# On multi-valued maps with images in the space of closed subsets of a metric space

- Evgeny S Zhukovskiy
^{1}and - Elena A Panasenko
^{1}Email author

**2013**:10

https://doi.org/10.1186/1687-1812-2013-10

© Zhukovskiy and Panasenko; licensee Springer 2013

**Received: **27 August 2012

**Accepted: **22 December 2012

**Published: **10 January 2013

## Abstract

A new metric in the space $clos(X)$ of all closed subsets of a metric space *X* is proposed. This metric, unlike the generalized Hausdorff metric, takes finite values only, and the convergence of a sequence of closed sets ${H}^{i}$, $i=1,2,\dots $ , with respect to this metric is equivalent to the convergence (in the sense of Hausdorff) for any $r\u2a7e0$ of the unions of ${H}^{i}$ with a closed ‘exterior ball’ of radius *r*. Using this metric allows one to investigate multi-valued maps that have images in $clos(X)$ and are not continuous in the Hausdorff metric. In the work, the necessary and sufficient conditions for a multi-valued map to be continuous and Lipschitz with respect to the metric presented are studied, a connection of these properties with their analogues in the Hausdorff metric is derived, and a generalization of the Nadler fixed point theorem is obtained.

**MSC:**47H04, 47H10, 54E35.

### Keywords

metric space multi-valued map fixed point## 1 Introduction

*X*from a point

*x*to a set

*M*; $d(N,M)\doteq {sup}_{x\in N}\varrho (x,M)$ is the deviation in the sense of Hausdorff of a set

*N*from the set

*M*; ${S}_{r}({x}_{0})\doteq \{x\in X:{\varrho}_{X}(x,{x}_{0})=r\}$, ${O}_{r}({x}_{0})\doteq \{x\in X:{\varrho}_{X}(x,{x}_{0})\u2a7dr\}$, ${O}_{r}^{o}({x}_{0})\doteq \{x\in X:{\varrho}_{X}(x,{x}_{0})<r\}$ are, respectively, a sphere, a closed and an open ball of radius $r>0$ centered at ${x}_{0}$ in the space

*X*; ${S}_{0}({x}_{0})={O}_{0}({x}_{0})=\{{x}_{0}\}$; ${O}_{0}^{o}({x}_{0})=\mathrm{\varnothing}$; $\overline{{O}_{r}^{o}({x}_{0})}\doteq \{x\in X:{\varrho}_{X}(x,{x}_{0})\u2a7er\}$ is a closed ‘exterior ball’,

*i.e.*, the compliment to an open ball; ${O}_{r}^{o}(M)\doteq {\bigcup}_{{x}_{0}\in M}\{x\in X:{\varrho}_{X}(x,{x}_{0})<r\}$ is an

*r*-neighborhood of $M\subset X$. By $clbd(X)$ denote the space of all

*nonempty closed bounded*subsets of

*X*. This space we endow with the Hausdorff metric dist:

*nonempty closed*subsets of a given metric space

*X*are investigated. The necessity of studying such maps arises, for example, in the problems concerning asymptotic behavior of controlled systems trajectories [1, 2]. Treating the space $clos(X)$ in pair with the Hausdorff metric dist (called, in this case, the generalized Hausdorff metric, meaning it may equal infinity) does not lead to substantial results if, for instance, the corresponding maps have images that are infinitely distant from each other (such maps, obviously, are not continuous, and the classical methods of analysis are not applicable here). As shown in [1–3], a way to overcome this difficulty and make it possible to use the known methods and standard techniques is to construct in the space $clos(X)$ a metric satisfying the following conditions:

- (1)
the distance between any closed sets is finite;

- (2)
if a sequence of closed sets is convergent with respect to the Hausdorff metric, then it is convergent with respect to the ‘new’ metric;

- (3)
the convergence of a sequence ${\{{H}^{i}\}}_{i=1}^{\mathrm{\infty}}\subset clos(X)$ means the convergence for any $r>0$ (with respect to the Hausdorff metric) of the sequence of bounded subsets ${H}_{r}^{i}\subset {H}^{i}$, ${H}_{r}^{i}\subset {O}_{r}$, such that ${\bigcup}_{i}{H}_{r}^{i}={H}^{i}$ and ${H}_{{r}_{1}}^{i}\subset {H}_{{r}_{2}}^{i}$ as soon as ${r}_{1}<{r}_{2}$.

For the space of convex closed subsets of ${\mathbb{R}}^{n}$, in [2], a metric (called the Hausdorff-Bebutov metric) satisfying (1)-(3) was suggested. It allowed to investigate differential inclusions with convex unbounded right-hand sides. For studying multi-valued maps with non-convex unbounded images in ${\mathbb{R}}^{n}$, the Hausdorff-Bebutov metric turns out to be inefficient since, considered in the space $clos({\mathbb{R}}^{n})$, it does not meet requirements (2) and (3). Moreover, in the structure of the Hausdorff-Bebutov metric, the local compactness of ${\mathbb{R}}^{n}$ is fundamental, so the definition of this metric itself cannot be used for an arbitrary metric space *X*. In what follows, a different construction for a metric in the space $clos(X)$ is offered. It fulfills all the listed requirements and can be used in the case of arbitrary *X*.

The function ${\rho}_{cl}$ takes finite values only and, as it will be shown later, satisfies all the axioms of a metric. This metric first was introduced by the authors in [4] for the space $clos({\mathbb{R}}^{n})$ of all nonempty closed subsets of a finite dimensional space. In [5], the metric ${\rho}_{cl}$ was used for studying the dynamical system of translations in the space of multi-valued maps with images in $clos({\mathbb{R}}^{n})$.

In the work, we point out the benefits the metric ${\rho}_{cl}$ gives when it comes to dealing with continuous and Lipschitz multi-valued maps. We emphasize that a crucial role here is played by the construction (2) used in the metric ${\rho}_{cl}$, *i.e.*, associating to every set $H\in clos(X)$ the set ${\mathfrak{S}}_{r}H\in clos(X)$, $r\u2a7e0$ (representing the union of *H* with the closed ‘exterior ball’ of radius *r* centered at *θ*).^{a} For a map $F:X\to clos(X)$, such an ‘extension’ of the values $F(x)$ allows, in particular, to obtain a fixed point theorem (which will be proved in the last section of the paper) for multi-valued maps that in the Hausdorff metric may not be contracting or even continuous and to which the known fixed point principles (for example, Nadler’s theorem) are not applicable. One can also use this idea for a map $F:X\to clbd(X)$ that is not contracting or continuous. In many cases, after the corresponding extension, the map ${\mathfrak{S}}_{r}F:X\to clos(X)$ gets these properties.

## 2 The space $(\mathrm{clos}(X),{\rho}_{\mathrm{cl}})$

We start with studying the space $(clos(X),{\rho}_{cl})$.

*i.e.*, in the situation when ${R}^{\ast}=\mathrm{\infty}$, but all the results obtained from now on hold also in the case of ${R}^{\ast}<\mathrm{\infty}$. Set

(the latter occurs only if ${R}^{\ast}<\mathrm{\infty}$).

We will need the following properties of this function: *for every* $p\in P$, *the function* $D(p,\cdot ):{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ *is non-decreasing; for every* $r\in {\mathbb{R}}_{+}$, *the function* $D(\cdot ,r):P\to {\mathbb{R}}_{+}$ *is non-increasing*.

Finally, if ${r}_{1},{r}_{2}\notin P$, ${R}^{\ast}<\mathrm{\infty}$, then $D(p,{r}_{1})=D(p,{r}_{2})$.

Let us illustrate the definition of the function *D* on some concrete metric spaces.

**Example 1** For any linear normed space *X* with any choice of $\theta \in X$, the function $D:{\mathbb{R}}_{+}^{2}\to {\mathbb{R}}_{+}$ is given by $D(p,r)=r-p$, $r>p$.

If $X=\mathbb{Z}$, ${\varrho}_{\mathbb{Z}}(n,m)=|n-m|$ $\mathrm{\forall}n,m\in \mathbb{Z}$, then no matter what point is chosen as $\theta \in \mathbb{Z}$, for any $r,p\in {\mathbb{R}}_{+}$, $r>p$, the value of $D(p,r)$ is equal to the number of integers belonging to the interval $[p,r)$.

One gets the same values for $D:{\mathbb{R}}_{+}^{2}\to {\mathbb{R}}_{+}$ if $X=\mathbb{N}$, ${\varrho}_{\mathbb{N}}(n,m)=|n-m|$ $\mathrm{\forall}n,m\in \mathbb{N}$, and $\theta =1$. If $\theta \ne 1$, the value $D(p,r)$ coincides with the number of integers in the set $[p,r)$ when $\theta -1\u2a7er>p$ or $r>p>\theta -1$, and is equal to the number of integers in the set $[1,\theta +r)$ when $r>\theta -1\u2a7ep$.

- (1)if $0\u2a7dp\u2a7d1$, then$D(p,r)=\{\begin{array}{cc}r-p,\hfill & r\u2a7d1,\hfill \\ r+1,\hfill & 1<r\u2a7d2,\hfill \\ 3,\hfill & r>2;\hfill \end{array}$
- (2)if $1\u2a7dp\u2a7d2$, then$D(p,r)=\{\begin{array}{cc}r-p,\hfill & r\u2a7d2,\hfill \\ 2-p,\hfill & r>2.\hfill \end{array}$

For arbitrary closed $F,G\subset X$ and $r\u2a7e0$, we determine now $dist({\mathfrak{S}}_{r}F,{\mathfrak{S}}_{r}G)$. Next, we show that this distance is finite and also give the properties of the function ${\mathbb{R}}_{+}\ni r\mapsto dist({\mathfrak{S}}_{r}F,{\mathfrak{S}}_{r}G)\in {\mathbb{R}}_{+}$; these properties will be used in the sequel.

**Lemma 1**

*Let*$F,G\in clos(X)$

*be given*.

*Then*$dist({\mathfrak{S}}_{r}F,{\mathfrak{S}}_{r}G)<\mathrm{\infty}$

*for every*$r\u2a7e0$

*and the following statements hold true*:

- (1)
*the function*${\mathbb{R}}_{+}\ni r\mapsto dist({\mathfrak{S}}_{r}F,{\mathfrak{S}}_{r}G)\in {\mathbb{R}}_{+}$*is non*-*decreasing*; - (2)
*for every*$r\u2a7e0$,*the inequality*$dist({\mathfrak{S}}_{r}F,{\mathfrak{S}}_{r}G)\u2a7ddist(F,G)$(6)*and the relation*$\underset{r\to \mathrm{\infty}}{lim}dist({\mathfrak{S}}_{r}F,{\mathfrak{S}}_{r}G)=dist(F,G)$(7)*hold*; - (3)
*denoted*${r}_{\ast}\doteq min\{\varrho (\theta ,F),\varrho (\theta ,G)\}$,*the inequality*$dist({\mathfrak{S}}_{r}F,{\mathfrak{S}}_{r}G)\u2a7dD({r}_{\ast},r)$*holds for each*$r\u2a7e0$; - (4)
*denoted*${r}^{\ast}\doteq max\{\varrho (\theta ,F),\varrho (\theta ,G)\}$,*the inequality*${\rho}^{o}(F,G)\u2a7ddist({\mathfrak{S}}_{r}F,{\mathfrak{S}}_{r}G)$*holds for each*$r\u2a7e{r}^{\ast}$.

*Proof*First of all, note that for any sets $A,B,C\in clos(X)$, one has

for every $x\in C$, we get the inequality $d(A\cup C,B\cup C)\u2a7dd(A,B)$. Similarly, it can be checked that $d(B\cup C,A\cup C)\u2a7dd(A,B)$. So, (8) is proved.

Thus, the function $r\mapsto dist({\mathfrak{S}}_{r}F,{\mathfrak{S}}_{r}G)$ is non-decreasing.

Now, we prove equality (7). If ${R}^{\ast}<\mathrm{\infty}$, then ${\mathfrak{S}}_{r}F=F$, ${\mathfrak{S}}_{r}G=G$ for $r>{R}^{\ast}$, so (7) is true.

Let ${R}^{\ast}=\mathrm{\infty}$. First, suppose that $dist(F,G)<\mathrm{\infty}$. Since the function $r\mapsto dist({\mathfrak{S}}_{r}F,{\mathfrak{S}}_{r}G)$ is non-decreasing and bounded, there exists $\alpha \doteq {lim}_{r\to \mathrm{\infty}}dist({\mathfrak{S}}_{r}F,{\mathfrak{S}}_{r}G)$. Then for each *r*, the inequality $\alpha \u2a7edist({\mathfrak{S}}_{r}F,{\mathfrak{S}}_{r}G)$ takes place, and it follows that for any $\epsilon >0$, every $x\in F$, and $r>{\varrho}_{X}(\theta ,x)+\alpha +2\epsilon $, one has $x\in {O}_{\alpha +\epsilon}^{o}({\mathfrak{S}}_{r}G)$ and $x\notin \overline{{O}_{r-\epsilon -\alpha}^{o}}$. Thus, taking into account $\overline{{O}_{r-\epsilon -\alpha}^{o}}\supseteq {O}_{\alpha +\epsilon}^{o}(\overline{{O}_{r}^{o}})$, one gets $x\notin {O}_{\alpha +\epsilon}^{o}(\overline{{O}_{r}^{o}})$, *i.e.*, $x\in {O}_{\alpha +\epsilon}^{o}(G)$. As above, for each point $y\in G$, $y\in {O}_{\alpha +\epsilon}^{o}(F)$ holds. Hence, $dist(F,G)\u2a7d\alpha +\epsilon $ for any $\epsilon >0$, which means that $dist(F,G)\u2a7d\alpha $. The inequality $dist(F,G)\u2a7e\alpha $ is easily obtained by passing to the limit in (6).

Now, let $dist(F,G)=\mathrm{\infty}$. If (7) is not true, then there exists $\alpha \doteq {lim}_{r\to \mathrm{\infty}}dist({\mathfrak{S}}_{r}F,{\mathfrak{S}}_{r}G)<\mathrm{\infty}$. So, arguing as before, one gets the inequality $dist(F,G)\u2a7d\alpha <\mathrm{\infty}$, which contradicts the initial assumption. Thus, equality (7) is proved.

*e.g.*, [6])

□

**Theorem 1** *Equality* (5) *defines a metric in the space* $clos(X)$. *If* *X* *is a complete metric space*, *then* $(clos(X),{\rho}_{cl})$ *is also complete*.

*Proof*Show that ${\rho}_{cl}$ satisfies all the axioms of a metric,

*i.e.*, that for any $F,G,H\in clos(X)$, the following hold:

- (1)
$0\u2a7d{\rho}_{cl}(F,G)<\mathrm{\infty}$, and ${\rho}_{cl}(F,G)=0$ if and only if $F=G$;

- (2)
${\rho}_{cl}(F,G)={\rho}_{cl}(G,F)$;

- (3)
${\rho}_{cl}(F,G)\u2a7d{\rho}_{cl}(F,H)+{\rho}_{cl}(H,G)$.

Thus, $(clos(X),{\rho}_{cl})$ is a metric space.

Now, let *X* be complete; show that $(clos(X),{\rho}_{cl})$ is also a complete metric space.

From (10), it follows that the number sequence ${\{\varrho (\theta ,{F}^{i})\}}_{i=1}^{\mathrm{\infty}}$ is fundamental, so it is convergent; denote its limit by ${r}_{0}$; obviously, ${r}_{0}\u2a7e0$.

Similarly, $d({\mathfrak{S}}_{r}{F}^{j}\cap {O}_{r+1},{\mathfrak{S}}_{r}{F}^{i}\cap {O}_{r+1})<\epsilon $ and relation (11) is proved.

and therefore ${\mathfrak{S}}_{r}{F}^{i}\to {\mathfrak{S}}_{r}{\mathfrak{F}}_{r}$. Moreover, $\varrho (\theta ,{\mathfrak{F}}_{r})=\varrho (\theta ,{\mathfrak{S}}_{r}{\mathfrak{F}}_{r})={r}_{0}$ (this equality follows from statement (4) of Lemma 1).

For every $r>{r}_{0}$, the set ${F}_{r}\doteq {\mathfrak{S}}_{r}{\mathfrak{F}}_{r}\cap {O}_{r}^{o}$ is not empty for $\varrho (\theta ,{\mathfrak{S}}_{r}{\mathfrak{F}}_{r})={r}_{0}$. Prove that for any $\overline{r}>r$, ${\mathfrak{S}}_{\overline{r}}{\mathfrak{F}}_{\overline{r}}\cap {O}_{r}^{o}={F}_{r}$ holds. Let $x\in {\mathfrak{S}}_{\overline{r}}{\mathfrak{F}}_{\overline{r}}\cap {O}_{r}^{o}$. Then there exists a sequence ${\{{x}_{i}\}}_{i=1}^{\mathrm{\infty}}$, ${x}_{i}\in {\mathfrak{S}}_{\overline{r}}{F}^{i}\subset {\mathfrak{S}}_{r}{F}^{i}$, convergent to *x*. This means that $x\in {\mathfrak{S}}_{r}{\mathfrak{F}}_{r}$; moreover, $x\in {O}_{r}^{o}$. Conversely, let $x\in {\mathfrak{S}}_{r}{\mathfrak{F}}_{r}\cap {O}_{r}^{o}$, then $x\in {\mathfrak{F}}_{r}$, $x\notin \overline{{O}_{r}^{o}}$, and hence there is a sequence ${\{{x}_{i}\}}_{i=1}^{\mathrm{\infty}}$, ${x}_{i}\in {\mathfrak{S}}_{r}{F}^{i}$, ${x}_{i}\to x$. Starting with some index *I*, all the members of this sequence satisfy the condition ${x}_{i}\notin \overline{{O}_{r}^{o}}$, and therefore ${x}_{i}\in {F}^{i}\subset {\mathfrak{S}}_{\overline{r}}{F}^{i}$ $\mathrm{\forall}i>I$. Since ${\mathfrak{S}}_{\overline{r}}{F}^{i}$ converges to ${\mathfrak{S}}_{\overline{r}}{\mathfrak{F}}_{\overline{r}}$, it follows that $x\in {\mathfrak{S}}_{\overline{r}}{\mathfrak{F}}_{\overline{r}}$. So, $x\in {\mathfrak{S}}_{\overline{r}}{\mathfrak{F}}_{\overline{r}}\cap {O}_{p}^{o}$.

From what has been proved, one may conclude that as *r* increases, the sets ${F}_{r}$ ‘expand’ in the following way: if $\overline{r}\u2a7er$, then ${F}_{r}={\mathfrak{S}}_{r}{\mathfrak{F}}_{r}\cap {O}_{r}^{o}={\mathfrak{S}}_{\overline{r}}{\mathfrak{F}}_{\overline{r}}\cap {O}_{r}^{o}={\mathfrak{S}}_{\overline{r}}{\mathfrak{F}}_{\overline{r}}\cap {O}_{\overline{r}}^{o}\cap {O}_{r}^{o}={F}_{\overline{r}}\cap {O}_{r}^{o}$ (*i.e.*, ${F}_{r}$ is a subset of ${F}_{\overline{r}}$ containing elements *x* such that ${\varrho}_{X}(\theta ,x)<r$.) Define now the set $F\doteq {\bigcup}_{r>{r}_{0}}{F}_{r}$ and show it is closed. Note that for every *r*, $F\cap {O}_{r}^{o}={F}_{r}$ holds and the set ${F}_{r}\cup \overline{{O}_{r}^{o}}={\mathfrak{S}}_{r}F$ is closed. Consider a sequence ${\{{y}_{i}\}}_{i=1}^{\mathrm{\infty}}\subset F$ convergent to *y*. This sequence is bounded, so starting with some index *i*, the inclusions ${y}_{i}\in {O}_{p}^{o}$, ${y}_{i}\in {\mathfrak{S}}_{p}F$, where $p={\varrho}_{X}(\theta ,y)+1$, hold. Thus, since ${\mathfrak{S}}_{p}F$ is closed, it follows that $y\in {\mathfrak{S}}_{p}F$; on the other hand, $y\notin \overline{{O}_{r}^{o}}$. So, $y\in {F}_{p}\subset F$ and *F* is closed.

*F*with respect to the metric ${\rho}_{cl}$. For an arbitrary $\epsilon >0$, put $\overline{r}(\epsilon )=2/\epsilon $. Since the sequence ${\{{\mathfrak{S}}_{\overline{r}(\epsilon )}{F}^{i}\}}_{i=1}^{\mathrm{\infty}}$ converges (in the Hausdorff metric) to the set ${\mathfrak{S}}_{\overline{r}(\epsilon )}{\mathfrak{F}}_{\overline{r}(\epsilon )}={\mathfrak{S}}_{\overline{r}(\epsilon )}F$, for every

*i*, starting with some

*I*, the following inequalities hold:

According to Lemma 1, the function $r\mapsto dist({\mathfrak{S}}_{r}{F}^{i},{\mathfrak{S}}_{r}F)$ is non-decreasing. Hence, from (12), it follows that $dist({\mathfrak{S}}_{r}{F}^{i},{\mathfrak{S}}_{r}F)<\epsilon /2$ for every $r\u2a7d\overline{r}(\epsilon )$. Next, for $r>\overline{r}(\epsilon )$, one has $1/r<\epsilon /2$. Thus, ${\rho}^{\mathfrak{S}}({F}^{i},F)\u2a7d\epsilon /2$ for all $i>I$, and therefore ${\rho}_{cl}({F}^{i},F)<\epsilon $, *i.e.*, ${lim}_{i\to \mathrm{\infty}}{\rho}_{cl}({F}^{i},F)=0$. □

**Remark 1**The function ${\rho}^{\mathfrak{S}}:clos(X)\times clos(X)\to {\mathbb{R}}_{+}$, given by (4), also defines a metric in the space $clos(X)$, but for a complete metric space

*X*, the space $(clos(X),{\rho}^{\mathfrak{S}})$, unlike $(clos(X),{\rho}_{cl})$, may not be complete. For example, the sequence of the sets ${F}^{i}\doteq \{i\}$ (each consisting of a natural number) in the space $(clos(\mathbb{R}),{\rho}^{\mathfrak{S}})$ is fundamental [4]:

holds, which contradicts relation (13).

*ε*-neighborhood of a set

*A*is given by

*i.e.*, by the

*ε*-neighborhood of the set ${\mathfrak{A}}_{\epsilon}$). This means that the ‘main part’ ${\rho}^{\mathfrak{S}}$ of the distance ${\rho}_{cl}$ can be defined by the equality

Let us now give a criterion for a sequence of closed sets to be convergent with respect to the metric ${\rho}_{cl}$.

**Lemma 2** *Given* $F,{F}_{i}\in clos(X)$, $i=1,2,\dots $ , *the convergence* ${\rho}_{cl}({F}_{i},F)\to 0$ *implies* $\varrho (\theta ,{F}^{i})\to \varrho (\theta ,F)$ *and* $dist({\mathfrak{S}}_{r}{F}^{i},{\mathfrak{S}}_{r}F)\to 0$ *for any* $r>0$.

*Conversely*, *let* ${\{{F}^{i}\}}_{i=1}^{\mathrm{\infty}}\subset clos(X)$. *If there exist* ${\varrho}_{0}\doteq {lim}_{i\to \mathrm{\infty}}\varrho (\theta ,{F}^{i})$ *and* ${r}_{0}\u2a7e0$ *such that for every* $r>{r}_{0}$, *the sequence* ${\{{\mathfrak{S}}_{r}{F}^{i}\}}_{i=1}^{\mathrm{\infty}}$ *converges in the metric* dist *to some set* ${F}_{r}\in clos(X)$, *then for any* *r*, $\overline{r}$ *such that* $\overline{r}>r>{r}_{0}$, ${F}_{\overline{r}}\cap {O}_{\overline{r}}^{o}\supset {F}_{r}\cap {O}_{r}^{o}$ *holds*, *and in the space* $clos(X)$ *the sequence* ${\{{F}^{i}\}}_{i=1}^{\mathrm{\infty}}$ *converges* (*with respect to* ${\rho}_{cl}$) *to the set* $F\doteq {\bigcup}_{r>{r}_{0}}({F}_{r}\cap {O}_{r}^{o})\in clos(X)$; *moreover*, $\varrho (\theta ,F)={\varrho}_{0}$.

We omit the proof of this statement since it repeats that of the corresponding result for the finite dimensional case [4].

We conclude this section by proving a theorem on the connection between convergence in the metric ${\rho}_{cl}$ and convergence in the metric dist. Note that if a sequence ${\{{F}^{i}\}}_{i=1}^{\mathrm{\infty}}\subset clos(X)$ converges in the Hausdorff metric, then by statement (2) of Lemma 1, it converges (to the same limit) in the metric ${\rho}^{\mathfrak{S}}$; moreover, as it will be shown later, in the metric ${\rho}_{cl}$. The converse is not true. For example, in ℝ the sequence of sets ${F}^{i}\doteq \{0,i\}$, $i=1,2,\dots $ , converges in the metric ${\rho}_{cl}$ to the set $F\doteq \{0\}$. On the other hand, for any *i*, *j*, one has $dist({F}^{i},{F}^{j})=|i-j|$, *i.e.*, the sequence ${\{{F}^{i}\}}_{i=1}^{\mathrm{\infty}}$ is divergent in the Hausdorff metric.

**Theorem 2**

*Let*$F,{F}^{i}\in clos(X)$, $i\in \mathbb{N}$.

*Then the following statements hold*:

- (1)
*if*$dist({F}^{i},F)\to 0$,*then*${\rho}_{cl}({F}^{i},F)\to 0$; - (2)
*if*${\rho}_{cl}({F}^{i},F)\to 0$*and there is a*$p>0$*such that*${F}^{i}\subset {O}_{p}$, $i\in \mathbb{N}$,*then*$F\in {O}_{p}$*and*$dist({F}^{i},F)\to 0$.

*Proof* (1) Suppose $dist({F}^{i},F)\to 0$, $i\to \mathrm{\infty}$. Then, for an arbitrary $\epsilon >0$, there is a number $I=I(\epsilon )$ such that $dist({F}^{i},F)<\epsilon $ for every $i>I$. So, according to inequality (9), one gets $|\varrho (\theta ,{F}^{i})-\varrho (\theta ,F)|<\epsilon $, and from (6) it follows that the inequality $dist({\mathfrak{S}}_{r}{F}^{i},{\mathfrak{S}}_{r}F)<\epsilon $ holds for every $r\u2a7e0$. Thus, by Lemma 2, ${\rho}_{cl}({F}^{i},F)\to 0$.

that contradicts the relation $dist({\mathfrak{S}}_{r}F,{\mathfrak{S}}_{r}{F}^{i})\to 0$. Hence $F\subset {O}_{p}$.

*i*one has

*i.e.*, $\varrho (f,F)=\varrho (f,F\cup \overline{{O}_{r}^{o}})$. Thus, $d({F}^{i},F\cup \overline{{O}_{r}^{o}})=d({F}^{i},F)$ and

hence $dist({F}^{i},F)\to 0$. □

## 3 Continuous multi-valued maps

Let *T* be a topological space. Recall that a map $F:T\to clos(X)$ is said to be continuous at a point ${t}_{0}\in T$ if for any $\epsilon >0$, one can find a neighborhood $V({t}_{0})$ of ${t}_{0}$ such that $dist(F({t}_{0}),F(t))<\epsilon $ for every $t\in V({t}_{0})$. This definition corresponds to endowing the space $clos(X)$ with the Hausdorff metric. In what follows, we call a map satisfying the listed requirements continuous in the Hausdorff metric in order to distinguish it from the one considered in the space $(clos(X),{\rho}_{cl})$.

**Definition 1** A map $F:T\to clos(X)$ is said to be *continuous at a point* ${t}_{0}\in T$ *in the metric* ${\rho}_{cl}$ if for any $\epsilon >0$, there is a neighborhood $V({t}_{0})$ of ${t}_{0}$ such that ${\rho}_{cl}(F({t}_{0}),F(t))<\epsilon $ for every $t\in V({t}_{0})$.

Since the space $clbd(X)$ with the metric ${\rho}_{cl}$ is a subspace of $clos(X)$, one can also apply the given definition to a map $F:T\to clbd(X)$, *i.e.*, to a map having closed bounded images.

Following the standard terminology, the map *F* is called *continuous on a set* ${T}_{0}\subseteq T$ in the metric dist or metric ${\rho}_{cl}$ if it is continuous in the corresponding metric at every point of ${T}_{0}$.

**Theorem 3**

- (1)
*If*$F:T\to clos(X)$*is continuous at a point*${t}_{0}\in T$*in the Hausdorff metric*,*then it is continuous at*${t}_{0}$*in the metric*${\rho}_{cl}$. - (2)
*If*$F:T\to clos(X)$*is continuous at a point*${t}_{0}\in T$*in the metric*${\rho}_{cl}$*and there exist a number*$p>0$*and a neighborhood*$V({t}_{0})$*of*${t}_{0}$*such that*$F(t)\subset {O}_{p}$*for every*$t\in V({t}_{0})$,*then**F**is continuous at*${t}_{0}$*in the Hausdorff metric*.

*Proof* (1) Suppose *F* is continuous at some point ${t}_{0}\in T$ in the Hausdorff metric and let $\epsilon >0$. Then there is a neighborhood $V({t}_{0})$ of ${t}_{0}$ such that $dist(F({t}_{0}),F(t))<\epsilon /2$ for every $t\in V({t}_{0})$. So, for every $t\in V({t}_{0})$, according to estimate (9), one gets the inequality ${\rho}^{o}(F({t}_{0}),F(t))<\epsilon /2$ and, according to (6), the inequality ${\rho}^{\mathfrak{S}}(F({t}_{0}),F(t))<\epsilon /2$. Therefore, ${\rho}_{cl}(F({t}_{0}),F(t))<\epsilon $, $t\in V({t}_{0})$.

*i.e.*, $\varrho (f,F(t)\cup \overline{{O}_{{r}_{0}}^{o}})=\varrho (f,F(t))$. Similarly, $\epsilon >d(F(t),F({t}_{0}))$.

Thus, the map *F* is continuous at a point ${t}_{0}$ in the Hausdorff metric. □

So, the continuity of $F:T\to clos(X)$ in the Hausdorff metric implies its continuity in the metric ${\rho}_{cl}$. The following example shows that the converse is not true.

**Example 2**Consider the map $F:[1,2]\to clos(\mathbb{R})$ given by $F(t)\doteq \{{3}^{n}t,n=0,1,2,\dots \}$. The map

*F*is not only not continuous in the metric dist, but for any ${t}_{1},{t}_{2}\in [1,2]$, ${t}_{1}\ne {t}_{2}$, the following holds:

*F*, nevertheless, is continuous with respect to the metric ${\rho}_{cl}$ in $clos(\mathbb{R})$ with $\theta =0$. For arbitrary ${t}_{1},{t}_{2}\in [1,2]$ (assume for definiteness ${t}_{1}<{t}_{2}$), one has

So, due to relations (15) and (17), *F* is continuous in the metric ${\rho}_{cl}$.

## 4 Lipschitz multi-valued maps

We consider now multi-valued maps defined on a metric space.

*F*is said to be

*q*-

*Lipschitz*(or

*Lipschitz with a constant*

*q*) if for any ${\omega}_{1},{\omega}_{2}\in \mathrm{\Omega}$, the following holds:

Naturally, this definition is applicable to a map $F:\mathrm{\Omega}\to clbd(X)$. In what follows, a map satisfying inequality (18) will be called *q*-*Lipschitz in the Hausdorff metric*.

To give the definition of a Lipschitz map in the case when $clos(X)$ is endowed with the metric ${\rho}_{cl}$, note that this metric is given by a pair of functions ${\rho}^{o},{\rho}^{\mathfrak{S}}:clos(X)\times clos(X)\to {\mathbb{R}}_{+}$ defined by (3), (4) (the second of which is a metric itself (see Remark 1)).

**Definition 2**Let $q\u2a7e0$. A map $F:\mathrm{\Omega}\to clos(X)$ is said to be

*q*-

*Lipschitz in the metric*${\rho}^{\mathfrak{S}}$ if for any ${\omega}_{1},{\omega}_{2}\in \mathrm{\Omega}$,

*q*-

*Lipschitz in the metric*${\rho}_{cl}$ if for any ${\omega}_{1},{\omega}_{2}\in X$, inequality (19) and the inequality

hold.

In the case of $\mathrm{\Omega}=X$ and $q<1$, a map *F* satisfying condition (18) is called *contracting* or *q*-*contracting* (*in the metric* dist). Similarly, a map $F:X\to clos(X)$ will be called *contracting in the metric* ${\rho}^{\mathfrak{S}}$ implying it satisfies inequality (19) with a constant $q<1$, and *contracting in the metric* ${\rho}_{cl}$ if it satisfies inequalities (19), (20).

In what follows, the conditions for a map *F* to be Lipschitz in the metrics dist, ${\rho}^{\mathfrak{S}}$, and ${\rho}_{cl}$, as well as connections between these properties are derived.

We begin with the criteria of the *q*-Lipschitzness of *F* in the metrics dist and ${\rho}_{\mathfrak{S}}$.

**Theorem 4**

- (1)
*A map*$F:\mathrm{\Omega}\to clos(X)$*is**q*-*Lipschitz in the metric*dist*if and only if the map*${\mathfrak{S}}_{r}F:\mathrm{\Omega}\to clos(X)$*is**q*-*Lipschitz in the metric*dist*for each*$r\u2a7e0$. - (2)
*A map*$F:\mathrm{\Omega}\to clos(X)$*is**q*-*Lipschitz in the metric*${\rho}^{\mathfrak{S}}$*if and only if for every*${r}_{0}>0$*and any*${\omega}_{1},{\omega}_{2}\in X$*such that*${\varrho}_{\mathrm{\Omega}}({\omega}_{1},{\omega}_{2})\u2a7d{(q{r}_{0})}^{-1}$,*the following holds*:$dist({\mathfrak{S}}_{r}F({\omega}_{1}),{\mathfrak{S}}_{r}F({\omega}_{2}))\u2a7dq{\varrho}_{\mathrm{\Omega}}({\omega}_{1},{\omega}_{2})\phantom{\rule{1em}{0ex}}\mathrm{\forall}r\u2a7d{r}_{0}.$(21)

*Proof* Statement (1) is a direct consequence of the relations between $dist(F({\omega}_{1}),F({\omega}_{2}))$ and $dist({\mathfrak{S}}_{r}F({\omega}_{1}),{\mathfrak{S}}_{r}F({\omega}_{2}))$ obtained in paragraph (2) of Lemma 1: the necessity follows from estimate (6) and sufficiency from equality (7).

*F*be

*q*-Lipschitz in the metric ${\rho}^{\mathfrak{S}}$,

*i.e.*,

Therefore, inequality (22) takes place for all ${\omega}_{1}$, ${\omega}_{2}$ such that ${\varrho}_{\mathrm{\Omega}}({\omega}_{1},{\omega}_{2})\u2a7d{(q{r}_{0})}^{-1}$. Since, according to paragraph (1) of Lemma 1, the function $r\mapsto dist({\mathfrak{S}}_{r}F({\omega}_{1}),{\mathfrak{S}}_{r}F({\omega}_{2}))$ does not decrease for any ${\omega}_{1}$, ${\omega}_{2}$, inequality (22) remains valid for every $r\u2a7d{r}_{0}$.

and thus ${\rho}^{\mathfrak{S}}(F({\omega}_{1}),F({\omega}_{2}))\u2a7dq{\varrho}_{\mathrm{\Omega}}({\omega}_{1},{\omega}_{2})$. □

**Remark 2** A multi-valued map $G:\mathrm{\Omega}\to clos(X)$ is said to be $(\epsilon ,q)$-*uniformly locally Lipschitz* (see, *e.g.*, [8]) if for any ${\omega}_{1},{\omega}_{2}\in \mathrm{\Omega}$ such that ${\varrho}_{\mathrm{\Omega}}({\omega}_{1},{\omega}_{2})\u2a7d\epsilon $, the inequality $dist(G({\omega}_{1}),G({\omega}_{2}))\u2a7dq{\varrho}_{\mathrm{\Omega}}({\omega}_{1},{\omega}_{2})$ holds. So, according to Theorem 4, a map $F:\mathrm{\Omega}\to clos(X)$ is *q*-Lipschitz in the metric ${\rho}^{\mathfrak{S}}$ if and only if for any $\epsilon >0$, the map ${\mathfrak{S}}_{r}F:\mathrm{\Omega}\to clos(X)$ is $(\epsilon ,q)$-uniformly locally Lipschitz for all $r\u2a7d{(q\epsilon )}^{-1}$.

The following definition allows to consider an important class of metric spaces often appearing in applications and to which the use of Theorem 4 (and the statements proved further on) is of particular interest.

**Definition 3** [8]

*convex*if for any points $\mu ,\nu \in \mathrm{\Omega}$, $\mu \ne \nu $, there exists a point $\omega \in \mathrm{\Omega}$ such that

*if*$(\mathrm{\Omega},{\varrho}_{\mathrm{\Omega}})$

*is complete and convex, then*

This property of convex metric spaces follows immediately from [[9], Theorems 14.1, 15.1].

If $(\mathrm{\Omega},{\varrho}_{\mathrm{\Omega}})$ is complete and convex, then paragraph (2) of Theorem 4 can be specified.

**Corollary 1** *Let* Ω *be a complete*, *convex metric space*. *Then a map* $F:\mathrm{\Omega}\to clos(X)$ *is* *q*-*Lipschitz in the metric* ${\rho}^{\mathfrak{S}}$ *if and only if inequality* (21) *holds for every* $r\u2a7e0$ *and any* ${\omega}_{1},{\omega}_{2}\in \mathrm{\Omega}$ (*i*.*e*., *if and only if the map* ${\mathfrak{S}}_{r}F:\mathrm{\Omega}\to clos(X)$ *is* *q*-*Lipschitz in the metric* dist *for every* $r\u2a7e0$).

*Proof* Necessity. For $r=0$ and any ${\omega}_{1},{\omega}_{2}\in \mathrm{\Omega}$, one has ${\mathfrak{S}}_{0}F({\omega}_{1})={\mathfrak{S}}_{0}F({\omega}_{2})=\mathrm{\Omega}$, so $dist({\mathfrak{S}}_{0}F({\omega}_{1}),{\mathfrak{S}}_{0}F({\omega}_{2}))=0$ and (21) holds for all ${\omega}_{1}$, ${\omega}_{2}$.

*F*is

*q*-Lipschitz in the metric ${\rho}^{\mathfrak{S}}$, so according to Theorem 4, for all ${\omega}_{1},{\omega}_{2}\in \mathrm{\Omega}$, the inequality ${\varrho}_{\mathrm{\Omega}}({\omega}_{1},{\omega}_{2})\u2a7d{(qr)}^{-1}$ implies relation (21). Fix some ${\omega}_{1}$, ${\omega}_{2}$ satisfying ${\varrho}_{\mathrm{\Omega}}({\omega}_{1},{\omega}_{2})>{(qr)}^{-1}$ and show that (21) remains valid. By the property (23), there exists a finite set of points ${\omega}_{1}={\xi}_{1},{\xi}_{2},\dots ,{\xi}_{n}={\omega}_{2}$ such that ${\varrho}_{\mathrm{\Omega}}({\xi}_{i},{\xi}_{i+1})\u2a7d{(qr)}^{-1}$, $i=1,\dots ,n-1$, and

The sufficiency part of the theorem follows directly from Theorem 4. □

Note that the hypotheses of completeness and convexity of Ω in Corollary 1 are crucial. Consider some examples.

**Example 3**Let $X={\bigcup}_{i\in \mathbb{Z}}{X}_{i}$, ${\varrho}_{X}({x}_{1},{x}_{2})=|{x}_{1}-{x}_{2}|$, $\theta =0$, where ${X}_{0}=[-1;1]$, ${X}_{i}=[2i;2i+1]$, ${X}_{-i}=[-2i-1;-2i]$, $i\in \mathbb{N}$. The space

*X*is complete and, obviously, not convex. Define the map $F:X\to clos(X)$ by

Estimate ${\rho}^{\mathfrak{S}}(F({x}_{1}),F({x}_{2}))$ for arbitrary ${x}_{1},{x}_{2}\in X$.

Evidently, if ${x}_{1},{x}_{2}\in {X}_{i}$, $i\in \mathbb{Z}$, then ${\rho}^{\mathfrak{S}}(F({x}_{1}),F({x}_{2}))=0$.

*r*, $1/r<1$ holds, one gets

In all the other cases, *i.e.*, when ${x}_{1}\in {X}_{-{i}_{1}}$, ${x}_{2}\in {X}_{-{i}_{2}}$, ${i}_{1},{i}_{2}=0,1,2,\dots $ , ${i}_{1}\ne {i}_{2}$, or when ${x}_{1}\in {X}_{-{i}_{1}}$, ${x}_{2}\in {X}_{{i}_{2}}$, ${i}_{1},{i}_{2}=0,1,2,\dots $ , ${i}_{1}^{2}+{i}_{2}^{2}\ne 0$, doing the analogous calculations one also gets estimation (25).

So, the map *F* is contracting in the metric ${\rho}^{\mathfrak{S}}$ with a constant $q=1/3$. Moreover, since ${\rho}^{o}(F({x}_{1}),F({x}_{2}))=0$ for any ${x}_{1}$, ${x}_{2}$, the map *F* is contracting in the metric ${\rho}_{cl}$.

*i.e.*, in this case, the Lipschitz constant $q\u2a7e1$. For the same values ${x}_{1}=1$, ${x}_{2}=2$ and $r=5$, the following hold:

Note that the map considered does not have fixed points in *X*. This means that, for example, in the Nadler theorem, one cannot replace the classical assumption of *F* be contracting in the metric dist with the one of *F* be contracting in the metric ${\rho}^{\mathfrak{S}}$. Nevertheless, using the map ${\mathfrak{S}}_{r}F$ instead of the map *F* allows to weaken the known conditions of fixed points existence. The corresponding result will be given in the next section.

**Example 4** Let $X={\bigcup}_{i\in \mathbb{Z}}{X}_{i}$, ${\varrho}_{X}({x}_{1},{x}_{2})=|{x}_{1}-{x}_{2}|$, $\theta =0$, where ${X}_{0}=(-1.1;1.1)$, ${X}_{i}=(2i;2i+1.1)$, ${X}_{-i}=(-2i-1.1;-2i)$, $i\in \mathbb{N}$. This space is convex, but not complete. Define the map $F:X\to clos(X)$ by (24).

The next theorem establishes a connection between the concepts of Lipschitzness considered in different metrics of the space $clos(X)$.

**Theorem 5**

- (1)
*If a map*$F:\mathrm{\Omega}\to clos(X)$*is**q*-*Lipschitz in the Hausdorff metric*,*then it is**q*-*Lipschitz in the metric*${\rho}_{cl}$*and all the more in the metric*${\rho}^{\mathfrak{S}}$. - (2)
*In the case when*Ω*is a complete and convex metric space*,*the concepts of**q*-*Lipschitzness of*$F:\mathrm{\Omega}\to clos(X)$*in the metrics*${\rho}^{\mathfrak{S}}$, ${\rho}_{cl}$,*and*dist*are equivalent*.

Here, a solid arrow means that the Lipschitzness of *F* in the first metric implies its Lipschitzness in the second metric; a dashed arrow corresponds to a relation between these properties that takes place only if the metric space Ω is complete and convex.

*Proof* (1) The correctness of the implication $dist\to {\rho}_{cl}$ follows from statements (2) and (4) of Lemma 1. Further, by Definition 2, the *q*-Lipschitzness of the map *F* in the metric ${\rho}_{cl}$ implies its *q*-Lipschitzness in the metric ${\rho}^{\mathfrak{S}}$.

(2) Now, let Ω be complete, convex and $F:\mathrm{\Omega}\to clos(X)$ be *q*-Lipschitz in the metric ${\rho}^{\mathfrak{S}}$. Then, according to Corollary 1, the map ${\mathfrak{S}}_{r}F$ is *q*-Lipschitz for every $r\u2a7e0$ and hence, by statement (1) of Theorem 4, the map *F* is *q*-Lipschitz in the metric dist. Therefore, *F* is *q*-Lipschitz in the metric ${\rho}_{cl}$ (see paragraph (1) of the theorem). □

Let us show that the convexity of a metric space is essential for equivalence of the properties of *q*-Lipschitzness in the metrics dist, ${\rho}_{cl}$, ${\rho}^{\mathfrak{S}}$.

Note that the maps considered in Examples 3 and 4 are contracting (with a constant $q=1/3$) in the metrics ${\rho}^{\mathfrak{S}}$ and ${\rho}_{cl}$. With respect to the metric dist, they are only Lipschitz with a constant $q>1$. Give an example of a map that is contracting in the metric ${\rho}_{cl}$ and not Lipschitz in the metric dist.

**Example 5**On the set ℕ, define the metric ${\varrho}_{\mathbb{N}}(n,m)=|n-m|$ and choose $\theta =1$. Consider $F:\mathbb{N}\to clos(\mathbb{N})$ given by $F(n)\doteq \{1,{(n+1)}^{2}\}$. Let $n,m\in \mathbb{N}$, $n<m$. Obviously, ${\rho}^{o}(F(n),F(m))=0$; determine $dist({\mathfrak{S}}_{r}F(n),{\mathfrak{S}}_{r}F(m))$ for different $r\u2a7e0$. If $r\u2a7d{(n+1)}^{2}$, then

*r*, $1/r<1$ holds, it follows that

*F*is

*q*-contracting, $q=1/4$, in the metric ${\rho}_{cl}$. On the other hand,

and therefore, there is no constant *q* for *F* to be *q*-Lipschitz in the Hausdorff metric.

Next, consider a map that is *q*-Lipschitz in the metric ${\rho}^{\mathfrak{S}}$ and not *q*-Lipschitz in the metric ${\rho}_{cl}$.

**Example 6**Let $F:\mathbb{N}\to clos(\mathbb{N})$ be given by $F(n)\doteq \{{(n+1)}^{2}\}$ (assume ${\varrho}_{\mathbb{N}}(n,m)=|n-m|$ $\mathrm{\forall}n,m\in \mathbb{N}$, $\theta =1$). Then, like in Example 5, for any $n,m\in \mathbb{N}$ ($n<m$), one has

*i.e.*, inequality (19) is valid with $q=1/4$. But

and thus estimate (20) does not hold.

## 5 Fixed point theorems

In this section, we prove a generalization of the Nadler fixed point theorem for a multi-valued map $F:X\to clos(X)$. The idea of the generalization consists in replacing the image $F(x)\in clos(X)$ by the set ${\mathfrak{S}}_{r}F(x)\in clos(X)$. As it has been mentioned already, this transformation allows to reduce the distance between the images of a map and, in a number of cases, to turn a multi-valued map that is not even continuous into a contracting one. And if it can be proved that a fixed point of the ‘new’ map ${\mathfrak{S}}_{r}F:X\to clos(X)$ does not belong to $\overline{{O}_{r}^{o}}$, then this point becomes a fixed one also for the initial map *F*.

**Theorem 6**

*Let*

*X*

*be a complete metric space and*$F:X\to clos(X)$.

*Let there exist*${r}_{0}>0$

*and*$q\in [0,1)$

*such that*:

- (1)
*the map*${\mathfrak{S}}_{{r}_{0}}F$*is**q*-*contracting*(*in the metric*dist),*i*.*e*.,$dist({\mathfrak{S}}_{{r}_{0}}F({x}_{1}),{\mathfrak{S}}_{{r}_{0}}F({x}_{2}))\u2a7dq{\varrho}_{X}({x}_{1},{x}_{2})\phantom{\rule{1em}{0ex}}\mathrm{\forall}{x}_{1},{x}_{2}\in X;$(26) - (2)
*there exists*${x}_{0}\in X$*such that*${\varrho}_{X}(\theta ,{x}_{0})+\frac{\varrho ({x}_{0},{\mathfrak{S}}_{{r}_{0}}F({x}_{0}))}{1-q}<{r}_{0}.$(27)

*Then there is a point* $\overline{x}\in X$ *satisfying the inclusion* $\overline{x}\in F(\overline{x})$ *and the estimate* ${\varrho}_{X}(\theta ,\overline{x})<{r}_{0}$.

*Proof*From assumption (1), by the Nadler theorem (see [6, 8]), it follows that for every ${x}_{0}\in X$ and any $\epsilon >0$, there is a fixed point $\overline{x}$ of the map ${\mathfrak{S}}_{{r}_{0}}F$ satisfying the inequality

*ε*so that

*ε*does exist. Then, from estimates (28), (29), one gets

So, $\overline{x}\in {\mathfrak{S}}_{{r}_{0}}F(\overline{x})$ and $\overline{x}\notin \overline{{O}_{{r}_{0}}^{o}}$, and it follows that $\overline{x}$ is a fixed point of the map *F*. □

In the classical theorem of Nadler, if a map $F:X\to clos(X)$ is contracting (in the Hausdorff metric) with a constant $q<1/2$ and with a fixed point $\overline{x}\in X$ satisfying the condition $F(\overline{x})\ne \{\overline{x}\}$, then this map has at least one more fixed point different from $\overline{x}$ (see [6]). For maps meeting the requirements of Theorem 6, the analogous statement may not be true. In fact, it is easy to explain: arguing as in [[6], Theorem 2.1.3], take ${u}_{0}\in {\mathfrak{S}}_{{r}_{0}}F(\overline{x})$, ${u}_{0}\notin \overline{x}$, as an initial point and, using the iteration procedure, find a fixed point $\overline{u}$ of the map ${\mathfrak{S}}_{{r}_{0}}F$. Since $q<1/2$, it is easy to show that $\overline{u}\ne \overline{x}$. It may turn out, however, that $\overline{u}\in \overline{{O}_{{r}_{0}}^{o}}$ and $\overline{u}\notin F(\overline{u})$, *i.e.*, the point $\overline{u}$ is not a fixed point of the map *F*. This situation does not occur if the element ${u}_{0}\in F(\overline{x})$ satisfies condition (2) of Theorem 6. Thus, the following statement holds.

**Corollary 2**

*Let*${\mathfrak{S}}_{{r}_{0}}F$

*be a contracting map with*$q<1/2$,

*and let*$\overline{x}$

*satisfying*${\varrho}_{X}(\theta ,\overline{x})<{r}_{0}$

*be a fixed point of the map*

*F*.

*Then*,

*if there is a point*${u}_{0}\in F(\overline{x})$, ${u}_{0}\ne \overline{x}$,

*such that*

*then* *F* *has at least one more fixed point* $\overline{u}\ne \overline{x}$.

**Remark 3**An attempt to ‘make’ the map

*F*a contracting one replacing it by the map $F\cap {O}_{{r}_{0}}:X\to clbd(X)$ turns out to be far less efficient than using the map ${\mathfrak{S}}_{{r}_{0}}F$. This follows from the obvious estimate (see inequality (8))

**Example 7** Consider the map $F:\mathbb{R}\to clos(\mathbb{R})$ given by $F(x)\doteq \{{e}^{x+i-3},i\in \mathbb{N}\}$ (assume ${\varrho}_{\mathbb{R}}({x}_{1},{x}_{2})=|{x}_{1}-{x}_{2}|$, $\theta =0$). We show first that this map is not *q*-Lipschitz (for no *q*) and, moreover, at any point is neither upper semicontinuous nor lower semicontinuous. More precisely, we prove that for any ${x}_{1},{x}_{2}\in \mathbb{R}$ close enough, $d(F({x}_{1}),F({x}_{2}))=\mathrm{\infty}$, $d(F({x}_{2}),F({x}_{1}))=\mathrm{\infty}$ hold.

Consider two situations: $i<j$ and $i>j$.

So, inequality (31) is correct.

If $i>j$, then ${e}^{{x}_{2}+i-3}>{e}^{{x}_{2}+j-3}>{e}^{{x}_{1}+j-3}$, and inequality (31) holds as well.

*i*,

*j*, the inequality

Despite the fact that $d(F({x}_{1}),F({x}_{2}))=d(F({x}_{2}),F({x}_{1}))=\mathrm{\infty}$ for any close enough ${x}_{1}$, ${x}_{2}$, Theorem 6 can be applied to the map *F*. Put ${r}_{0}=1/2$, $\theta =0$, ${x}_{0}=0$ and prove that relations (26) and (27) are valid. To determine the set ${\mathfrak{S}}_{{r}_{0}}F(x)$, we will use equality (2^{′}) since in the space ℝ definitions (2) and (2^{′}) lead to the same values of ${\rho}^{\mathfrak{S}}$, ${\rho}_{cl}$ (see the footnote on page 2), but the calculations by means of (2^{′}) are less ponderous.

*i.e.*, the map ${\mathfrak{S}}_{{r}_{0}}F$ is contracting with a constant $q=1/2$.

so condition (2) of Theorem 6 is also satisfied. Thus, there exists a fixed point $\overline{x}$ of the map *F* such that $|\overline{x}|<1/2$ (it is easy to check that $\overline{x}$ is the minimal solution of the equation ${e}^{x-2}=x$).

## Endnote

*X*, the values of ${\rho}^{\mathfrak{S}}$ and ${\rho}_{cl}$ calculated by means of (4) and (5) do not change if one defines the operator ${\mathfrak{S}}_{r}:clos(X)\to clos(X)$ by the equality

This very definition was used in [4] for $X={\mathbb{R}}^{n}$. Using (2^{′}) for an arbitrary metric space does not seem possible since, first of all, a sphere ${S}_{r}$ may result in an empty set and as a consequence, the set ${\mathfrak{S}}_{r}H$ will be empty when $\varrho (\theta ,H)>r$.

## Declarations

## Authors’ Affiliations

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