) \). Since \(g_{0}(p)\in F(p)=\{p\} \), we have
Consequently, since \(g_{1}(p)\in F\circ s_{1} (p)=\{p\}\),
Inductively, for each \(n\in \mathbb{N}\), \(s_{n}(p)=p \). So, \(p\in\bigcap_{n\in \mathbb{N}}\operatorname{Fix}(s_{n})=\operatorname{Fix}(\mathcal{S})\).
Now, suppose \(p\in r(\mathrm{C}(\mathcal{S})) \). Then \(p = r(x)\) for some \(x\in \mathrm{C}(\mathcal{S})\).
Since \(s_{n+1}(x)s_{n}(x)=\alpha_{n}(g_{n} (x)s_{n}(x))\), \((\alpha_{n})\) is a sequence in \([a,b] \subseteq(0,1)\) and \((s_{n}(x))\) is a convergent sequence, we have
Thus, \(\lim_{n\rightarrow\infty} \Vert g_{n}(x)s_{n}(x)\Vert =0 \). For each \(\varepsilon>0\), since \((s_{n}(x))\) converges to p and F is Hcontinuous, there is \(N\in \mathbb{N}\) such that \(H( F\circ s_{n}(x) , F(p) )< \varepsilon\) for all \(n\geq N\). Consequently, \(g_{n}(x)\in F\circ s_{n}(x) \subseteq\eta( F(p) ; \varepsilon)\) for all \(n\geq N\). That is, \(\lim_{n\rightarrow\infty} d(g_{n}(x), F(p))=0\). Since, by Lemma 2.1(1),
it follows that \(d( p, F(p)) =0\), and hence \(p\in \operatorname{Fix}(F)\) since \(F(p)\) is closed. Therefore, \(r(\mathrm{C}(\mathcal{S}))\subseteq \operatorname{Fix}(F)\). □
Lemma 3.8
Suppose that \(F:X\rightarrow \operatorname{CCB}(X)\) is quasinonexpansive and satisfies the endpoint condition. Then, for each \(x\in X\), \(p\in \operatorname{Fix}(F)\) and \(m \geq n \),
and r is continuous.
Proof
For each \(x\in X \), \(p\in \operatorname{Fix}(F)\) and \(n\in \mathbb{N}\), we have
and hence
Inductively, we have \(\Vert s_{m}(x)p\Vert \leq \Vert s_{n}(x)p\Vert \) whenever \(m \geq n\).
For the continuity of r, let \(x\in \mathrm{C}(\mathcal{S})\) and \(\varepsilon>0\). Then there exists \(N\in \mathbb{N}\) such that \(\Vert s_{N}(x) r(x)\Vert < \frac{\varepsilon}{3}\), and by the continuity of \(s_{N}\), there exists \(\delta>0 \) such that \(\Vert s_{N}(x)s_{N}(y) \Vert < \frac{ \varepsilon}{3}\) whenever \(\Vert xy\Vert <\delta\). Hence, for each \(y\in \mathrm{C}(\mathcal{S}) \) with \(\Vert xy\Vert <\delta\), there is \(M\geq N\) such that \(\Vert s_{M}(y)r(y)\Vert <\frac{\varepsilon}{3}\). Observe that since \(r(x)\in \operatorname{Fix}(F)\) and by the above property of \((s_{n})\), we have
Then it follows that
which proves the continuity of r. □
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
Consider \(X= [1,1]\times[0,1] \subseteq \mathbb{R}^{2}\) with the maximum norm \(\Vert (x,y)\Vert = \max\{\vert x\vert ,\vert y\vert \} \). Define \(f: X \rightarrow X\) by
Then f is a nonexpansive mapping which is also a nonexpansive setvalued mapping satisfying the endpoint condition. Observe that \(\operatorname{Fix}(f)= \{(x,\vert x\vert ): x\in \mathbb{R}\}\) is not convex.
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
For a (nonempty) Hausdorff space X, a sequence \(\mathcal{S}=(s_{n})\) of mappings of X is said to be a fixed point resolution, or a resolution in short, if \(r: \mathrm{C}(\mathcal{S}) \rightarrow X \) given by
is continuous and \(r(\mathrm{C}(\mathcal{S})) \subseteq \operatorname{Fix}(\mathcal{S})\).
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
Suppose that X is a subset of a Banach space E and \(F : X \to Y\subseteq2^{X}\) is a setvalued mapping. Then there exists a resolution \(\mathcal{S}\) with \(\operatorname{Fix}(\mathcal{S}) = \operatorname{Fix}(F)\) if one the following conditions holds:

(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.
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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.
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Chanthorn, P., Chaoha, P. Fixed point sets of setvalued mappings. Fixed Point Theory Appl 2015, 56 (2015). https://doi.org/10.1186/s1366301503056
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DOI: https://doi.org/10.1186/s1366301503056
MSC
 47H04
 47H09
Keywords
 fixed point set
 setvalued mapping