# Startpoints and(

Authors: Yaé Ulrich Gaba
Journal: Journal of Mathematics
 (2014)

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## Abstract

We introduce the concept ofstartpoint and endpoint for multivalued maps defined on a quasi-pseudometric space. We investigate the relation between these new concepts and the existence of fixed points for these set valued maps.


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## Body

## 1. Introduction

In the last few years there has been a growing interest in the theory of quasi-metric spaces and other related structures such as quasi-normed cones and asymmetric normed linear spaces (see, e.g., [1]), because such a theory provides an important tool and a convenient framework in the study of several problems in theoretical computer science, applied physics, approximation theory, and convex analysis. Many works on general topology have been done in order to extend the well-known results of the classical theory. In particular, various types of completeness are studied in [2], showing, for instance, that the classical concept ofCauchy sequences can be accordingly modified. In the same reference, which uses an approach based on uniformities, thebicompletion of a T
0-quasi-pseudometric has been explored. It is worth mentioning that, in the fixed point theory,completeness is a key element, since most of the constructed sequences will be assumed to have aCauchy type property.It is the aim of this paper to continue the study of quasi-pseudometric spaces by proving some fixed point results and investigating a bit more the behaviour of set-valued mappings. Thus, in Section3 a suitable notion of (
α
,
γ
)-contractive mapping is given for self-mappings defined on quasi-pseudometric spaces and some fixed point results are discussed. In Sections 4 and 5, the notions ofstartpoint andendpoint for set-valued mappings are introduced and different variants of such concepts, as well as their connections with the fixed point of a multivalued map, are characterized.For recent results in the theory of asymmetric spaces, the reader is referred to [3–8].


## 2. Preliminaries

Definition 1.
LetX be a nonempty set. A function d
:
X
×
X
→
[
0
,
∞
) is called aquasi-pseudometric on X if (i)
d
(
x
,
x
)
=
0 for all x
∈
X,
(ii)
d
(
x
,
z
)
≤
d
(
x
,
y
)
+
d
(
y
,
z
) for all x
,
y
,
z
∈
X.Moreover, if d
(
x
,
y
)
=
0
=
d
(
y
,
x
)
⇒
x
=
y, then d is said to be a T
0-quasi-pseudometric. The latter condition is referred to as the T
0-condition.Remark 2.
(i) Letd be a quasi-pseudometric on X; then the map d
-
1 defined by d
-
1
(
x
,
y
)
=
d
(
y
,
x
) whenever x
,
y
∈
X is also a quasi-pseudometric on X, called theconjugate of d. In the literature, d
-
1 is also denoted by d
t or d
¯.
(ii) It is easy to verify that the functiond
s defined by d
s
∶
=
d
∨
d
-
1, that is, d
s
(
x
,
y
)
=
max
⁡
⁡
{
d
(
x
,
y
)
,
d
(
y
,
x
)
}, defines a metric on X whenever d is a T
0-quasi-pseudometric on X.Let(
X
,
d
) be a quasi-pseudometric space. For x
∈
X and ɛ
>
0,
(1)
B
d
(
x
,
ɛ
)
=
{
y
∈
X
:
d
(
x
,
y
)
<
ɛ
}
denotes the open ɛ-ball at x. The collection of all such balls yields a base for the topology τ
(
d
) induced by d on X. Hence, for any A
∈
X, we will, respectively, denote by int
⁡
τ
(
d
)
A and cl
⁡
τ
(
d
)
A the interior and the closure of the set A with respect to the topology τ
(
d
).Similarly, forx
∈
X and ɛ
≥
0,
(2)
C
d
(
x
,
ɛ
)
=
{
y
∈
X
:
d
(
x
,
y
)
≤
ɛ
}
denotes the closed ɛ-ball at x. We will say that a subset E
⊂
X isjoin-closed if it is τ
(
d
s
)-closed, that is, closed with respect to the topology generated by d
s. The topology τ
(
d
s
) is finer than the topologies τ
(
d
) and τ
(
d
-
1
).Definition 3.
Let(
X
,
d
) be a quasi-pseudometric space. The convergence of a sequence (
x
n
) to x with respect to τ
(
d
), called d-convergence orleft-convergence and denoted by x
n
→
d
x, is defined in the following way:
(3)
x
n
→
d
x
⟺
d
(
x
,
x
n
)
⟶
0
.
Similarly, the convergence of a sequence(
x
n
) to x with respect to τ
(
d
-
1
), called d
-
1-convergence orright-convergence and denoted by x
n
→
d
-
1
x, is defined in the following way:
(4)
x
n
→
d
-
1
x
⟺
d
(
x
n
,
x
)
⟶
0
.
Finally, in a quasi-pseudometric space(
X
,
d
), we will say that a sequence (
x
n
)
d
s-converges to x if it is both left and right convergent to x, and we denote it by x
n
→
d
s
x or x
n
→
x when there is no confusion. Hence,
(5)
x
n
→
d
s
x
⟺
x
n
→
d
x
,
x
n
→
d
-
1
x
.Definition 4.
A sequence(
x
n
) in a quasi-pseudometric (
X
,
d
) is called(a)
left
d
-Cauchy if for every ϵ
>
0, there exist x
∈
X and n
0
∈
N such that
(6)
∀
n
≥
n
0
d
(
x
,
x
n
)
<
ϵ
;
(b)
left
K
-Cauchy if for every ϵ
>
0, there exists n
0
∈
N such that
(7)
∀
n
,
k
:
n
0
≤
k
≤
n
d
(
x
k
,
x
n
)
<
ϵ
;
(c)
d
s-Cauchy if for every ϵ
>
0, there exists n
0
∈
N such that
(8)
∀
n
,
k
≥
n
0
d
(
x
n
,
x
k
)
<
ϵ
.
Dually, we define, in the same way,right
d
-Cauchy andright
K
-Cauchy sequences.Remark 5.
Consider the following:
(i)d
s-Cauchy ⇒ left K-Cauchy ⇒ left d-Cauchy. The same implications hold for the corresponding right notions. None of the above implications is reversible.
(ii) A sequence is leftd-Cauchy with respect to d if and only if it is right K-Cauchy with respect to d
-
1.
(iii) A sequence is leftK-Cauchy with respect to d if and only if it is right K-Cauchy with respect to d
-
1.
(iv) A sequence isd
s-Cauchy if and only if it is both left and right K-Cauchy.Definition 6.
A quasi-pseudometric space(
X
,
d
) is called(i)
left-
K
-complete provided that any left K-Cauchy sequence is d-convergent,
(ii)
left Smyth sequentially complete if any left K-Cauchy sequence is d
s-convergent.The dual notions ofright-completeness are easily derived from the above definition.Definition 7.
AT
0-quasi-pseudometric space (
X
,
d
) is calledbicomplete provided that the metric d
s on X is complete.As usual, a subsetA of a quasi-pseudometric space (
X
,
d
) will be calledbounded provided that there exists a positive real constant M such that d
(
x
,
y
)
<
M whenever x
,
y
∈
A. This is equivalent to saying that there exist x
0
∈
X, r
,
s
≥
0 such that A
⊆
C
d
(
x
0
,
r
)
∩
C
d
-
1
(
x
0
,
r
).We also define thediameter
δ
(
A
) of A by δ
(
A
)
∶
=
sup
⁡
⁡
{
d
(
x
,
y
)
:
x
,
y
∈
A
}. Hence, A is bounded if and only if δ
(
A
)
<
∞. It is not difficult to see that this definition coincides with that of a bounded set in a metric space.Let(
X
,
d
) be a quasi-pseudometric space. We set P
0
(
X
)
∶
=
2
X
∖
{
∅
} where 2
X denotes the power set of X. For x
∈
X and A
,
B
∈
P
0
(
X
), we define
(9)
d
(
x
,
A
)
=
inf
⁡
⁡
{
d
(
x
,
a
)
,
a
∈
A
}
,
d
(
A
,
x
)
=
inf
⁡
⁡
{
d
(
a
,
x
)
,
a
∈
A
}
,
and we define H
(
A
,
B
) by
(10)
H
(
A
,
B
)
=
max
⁡
⁡
{
sup
⁡
a
∈
A
d
(
a
,
B
)
,
sup
⁡
b
∈
B
d
(
A
,
b
)
}
.ThenH is an extended quasi-pseudometric on P
0
(
X
). Moreover, we know from [9] that the restriction of H to S
cl
⁡
(
X
)
=
{
A
⊆
X
:
A
=
(
cl
⁡
τ
(
d
)
A
)
∩
(
cl
⁡
τ
(
d
-
1
)
A
)
} is an extended T
0-quasi-pseudometric. We will denote by C
B
(
X
) the collection of all nonempty bounded and τ
(
d
s
)-closed subsets of X.We complete this section by the following lemma.Lemma 8.
Let(
X
,
d
) be a quasi-pseudometric space. For every fixed x
∈
X, the mapping y
↦
d
(
x
,
y
) is τ
(
d
)-upper semicontinuous (
τ
(
d
)-usc in short) and τ
(
d
-
1
)-lower semicontinuous (
τ
(
d
-
1
)-lsc in short). For every fixed y
∈
X, the mapping x
↦
d
(
x
,
y
) is τ
(
d
)-lsc and τ
(
d
-
1
)-usc.Proof.
To prove thatd
(
x
,
·
) is τ
(
d
)-usc and τ
(
d
-
1
)-lsc, we have to show that the set {
y
∈
X
:
d
(
x
,
y
)
<
α
} is τ
(
d
)-open and {
y
∈
X
:
d
(
x
,
y
)
>
α
} is τ
(
d
-
1
)-open, for every α
∈
R, properties that are easy to check.
Indeed, fory
∈
X such that d
(
x
,
y
)
<
α, let r
:
=
α
-
d
(
x
,
y
)
>
0. If z
∈
Xis such that d
(
y
,
z
)
<
r, then
(11)
d
(
x
,
z
)
≤
d
(
x
,
y
)
+
(
y
,
z
)
<
d
(
x
,
y
)
+
r
=
α
,
showing that B
d
(
y
,
r
)
⊂
{
y
∈
X
:
d
(
x
,
y
)
<
α
}.
Similarly, fory
∈
X with d
(
x
,
y
)
>
α take r
:
=
d
(
x
,
y
)
-
α
>
0. If z
∈
X satisfies d
(
z
,
y
)
=
d
-
1
(
y
,
z
)
<
r, then
(12)
d
(
x
,
y
)
≤
d
(
x
,
z
)
+
(
z
,
y
)
<
d
(
x
,
z
)
+
r
,
so that d
(
x
,
z
)
>
d
(
x
,
y
)
-
r
=
α. Consequently, B
d
-
1
(
y
,
r
)
⊂
{
y
∈
X
:
d
(
x
,
y
)
>
α
}.


## 3. Some First Results

We begin by recalling the following.Definition 9.
A functionγ
:
[
0
,
∞
)
→
[
0
,
∞
) is called a (
c
)-comparison function if (γ
1)
γ is nondecreasing;
(γ
2)
∑
n
=
1
∞
γ
n
(
t
)
<
∞ for all t
>
0, where γ
n is the nth iterate of γ.We will denote by Γ the set of such functions. Note that for any γ
∈
Γ,   γ
(
t
)
<
t for any t
>
0.We then introduce the following definitions.Definition 10.
Let(
X
,
d
) be a quasi-pseudometric type space. A function T
:
X
→
X is called d-sequentially continuous orleft-sequentially continuous if for any d-convergent sequence (
x
n
) with x
n
→
d
x, the sequence (
T
x
n
)
d-converges to T
x; that is, T
x
n
→
d
T
x.
Similarly, we define ad
-
1-sequentially continuous orright-sequentially continuous function.Definition 11.
Let(
X
,
d
) be a quasi-pseudometric space, and let T
:
X
→
X and α
:
X
×
X
→
[
0
,
∞
) be mappings. We say that T is α-admissible if
(13)
α
(
x
,
y
)
≥
1
⟹
α
(
T
x
,
T
y
)
≥
1
,
whenever x
,
y
∈
X.Definition 12.
Let(
X
,
d
) be a quasi-pseudometric space and let T
:
X
→
X be a mapping. We say that T is an (
α
,
γ
)-contractive mapping if there exist two functions α
:
X
×
X
→
[
0
,
∞
) and γ
∈
Γ such that
(14)
α
(
x
,
y
)
d
(
T
x
,
T
y
)
≤
γ
(
d
(
x
,
y
)
)
,
whenever x
,
y
∈
X.We now state the first fixed point theorem.Theorem 13.
Let(
X
,
d
) be a Hausdorff left K-complete T
0-quasi-pseudometric space. Suppose that T
:
X
→
X is an (
α
,
γ
)-contractive mapping which satisfies the following conditions: (i)
T is α-admissible;
(ii)
there existsx
0
∈
X such that α
(
x
0
,
T
x
0
)
≥
1;
(iii)
T is d-sequentially continuous.Then T has a fixed point.Proof.
By (ii), there existsx
0
∈
X such that α
(
x
0
,
T
x
0
)
≥
1. Let us define the sequence (
x
n
) in X by x
n
+
1
=
T
x
n for all n
=
0,1
,
2
,
…. Without loss of generality, we can always assume that x
n
≠
x
n
+
1 for all n
∈
N, since if x
n
0
=
x
n
0
+
1 for some n
0
∈
N, the proof is complete.From assumption (i), we derive(15)
α
(
x
0
,
T
x
0
)
=
α
(
x
0
,
x
1
)
≥
1
⟹
α
(
T
x
0
,
T
x
1
)
=
α
(
x
1
,
x
2
)
≥
1
.
Recursively, we get
(16)
α
(
x
n
,
x
n
+
1
)
≥
1
∀
n
=
0,1
,
2
,
…
.
Since T is (
α
,
γ
)-contractive, we can write
(17)
d
(
x
n
,
x
n
+
1
)
=
d
(
T
x
n
-
1
,
T
x
n
)
≤
α
(
x
n
-
1
,
x
n
)
d
(
T
x
n
-
1
,
T
x
n
)
≤
γ
(
d
(
x
n
-
1
,
x
n
)
)
,
for all n
=
0,1
,
2
,
…. Inductively, we obtain
(18)
d
(
x
n
,
x
n
+
1
)
≤
γ
n
(
d
(
x
0
,
x
1
)
)
,
n
=
1,2
,
…
.
Therefore, for any k
≥
1, using the triangle inequality, we get
(19)
d
(
x
n
,
x
n
+
k
)
≤
d
(
x
n
,
x
n
+
1
)
+
d
(
x
n
+
1
,
x
n
+
2
)
+
⋯
+
d
(
x
n
+
k
-
1
,
x
n
+
k
)
≤
∑
i
=
n
n
+
k
-
1
γ
i
(
d
(
x
0
,
x
1
)
)
≤
∑
i
=
n
∞
γ
i
(
d
(
x
0
,
x
1
)
)
.
Letting n
→
∞, we derive that d
(
x
n
,
x
n
+
k
)
→
0. Hence, (
x
n
) is a left K-Cauchy sequence. Since (
X
,
d
) is left K-complete and T
d-sequentially continuous, there exists x
* such that x
n
→
d
x
* and x
n
+
1
→
d
T
x
*. Since X is Hausdorff, we have that x
*
=
T
x
*.Corollary 14.
Let(
X
,
d
) be a Hausdorff right K-complete T
0-quasi-pseudometric space. Suppose that T
:
X
→
X is an (
α
,
γ
)-contractive mapping which satisfies the following conditions: (i)
T is α-admissible;
(ii)
there existsx
0
∈
X such that α
(
T
x
0
,
x
0
)
≥
1;
(iii)
T is d
-
1-sequentially continuous.Then T has a fixed point.Corollary 15.
Let(
X
,
d
) be a bicomplete quasi-pseudometric space. Suppose that T
:
X
→
X is an (
α
,
γ
)-contractive mapping which satisfies the following conditions: (i)
T is α-admissible and the function α is symmetric; that is, α
(
a
,
b
)
=
α
(
b
,
a
) for any a
,
b
∈
X;
(ii)
there existsx
0
∈
X such that α
(
T
x
0
,
x
0
)
≥
1;
(iii)
T is d
s-sequentially continuous.Then T has a fixed point.Proof.
Following the proof of Theorem13, it is clear that the sequence (
x
n
) in X defined by x
n
+
1
=
T
x
n for all n
=
0,1
,
2
,
… is d
s-Cauchy. Since (
X
,
d
s
) is complete and T sequentially continuous, there exists x
* such that x
n
→
d
s
x
* and x
n
+
1
→
d
s
T
x
*. Since (
X
,
d
s
) is Hausdorff, we have that x
*
=
T
x
*.Remark 16.
In fact, we do not needα to be symmetric. It is enough, for the result to be true, to have a point x
0
∈
X for which α
(
x
0
,
T
x
0
)
≥
1 and α
(
T
x
0
,
x
0
)
≥
1.We conclude this section by the following results which are in fact consequences of Theorem13.Theorem 17.
Let(
X
,
d
) be a Hausdorff left K-complete T
0-quasi-pseudometric space. Suppose that T
:
X
→
X is an (
α
,
γ
)-contractive mapping which satisfies the following conditions:(i)
T is α-admissible;
(ii)
there existsx
0
∈
X such that α
(
x
0
,
T
x
0
)
≥
1;
(iii)
if(
x
n
) is a sequence in X such that α
(
x
n
,
x
n
+
1
)
≥
1 for all n
=
1,2
,
… and x
n
→
d
x, then there exists a subsequence (
x
n
(
k
)
) of (
x
n
) such that α
(
x
n
(
k
)
,
x
)
≥
1 for all k.Then T has a fixed point.Proof.
Following the proof of Theorem13, we know that the sequence (
x
n
) defined by x
n
+
1
=
T
x
n for all n
=
0,1
,
2
,
…
d-converges to some x
* and satisfies α
(
x
n
,
x
n
+
1
)
≥
1 for n
≥
1. From condition (iii), we know that there exists a subsequence (
x
n
(
k
)
) of (
x
n
) such that α
(
x
n
(
k
)
,
x
*
)
≥
1 for all k. Since T is an (
α
,
γ
)-contractive mapping, we get
(20)
d
(
x
n
(
k
)
+
1
,
T
x
*
)
=
d
(
T
x
n
(
k
)
,
T
x
*
)
≤
α
(
x
n
(
k
)
,
x
*
)
d
(
T
x
n
(
k
)
,
T
x
*
)
≤
γ
(
d
(
x
n
(
k
)
,
x
*
)
)
.
Letting k
→
∞, we obtain d
(
x
n
(
k
)
+
1
,
T
x
*
)
→
0. Since X is Hausdorff, we have that T
x
*
=
x
*.
This completes the proof.Corollary 18.
Let(
X
,
d
) be a Hausdorff right K-complete T
0-quasi-pseudometric space. Suppose that T
:
X
→
X is an (
α
,
γ
)-contractive mapping which satisfies the following conditions:(i)
T is α-admissible;
(ii)
there existsx
0
∈
X such that α
(
T
x
0
,
x
0
)
≥
1;
(iii)
if(
x
n
) is a sequence in X such that α
(
x
n
+
1
,
x
n
)
≥
1 for all n
=
1,2
,
… and x
n
→
d
-
1
x, then there exists a subsequence (
x
n
(
k
)
) of (
x
n
) such that α
(
x
,
x
n
(
k
)
)
≥
1 for all k.Then T has a fixed point.Corollary 19.
Let(
X
,
d
) be a bicomplete quasi-pseudometric space. Suppose that T
:
X
→
X is an (
α
,
γ
)-contractive mapping which satisfies the following conditions:(i)
T is α-admissible and the function α is symmetric;
(ii)
there existsx
0
∈
X such that α
(
T
x
0
,
x
0
)
≥
1;
(iii)
if(
x
n
) is a sequence in X such that α
(
x
m
,
x
n
)
≥
1 for all n
,
m
∈
N and x
n
→
d
s
x, then there exists a subsequence (
x
n
(
k
)
) of (
x
n
) such that α
(
x
,
x
n
(
k
)
)
≥
1 for all k.Then T has a fixed point.


## 4. Startpoint Theory

It is important to mention that there are a variety of endpoint concepts in the literature (see, e.g., [10]), each of them corresponding to a specified setting. Here we introduce a similar notion for set-valued maps defined on quasi-pseudometric spaces. Let (
X
,
d
) be a T
0-quasi-pseudometric space.Definition 20.
LetF
:
X
→
2
X be a set-valued map. An element x
∈
X is said to be (i)
a fixed point ofF if x
∈
F
x,
(ii)
a startpoint ofF if H
(
{
x
}
,
F
x
)
=
0,
(iii)
an endpoint ofF if H
(
F
x
,
{
x
}
)
=
0,
(iv)
anɛ-startpoint of F for some ɛ
∈
(
0,1
) if H
(
{
x
}
,
F
x
)
<
ɛ,
(v)
anɛ-endpoint of F for some ɛ
∈
(
0,1
) if H
(
F
x
,
{
x
}
)
<
ɛ.Remark 21.
It is therefore obvious that ifx is both a startpoint of F and an endpoint of F, then x is a fixed point of F. In fact, F
x is a singleton. But a fixed point need not be a startpoint nor an endpoint.
Indeed, consider theT
0-quasi-pseudometric space (
X
,
d
), where X
=
{
0,1
} and d is defined by d
(
0,1
)
=
0, d
(
1,0
)
=
1, and d
(
x
,
x
) for x
=
0,1. We define on X the set-valued map F
:
X
→
2
X by F
x
=
X. Obviously, 1 is a fixed point, but H
(
{
1
}
,
F
1
)
=
H
(
{
1
}
,
X
)
=
max
⁡
{
d
(
1,1
)
,
d
(
1,0
)
}
=
1
≠
0.Lemma 22.
Let(
X
,
d
) be a T
0-quasi-pseudometric space and let F
:
X
→
2
X be a set-valued map. An element x
∈
X is a startpoint of F if and only if it is an ɛ-startpoint of F for every ɛ
∈
(
0,1
).Lemma 23.
Let(
X
,
d
) be a T
0-quasi-pseudometric space and let F
:
X
→
2
X be a set-valued map. An element x
∈
X is an endpoint of F if and only if it is an ɛ-endpoint of F for every ɛ
∈
(
0,1
).Definition 24.
Let(
X
,
d
) be a T
0-quasi-pseudometric space. We say that a set-valued map F
:
X
→
2
X has theapproximate startpoint property (resp.,approximate endpoint property) if
(21)
inf
⁡
x
∈
X
sup
⁡
y
∈
F
x
d
(
x
,
y
)
=
0
(
resp
.
,
inf
⁡
x
∈
X
sup
⁡
y
∈
F
x
d
(
y
,
x
)
=
0
)
.Definition 25.
Let(
X
,
d
) be a T
0-quasi-pseudometric space. We say that a set-valued map F
:
X
→
2
X has theapproximate mix-point property if
(22)
inf
⁡
x
∈
X
sup
⁡
y
∈
F
x
d
s
(
x
,
y
)
=
0
.
Here, it is also very clear that F has approximate mix-point property if and only if F has both the approximate startpoint and the approximate endpoint properties.We are therefore naturally led to this definition.Definition 26.
Letf
:
X
→
X be a single-valued map on a T
0-quasi-pseudometric space (
X
,
d
). Then f has theapproximate startpoint property (resp.,approximate endpoint property) if and only if
(23)
inf
⁡
x
∈
X
d
(
x
,
f
x
)
=
0
(
resp
.
,
inf
⁡
x
∈
X
d
(
f
x
,
x
)
=
0
)
.We motivate our coming results by the following examples. We basically show that the concepts defined above are independent and do not necessarily coincide. The list of examples presented is not exhaustive and more can be constructed, showing the connection between the notions defined above.Example 27.
LetX
=
{
0,1
,
2
}. The map d
:
X
×
X
→
[
0
,
∞
) 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
∈
X is a T
0-quasi-pseudometric on X. Let F
:
X
→
2
X be the set mapping defined by F
a
=
X
∖
{
a
} for any a
∈
X. By definition, F does not have any fixed point. Nevertheless, a simple computation shows that H
(
{
0
}
,
F
0
)
=
0, and hence 0 is a startpoint and it is the only one. Also there is no endpoint. Again, with a direct computation, we have inf
⁡
x
∈
X
sup
⁡
y
∈
F
x
d
(
x
,
y
)
=
0, showing that F has the approximate startpoint property, but inf
⁡
x
∈
X
sup
⁡
y
∈
F
x
d
(
y
,
x
)
=
1, showing that F does not have the approximate endpoint property.Example 28.
LetX
=
{
1
/
n
,
n
=
1,2
,
…
}. The map d
:
X
×
X
→
[
0
,
∞
) defined by d
(
1
/
n
,
1
/
m
)
=
max
⁡
{
1
/
n
-
1
/
m
,
0
} is a T
0-quasi-pseudometric on X. Let F
:
X
→
2
X be the set-valued mapping defined by F
a
=
X
∖
{
a
} for any a
∈
X. By definition, F does not have any fixed point.
For a fixedn
0
∈
N,
(24)
d
(
1
n
0
,
1
m
)
=
{
0
if
m
≤
n
0
,
1
n
0
-
1
m
if
m
>
n
0
.
Similarly for a fixed n
0
∈
N,
(25)
d
(
1
m
,
1
n
0
)
=
{
0
if
m
≥
n
0
,
1
m
-
1
n
0
if
m
<
n
0
.
Hence, F does not have any startpoint nor endpoint (which also implies that F does not have any fixed point).But for a givenɛ
∈
(
0,1
), there exists k
0
∈
N such that 1
/
k
0
<
ɛ. We also know from definition that H
(
{
1
/
k
0
}
,
F
(
1
/
k
0
)
)
=
1
/
k
0, so, 1
/
k
0 is an ɛ-startpoint of F. More generally, for a given ɛ
∈
(
0,1
), there exists k
ɛ
∈
N such that 1
/
k
ɛ is an ɛ-startpoint of F. Moreover, for any k
≥
k
ɛ, 1
/
k is an ɛ-startpoint of F.Similarly, we can show thatF admits an ɛ-endpoint.We can now state our first result.Theorem 29.
Let(
X
,
d
) be a bicomplete quasi-pseudometric space. Let F
:
X
→
C
B
(
X
) be a set-valued map that satisfies
(26)
H
(
F
x
,
F
y
)
≤
ψ
(
d
(
x
,
y
)
)
,
f
o
r
e
a
c
h
x
,
y
∈
X
,
where ψ
:
[
0
,
∞
)
→
[
0
,
∞
) is upper semicontinuous, ψ
(
t
)
<
t for each t
>
0, and lim
⁡
inf
⁡
t
→
∞
⁡
(
t
-
ψ
(
t
)
)
>
0. Then there exists a unique x
0
∈
X which is both a startpoint and an endpoint of F if and only if F has the approximate mix-point property.Proof.
It is clear that ifF admits a point which is both a startpoint and an endpoint, then F has the approximate startpoint property and the approximate endpoint property. Just observe that H
(
{
x
}
,
F
x
)
=
sup
⁡
y
∈
F
x
d
(
x
,
y
) and H
(
F
x
,
{
x
}
)
=
sup
⁡
y
∈
F
x
d
(
y
,
x
). Conversely, suppose F has the approximate mix-point property. Then
(27)
C
n
=
{
x
∈
X
:
sup
⁡
y
∈
F
x
d
s
(
x
,
y
)
≤
1
n
}
≠
∅
,
for each n
∈
N. Also it is clear that for each n
∈
N, C
n
+
1
⊆
C
n. The map x
↦
sup
⁡
y
∈
F
x
d
s
(
x
,
y
) is τ
(
d
s
)-lower semicontinuous (as supremum of τ
(
d
s
)-continuous mappings); we have that C
n is τ
(
d
s
)-closed.
Next we prove that, for eachn
∈
N
,
C
n is bounded.
Assume by the way of contradiction thatδ
(
C
n
)
=
∞ for each n
∈
N. Then there exist x
n
,
y
n
∈
C
n such that d
(
x
n
,
y
n
)
≥
n. From (26), we obtain that
(28)
d
(
x
,
y
)
=
H
(
{
x
}
,
{
y
}
)
≤
H
(
{
x
}
,
F
x
)
+
H
(
F
x
,
F
y
)
+
H
(
F
y
,
{
y
}
)
=
2
n
+
ψ
(
d
(
x
,
y
)
)
,
whenever x
,
y
∈
C
n.
Therefore,(29)
d
(
x
,
y
)
-
ψ
(
d
(
x
,
y
)
)
≤
2
n
,
whenever x
,
y
∈
C
n. Hence,
(30)
lim
⁡
n
→
∞
(
d
(
x
n
,
y
n
)
-
ψ
(
d
(
x
n
,
y
n
)
)
)
=
0
,
lim
⁡
n
→
∞
d
(
x
n
,
y
n
)
=
∞
.
This contradicts our assumption. Now we show that lim
⁡
n
→
∞
δ
(
C
n
)
=
0. On the contrary, assume lim
⁡
n
→
∞
δ
(
C
n
)
=
r
0
>
0 (note that the sequence (
δ
(
C
n
)
) is nonincreasing and bounded below and then has a limit). Let
(31)
β
=
inf
⁡
n
∈
N
inf
⁡
{
liminf
⁡
k
→
∞
(
r
n
,
k
-
ψ
(
r
n
,
k
)
)
:
(
x
n
,
k
,
y
n
,
k
)
∈
C
n
,
a
a
a
a
a
a
a
r
n
,
k
=
d
(
x
n
,
k
,
y
n
,
k
)
⟶
δ
(
C
n
)
as
k
⟶
∞
L
L
}
.
Now we show that β
>
0 (notice β
≥
0). Arguing by contradiction, we assume β
=
0; then by the definition of β, there exists a sequence r
n such that r
n
→
r
0 and lim
⁡
n
→
∞
(
r
n
-
ψ
(
r
n
)
)
=
0. Then lim
⁡
n
→
∞
ψ
(
r
n
)
=
r
0. But since ψ is upper semicontinuous and r
0
>
0, then
(32)
r
0
=
lim
⁡
n
→
∞
ψ
(
r
n
)
≤
ψ
(
r
0
)
<
r
0
.
This contradiction shows that β
>
0. For each n
∈
N, let (
x
k
,
y
k
)
∈
C
n be a sequence such that d
(
x
k
,
y
k
)
→
δ
(
C
n
), as k
→
∞. Then from (29) we get
(33)
β
≤
liminf
⁡
k
→
∞
(
d
(
x
k
,
y
k
)
-
ψ
(
d
(
x
k
,
y
k
)
)
)
≤
2
n
,
aaaaaaaaaaaaaiiaaaaaaa
for
each
n
∈
N
.
Hence, β
=
0. This contradiction shows that lim
⁡
n
→
∞
⁡
δ
(
C
n
)
=
0. It follows from the Cantor intersection theorem that ⋂
n
∈
N
‍
C
n
=
{
x
0
}.
Thus,H
(
{
x
0
}
,
F
x
0
)
=
sup
⁡
y
∈
F
x
0
d
(
x
,
y
)
=
0
=
sup
⁡
y
∈
F
x
0
d
(
y
,
x
)
=
H
(
F
x
0
,
{
x
0
}
). For uniqueness, if z
0 is an arbitrary startpoint of F, then H
(
{
z
0
}
,
F
z
0
)
=
0
=
H
(
F
z
0
,
{
z
0
}
), and so z
0
∈
⋂
n
∈
N
‍
C
n
=
{
x
0
}.
This completes the proof.Corollary 30.
Let(
X
,
d
) be a bicomplete quasi-pseudometric space. Let F
:
X
→
C
B
(
X
) be a set-valued map that satisfies
(34)
H
(
F
x
,
F
y
)
≤
ψ
(
d
(
x
,
y
)
)
,
f
o
r
e
a
c
h
x
,
y
∈
X
,
where ψ
:
[
0
,
∞
)
→
[
0
,
∞
) is upper semicontinuous, ψ
(
t
)
<
t for each t
>
0, and lim
⁡
⁡
inf
⁡
t
→
∞
⁡
(
t
-
ψ
(
t
)
)
>
0. If F has the approximate mix-point property then F has a fixed point.Proof.
From Theorem29, we conclude that there exists x
0 which is both a startpoint and an endpoint; that is, H
(
{
x
0
}
,
F
x
0
)
=
0
=
H
(
F
x
0
,
{
x
0
}
). The T
0-condition therefore guarantees the desired result.Theorem 31.
Let(
X
,
d
) be a bicomplete quasi-pseudometric space. Let F
:
X
→
C
B
(
X
) be a set-valued map that satisfies
(35)
H
(
F
x
,
F
y
)
≤
k
d
(
x
,
y
)
,
f
o
r
e
a
c
h
x
,
y
∈
X
,
where 0
≤
k
<
1. Then there exists a unique x
0
∈
X which is both a startpoint and an endpoint of F if and only if F has the approximate mix-point property.Proof.
Takeψ
(
t
)
=
k
t in Theorem 29.We then deduce the following result for single-valued maps.Theorem 32.
Let(
X
,
d
) be a bicomplete quasi-pseudometric space. Let f
:
X
→
X be a map that satisfies
(36)
d
(
f
x
,
f
y
)
≤
ψ
(
d
(
x
,
y
)
)
,
f
o
r
e
a
c
h
x
,
y
∈
X
,
where ψ
:
[
0
,
∞
)
→
[
0
,
∞
) is upper semicontinuous, ψ
(
t
)
<
t for each t
>
0, and lim
⁡
⁡
inf
⁡
t
→
∞
(
t
-
ψ
(
t
)
)
>
0. Then f has the approximate startpoint property.Proof.
By the way of contradiction, suppose thatinf
⁡
x
∈
X
d
(
x
,
f
x
)
>
0. Then
(37)
inf
⁡
x
∈
X
d
(
x
,
f
x
)
≤
inf
⁡
y
∈
f
(
X
)
d
(
y
,
f
y
)
=
inf
⁡
x
∈
X
d
(
f
x
,
f
2
x
)
≤
inf
⁡
x
∈
X
ψ
(
d
(
x
,
f
x
)
)
.
Since ψ
(
d
(
x
,
y
)
)
≤
d
(
x
,
y
), then inf
⁡
x
∈
X
ψ
(
d
(
x
,
f
x
)
)
≤
inf
⁡
x
∈
X
d
(
x
,
f
x
).
Now, on the contrary, suppose again that(38)
inf
⁡
x
∈
X
ψ
(
d
(
x
,
f
x
)
)
=
inf
⁡
x
∈
X
d
(
x
,
f
x
)
.
Let (
x
n
)
⊂
X be a sequence such that lim
⁡
n
→
∞
d
(
x
n
,
f
(
x
n
)
)
=
inf
⁡
x
∈
X
d
(
x
,
f
x
). By passing to subsequences if necessary, we may assume that lim
⁡
n
→
∞
ψ
(
d
(
x
n
,
f
(
x
n
)
)
) exists. Then from (37) we have
(39)
inf
⁡
x
∈
X
d
(
x
,
f
x
)
≤
inf
⁡
x
∈
X
ψ
(
d
(
x
,
f
x
)
)
≤
lim
⁡
n
→
∞
ψ
(
d
(
x
n
,
f
(
x
n
)
)
)
≤
ψ
(
inf
⁡
x
∈
X
d
(
x
,
f
x
)
)
<
inf
⁡
x
∈
X
d
(
x
,
f
x
)
.
We get a contradiction, soinf
⁡
x
∈
X
ψ
(
d
(
x
,
f
x
)
)
<
inf
⁡
x
∈
X
d
(
x
,
f
x
) which again contradicts (37).
This completes the proof.Corollary 33.
Let(
X
,
d
) be a bicomplete quasi-pseudometric space. Let f
:
X
→
X be a map that satisfies
(40)
d
(
f
x
,
f
y
)
≤
ψ
(
d
(
x
,
y
)
)
,
f
o
r
e
a
c
h
x
,
y
∈
X
,
where ψ
:
[
0
,
∞
)
→
[
0
,
∞
) is upper semicontinuous, ψ
(
t
)
<
t for each t
>
0, and lim
⁡
⁡
inf
⁡
t
→
∞
⁡
(
t
-
ψ
(
t
)
)
>
0. Then f has the approximate endpoint property.We finish this section by the following fixed point result.Corollary 34.
Let(
X
,
d
) be a bicomplete quasi-pseudometric space. Let f
:
X
→
X be a map that satisfies
(41)
d
(
f
x
,
f
y
)
≤
ψ
(
d
(
x
,
y
)
)
,
f
o
r
e
a
c
h
x
,
y
∈
X
,
where ψ
:
[
0
,
∞
)
→
[
0
,
∞
) is upper semicontinuous, ψ
(
t
)
<
t for each t
>
0, and lim
⁡
⁡
inf
⁡
t
→
∞
⁡
(
t
-
ψ
(
t
)
)
>
0. Then f has a fixed point.Proof.
From Theorem32 and Corollary 33, we conclude that f has the approximate mix-point property. Hence, by Corollary 30, we have the desired result.


## 5. More Results

The following theorem is the main result of this section.Theorem 35.
Let(
X
,
d
) be a left K-complete quasi-pseudometric space. Let F
:
X
→
C
B
(
X
) be a set-valued map and f
:
X
→
R as f
(
x
)
=
H
(
{
x
}
,
F
x
). If there exists c
∈
(
0,1
) such that for all x
∈
X there exists y
∈
F
x satisfying
(42)
H
(
{
y
}
,
F
y
)
≤
c
(
d
(
x
,
y
)
)
,
then T has a startpoint.Proof.
For any initialx
0
∈
X, there exists x
1
∈
F
x
0
⊆
X such that
(43)
H
(
{
x
1
}
,
F
x
1
)
≤
c
(
d
(
x
0
,
x
1
)
)
,
and for x
1
∈
X, there is x
2
∈
F
x
1
⊆
X such that
(44)
H
(
{
x
2
}
,
F
x
2
)
≤
c
(
d
(
x
1
,
x
2
)
)
.
Continuing this process, we can get an iterative sequence (
x
n
) where x
n
+
1
∈
F
x
n
⊆
X and
(45)
H
(
{
x
n
+
1
}
,
F
x
n
+
1
)
≤
c
(
d
(
x
n
,
x
n
+
1
)
)
,
n
=
0,1
,
2
,
…
.
Claim
1.
(
x
n
) is a left K-Cauchy sequence.
On one hand,(46)
H
(
{
x
n
+
1
}
,
F
x
n
+
1
)
≤
c
(
d
(
x
n
,
x
n
+
1
)
)
,
n
=
0,1
,
2
,
…
.
On the other hand, x
n
+
1
∈
F
x
n implies
(47)
d
(
x
n
,
x
n
+
1
)
≤
H
(
{
x
n
}
,
F
x
n
)
,
n
=
0,1
,
2
,
…
.
By the two above inequalities, we have
(48)
d
(
x
n
+
1
,
x
n
+
2
)
≤
c
d
(
x
n
,
x
n
+
1
)
n
=
0,1
,
2
,
…
,
H
(
{
x
n
+
1
}
,
F
x
n
+
1
)
≤
c
H
(
{
x
n
}
,
F
x
n
)
n
=
0,1
,
2
,
…
.
We then deduce by iteration that
(49)
d
(
x
n
,
x
n
+
1
)
≤
c
n
d
(
x
0
,
x
1
)
n
=
0,1
,
2
,
…
,
H
(
{
x
n
+
1
}
,
F
x
n
+
1
)
≤
c
n
H
(
{
x
0
}
,
F
x
0
)
n
=
0,1
,
2
,
…
.
Then for n
,
m
∈
N, n
<
m,
(50)
d
(
x
n
,
x
m
)
≤
d
(
x
n
,
x
n
+
1
)
+
d
(
x
n
+
1
,
x
n
+
2
)
+
⋯
+
d
(
x
m
-
1
,
x
m
)
≤
[
c
n
+
c
n
+
1
+
⋯
+
c
m
-
1
]
d
(
x
0
,
x
1
)
≤
c
n
1
-
c
d
(
x
0
,
x
1
)
,
and since c
n
→
0 as n
→
∞ we conclude that (
x
n
) is a left K-Cauchy sequence. According to the left K-completeness of (
X
,
d
), there exists x
*
∈
X such that x
n
→
d
x
*.
Claim
2.
x
* is a startpoint of F.
Observe that the sequence(
f
x
n
)
=
(
H
(
{
x
n
}
,
F
x
n
)
) is decreasing and hence converges to 0. Since f is τ
(
d
)-lower semicontinuous (as supremum of τ
(
d
)-lower semicontinuous functions), we have
(51)
0
≤
f
(
x
*
)
≤
liminf
⁡
n
→
∞
f
(
x
n
)
=
0
.
Hence, f
(
x
*
)
=
0; that is, H
(
{
x
*
}
,
F
x
*
)
=
0.
This completes the proof.Example 36.
LetX
=
{
1
,
1
/
2
,
1
/
3
}. The map d
:
X
×
X
→
[
0
,
∞
) defined by d
(
1
/
n
,
1
/
m
)
=
max
⁡
⁡
{
1
/
n
-
1
/
m
,
0
} is a T
0-quasi-pseudometric on X. Let F
:
X
→
2
X be the set mapping defined by F
a
=
X
∖
{
a
} for any a
∈
X. With c
=
1
/
2, the map F satisfies the assumptions of our theorem, so it has a startpoint, which in the present case is 1
/
3.
More generally, if we setX
n
=
{
1
/
i
,
i
=
1,2
,
…
,
n
} and d as defined above, with c
=
1
/
2, the map F defined by F
a
=
X
∖
{
a
} for any a
∈
X satisfies the assumptions of our theorem, so it has a startpoint, which in this case is 1
/
n.Corollary 37.
Let(
X
,
d
) be a right K-complete quasi-pseudometric space. Let F
:
X
→
C
B
(
X
) be a set-valued map and f
:
X
→
R defined by f
(
x
)
=
H
(
F
x
,
{
x
}
). If there exists c
∈
(
0,1
) such that for all x
∈
X there exists y
∈
F
x satisfying
(52)
H
(
F
y
,
{
y
}
)
≤
c
(
d
(
y
,
x
)
)
,
then T has an endpoint.Corollary 38.
Let(
X
,
d
) be a bicomplete quasi-pseudometric space. Let F
:
X
→
C
B
(
X
) be a set-valued map and f
:
X
→
R defined by f
(
x
)
=
H
s
(
F
x
,
{
x
}
)
=
max
⁡
⁡
{
H
(
F
x
,
{
x
}
)
,
H
(
{
x
}
,
F
x
)
}. If there exists c
∈
(
0,1
) such that for all x
∈
X there exists y
∈
F
x satisfying
(53)
H
s
(
{
y
}
,
F
y
)
≤
c
(
min
⁡
⁡
{
d
(
y
,
x
)
,
d
(
y
,
x
)
}
)
,
then T has a fixed point.Proof.
We give here the main idea of the proof.
Observe that inequality (53) guarantees that the sequence (
x
n
) constructed in the proof of Theorem 35 is a d
s-Cauchy sequence and hence d
s-converges to some x
*. Using the fact that f is τ
(
d
s
)-lower semicontinuous (as supremum of τ
(
d
s
)-continuous functions), we have
(54)
0
≤
f
(
x
*
)
≤
liminf
⁡
n
→
∞
f
(
x
n
)
=
0
.
Hence, f
(
x
*
)
=
0; that is, H
(
{
x
*
}
,
F
x
*
)
=
0
=
H
(
F
x
*
,
{
x
*
}
), and we are done.Remark 39.
All the results given remain true when we replace accordingly the bicomplete quasi-pseudometric space(
X
,
d
) with a left Smyth sequentially complete/left K-complete or a right Smyth sequentially complete/right K-complete space.



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*Source: 709253-2014-07-02.xml*