Mix-point property in quasi-pseudometric spaces

TL;DR

This paper introduces new results on startpoint theory in bicomplete quasi-pseudometric spaces, characterizing the existence of startpoints and endpoints via the mix-point property, extending previous findings in the field.

Contribution

It provides novel characterizations and existence results for startpoints and endpoints in quasi-pseudometric spaces using the mix-point property.

Findings

Existence of startpoints and endpoints under certain conditions

Characterization of startpoints and endpoints via mix-point property

Extension of known results in quasi-pseudometric space theory

Abstract

In this article, we give new results in the startpoint theory for quasi-pseudometric spaces. The results we present provide us with the existence of startpoint (endpoint, fixed point) for multi-valued maps defined on a bicomplete quasi-pseudometric space. We characterise the existence of startpoint and endpoint by the so-called \textit{mix-point property}. The present results extend known ones in the area.

Equations

$$d(x,A)=\inf \{d(x,a),a\in A\},d(A,x)=\inf \{d(a,x),a\in A\}.$$

$$H(A,B)=\max \left\{\underset{a\in A}{\sup }d(a,B),\underset{b\in B}{\sup }d(A,b)\right\}\text{ whenever }A,B,\in {\mathcal{P}}_0(X).$$

$$\underset{x\in X}{\inf }\underset{y\in Fx}{\sup }d(x,y)=0(\text{resp.}\underset{x\in X}{\inf }\underset{y\in Fx}{\sup }d(y,x)=0).$$

$$\underset{x\in X}{\inf }\underset{y\in Fx}{\sup }{d}^s(x,y)=0.$$

$$H(Fx,Fy)\le \psi (d(x,y)),\text{ for each }x,y\in X,$$

$$\underset{x\in X}{\inf }\underset{y\in Fx}{\sup }d(Jx,y)=0(\text{resp.}\underset{x\in X}{\inf }\underset{y\in Fx}{\sup }d(y,Jx)=0).$$

$$\underset{x\in X}{\inf }\underset{y\in Fx}{\sup }{d}^s(Jx,y)=0.$$

$$H(Fx,Fy)\le \psi (d(Jx,Jy)),\text{ for each }x,y\in X,$$

$$H(Fx,Fy)\le k(d(Jx,Jy)),\text{ for each }x,y\in X,$$

$$H(Fx,Fy)\le \psi (d(Jx,Jy)),\text{ for each }x,y\in X,$$

$$C_n=\left\{x\in X:\underset{y\in Fx}{\sup }{d}^s(x,y)\le \frac{1}{n}\right\}\ne \varnothing ,\text{ for }n\in \mathbb{N}=\{1,2,\cdots \},$$

$$H(Fx,Fy)\le \alpha d(Jx,Jy),\text{ for each }x,y\in X,$$

$$C_n=\left\{x\in X:\underset{y\in Fx}{\sup }{d}^s(Jx,y)\le \frac{1}{n}\right\}\ne \varnothing ,$$

$$d(Jx,Jy)$$

$$\le H(\{Jx\},Fx)+H(Fx,Fy)+H(Fy,\{Jy\})$$

$$\le \frac{2}{n}+\alpha d(Jx,Jy).$$

$$d(Jx,Jy)\le \frac{2}{n(1-\alpha )},$$

$$\delta (C_n)\le \frac{2}{rn(1-\alpha )}.$$

$$\bigcap _{n\in N}{C}_n=\{x_0\}.$$

$$H(\{J{x}_0\},F{x}_0)=\underset{y\in F{x}_0}{\sup }d(J{x}_0,y)=0=\underset{y\in F{x}_0}{\sup }d(y,J{x}_0)=H(F{x}_0,\{J{x}_0\}).$$

$$H(\{J{z}_0\},F{z}_0)=0=H(F{z}_0,\{J{z}_0\}),$$

$$rd(x,y)\le d(Jx,Jy).$$

$$\underset{x\in X}{\inf }\underset{y\in Fx}{\sup }{d}^s(Jx,y)=0.$$

$$H(Fx,Fy)\le \alpha d(Jx,Jy),\text{ for each }x,y\in X,$$

$$H(Fx,Fy)\le \alpha [d(Jx,Fx)+d(Jy,Fy)],\text{ for each }x,y\in X,$$

$$C_n=\left\{x\in X:\underset{y\in Fx}{\sup }{d}^s(Jx,y)\le \frac{1}{n}\right\}\ne \varnothing ,$$

$$d(Jx,Jy)$$

$$\le H(\{Jx\},Fx)+H(Fx,Fy)+H(Fy,\{Jy\})$$

$$\le \frac{2}{n}+\alpha [d(Jx,Fx)+d(Jy,Fy)]$$

$$\le \frac{1}{n}(2+2\alpha ).$$

$$d(Jx,Jy)\le \frac{1}{n}(2+2\alpha ),$$

$$\delta (C_n)\le \frac{1}{rn}(2+2\alpha ).$$

$$\bigcap _{n\in N}{C}_n=\{x_0\}.$$

Full text

1. Introduction and preliminaries

The theory of startpoint, first introduced in [1], came to extend the idea of fixed points for multi-valued mappings defined on quasi-pseudometric spaces. A series of three papers, see [1, 2, 3] has given a more or less detailed introduction to the subject. The aim of the present article is to continue this study by introducing the idea of mix-point property, which is used to characterise the existence of startpoints.

Definition 1.1**.**

Let $X$ be a non empty set. A function $d:X\times X\to[0,\infty)$ is called a quasi-pseudometric on $X$ if:

1. i)

$d(x,x)=0\quad\forall\ x\in X$, 2. ii)

$d(x,z)\leq d(x,y)+d(y,z)\quad\forall\ x,y,z\in X$.

Moreover, if

1. iii)

$d(x,y)=0=d(y,x)\Longrightarrow x=y$, then $d$ is said to be a $T_{0}$-quasi-metric.

The latter condition is referred to as the $T_{0}$-condition.

Remark 1.2*.*

• •

Let $d$ be a quasi-pseudometric on $X$, then the function $d^{-1}$ defined by $d^{-1}(x,y)=d(y,x)$ whenever $x,y\in X$ is also a quasi-pseudometric on $X$, called the conjugate of $d$. In the literature, $d^{-1}$ is also denoted $d^{t}$ or $\bar{d}$.

• •

It is easy to verify that the function $d^{s}$ defined by $d^{s}:=d\vee d^{-1}$, i.e. $d^{s}(x,y)=\max\{d(x,y),d(y,x)\}$ defines a metric on $X$ whenever $d$ is a $T_{0}$-quasi-metric on $X$.

Definition 1.3**.**

[1] A $T_{0}$-quasi-metric space $(X,d)$ is called bicomplete provided that the metric $d^{s}$ on $X$ is complete.

Let $(X,d)$ be a quasi-pseudometric space. We set $\mathscr{P}_{0}(X):=2^{X}\setminus\{\emptyset\}$ where $2^{X}$ denotes the power set of $X$. For $x\in X$ and $A\in\mathscr{P}_{0}(X)$, we define:

[TABLE]

We also define the map $H:\mathscr{P}_{0}(X)\times\mathscr{P}_{0}(X)\to[0,\infty]$ by

[TABLE]

Then $H$ is an extended111This means that $H$ can attain the value $\infty$ as it appears in the definition. quasi-pseudometric on $\mathscr{P}_{0}(X)$.

2. Some first results

We briefly recall the idea of a startpoint, as initially intended in [1].

Definition 2.1**.**

(Compare [1]) Let $(X,d)$ be a $T_{0}$-quasi-metric space.

Let $F:X\to 2^{X}$ be a set-valued map. An element $x\in X$ is said to be

1. (i)

a fixed point of $F$ if $x\in Fx$, 2. (ii)

a startpoint of $F$ if $H(\{x\},Fx)=0$, 3. (iii)

an endpoint of $F$ if $H(Fx,\{x\})=0$. 4. (iv)

an $\varepsilon$-startpoint of $F$ for some $\varepsilon\in(0,1)$ if $H(\{x\},Fx)<\varepsilon$. 5. (v)

an $\varepsilon$-endpoint of $F$ for some $\varepsilon\in(0,1)$ if $H(Fx,\{x\})<\varepsilon$.

Remark 2.2*.*

( See [1]) It is therefore obvious that if $x$ is both a startpoint of $F$ and an endpoint of $F$, then $x$ is a fixed point of $F$. In fact, $Fx$ is a singleton. Observe that a fixed point need not to be a startpoint nor an endpoint.

Indeed, consider the $T_{0}$-quasi-metric space $(X,d)$ where $X=\{0,1\}$ and $d$ defined by $d(0,1)=0,\ d(1,0)=1$ and $d(x,x)=0$ for $x=0,1$. We define on $X$ the set-valued map $F:X\to 2^{X}$ by $Fx=X$. Obviously, $1$ is a fixed point, but $H(\{1\},F1)=H(\{1\},X)=\max\{d(1,1),d(1,0)\}=1\neq 0$.

Lemma 2.3**.**

[1]** Let $(X,d)$ be a $T_{0}$-quasi-metric space and $F:X\to 2^{X}$ be a set-valued map. An element $x\in X$ is a startpoint of $F$ if and only if it is an $\varepsilon$-startpoint of $F$ for every $\varepsilon\in(0,1)$.

Lemma 2.4**.**

[1]** Let $(X,d)$ be a $T_{0}$-quasi-metric space and $F:X\to 2^{X}$ be a set-valued map. An element $x\in X$ is an endpoint of $F$ if and only if it is an $\varepsilon$-endpoint of $F$ for every $\varepsilon\in(0,1)$.

Definition 2.5**.**

[1] Let $(X,d)$ be a $T_{0}$-quasi-metric space. We say that a set-valued map $F:X\to 2^{X}$ has the approximate startpoint property (resp. approximate endpoint property ), if

[TABLE]

Definition 2.6**.**

[1] Let $(X,d)$ be a $T_{0}$-quasi-metric space. We say that a set-valued map $F:X\to 2^{X}$ has the approximate mix-point property if

[TABLE]

Here, it is also very clear that $F$ has the approximate mix-point property if and only if $F$ has both the approximate startpoint and the approximate endpoint properties.

We illustrate our definitions by giving the following example.

Example 2.7**.**

(Compare [1]) Let $X=\{0,1,2\}$. The map $d:X\times X\to[0,\infty)$ defined by $d(0,1)=d(0,2)=0,\ d(1,0)=d(1,2)=1$, $d(2,0)=d(2,1)=2$ and $d(x,x)=0$ for all $x\in X$ is a $T_{0}$-quasi-pseudometric on $X$. Let $F:X\to 2^{X}$ be the set mapping defined by $Fa=X\setminus\{a\}$ for any $a\in X$. Then $F$ admits [math] as unique startpoint, no endpoint and no fixed point. Moreover, $F$ has the approximate startpoint property but does not have the approximate endpoint property.

We recall below the main theorem in the startpoint theory that appeared in [1].

Theorem 2.8**.**

[1, Theorem 29]** Let $(X,d)$ be a bicomplete quasi-pseudometric space. Let $F:X\to CB(X)$ be a set-valued map that satisfies

[TABLE]

where $\psi:[0,\infty)\to[0,\infty)$ is upper semicontinuous, $\psi(t)<t$ for each $t>0$ and $\underset{t\to\infty}{\liminf}(t-\psi(t))>0$. Then there exists a unique $x_{0}\in X$ which is both a startpoint and an endpoint of $F$ if and only if $F$ has the approximate mix-point property.

We introduce the following definitions:

Definition 2.9**.**

Let $(X,d)$ be a quasi-pseudometric space, $J:X\to X$ be a single valued mapping and $F:X\to 2^{X}$ be a multi-valued mapping. We say that the mappings $J$ and $F$ have the approximate startpoint property (resp. approximate endpoint property ), if

[TABLE]

Definition 2.10**.**

Let $(X,d)$ be a $T_{0}$-quasi-pseudometric space, $J:X\to X$ be a single valued mapping. We say that $J$ and the set-valued map $F:X\to 2^{X}$ have the approximate mix-point property if

[TABLE]

Definition 2.11**.**

(Compare [1]) Let $(X,d)$ be a quasi-pseudometric space, $J:X\to X$ be a single valued mapping. Let $F:X\to 2^{X}$ be a set-valued map. An element $x\in X$ is said to be

1. (i)

a $J$-fixed point of $F$ if $Jx\in Fx$, 2. (ii)

a startpoint of $J$ and $F$ if $H(\{Jx\},Fx)=0$, 3. (iii)

an endpoint of $J$ and $F$ if $H(Fx,\{Jx\})=0$.

The next three results are the first results of this paper. We shall not give any proof, since the proofs follow the same arguments as the proofs in [1].

Theorem 2.12**.**

(Compare[1, Theorem 29]) Let $(X,d)$ be a bicomplete quasi-pseudometric space. Assume $J:X\to X$ is a continuous single-valued map and let $F:X\to CB(X)$ be a set-valued map that satisfiy

[TABLE]

where $\psi:[0,\infty)\to[0,\infty)$ is usc222for upper semicontinuous, $\psi(t)<t$ for each $t>0$ and $\underset{t\to\infty}{\liminf}(t-\psi(t))>0$. Then there exists a unique $x_{0}\in X$ which is both a startpoint and an endpoint of $J$ and $F$ if and only if $J$ and $F$ have the approximate mix-point property.

Theorem 2.13**.**

(Compare[1, Theorem 31]) Let $(X,d)$ be a bicomplete quasi-pseudometric space. Assume $J:X\to X$ is a continuous single-valued map and let $F:X\to CB(X)$ be a set-valued map that satisfy

[TABLE]

where $k\in[0,1)$. Then there exists a unique $x_{0}\in X$ which is both a startpoint and an endpoint of $J$ and $F$ if and only if $J$ and $F$ have the approximate mix-point property.

Theorem 2.14**.**

(Compare[1, Corollary 30]) Let $(X,d)$ be a bicomplete quasi-pseudometric space. Assume $J:X\to X$ is a continuous single-valued map. Let $F:X\to CB(X)$ be a set-valued map that satisfies

[TABLE]

where $\psi:[0,\infty)\to[0,\infty)$ is an upper semicontinuous map that satisfies $\psi(t)<t$ for each $t>0$ and $\underset{t\to\infty}{\liminf}(t-\psi(t))>0$. If $J$ and $F$ have the approximate mix-point property then $F$ has a $J$-fixed point.

Remark 2.15*.*

Observe that if we put $J=I_{X}$(identity map on $X$) in Theorems 2.12, 2.13 and 2.14 respectively, we obtain [1, Theorem 29, Theorem 31, Corollary 30] respectively.

3. More results

In [1], the proof of Theorem 2.8 basically establishes that the sets

[TABLE]

form a non-increasing sequence of bounded and $\tau(d^{s})$-closed sets. The conclusion follows from the Cantor intersection theorem. We shall use a similar approach in proving the next two results, with the difference that we present simpler and shorter arguments.

We now present the first non trivial generalisation of [1, Theorem 29].

Theorem 3.1**.**

Let $(X,d)$ be a bicomplete quasi-pseudometric space. Assume $J:X\to X$ is a continuous single-valued map such that $rd(x,y)\leq d(Jx,Jy)$ for some constant $r>0$. Let $F:X\to CB(X)$ be a set-valued map that satisfies

[TABLE]

where $\alpha\in(0,1)$ and $r\alpha<1$. Then there exists a unique $x_{0}\in X$ which is both a startpoint and an endpoint of $J$ and $F$ if and only if $J$ and $F$ have the approximate mix-point property.

Proof.

It is clear that if $J$ and $F$ admit a point which is both a startpoint and an endpoint, then $J$ and $F$ have the approximate startpoint property and the approximate endpoint property, i.e the approximate mix-point property. Conversely, suppose $J$ and $F$ have the approximate mix-point property. Then

[TABLE]

for each $n\in\mathbb{N}$. Moreover for each $n\in\mathbb{N}$, $C_{n+1}\subseteq C_{n}$. Since the map $x\mapsto\underset{y\in Fx}{\sup}d^{s}(Jx,y)$ is $\tau(d^{s})$-lsc333For lower semicontinuous (as supremum of $\tau(d^{s})$-continuous mappings), then $C_{n}$ is $\tau(d^{s})$-closed.

Next we prove that for each $n\in\mathbb{N},$ $C_{n}$ is bounded. Indeed, for any $x,y\in C_{n},$

[TABLE]

So

[TABLE]

and since $rd(x,y)\leq d(Jx,Jy)$, we have

[TABLE]

Therefore $\underset{n\rightarrow\infty}{\lim}\delta(C_{n})=0$. It follows from the Cantor intersection theorem that

[TABLE]

Thus

[TABLE]

For uniqueness, if $z_{0}$ is an arbitrary startpoint and endpoint of $J$ and $F$, then

[TABLE]

and so $z_{0}\in\underset{n\in N}{\bigcap}C_{n}=\{x_{0}\}$. ∎

We give the following example to illustrate our result.

Example 3.2**.**

Consider the $T_{0}$-quasi-metric space $(X,d)$ where $X=\{0,1\}$ and $d$ be defined by $d(0,1)=0,\ d(1,0)=1$ and $d(x,x)=0$ for $x=0,1$. Note that $(X,d)$ is bicomplete. We define on $X$ the set-valued map $F:X\to 2^{X}$ by $Fx=\{0\}$ and the single-valued continuous mapping $J:X\to X$ by $Jx=x^{2}$. It is clear that for all $x,y\in X,H(Fx,Fy)=0$.

For $x=0,y=1,\ d(x,y)=d(0,1)=0,$ and $d(Jx,Jy)=d(0,1)=0.$

For $x=1,y=0,\ d(x,y)=d(1,0)=1,$ and $d(Jx,Jy)=d(1,0)=1.$

So if we set $r=\frac{1}{2}$ and $\alpha=\frac{1}{3}$, we have that $r\alpha=\frac{1}{6}<1$ and

[TABLE]

Moreover, the condition (3.1) is satisfied. Since $H(\{J0\},F0)=0=H(F0,\{J0\})$, then [math] is both a startpoint and an endpoint of $J$ and $F$.

Observe also that:

-for $x=0,Fx=\{0\},\underset{y\in Fx}{\sup}d^{s}(Jx,y)=d^{s}(0,0)=0,$

-for $x=1,Fx=\{0\},\underset{y\in Fx}{\sup}d^{s}(Jx,y)=d^{s}(1,0)=1,$ and hence

[TABLE]

i.e. $J$ and $F$ have the approximate mix-point property.

Corollary 3.3**.**

Let $(X,d)$ be a bicomplete quasi-pseudometric space. Assume $J:X\to X$ is a continuous single-valued map such that $rd(x,y)\leq d(Jx,Jy)$ for some constant $r>0$ and $\text{ for each }x,y\in X$. Let $F:X\to CB(X)$ be a set-valued map that satisfies

[TABLE]

where $\alpha\in(0,1)$ and $r\alpha<1$. If $J$ and $F$ have the approximate mix-point property then $F$ has a $J$-fixed point.

Proof.

From Theorem 3.1, we conclude that there exists $x_{0}\in X$ which is both a startpoint and an endpoint for $J$ and $F$, i.e $H(\{Jx_{0}\},Fx_{0})=0=H(Fx_{0},\{Jx_{0}\})$. The $T_{0}$-condition therefore guarantees the desired result.

∎

Theorem 3.4**.**

Let $(X,d)$ be a bicomplete quasi-pseudometric space. Assume $J:X\to X$ is a continuous single-valued map such that $rd(x,y)\leq d(Jx,Jy)$ for some constant $r>0$ and $\text{ for each }x,y\in X$. Let $F:X\to CB(X)$ be a set-valued map that satisfies

[TABLE]

where $\alpha\in(0,1/2)$. Then there exists a unique $x_{0}\in X$ which is both a startpoint and an endpoint of $J$ and $F$ if and only if $J$ and $F$ have the approximate mix-point property.

Proof.

Once again, only one implication will be of interest to us, since the other one is trivial. So suppose $J$ and $F$ have the approximate mix-point property. Then we already know that the sets

[TABLE]

for each $n\in\mathbb{N}$ are $\tau(d^{s})$-closed and that $C_{n+1}\subseteq C_{n}$.

Next we prove that for each $n\in\mathbb{N},$ $C_{n}$ is bounded. Indeed, for any $x,y\in C_{n},$

[TABLE]

So

[TABLE]

and since $rd(x,y)\leq d(Jx,Jy)$, we have

[TABLE]

Therefore $\underset{n\rightarrow\infty}{\lim}\delta(C_{n})=0$. It follows from the Cantor intersection theorem that

[TABLE]

Thus

[TABLE]

For uniqueness, if $z_{0}$ is an arbitrary startpoint and endpoint of $J$ and $F$, then

[TABLE]

and so $z_{0}\in\underset{n\in N}{\bigcap}C_{n}=\{x_{0}\}$.

∎

Example 3.5**.**

Consider the $T_{0}$-quasi-metric space $(X,d)$ where $X=\{0,1\}$ and $d$ defined by $d(0,1)=0,\ d(1,0)=1$ and $d(x,x)=0$ for $x=0,1$. Note that $(X,d)$ is bicomplete. We define on $X$ the set-valued map $F:X\to 2^{X}$ by $Fx=\{0\}$ and the single valued continuous map $J:X\to X$ by $Jx=x^{3}$. It is clear that for all $x,y\in X,H(Fx,Fy)=0$.

For $x=0,y=1,d(x,y)=d(0,1)=0,d(Jx,Jy)=d(0,1)=0.$

For $x=1,y=0,d(x,y)=d(1,0)=1,d(Jx,Jy)=d(1,0)=1.$ So if we set $r=\frac{1}{2}$, we have that

[TABLE]

So if we set $\alpha=\frac{1}{3}$, we have that $0<\alpha<\frac{1}{2}$. Since $H(\{J0\},F0)=0=H(F0,\{J0\})$, then [math] is both a startpoint and an endpoint of $J$ and $F$.

For $x=0,y=1,\alpha[d(Jx,Fx)+d(Jy,Fy)]=\frac{1}{3}$ and

for $x=1,y=0,\alpha[d(Jx,Fx)+d(Jy,Fy)]=\frac{1}{3}$, so the condition (3.3) is satisfied.

Observe also that

[TABLE]

i.e. $J$ and $F$ have the approximate mix-point property.

Corollary 3.6**.**

Let $(X,d)$ be a bicomplete quasi-pseudometric space. Assume $J:X\to X$ is a continuous single-valued map such that $rd(x,y)\leq d(Jx,Jy)$ for some constant $r>0$ and $\text{ for each }x,y\in X$. Let $F:X\to CB(X)$ be a set-valued map that satisfies

[TABLE]

where $\alpha\in(0,1/2)$. If $J$ and $F$ have the approximate mix-point property then $F$ has a $J$-fixed point.

Using the same idea as in the proof of Theorem 3.1 and Theorem 3.4, one can establish the following results:

Theorem 3.7**.**

Let $(X,d)$ be a bicomplete quasi-pseudometric space. Assume $J:X\to X$ is a continuous single-valued map such that $rd(x,y)\leq d(Jx,Jy)$ for some constant $r>0$ and $\text{ for each }x,y\in X$. Let $F:X\to CB(X)$ be a set-valued map that satisfies

[TABLE]

where $0<\alpha<1/2$ with $2r\alpha<1$. Then there exists a unique $x_{0}\in X$ which is both a startpoint and an endpoint of $J$ and $F$ if and only if $J$ and $F$ have the approximate mix-point property.

Theorem 3.8**.**

Let $(X,d)$ be a bicomplete quasi-pseudometric space. Assume $J:X\to X$ is a continuous single-valued map such that $rd(x,y)\leq d(Jx,Jy)$ for some constant $r>0$ and $\text{ for each }x,y\in X$. Let $F:X\to CB(X)$ be a set-valued map that satisfies

[TABLE]

where $\alpha>0$ and $L\geq 0$ such that $r(\alpha+L)<1$. Then there exists a unique $x_{0}\in X$ which is both a startpoint and an endpoint of $J$ and $F$ if and only if $J$ and $F$ have the approximate mix-point property.

4. Concluding remarks

All the above results remain true if instead we consider a quasi-pseudometric type space $(X,d,b)$ (see [4]). On the other side, the sets $C_{n}$ considered in the investigation can be made more general in the sense that we could consider the net $\{C_{\varepsilon}\}_{\varepsilon>0}$ where the $C_{\varepsilon}$ are sets of the form:

[TABLE]

Therefore, the Theorem 3.1 could be reformulated as follows:

Theorem 4.1**.**

Let $(X,d,b)$ be a bicomplete quasi-pseudometric type space. Assume $J:X\to X$ is a continuous single-valued map such that $rd(x,y)\leq d(Jx,Jy)$ for some constant $r>0$ and $\text{ for each }x,y\in X$. Let $F:X\to CB(X)$ be a set-valued map that satisfies

[TABLE]

where $\alpha\in(0,1)$ such that $r\alpha b^{2}<1$. Then there exists a unique $x_{0}\in X$ which is both a startpoint and an endpoint of $J$ and $F$ if and only if $J$ and $F$ have the approximate mix-point property.

In proving this Theorem 4.1, the following lemma is key:

Lemma 4.2**.**

Let $(X,d,b)$ be a bicomplete quasi-pseudometric type space. Assume $J:X\to X$ is a continuous single-valued map such that $rd(x,y)\leq d(Jx,Jy)$ for some constant $r>0$ such that $r\alpha b^{2}<1$ and $\text{ for each }x,y\in X$. Let $F:X\to CB(X)$ be a set-valued map that satisfies

[TABLE]

where $\alpha\in(0,1)$. Then

[TABLE]

Similarly, the Theorem 3.4 could be reformulated as follows:

Theorem 4.3**.**

Let $(X,d,b)$ be a bicomplete quasi-pseudometric type space. Assume $J:X\to X$ is a continuous single-valued map such that $rd(x,y)\leq d(Jx,Jy)$ for some constant $r>0$ and $\text{ for each }x,y\in X$. Let $F:X\to CB(X)$ be a set-valued map that satisfies

[TABLE]

where $\alpha\in(0,1/2)$. Then there exists a unique $x_{0}\in X$ which is both a startpoint and an endpoint of $J$ and $F$ if and only if $J$ and $F$ have the approximate mix-point property.

The key lemma for Theorem 4.3 is

Lemma 4.4**.**

Let $(X,d,b)$ be a bicomplete quasi-pseudometric type space. Assume $J:X\to X$ is a continuous single-valued map such that $rd(x,y)\leq d(Jx,Jy)$ for some constant $r>0$ and $\text{ for each }x,y\in X$. Let $F:X\to CB(X)$ be a set-valued map that satisfies

[TABLE]

where $\alpha\in(0,1/2)$. Then

[TABLE]

In a similar manner, the Theorem 3.7 could be reformulated as follows:

Theorem 4.5**.**

Let $(X,d,b)$ be a bicomplete quasi-pseudometric type space. Assume $J:X\to X$ is a continuous single-valued map such that $rd(x,y)\leq d(Jx,Jy)$ for some constant $r>0$ and $\text{ for each }x,y\in X$. Let $F:X\to CB(X)$ be a set-valued map that satisfies

[TABLE]

where $0<\alpha<1/2$ with $2b^{2}r\alpha<1$. Then there exists a unique $x_{0}\in X$ which is both a startpoint and an endpoint of $J$ and $F$ if and only if $J$ and $F$ have the approximate mix-point property.

The key lemma for Theorem 4.5 is therefore:

Lemma 4.6**.**

Let $(X,d,b)$ be a bicomplete quasi-pseudometric type space. Assume $J:X\to X$ is a continuous single-valued map such that $rd(x,y)\leq d(Jx,Jy)$ for some constant $r>0$ and $\text{ for each }x,y\in X$. Let $F:X\to CB(X)$ be a set-valued map that satisfies

[TABLE]

where $0<\alpha<1/2$ with $2b^{2}r\alpha<1$. Then

[TABLE]

Finally the Theorem 3.8 could be reformulated as follows:

Theorem 4.7**.**

Let $(X,d,b)$ be a bicomplete quasi-pseudometric type space. Assume $J:X\to X$ is a continuous single-valued map such that $rd(x,y)\leq d(Jx,Jy)$ for some constant $r>0$and $\text{ for each }x,y\in X$. Let $F:X\to CB(X)$ be a set-valued map that satisfies

[TABLE]

where $\alpha>0$ and $L\geq 0$ such that $rb^{2}(\alpha+bL)<1$. Then there exists a unique $x_{0}\in X$ which is both a startpoint and an endpoint of $J$ and $F$ if and only if $J$ and $F$ have the approximate mix-point property.

The proof of Theorem 4.7 will be done with the use of the following lemma:

Lemma 4.8**.**

Let $(X,d,b)$ be a bicomplete quasi-pseudometric type space. Assume $J:X\to X$ is a continuous single-valued map such that $rd(x,y)\leq d(Jx,Jy)$ for some constant $r>0$ and $\text{ for each }x,y\in X$. Let $F:X\to CB(X)$ be a set-valued map that satisfies

[TABLE]

where $\alpha>0$ and $L\geq 0$ such that $rb^{2}(\alpha+bL)<1$. Then

[TABLE]

Conflict of interest.

The author declares that there is no conflict of interests regarding the publication of this article.