Startpoints and Contractions in Qua
Hindawi Publishing Corporation
Journal of Mathematics
Volume 2014, Article ID 709253, 8 pages
http://dx.doi.org/10.1155/2014/709253
Research Article
Startpoints and (�, �)-Contractions in
Quasi-Pseudometric Spaces
Yaé Ulrich Gaba
Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7701, South Africa
Correspondence should be addressed to Yaé Ulrich Gaba; gabayae2@gmail.com
Received 7 May 2014; Accepted 18 June 2014; Published 2 July 2014
Academic Editor: Bruce A. Watson
Copyright © 2014 Yaé Ulrich Gaba. his is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We introduce the concept of startpoint and endpoint for multivalued maps deined on a quasi-pseudometric space. We investigate
the relation between these new concepts and the existence of ixed points for these set valued maps.
Dedicated to my beloved Clémence on the occasion of her 25th birthday
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 of Cauchy sequences
can be accordingly modiied. In the same reference, which
uses an approach based on uniformities, the bicompletion
of a �0 -quasi-pseudometric has been explored. It is worth
mentioning that, in the ixed point theory, completeness is a
key element, since most of the constructed sequences will be
assumed to have a Cauchy type property.
It is the aim of this paper to continue the study of quasipseudometric spaces by proving some ixed point results
and investigating a bit more the behaviour of set-valued
mappings. hus, in Section 3 a suitable notion of (�, �)contractive mapping is given for self-mappings deined on
quasi-pseudometric spaces and some ixed point results are
discussed. In Sections 4 and 5, the notions of startpoint and
endpoint for set-valued mappings are introduced and diferent variants of such concepts, as well as their connections with
the ixed 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
Deinition 1. Let � be a nonempty set. A function � : � ×
� → [0, ∞) is called a quasi-pseudometric on � if
(i) �(�, �) = 0 for all � ∈ �,
(ii) �(�, �) ≤ �(�, �) + �(�, �) for all �, �, � ∈ �.
Moreover, if �(�, �) = 0 = �(�, �) ⇒ � = �, then � is said to
be a �0 -quasi-pseudometric. he latter condition is referred to
as the �0 -condition.
Remark 2. (i) Let � be a quasi-pseudometric on �; then the
map �−1 deined by �−1 (�, �) = �(�, �) whenever �, � ∈ � is
also a quasi-pseudometric on �, called the conjugate of �. In
the literature, �−1 is also denoted by �� or �.
(ii) It is easy to verify that the function �� deined by �� :=
� ∨ �−1 , that is, �� (�, �) = max{�(�, �), �(�, �)}, deines a
metric on � whenever � is a �0 -quasi-pseudometric on �.
2
Journal of Mathematics
Let (�, �) be a quasi-pseudometric space. For � ∈ � and
� > 0,
�� (�, �) = {� ∈ � : � (�, �) < �}
(1)
�� (�, �) = {� ∈ � : � (�, �) ≤ �}
(2)
denotes the open �-ball at �. he collection of all such balls
yields a base for the topology �(�) induced by � on �. Hence,
for any � ∈ �, we will, respectively, denote by int�(�) � and
cl�(�) � the interior and the closure of the set � with respect
to the topology �(�).
Similarly, for � ∈ � and � ≥ 0,
denotes the closed �-ball at �. We will say that a subset � ⊂ �
is join-closed if it is �(�� )-closed, that is, closed with respect to
the topology generated by �� . he topology �(�� ) is iner than
the topologies �(�) and �(�−1 ).
Deinition 3. Let (�, �) be a quasi-pseudometric space. he
convergence of a sequence (�� ) to � with respect to �(�),
called �-convergence or let-convergence and denoted by �� →
�
�, is deined in the following way:
�
�� →
� � ⇐⇒ � (�, �� ) �→ 0.
�
(3)
Similarly, the convergence of a sequence (�� ) to � with
respect to �(�−1 ), called �−1 -convergence or right-convergence
and denoted by �� ��→ �, is deined in the following way:
�−1
�� ��→ � ⇐⇒ � (�� , �) �→ 0.
�−1
(4)
Finally, in a quasi-pseudometric space (�, �), we will say
that a sequence (�� ) �� -converges to � if it is both let and right
convergent to �, and we denote it by �� �→ � or �� → �
when there is no confusion. Hence,
��
� �,
�� �→ � ⇐⇒ �� →
��
�
�� ��→ �.
�−1
(5)
Deinition 4. A sequence (�� ) in a quasi-pseudometric (�, �)
is called
(a) let �-Cauchy if for every � > 0, there exist � ∈ � and
�0 ∈ N such that
∀� ≥ �0
� (�, �� ) < �;
(6)
(b) let �-Cauchy if for every � > 0, there exists �0 ∈ N
such that
∀�, � : �0 ≤ � ≤ � � (�� , �� ) < �;
(7)
(c) �� -Cauchy if for every � > 0, there exists �0 ∈ N such
that
∀�, � ≥ �0
� (�� , �� ) < �.
(8)
Dually, we deine, in the same way, right �-Cauchy and
right �-Cauchy sequences.
Remark 5. Consider the following:
(i) �� -Cauchy ⇒ let �-Cauchy ⇒ let �-Cauchy. he
same implications hold for the corresponding right notions.
None of the above implications is reversible.
(ii) A sequence is let �-Cauchy with respect to � if and
only if it is right �-Cauchy with respect to �−1 .
(iii) A sequence is let �-Cauchy with respect to � if and
only if it is right �-Cauchy with respect to �−1 .
(iv) A sequence is �� -Cauchy if and only if it is both let
and right �-Cauchy.
Deinition 6. A quasi-pseudometric space (�, �) is called
(i) let-�-complete provided that any let �-Cauchy
sequence is �-convergent,
(ii) let Smyth sequentially complete if any let �-Cauchy
sequence is �� -convergent.
he dual notions of right-completeness are easily derived from
the above deinition.
Deinition 7. A �0 -quasi-pseudometric space (�, �) is called
bicomplete provided that the metric �� on � is complete.
As usual, a subset � of a quasi-pseudometric space (�, �)
will be called bounded provided that there exists a positive
real constant � such that �(�, �) < � whenever �, � ∈ �.
his is equivalent to saying that there exist �0 ∈ �, �, � ≥ 0
such that � ⊆ �� (�0 , �) ∩ ��−1 (�0 , �).
We also deine the diameter �(�) of � by �(�) :=
sup{�(�, �) : �, � ∈ �}. Hence, � is bounded if and only
if �(�) < ∞. It is not diicult to see that this deinition
coincides with that of a bounded set in a metric space.
Let (�, �) be a quasi-pseudometric space. We set
P0 (�) := 2� \ {0} where 2� denotes the power set of �. For
� ∈ � and �, � ∈ P0 (�), we deine
� (�, �) = inf {� (�, �) , � ∈ �} ,
� (�, �) = inf {� (�, �) , � ∈ �} ,
(9)
and we deine �(�, �) by
� (�, �) = max {sup � (�, �) , sup � (�, �)} .
�∈�
�∈�
(10)
hen � is an extended quasi-pseudometric on P0 (�).
Moreover, we know from [9] that the restriction of � to
�cl (�) = {� ⊆ � : � = (cl�(�) �) ∩ (cl�(�−1 ) �)} is an
extended �0 -quasi-pseudometric. We will denote by ��(�)
the collection of all nonempty bounded and �(�� )-closed
subsets of �.
We complete this section by the following lemma.
Lemma 8. Let (�, �) be a quasi-pseudometric space. For
every ixed � ∈ �, the mapping � �→ �(�, �) is �(�)upper semicontinuous (�(�)-usc in short) and �(�−1 )-lower
semicontinuous (�(�−1 )-lsc in short). For every ixed � ∈ �,
the mapping � →
� �(�, �) is �(�)-lsc and �(�−1 )-usc.
Journal of Mathematics
3
Proof. To prove that �(�, ⋅) is �(�)-usc and �(�−1 )-lsc, we have
to show that the set {� ∈ � : �(�, �) < �} is �(�)-open and
{� ∈ � : �(�, �) > �} is �(�−1 )-open, for every � ∈ R,
properties that are easy to check.
Indeed, for � ∈ � such that �(�, �) < �, let � := � −
�(�, �) > 0. If � ∈ � is such that �(�, �) < �, then
� (�, �) ≤ � (�, �) + (�, �) < � (�, �) + � = �,
(11)
showing that �� (�, �) ⊂ {� ∈ � : �(�, �) < �}.
Similarly, for � ∈ � with �(�, �) > � take � := �(�, �) −
� > 0. If � ∈ � satisies �(�, �) = �−1 (�, �) < �, then
� (�, �) ≤ � (�, �) + (�, �) < � (�, �) + �,
(12)
so that �(�, �) > �(�, �) − � = �. Consequently, ��−1 (�, �) ⊂
{� ∈ � : �(�, �) > �}.
heorem 13. Let (�, �) be a Hausdorf let �-complete �0 quasi-pseudometric space. Suppose that � : � → � is
an (�, �)-contractive mapping which satisies the following
conditions:
(i) � is �-admissible;
(ii) there exists �0 ∈ � such that �(�0 , ��0 ) ≥ 1;
(iii) � is �-sequentially continuous.
hen � has a ixed point.
Proof. By (ii), there exists �0 ∈ � such that �(�0 , ��0 ) ≥ 1.
Let us deine the sequence (�� ) in � by ��+1 = ��� for all � =
0, 1, 2, . . .. Without loss of generality, we can always assume
that �� ≠ ��+1 for all � ∈ N, since if ��0 = ��0 +1 for some
�0 ∈ N, the proof is complete.
From assumption (i), we derive
3. Some First Results
� (�0 , ��0 ) = � (�0 , �1 ) ≥ 1
We begin by recalling the following.
�⇒ � (��0 , ��1 ) = � (�1 , �2 ) ≥ 1.
Deinition 9. A function � : [0, ∞) → [0, ∞) is called a (�)comparison function if
(�1 ) � is nondecreasing;
Recursively, we get
�
�
(�2 ) ∑∞
�=1 � (�) < ∞ for all � > 0, where � is the �th iterate
of �.
We will denote by Γ the set of such functions. Note that for
any � ∈ Γ, �(�) < � for any � > 0.
� (�� , ��+1 ) ≥ 1 ∀� = 0, 1, 2, . . . .
Deinition 10. Let (�, �) be a quasi-pseudometric type space.
A function � : � → � is called �-sequentially continuous or
let-sequentially continuous if for any �-convergent sequence
� �, the sequence (��� ) �-converges to ��; that
(�� ) with �� →
�
is, ��� →
� ��.
Similarly, we deine a �−1 -sequentially continuous or rightsequentially continuous function.
�
Deinition 11. Let (�, �) be a quasi-pseudometric space, and
let � : � → � and � : � × � → [0, ∞) be mappings. We
say that � is �-admissible if
� (�� , ��+1 ) = � (���−1 , ��� )
≤ � (��−1 , �� ) � (���−1 , ��� )
for all � = 0, 1, 2, . . .. Inductively, we obtain
� (�� , ��+1 ) ≤ �� (� (�0 , �1 )) ,
whenever �, � ∈ �.
We now state the irst ixed point theorem.
(14)
� = 1, 2, . . . .
(18)
herefore, for any � ≥ 1, using the triangle inequality, we get
� (�� , ��+� ) ≤ � (�� , ��+1 ) + � (��+1 , ��+2 )
+ ⋅ ⋅ ⋅ + � (��+�−1 , ��+� )
�+�−1
≤ ∑ �� (� (�0 , �1 ))
whenever �, � ∈ �.
� (�, �) � (��, ��) ≤ � (� (�, �)) ,
(17)
≤ � (� (��−1 , �� )) ,
(13)
Deinition 12. Let (�, �) be a quasi-pseudometric space and
let � : � → � be a mapping. We say that � is an (�, �)contractive mapping if there exist two functions � : � × � →
[0, ∞) and � ∈ Γ such that
(16)
Since � is (�, �)-contractive, we can write
We then introduce the following deinitions.
� (�, �) ≥ 1 �⇒ � (��, ��) ≥ 1,
(15)
(19)
�=�
∞
≤ ∑�� (� (�0 , �1 )) .
�=�
Letting � → ∞, we derive that �(�� , ��+� ) → 0. Hence, (�� )
is a let �-Cauchy sequence. Since (�, �) is let �-complete
and � �-sequentially continuous, there exists �∗ such that
�� →
� �∗ and ��+1 →
� ��∗ . Since � is Hausdorf, we have
∗
∗
that � = �� .
�
�
4
Journal of Mathematics
Corollary 14. Let (�, �) be a Hausdorf right �-complete
�0 -quasi-pseudometric space. Suppose that � : � → �
is an (�, �)-contractive mapping which satisies the following
conditions:
(i) � is �-admissible;
(ii) there exists �0 ∈ � such that �(��0 , �0 ) ≥ 1;
(iii) � is �−1 -sequentially continuous.
hen � has a ixed point.
Corollary 15. Let (�, �) be a bicomplete quasi-pseudometric
space. Suppose that � : � → � is an (�, �)-contractive
mapping which satisies the following conditions:
(i) � is �-admissible and the function � is symmetric; that
is, �(�, �) = �(�, �) for any �, � ∈ �;
(ii) there exists �0 ∈ � such that �(��0 , �0 ) ≥ 1;
(iii) � is �� -sequentially continuous.
hen � has a ixed point.
Proof. Following the proof of heorem 13, it is clear that the
sequence (�� ) in � deined by ��+1 = ��� for all � = 0, 1, 2, . . .
is �� -Cauchy. Since (�, �� ) is complete and � sequentially
continuous, there exists �∗ such that �� �→ �∗ and ��+1 �→
��∗ . Since (�, �� ) is Hausdorf, we have that �∗ = ��∗ .
��
��
Remark 16. In fact, we do not need � to be symmetric. It is
enough, for the result to be true, to have a point �0 ∈ � for
which �(�0 , ��0 ) ≥ 1 and �(��0 , �0 ) ≥ 1.
We conclude this section by the following results which
are in fact consequences of heorem 13.
heorem 17. Let (�, �) be a Hausdorf let �-complete �0 quasi-pseudometric space. Suppose that � : � → � is
an (�, �)-contractive mapping which satisies the following
conditions:
(i) � is �-admissible;
(ii) there exists �0 ∈ � such that �(�0 , ��0 ) ≥ 1;
(iii) if (�� ) is a sequence in � such that �(�� , ��+1 ) ≥ 1
for all � = 1, 2, . . . and �� →
� �, then there exists a
subsequence (��(�) ) of (�� ) such that �(��(�) , �) ≥ 1 for
all �.
�
hen � has a ixed point.
Proof. Following the proof of heorem 13, we know that the
sequence (�� ) deined by ��+1 = ��� for all � = 0, 1, 2, . . . �converges to some �∗ and satisies �(�� , ��+1 ) ≥ 1 for � ≥ 1.
From condition (iii), we know that there exists a subsequence
(��(�) ) of (�� ) such that �(��(�) , �∗ ) ≥ 1 for all �. Since � is an
(�, �)-contractive mapping, we get
� (��(�)+1 , ��∗ ) = � (���(�) , ��∗ )
≤ � (��(�) , �∗ ) � (���(�) , ��∗ )
≤ � (� (��(�) , �∗ )) .
(20)
Letting � → ∞, we obtain �(��(�)+1 , ��∗ ) → 0. Since � is
Hausdorf, we have that ��∗ = �∗ .
his completes the proof.
Corollary 18. Let (�, �) be a Hausdorf right �-complete
�0 -quasi-pseudometric space. Suppose that � : � → �
is an (�, �)-contractive mapping which satisies the following
conditions:
(i) � is �-admissible;
(ii) there exists �0 ∈ � such that �(��0 , �0 ) ≥ 1;
(iii) if (�� ) is a sequence in � such that �(��+1 , �� ) ≥ 1
for all � = 1, 2, . . . and �� ��→ �, then there exists a
subsequence (��(�) ) of (�� ) such that �(�, ��(�) ) ≥ 1 for
all �.
�−1
hen � has a ixed point.
Corollary 19. Let (�, �) be a bicomplete quasi-pseudometric
space. Suppose that � : � → � is an (�, �)-contractive
mapping which satisies the following conditions:
(i) � is �-admissible and the function � is symmetric;
(ii) there exists �0 ∈ � such that �(��0 , �0 ) ≥ 1;
(iii) if (�� ) is a sequence in � such that �(�� , �� ) ≥ 1 for all
�, � ∈ N and �� �→ �, then there exists a subsequence
(��(�) ) of (�� ) such that �(�, ��(�) ) ≥ 1 for all �.
��
hen � has a ixed 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 speciied setting. Here we introduce a similar
notion for set-valued maps deined on quasi-pseudometric
spaces. Let (�, �) be a �0 -quasi-pseudometric space.
Deinition 20. Let � : � → 2� be a set-valued map. An
element � ∈ � is said to be
(i) a ixed point of � if � ∈ ��,
(ii) a startpoint of � if �({�}, ��) = 0,
(iii) an endpoint of � if �(��, {�}) = 0,
(iv) an �-startpoint of � for some � ∈ (0, 1) if �({�}, ��) <
�,
(v) an �-endpoint of � for some � ∈ (0, 1) if �(��, {�}) <
�.
Remark 21. It is therefore obvious that if � is both a startpoint
of � and an endpoint of �, then � is a ixed point of �. In fact,
�� is a singleton. But a ixed point need not be a startpoint
nor an endpoint.
Indeed, consider the �0 -quasi-pseudometric space (�, �),
where � = {0, 1} and � is deined by �(0, 1) = 0, �(1, 0) = 1,
and �(�, �) for � = 0, 1. We deine on � the set-valued map
� : � → 2� by �� = �. Obviously, 1 is a ixed point, but
�({1}, �1) = �({1}, �) = max{�(1, 1), �(1, 0)} = 1 ≠ 0.
Journal of Mathematics
5
Lemma 22. Let (�, �) be a �0 -quasi-pseudometric space and
let � : � → 2� be a set-valued map. An element � ∈ � is a
startpoint of � if and only if it is an �-startpoint of � for every
� ∈ (0, 1).
Lemma 23. Let (�, �) be a �0 -quasi-pseudometric space and
let � : � → 2� be a set-valued map. An element � ∈ � is
an endpoint of � if and only if it is an �-endpoint of � for every
� ∈ (0, 1).
Deinition 24. Let (�, �) be a �0 -quasi-pseudometric space.
We say that a set-valued map � : � → 2� has the
approximate startpoint property (resp., approximate endpoint
property) if
inf sup � (�, �) = 0
�∈��∈��
(resp., inf sup � (�, �) = 0) .
�∈��∈��
(21)
Deinition 25. Let (�, �) be a �0 -quasi-pseudometric space.
We say that a set-valued map � : � → 2� has the
approximate mix-point property if
inf sup �� (�, �) = 0.
�∈��∈��
(22)
Here, it is also very clear that � has approximate mix-point
property if and only if � has both the approximate startpoint
and the approximate endpoint properties.
We are therefore naturally led to this deinition.
Deinition 26. Let � : � → � be a single-valued map
on a �0 -quasi-pseudometric space (�, �). hen � has the
approximate startpoint property (resp., approximate endpoint
property) if and only if
inf � (�, ��) = 0
�∈�
(resp., inf � (��, �) = 0) .
�∈�
(23)
We motivate our coming results by the following examples. We basically show that the concepts deined above
are independent and do not necessarily coincide. he list
of examples presented is not exhaustive and more can be
constructed, showing the connection between the notions
deined above.
Example 27. Let � = {0, 1, 2}. he map � : � × � → [0, ∞)
deined by �(0, 1) = �(0, 2) = 0, �(1, 0) = �(1, 2) =
1, �(2, 0) = �(2, 1) = 2, and �(�, �) = 0 for all � ∈
� is a �0 -quasi-pseudometric on �. Let � : � → 2�
be the set mapping deined by �� = � \ {�} for any
� ∈ �. By deinition, � does not have any ixed point.
Nevertheless, a simple computation shows that �({0}, �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 �∈� sup�∈�� �(�, �) = 0, showing that � has the
approximate startpoint property, but inf �∈� sup�∈�� �(�, �) =
1, showing that � does not have the approximate endpoint
property.
Example 28. Let � = {1/�, � = 1, 2, . . .}. he map � : � ×
� → [0, ∞) deined by �(1/�, 1/�) = max{1/� − 1/�, 0}
is a �0 -quasi-pseudometric on �. Let � : � → 2� be the
set-valued mapping deined by �� = � \ {�} for any � ∈ �.
By deinition, � does not have any ixed point.
For a ixed �0 ∈ N,
�(
{0
1 1
1
, )={1
−
�0 �
{ �0 �
if � ≤ �0 ,
(24)
if � ≥ �0 ,
(25)
if � > �0 .
Similarly for a ixed �0 ∈ N,
�(
{0
1 1
1
, )={1
−
� �0
�
�
{
0
if � < �0 .
Hence, � does not have any startpoint nor endpoint (which
also implies that � does not have any ixed point).
But for a given � ∈ (0, 1), there exists �0 ∈ N such
that 1/�0 < �. We also know from deinition that �({1/�0 },
�(1/�0 )) = 1/�0 , so, 1/�0 is an �-startpoint of �. More
generally, for a given � ∈ (0, 1), there exists �� ∈ N such that
1/�� is an �-startpoint of �. Moreover, for any � ≥ �� , 1/� is
an �-startpoint of �.
Similarly, we can show that � admits an �-endpoint.
We can now state our irst result.
heorem 29. Let (�, �) be a bicomplete quasi-pseudometric
space. Let � : � → ��(�) be a set-valued map that satisies
� (��, ��) ≤ � (� (�, �)) ,
��� ���ℎ �, � ∈ �,
(26)
where � : [0, ∞) → [0, ∞) is upper semicontinuous, �(�) < �
for each � > 0, and lim inf � → ∞ (�−�(�)) > 0. hen there exists
a unique �0 ∈ � which is both a startpoint and an endpoint of
� if and only if � has the approximate mix-point property.
Proof. It is clear that if � admits a point which is both a
startpoint and an endpoint, then � has the approximate startpoint property and the approximate endpoint property. Just
observe that �({�}, ��) = sup�∈�� �(�, �) and �(��, {�}) =
sup�∈�� �(�, �). Conversely, suppose � has the approximate
mix-point property. hen
�� = {� ∈ � : sup �� (�, �) ≤
�∈��
1
} ≠ 0,
�
(27)
for each � ∈ N. Also it is clear that for each � ∈ N, ��+1 ⊆ �� .
he map � �→ sup�∈�� �� (�, �) is �(�� )-lower semicontinuous
(as supremum of �(�� )-continuous mappings); we have that
�� is �(�� )-closed.
Next we prove that, for each � ∈ N, �� is bounded.
6
Journal of Mathematics
Assume by the way of contradiction that �(�� ) = ∞ for
each � ∈ N. hen there exist �� , �� ∈ �� such that �(�� , �� ) ≥
�. From (26), we obtain that
� (�, �) = � ({�} , {�})
≤ � ({�} , ��) + � (��, ��) + � (��, {�})
=
2
+ � (� (�, �)) ,
�
(28)
whenever �, � ∈ �� . Hence,
2
,
�
(29)
(30)
lim � (�� , �� ) = ∞.
�→∞
his contradicts our assumption. Now we show that
lim� → ∞ �(�� ) = 0. On the contrary, assume lim� → ∞ �(�� ) =
�0 > 0 (note that the sequence (�(�� )) is nonincreasing and
bounded below and then has a limit). Let
�→∞
��,� = � (��,� , ��,� ) �→ � (�� ) as � �→ ∞} .
(31)
Now we show that � > 0 (notice � ≥ 0). Arguing by
contradiction, we assume � = 0; then by the deinition of �,
there exists a sequence �� such that �� → �0 and lim� → ∞ (�� −
�(�� )) = 0. hen lim� → ∞ �(�� ) = �0 . But since � is upper
semicontinuous and �0 > 0, then
�0 = lim � (�� ) ≤ � (�0 ) < �0 .
(32)
�→∞
his contradiction shows that � > 0. For each � ∈ �, let
(�� , �� ) ∈ �� be a sequence such that �(�� , �� ) → �(�� ), as
� → ∞. hen from (29) we get
� ≤ lim inf (� (�� , �� ) − � (� (�� , �� ))) ≤
�→∞
(34)
heorem 31. Let (�, �) be a bicomplete quasi-pseudometric
space. Let � : � → ��(�) be a set-valued map that satisies
2
,
�
for each � ∈ �.
��� ���ℎ �, � ∈ �,
(35)
where 0 ≤ � < 1. hen there exists a unique �0 ∈ � which is
both a startpoint and an endpoint of � if and only if � has the
approximate mix-point property.
Proof. Take �(�) = �� in heorem 29.
We then deduce the following result for single-valued
maps.
� = inf inf {lim inf (��,� − � (��,� )) : (��,� , ��,� ) ∈ �� ,
�∈N
��� ���ℎ �, � ∈ �,
where � : [0, ∞) → [0, ∞) is upper semicontinuous, �(�) < �
for each � > 0, and lim inf � → ∞ (� − �(�)) > 0. If � has the
approximate mix-point property then � has a ixed point.
� (��, ��) ≤ �� (�, �) ,
lim (� (�� , �� ) − � (� (�� , �� ))) = 0,
�→∞
� (��, ��) ≤ � (� (�, �)) ,
Proof. From heorem 29, we conclude that there exists
�0 which is both a startpoint and an endpoint; that is,
�({�0 }, ��0 ) = 0 = �(��0 , {�0 }). he �0 -condition therefore
guarantees the desired result.
whenever �, � ∈ �� .
herefore,
� (�, �) − � (� (�, �)) ≤
Corollary 30. Let (�, �) be a bicomplete quasi-pseudometric
space. Let � : � → ��(�) be a set-valued map that satisies
heorem 32. Let (�, �) be a bicomplete quasi-pseudometric
space. Let � : � → � be a map that satisies
� (��, ��) ≤ � (� (�, �)) ,
��� ���ℎ �, � ∈ �,
where � : [0, ∞) → [0, ∞) is upper semicontinuous, �(�) < �
for each � > 0, and lim inf � → ∞ (� − �(�)) > 0. hen � has the
approximate startpoint property.
Proof. By the way of
inf �∈� �(�, ��) > 0. hen
contradiction,
suppose
�∈�(�)
= inf � (��, �2 �)
(37)
≤ inf � (� (�, ��)) .
(33)
�) = �(��0 , {�0 }). For uniqueness, if �0 is an arbitrary
startpoint of �, then �({�0 }, ��0 ) = 0 = �(��0 , {�0 }), and
so �0 ∈ ⋂�∈� �� = {�0 }.
his completes the proof.
that
inf � (�, ��) ≤ inf � (�, ��)
�∈�
�∈�
Hence, � = 0. his contradiction shows that lim� → ∞ �(�� ) =
0. It follows from the Cantor intersection theorem that
⋂�∈� �� = {�0 }.
hus, �({�0 }, ��0 ) = sup�∈��0 �(�, �) = 0 = sup�∈��0 �(�,
(36)
�∈�
Since �(�(�, �)) ≤ �(�, �), then inf �∈� �(�(�, ��))
inf �∈� �(�, ��).
Now, on the contrary, suppose again that
inf � (� (�, ��)) = inf � (�, ��) .
�∈�
�∈�
≤
(38)
Let (�� ) ⊂ � be a sequence such that lim� → ∞ �(�� , �(�� )) =
inf �∈� �(�, ��). By passing to subsequences if necessary, we
Journal of Mathematics
7
may assume that lim� → ∞ �(�(�� , �(�� ))) exists. hen from
(37) we have
� ({��+1 } , ���+1 ) ≤ � (� (�� , ��+1 )) ,
inf � (�, ��) ≤ inf � (� (�, ��))
�∈�
�∈�
≤ lim � (� (�� , � (�� )))
�→∞
≤ � (inf � (�, ��))
(39)
�∈�
�∈�
<
Corollary 33. Let (�, �) be a bicomplete quasi-pseudometric
space. Let � : � → � be a map that satisies
� (��, ��) ≤ � (� (�, �)) ,
��� ���ℎ �, � ∈ �,
(40)
where � : [0, ∞) → [0, ∞) is upper semicontinuous, �(�) < �
for each � > 0, and lim inf � → ∞ (� − �(�)) > 0. hen � has the
approximate endpoint property.
We inish this section by the following ixed point result.
Corollary 34. Let (�, �) be a bicomplete quasi-pseudometric
space. Let � : � → � be a map that satisies
� (��, ��) ≤ � (� (�, �)) ,
for each �, � ∈ �,
(41)
where � : [0, ∞) → [0, ∞) is upper semicontinuous, �(�) < �
for each � > 0, and lim inf � → ∞ (� − �(�)) > 0. hen � has a
ixed point.
Proof. From heorem 32 and Corollary 33, we conclude
that � has the approximate mix-point property. Hence, by
Corollary 30, we have the desired result.
5. More Results
� = 0, 1, 2, . . . .
(46)
On the other hand, ��+1 ∈ ��� implies
� (�� , ��+1 ) ≤ � ({�� } , ��� ) ,
� = 0, 1, 2, . . . .
(47)
By the two above inequalities, we have
� (��+1 , ��+2 ) ≤ �� (�� , ��+1 )
� = 0, 1, 2, . . . ,
� ({��+1 } , ���+1 ) ≤ �� ({�� } , ��� )
� = 0, 1, 2, . . . .
(48)
We then deduce by iteration that
� (�� , ��+1 ) ≤ �� � (�0 , �1 )
� = 0, 1, 2, . . . ,
� ({��+1 } , ���+1 ) ≤ �� � ({�0 } , ��0 )
� = 0, 1, 2, . . . .
(49)
hen for �, � ∈ N, � < �,
� (�� , �� ) ≤ � (�� , ��+1 ) + � (��+1 , ��+2 )
+ ⋅ ⋅ ⋅ + � (��−1 , �� )
≤ [�� + ��+1 + ⋅ ⋅ ⋅ + ��−1 ] � (�0 , �1 )
≤
(50)
��
� (�0 , �1 ) ,
1−�
and since �� → 0 as � → ∞ we conclude that (�� ) is a let
�-Cauchy sequence. According to the let �-completeness of
(�, �), there exists �∗ ∈ � such that �� →
� �∗ .
�
he following theorem is the main result of this section.
heorem 35. Let (�, �) be a let �-complete quasipseudometric space. Let � : � → ��(�) be a set-valued
map and � : � → R as �(�) = �({�}, ��). If there exists
� ∈ (0, 1) such that for all � ∈ � there exists � ∈ �� satisfying
� ({�} , ��) ≤ � (� (�, �)) ,
� = 0, 1, 2, . . . .
(45)
Claim 1. (�� ) is a let �-Cauchy sequence.
On one hand,
� ({��+1 } , ���+1 ) ≤ � (� (�� , ��+1 )) ,
< inf � (�, ��) .
We get a contradiction, so inf �∈� �(�(�, ��))
inf �∈� �(�, ��) which again contradicts (37).
his completes the proof.
Continuing this process, we can get an iterative sequence (�� )
where ��+1 ∈ ��� ⊆ � and
(42)
then � has a startpoint.
Proof. For any initial �0 ∈ �, there exists �1 ∈ ��0 ⊆ � such
that
� ({�1 } , ��1 ) ≤ � (� (�0 , �1 )) ,
(43)
� ({�2 } , ��2 ) ≤ � (� (�1 , �2 )) .
(44)
and for �1 ∈ �, there is �2 ∈ ��1 ⊆ � such that
Claim 2. �∗ is a startpoint of �.
Observe that the sequence (��� ) = (�({�� }, ��� )) is
decreasing and hence converges to 0. Since � is �(�)-lower
semicontinuous (as supremum of �(�)-lower semicontinuous
functions), we have
0 ≤ � (�∗ ) ≤ lim inf � (�� ) = 0.
�→∞
(51)
Hence, �(�∗ ) = 0; that is, �({�∗ }, ��∗ ) = 0.
his completes the proof.
Example 36. Let � = {1, 1/2, 1/3}. he map � : � × � →
[0, ∞) deined by �(1/�, 1/�) = max{1/� − 1/�, 0} is a �0 quasi-pseudometric on �. Let � : � → 2� be the set
mapping deined by �� = � \ {�} for any � ∈ �. With � = 1/2,
the map � satisies the assumptions of our theorem, so it has
a startpoint, which in the present case is 1/3.
8
Journal of Mathematics
More generally, if we set �� = {1/�, � = 1, 2, . . . , �} and
� as deined above, with � = 1/2, the map � deined by �� =
�\{�} for any � ∈ � satisies the assumptions of our theorem,
so it has a startpoint, which in this case is 1/�.
Corollary 37. Let (�, �) be a right �-complete quasipseudometric space. Let � : � → ��(�) be a set-valued map
and � : � → R deined by �(�) = �(��, {�}). If there exists
� ∈ (0, 1) such that for all � ∈ � there exists � ∈ �� satisfying
� (��, {�}) ≤ � (� (�, �)) ,
(52)
then � has an endpoint.
Corollary 38. Let (�, �) be a bicomplete quasi-pseudometric
space. Let � : � → ��(�) be a set-valued map and
� : � → R deined by �(�) = �� (��, {�}) =
max{�(��, {�}), �({�}, ��)}. If there exists � ∈ (0, 1) such that
for all � ∈ � there exists � ∈ �� satisfying
�� ({�} , ��) ≤ � (min {� (�, �) , � (�, �)}) ,
(53)
then � has a ixed point.
Proof. We give here the main idea of the proof.
Observe that inequality (53) guarantees that the sequence
(�� ) constructed in the proof of heorem 35 is a �� -Cauchy
sequence and hence �� -converges to some �∗ . Using the fact
that � is �(�� )-lower semicontinuous (as supremum of �(�� )continuous functions), we have
0 ≤ � (�∗ ) ≤ lim inf � (�� ) = 0.
�→∞
(54)
Hence, �(�∗ ) = 0; that is, �({�∗ }, ��∗ ) = 0 = �(��∗ , {�∗ }),
and we are done.
Remark 39. All the results given remain true when we replace
accordingly the bicomplete quasi-pseudometric space (�, �)
with a let Smyth sequentially complete/let �-complete or a
right Smyth sequentially complete/right �-complete space.
Conflict of Interests
he author declares that there is no conlict of interests
regarding the publication of this paper.
References
[1] K. Włodarczyk and R. Plebaniak, “Asymmetric structures,
discontinuous contractions and iterative approximation of ixed
and periodic points,” Fixed Point heory and Applications, vol.
2013, article 128, 2013.
[2] H.-P. A. Künzi, “An introduction to quasi-uniform spaces,”
Contemporary Mathematics, vol. 486, pp. 239–304, 2009.
[3] K. Wlodarczyk and R. Plebaniak, “Generalized uniform spaces,
uniformly locally contractive set-valued dynamic systems and
ixed points,” Fixed Point heory and Applications, vol. 2012,
article 104, 2012.
[4] K. W. lodarczyk and R. Plebaniak, “Fixed points and endpoints
of contractive set-valued maps in cone uniform spaces with generalized pseudo distances,” Fixed Point heory and Applications,
vol. 2012, article 176, 15 pages, 2012.
[5] K. W. Włodarczyk and R. Plebaniak, “Leader type contractions,
periodic and ixed points and new completivity in quasi-gauge
spaces with generalized quasi-pseudodistances,” Topology and
its Applications, vol. 159, no. 16, pp. 3504–3512, 2012.
[6] K. Włodarczyk and R. Plebaniak, “Contractivity of Leader type
and ixed points in uniform spaces with generalized pseudodistances,” Journal of Mathematical Analysis and Applications, vol.
387, no. 2, pp. 533–541, 2012.
[7] K. Włodarczyk and R. Plebaniak, “Contractions of Banach,
Tarafdar, Meir-Keeler, Ćirić-Jachymski-Matkowski and Suzuki
types and ixed points in uniform spaces with generalized pseudodistances,” Journal of Mathematical Analysis and Applications,
vol. 404, no. 2, pp. 338–350, 2013.
[8] K. W lodarczyk and R. Plebaniak, “New completeness and
periodic points of discontinuous contractions of Banach type
in quasi-gauge spaces without Hausdorf property,” Fixed Point
heory and Applications, vol. 2013, article 289, 27 pages, 2013.
[9] H.-P. Künzi and C. Ryser, “he Bourbaki quasi-uniformity,”
Topology Proceedings, vol. 20, pp. 161–183, 1995.
[10] C. A. Agyingi, P. Haihambo, and H.-P. A. Künzi, “Endpoints in
�0 -quasimetric spaces: part II,” Abstract and Applied Analysis,
vol. 2013, Article ID 539573, 10 pages, 2013.
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Volume 2014, Article ID 709253, 8 pages
http://dx.doi.org/10.1155/2014/709253
Research Article
Startpoints and (�, �)-Contractions in
Quasi-Pseudometric Spaces
Yaé Ulrich Gaba
Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7701, South Africa
Correspondence should be addressed to Yaé Ulrich Gaba; gabayae2@gmail.com
Received 7 May 2014; Accepted 18 June 2014; Published 2 July 2014
Academic Editor: Bruce A. Watson
Copyright © 2014 Yaé Ulrich Gaba. his is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We introduce the concept of startpoint and endpoint for multivalued maps deined on a quasi-pseudometric space. We investigate
the relation between these new concepts and the existence of ixed points for these set valued maps.
Dedicated to my beloved Clémence on the occasion of her 25th birthday
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 of Cauchy sequences
can be accordingly modiied. In the same reference, which
uses an approach based on uniformities, the bicompletion
of a �0 -quasi-pseudometric has been explored. It is worth
mentioning that, in the ixed point theory, completeness is a
key element, since most of the constructed sequences will be
assumed to have a Cauchy type property.
It is the aim of this paper to continue the study of quasipseudometric spaces by proving some ixed point results
and investigating a bit more the behaviour of set-valued
mappings. hus, in Section 3 a suitable notion of (�, �)contractive mapping is given for self-mappings deined on
quasi-pseudometric spaces and some ixed point results are
discussed. In Sections 4 and 5, the notions of startpoint and
endpoint for set-valued mappings are introduced and diferent variants of such concepts, as well as their connections with
the ixed 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
Deinition 1. Let � be a nonempty set. A function � : � ×
� → [0, ∞) is called a quasi-pseudometric on � if
(i) �(�, �) = 0 for all � ∈ �,
(ii) �(�, �) ≤ �(�, �) + �(�, �) for all �, �, � ∈ �.
Moreover, if �(�, �) = 0 = �(�, �) ⇒ � = �, then � is said to
be a �0 -quasi-pseudometric. he latter condition is referred to
as the �0 -condition.
Remark 2. (i) Let � be a quasi-pseudometric on �; then the
map �−1 deined by �−1 (�, �) = �(�, �) whenever �, � ∈ � is
also a quasi-pseudometric on �, called the conjugate of �. In
the literature, �−1 is also denoted by �� or �.
(ii) It is easy to verify that the function �� deined by �� :=
� ∨ �−1 , that is, �� (�, �) = max{�(�, �), �(�, �)}, deines a
metric on � whenever � is a �0 -quasi-pseudometric on �.
2
Journal of Mathematics
Let (�, �) be a quasi-pseudometric space. For � ∈ � and
� > 0,
�� (�, �) = {� ∈ � : � (�, �) < �}
(1)
�� (�, �) = {� ∈ � : � (�, �) ≤ �}
(2)
denotes the open �-ball at �. he collection of all such balls
yields a base for the topology �(�) induced by � on �. Hence,
for any � ∈ �, we will, respectively, denote by int�(�) � and
cl�(�) � the interior and the closure of the set � with respect
to the topology �(�).
Similarly, for � ∈ � and � ≥ 0,
denotes the closed �-ball at �. We will say that a subset � ⊂ �
is join-closed if it is �(�� )-closed, that is, closed with respect to
the topology generated by �� . he topology �(�� ) is iner than
the topologies �(�) and �(�−1 ).
Deinition 3. Let (�, �) be a quasi-pseudometric space. he
convergence of a sequence (�� ) to � with respect to �(�),
called �-convergence or let-convergence and denoted by �� →
�
�, is deined in the following way:
�
�� →
� � ⇐⇒ � (�, �� ) �→ 0.
�
(3)
Similarly, the convergence of a sequence (�� ) to � with
respect to �(�−1 ), called �−1 -convergence or right-convergence
and denoted by �� ��→ �, is deined in the following way:
�−1
�� ��→ � ⇐⇒ � (�� , �) �→ 0.
�−1
(4)
Finally, in a quasi-pseudometric space (�, �), we will say
that a sequence (�� ) �� -converges to � if it is both let and right
convergent to �, and we denote it by �� �→ � or �� → �
when there is no confusion. Hence,
��
� �,
�� �→ � ⇐⇒ �� →
��
�
�� ��→ �.
�−1
(5)
Deinition 4. A sequence (�� ) in a quasi-pseudometric (�, �)
is called
(a) let �-Cauchy if for every � > 0, there exist � ∈ � and
�0 ∈ N such that
∀� ≥ �0
� (�, �� ) < �;
(6)
(b) let �-Cauchy if for every � > 0, there exists �0 ∈ N
such that
∀�, � : �0 ≤ � ≤ � � (�� , �� ) < �;
(7)
(c) �� -Cauchy if for every � > 0, there exists �0 ∈ N such
that
∀�, � ≥ �0
� (�� , �� ) < �.
(8)
Dually, we deine, in the same way, right �-Cauchy and
right �-Cauchy sequences.
Remark 5. Consider the following:
(i) �� -Cauchy ⇒ let �-Cauchy ⇒ let �-Cauchy. he
same implications hold for the corresponding right notions.
None of the above implications is reversible.
(ii) A sequence is let �-Cauchy with respect to � if and
only if it is right �-Cauchy with respect to �−1 .
(iii) A sequence is let �-Cauchy with respect to � if and
only if it is right �-Cauchy with respect to �−1 .
(iv) A sequence is �� -Cauchy if and only if it is both let
and right �-Cauchy.
Deinition 6. A quasi-pseudometric space (�, �) is called
(i) let-�-complete provided that any let �-Cauchy
sequence is �-convergent,
(ii) let Smyth sequentially complete if any let �-Cauchy
sequence is �� -convergent.
he dual notions of right-completeness are easily derived from
the above deinition.
Deinition 7. A �0 -quasi-pseudometric space (�, �) is called
bicomplete provided that the metric �� on � is complete.
As usual, a subset � of a quasi-pseudometric space (�, �)
will be called bounded provided that there exists a positive
real constant � such that �(�, �) < � whenever �, � ∈ �.
his is equivalent to saying that there exist �0 ∈ �, �, � ≥ 0
such that � ⊆ �� (�0 , �) ∩ ��−1 (�0 , �).
We also deine the diameter �(�) of � by �(�) :=
sup{�(�, �) : �, � ∈ �}. Hence, � is bounded if and only
if �(�) < ∞. It is not diicult to see that this deinition
coincides with that of a bounded set in a metric space.
Let (�, �) be a quasi-pseudometric space. We set
P0 (�) := 2� \ {0} where 2� denotes the power set of �. For
� ∈ � and �, � ∈ P0 (�), we deine
� (�, �) = inf {� (�, �) , � ∈ �} ,
� (�, �) = inf {� (�, �) , � ∈ �} ,
(9)
and we deine �(�, �) by
� (�, �) = max {sup � (�, �) , sup � (�, �)} .
�∈�
�∈�
(10)
hen � is an extended quasi-pseudometric on P0 (�).
Moreover, we know from [9] that the restriction of � to
�cl (�) = {� ⊆ � : � = (cl�(�) �) ∩ (cl�(�−1 ) �)} is an
extended �0 -quasi-pseudometric. We will denote by ��(�)
the collection of all nonempty bounded and �(�� )-closed
subsets of �.
We complete this section by the following lemma.
Lemma 8. Let (�, �) be a quasi-pseudometric space. For
every ixed � ∈ �, the mapping � �→ �(�, �) is �(�)upper semicontinuous (�(�)-usc in short) and �(�−1 )-lower
semicontinuous (�(�−1 )-lsc in short). For every ixed � ∈ �,
the mapping � →
� �(�, �) is �(�)-lsc and �(�−1 )-usc.
Journal of Mathematics
3
Proof. To prove that �(�, ⋅) is �(�)-usc and �(�−1 )-lsc, we have
to show that the set {� ∈ � : �(�, �) < �} is �(�)-open and
{� ∈ � : �(�, �) > �} is �(�−1 )-open, for every � ∈ R,
properties that are easy to check.
Indeed, for � ∈ � such that �(�, �) < �, let � := � −
�(�, �) > 0. If � ∈ � is such that �(�, �) < �, then
� (�, �) ≤ � (�, �) + (�, �) < � (�, �) + � = �,
(11)
showing that �� (�, �) ⊂ {� ∈ � : �(�, �) < �}.
Similarly, for � ∈ � with �(�, �) > � take � := �(�, �) −
� > 0. If � ∈ � satisies �(�, �) = �−1 (�, �) < �, then
� (�, �) ≤ � (�, �) + (�, �) < � (�, �) + �,
(12)
so that �(�, �) > �(�, �) − � = �. Consequently, ��−1 (�, �) ⊂
{� ∈ � : �(�, �) > �}.
heorem 13. Let (�, �) be a Hausdorf let �-complete �0 quasi-pseudometric space. Suppose that � : � → � is
an (�, �)-contractive mapping which satisies the following
conditions:
(i) � is �-admissible;
(ii) there exists �0 ∈ � such that �(�0 , ��0 ) ≥ 1;
(iii) � is �-sequentially continuous.
hen � has a ixed point.
Proof. By (ii), there exists �0 ∈ � such that �(�0 , ��0 ) ≥ 1.
Let us deine the sequence (�� ) in � by ��+1 = ��� for all � =
0, 1, 2, . . .. Without loss of generality, we can always assume
that �� ≠ ��+1 for all � ∈ N, since if ��0 = ��0 +1 for some
�0 ∈ N, the proof is complete.
From assumption (i), we derive
3. Some First Results
� (�0 , ��0 ) = � (�0 , �1 ) ≥ 1
We begin by recalling the following.
�⇒ � (��0 , ��1 ) = � (�1 , �2 ) ≥ 1.
Deinition 9. A function � : [0, ∞) → [0, ∞) is called a (�)comparison function if
(�1 ) � is nondecreasing;
Recursively, we get
�
�
(�2 ) ∑∞
�=1 � (�) < ∞ for all � > 0, where � is the �th iterate
of �.
We will denote by Γ the set of such functions. Note that for
any � ∈ Γ, �(�) < � for any � > 0.
� (�� , ��+1 ) ≥ 1 ∀� = 0, 1, 2, . . . .
Deinition 10. Let (�, �) be a quasi-pseudometric type space.
A function � : � → � is called �-sequentially continuous or
let-sequentially continuous if for any �-convergent sequence
� �, the sequence (��� ) �-converges to ��; that
(�� ) with �� →
�
is, ��� →
� ��.
Similarly, we deine a �−1 -sequentially continuous or rightsequentially continuous function.
�
Deinition 11. Let (�, �) be a quasi-pseudometric space, and
let � : � → � and � : � × � → [0, ∞) be mappings. We
say that � is �-admissible if
� (�� , ��+1 ) = � (���−1 , ��� )
≤ � (��−1 , �� ) � (���−1 , ��� )
for all � = 0, 1, 2, . . .. Inductively, we obtain
� (�� , ��+1 ) ≤ �� (� (�0 , �1 )) ,
whenever �, � ∈ �.
We now state the irst ixed point theorem.
(14)
� = 1, 2, . . . .
(18)
herefore, for any � ≥ 1, using the triangle inequality, we get
� (�� , ��+� ) ≤ � (�� , ��+1 ) + � (��+1 , ��+2 )
+ ⋅ ⋅ ⋅ + � (��+�−1 , ��+� )
�+�−1
≤ ∑ �� (� (�0 , �1 ))
whenever �, � ∈ �.
� (�, �) � (��, ��) ≤ � (� (�, �)) ,
(17)
≤ � (� (��−1 , �� )) ,
(13)
Deinition 12. Let (�, �) be a quasi-pseudometric space and
let � : � → � be a mapping. We say that � is an (�, �)contractive mapping if there exist two functions � : � × � →
[0, ∞) and � ∈ Γ such that
(16)
Since � is (�, �)-contractive, we can write
We then introduce the following deinitions.
� (�, �) ≥ 1 �⇒ � (��, ��) ≥ 1,
(15)
(19)
�=�
∞
≤ ∑�� (� (�0 , �1 )) .
�=�
Letting � → ∞, we derive that �(�� , ��+� ) → 0. Hence, (�� )
is a let �-Cauchy sequence. Since (�, �) is let �-complete
and � �-sequentially continuous, there exists �∗ such that
�� →
� �∗ and ��+1 →
� ��∗ . Since � is Hausdorf, we have
∗
∗
that � = �� .
�
�
4
Journal of Mathematics
Corollary 14. Let (�, �) be a Hausdorf right �-complete
�0 -quasi-pseudometric space. Suppose that � : � → �
is an (�, �)-contractive mapping which satisies the following
conditions:
(i) � is �-admissible;
(ii) there exists �0 ∈ � such that �(��0 , �0 ) ≥ 1;
(iii) � is �−1 -sequentially continuous.
hen � has a ixed point.
Corollary 15. Let (�, �) be a bicomplete quasi-pseudometric
space. Suppose that � : � → � is an (�, �)-contractive
mapping which satisies the following conditions:
(i) � is �-admissible and the function � is symmetric; that
is, �(�, �) = �(�, �) for any �, � ∈ �;
(ii) there exists �0 ∈ � such that �(��0 , �0 ) ≥ 1;
(iii) � is �� -sequentially continuous.
hen � has a ixed point.
Proof. Following the proof of heorem 13, it is clear that the
sequence (�� ) in � deined by ��+1 = ��� for all � = 0, 1, 2, . . .
is �� -Cauchy. Since (�, �� ) is complete and � sequentially
continuous, there exists �∗ such that �� �→ �∗ and ��+1 �→
��∗ . Since (�, �� ) is Hausdorf, we have that �∗ = ��∗ .
��
��
Remark 16. In fact, we do not need � to be symmetric. It is
enough, for the result to be true, to have a point �0 ∈ � for
which �(�0 , ��0 ) ≥ 1 and �(��0 , �0 ) ≥ 1.
We conclude this section by the following results which
are in fact consequences of heorem 13.
heorem 17. Let (�, �) be a Hausdorf let �-complete �0 quasi-pseudometric space. Suppose that � : � → � is
an (�, �)-contractive mapping which satisies the following
conditions:
(i) � is �-admissible;
(ii) there exists �0 ∈ � such that �(�0 , ��0 ) ≥ 1;
(iii) if (�� ) is a sequence in � such that �(�� , ��+1 ) ≥ 1
for all � = 1, 2, . . . and �� →
� �, then there exists a
subsequence (��(�) ) of (�� ) such that �(��(�) , �) ≥ 1 for
all �.
�
hen � has a ixed point.
Proof. Following the proof of heorem 13, we know that the
sequence (�� ) deined by ��+1 = ��� for all � = 0, 1, 2, . . . �converges to some �∗ and satisies �(�� , ��+1 ) ≥ 1 for � ≥ 1.
From condition (iii), we know that there exists a subsequence
(��(�) ) of (�� ) such that �(��(�) , �∗ ) ≥ 1 for all �. Since � is an
(�, �)-contractive mapping, we get
� (��(�)+1 , ��∗ ) = � (���(�) , ��∗ )
≤ � (��(�) , �∗ ) � (���(�) , ��∗ )
≤ � (� (��(�) , �∗ )) .
(20)
Letting � → ∞, we obtain �(��(�)+1 , ��∗ ) → 0. Since � is
Hausdorf, we have that ��∗ = �∗ .
his completes the proof.
Corollary 18. Let (�, �) be a Hausdorf right �-complete
�0 -quasi-pseudometric space. Suppose that � : � → �
is an (�, �)-contractive mapping which satisies the following
conditions:
(i) � is �-admissible;
(ii) there exists �0 ∈ � such that �(��0 , �0 ) ≥ 1;
(iii) if (�� ) is a sequence in � such that �(��+1 , �� ) ≥ 1
for all � = 1, 2, . . . and �� ��→ �, then there exists a
subsequence (��(�) ) of (�� ) such that �(�, ��(�) ) ≥ 1 for
all �.
�−1
hen � has a ixed point.
Corollary 19. Let (�, �) be a bicomplete quasi-pseudometric
space. Suppose that � : � → � is an (�, �)-contractive
mapping which satisies the following conditions:
(i) � is �-admissible and the function � is symmetric;
(ii) there exists �0 ∈ � such that �(��0 , �0 ) ≥ 1;
(iii) if (�� ) is a sequence in � such that �(�� , �� ) ≥ 1 for all
�, � ∈ N and �� �→ �, then there exists a subsequence
(��(�) ) of (�� ) such that �(�, ��(�) ) ≥ 1 for all �.
��
hen � has a ixed 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 speciied setting. Here we introduce a similar
notion for set-valued maps deined on quasi-pseudometric
spaces. Let (�, �) be a �0 -quasi-pseudometric space.
Deinition 20. Let � : � → 2� be a set-valued map. An
element � ∈ � is said to be
(i) a ixed point of � if � ∈ ��,
(ii) a startpoint of � if �({�}, ��) = 0,
(iii) an endpoint of � if �(��, {�}) = 0,
(iv) an �-startpoint of � for some � ∈ (0, 1) if �({�}, ��) <
�,
(v) an �-endpoint of � for some � ∈ (0, 1) if �(��, {�}) <
�.
Remark 21. It is therefore obvious that if � is both a startpoint
of � and an endpoint of �, then � is a ixed point of �. In fact,
�� is a singleton. But a ixed point need not be a startpoint
nor an endpoint.
Indeed, consider the �0 -quasi-pseudometric space (�, �),
where � = {0, 1} and � is deined by �(0, 1) = 0, �(1, 0) = 1,
and �(�, �) for � = 0, 1. We deine on � the set-valued map
� : � → 2� by �� = �. Obviously, 1 is a ixed point, but
�({1}, �1) = �({1}, �) = max{�(1, 1), �(1, 0)} = 1 ≠ 0.
Journal of Mathematics
5
Lemma 22. Let (�, �) be a �0 -quasi-pseudometric space and
let � : � → 2� be a set-valued map. An element � ∈ � is a
startpoint of � if and only if it is an �-startpoint of � for every
� ∈ (0, 1).
Lemma 23. Let (�, �) be a �0 -quasi-pseudometric space and
let � : � → 2� be a set-valued map. An element � ∈ � is
an endpoint of � if and only if it is an �-endpoint of � for every
� ∈ (0, 1).
Deinition 24. Let (�, �) be a �0 -quasi-pseudometric space.
We say that a set-valued map � : � → 2� has the
approximate startpoint property (resp., approximate endpoint
property) if
inf sup � (�, �) = 0
�∈��∈��
(resp., inf sup � (�, �) = 0) .
�∈��∈��
(21)
Deinition 25. Let (�, �) be a �0 -quasi-pseudometric space.
We say that a set-valued map � : � → 2� has the
approximate mix-point property if
inf sup �� (�, �) = 0.
�∈��∈��
(22)
Here, it is also very clear that � has approximate mix-point
property if and only if � has both the approximate startpoint
and the approximate endpoint properties.
We are therefore naturally led to this deinition.
Deinition 26. Let � : � → � be a single-valued map
on a �0 -quasi-pseudometric space (�, �). hen � has the
approximate startpoint property (resp., approximate endpoint
property) if and only if
inf � (�, ��) = 0
�∈�
(resp., inf � (��, �) = 0) .
�∈�
(23)
We motivate our coming results by the following examples. We basically show that the concepts deined above
are independent and do not necessarily coincide. he list
of examples presented is not exhaustive and more can be
constructed, showing the connection between the notions
deined above.
Example 27. Let � = {0, 1, 2}. he map � : � × � → [0, ∞)
deined by �(0, 1) = �(0, 2) = 0, �(1, 0) = �(1, 2) =
1, �(2, 0) = �(2, 1) = 2, and �(�, �) = 0 for all � ∈
� is a �0 -quasi-pseudometric on �. Let � : � → 2�
be the set mapping deined by �� = � \ {�} for any
� ∈ �. By deinition, � does not have any ixed point.
Nevertheless, a simple computation shows that �({0}, �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 �∈� sup�∈�� �(�, �) = 0, showing that � has the
approximate startpoint property, but inf �∈� sup�∈�� �(�, �) =
1, showing that � does not have the approximate endpoint
property.
Example 28. Let � = {1/�, � = 1, 2, . . .}. he map � : � ×
� → [0, ∞) deined by �(1/�, 1/�) = max{1/� − 1/�, 0}
is a �0 -quasi-pseudometric on �. Let � : � → 2� be the
set-valued mapping deined by �� = � \ {�} for any � ∈ �.
By deinition, � does not have any ixed point.
For a ixed �0 ∈ N,
�(
{0
1 1
1
, )={1
−
�0 �
{ �0 �
if � ≤ �0 ,
(24)
if � ≥ �0 ,
(25)
if � > �0 .
Similarly for a ixed �0 ∈ N,
�(
{0
1 1
1
, )={1
−
� �0
�
�
{
0
if � < �0 .
Hence, � does not have any startpoint nor endpoint (which
also implies that � does not have any ixed point).
But for a given � ∈ (0, 1), there exists �0 ∈ N such
that 1/�0 < �. We also know from deinition that �({1/�0 },
�(1/�0 )) = 1/�0 , so, 1/�0 is an �-startpoint of �. More
generally, for a given � ∈ (0, 1), there exists �� ∈ N such that
1/�� is an �-startpoint of �. Moreover, for any � ≥ �� , 1/� is
an �-startpoint of �.
Similarly, we can show that � admits an �-endpoint.
We can now state our irst result.
heorem 29. Let (�, �) be a bicomplete quasi-pseudometric
space. Let � : � → ��(�) be a set-valued map that satisies
� (��, ��) ≤ � (� (�, �)) ,
��� ���ℎ �, � ∈ �,
(26)
where � : [0, ∞) → [0, ∞) is upper semicontinuous, �(�) < �
for each � > 0, and lim inf � → ∞ (�−�(�)) > 0. hen there exists
a unique �0 ∈ � which is both a startpoint and an endpoint of
� if and only if � has the approximate mix-point property.
Proof. It is clear that if � admits a point which is both a
startpoint and an endpoint, then � has the approximate startpoint property and the approximate endpoint property. Just
observe that �({�}, ��) = sup�∈�� �(�, �) and �(��, {�}) =
sup�∈�� �(�, �). Conversely, suppose � has the approximate
mix-point property. hen
�� = {� ∈ � : sup �� (�, �) ≤
�∈��
1
} ≠ 0,
�
(27)
for each � ∈ N. Also it is clear that for each � ∈ N, ��+1 ⊆ �� .
he map � �→ sup�∈�� �� (�, �) is �(�� )-lower semicontinuous
(as supremum of �(�� )-continuous mappings); we have that
�� is �(�� )-closed.
Next we prove that, for each � ∈ N, �� is bounded.
6
Journal of Mathematics
Assume by the way of contradiction that �(�� ) = ∞ for
each � ∈ N. hen there exist �� , �� ∈ �� such that �(�� , �� ) ≥
�. From (26), we obtain that
� (�, �) = � ({�} , {�})
≤ � ({�} , ��) + � (��, ��) + � (��, {�})
=
2
+ � (� (�, �)) ,
�
(28)
whenever �, � ∈ �� . Hence,
2
,
�
(29)
(30)
lim � (�� , �� ) = ∞.
�→∞
his contradicts our assumption. Now we show that
lim� → ∞ �(�� ) = 0. On the contrary, assume lim� → ∞ �(�� ) =
�0 > 0 (note that the sequence (�(�� )) is nonincreasing and
bounded below and then has a limit). Let
�→∞
��,� = � (��,� , ��,� ) �→ � (�� ) as � �→ ∞} .
(31)
Now we show that � > 0 (notice � ≥ 0). Arguing by
contradiction, we assume � = 0; then by the deinition of �,
there exists a sequence �� such that �� → �0 and lim� → ∞ (�� −
�(�� )) = 0. hen lim� → ∞ �(�� ) = �0 . But since � is upper
semicontinuous and �0 > 0, then
�0 = lim � (�� ) ≤ � (�0 ) < �0 .
(32)
�→∞
his contradiction shows that � > 0. For each � ∈ �, let
(�� , �� ) ∈ �� be a sequence such that �(�� , �� ) → �(�� ), as
� → ∞. hen from (29) we get
� ≤ lim inf (� (�� , �� ) − � (� (�� , �� ))) ≤
�→∞
(34)
heorem 31. Let (�, �) be a bicomplete quasi-pseudometric
space. Let � : � → ��(�) be a set-valued map that satisies
2
,
�
for each � ∈ �.
��� ���ℎ �, � ∈ �,
(35)
where 0 ≤ � < 1. hen there exists a unique �0 ∈ � which is
both a startpoint and an endpoint of � if and only if � has the
approximate mix-point property.
Proof. Take �(�) = �� in heorem 29.
We then deduce the following result for single-valued
maps.
� = inf inf {lim inf (��,� − � (��,� )) : (��,� , ��,� ) ∈ �� ,
�∈N
��� ���ℎ �, � ∈ �,
where � : [0, ∞) → [0, ∞) is upper semicontinuous, �(�) < �
for each � > 0, and lim inf � → ∞ (� − �(�)) > 0. If � has the
approximate mix-point property then � has a ixed point.
� (��, ��) ≤ �� (�, �) ,
lim (� (�� , �� ) − � (� (�� , �� ))) = 0,
�→∞
� (��, ��) ≤ � (� (�, �)) ,
Proof. From heorem 29, we conclude that there exists
�0 which is both a startpoint and an endpoint; that is,
�({�0 }, ��0 ) = 0 = �(��0 , {�0 }). he �0 -condition therefore
guarantees the desired result.
whenever �, � ∈ �� .
herefore,
� (�, �) − � (� (�, �)) ≤
Corollary 30. Let (�, �) be a bicomplete quasi-pseudometric
space. Let � : � → ��(�) be a set-valued map that satisies
heorem 32. Let (�, �) be a bicomplete quasi-pseudometric
space. Let � : � → � be a map that satisies
� (��, ��) ≤ � (� (�, �)) ,
��� ���ℎ �, � ∈ �,
where � : [0, ∞) → [0, ∞) is upper semicontinuous, �(�) < �
for each � > 0, and lim inf � → ∞ (� − �(�)) > 0. hen � has the
approximate startpoint property.
Proof. By the way of
inf �∈� �(�, ��) > 0. hen
contradiction,
suppose
�∈�(�)
= inf � (��, �2 �)
(37)
≤ inf � (� (�, ��)) .
(33)
�) = �(��0 , {�0 }). For uniqueness, if �0 is an arbitrary
startpoint of �, then �({�0 }, ��0 ) = 0 = �(��0 , {�0 }), and
so �0 ∈ ⋂�∈� �� = {�0 }.
his completes the proof.
that
inf � (�, ��) ≤ inf � (�, ��)
�∈�
�∈�
Hence, � = 0. his contradiction shows that lim� → ∞ �(�� ) =
0. It follows from the Cantor intersection theorem that
⋂�∈� �� = {�0 }.
hus, �({�0 }, ��0 ) = sup�∈��0 �(�, �) = 0 = sup�∈��0 �(�,
(36)
�∈�
Since �(�(�, �)) ≤ �(�, �), then inf �∈� �(�(�, ��))
inf �∈� �(�, ��).
Now, on the contrary, suppose again that
inf � (� (�, ��)) = inf � (�, ��) .
�∈�
�∈�
≤
(38)
Let (�� ) ⊂ � be a sequence such that lim� → ∞ �(�� , �(�� )) =
inf �∈� �(�, ��). By passing to subsequences if necessary, we
Journal of Mathematics
7
may assume that lim� → ∞ �(�(�� , �(�� ))) exists. hen from
(37) we have
� ({��+1 } , ���+1 ) ≤ � (� (�� , ��+1 )) ,
inf � (�, ��) ≤ inf � (� (�, ��))
�∈�
�∈�
≤ lim � (� (�� , � (�� )))
�→∞
≤ � (inf � (�, ��))
(39)
�∈�
�∈�
<
Corollary 33. Let (�, �) be a bicomplete quasi-pseudometric
space. Let � : � → � be a map that satisies
� (��, ��) ≤ � (� (�, �)) ,
��� ���ℎ �, � ∈ �,
(40)
where � : [0, ∞) → [0, ∞) is upper semicontinuous, �(�) < �
for each � > 0, and lim inf � → ∞ (� − �(�)) > 0. hen � has the
approximate endpoint property.
We inish this section by the following ixed point result.
Corollary 34. Let (�, �) be a bicomplete quasi-pseudometric
space. Let � : � → � be a map that satisies
� (��, ��) ≤ � (� (�, �)) ,
for each �, � ∈ �,
(41)
where � : [0, ∞) → [0, ∞) is upper semicontinuous, �(�) < �
for each � > 0, and lim inf � → ∞ (� − �(�)) > 0. hen � has a
ixed point.
Proof. From heorem 32 and Corollary 33, we conclude
that � has the approximate mix-point property. Hence, by
Corollary 30, we have the desired result.
5. More Results
� = 0, 1, 2, . . . .
(46)
On the other hand, ��+1 ∈ ��� implies
� (�� , ��+1 ) ≤ � ({�� } , ��� ) ,
� = 0, 1, 2, . . . .
(47)
By the two above inequalities, we have
� (��+1 , ��+2 ) ≤ �� (�� , ��+1 )
� = 0, 1, 2, . . . ,
� ({��+1 } , ���+1 ) ≤ �� ({�� } , ��� )
� = 0, 1, 2, . . . .
(48)
We then deduce by iteration that
� (�� , ��+1 ) ≤ �� � (�0 , �1 )
� = 0, 1, 2, . . . ,
� ({��+1 } , ���+1 ) ≤ �� � ({�0 } , ��0 )
� = 0, 1, 2, . . . .
(49)
hen for �, � ∈ N, � < �,
� (�� , �� ) ≤ � (�� , ��+1 ) + � (��+1 , ��+2 )
+ ⋅ ⋅ ⋅ + � (��−1 , �� )
≤ [�� + ��+1 + ⋅ ⋅ ⋅ + ��−1 ] � (�0 , �1 )
≤
(50)
��
� (�0 , �1 ) ,
1−�
and since �� → 0 as � → ∞ we conclude that (�� ) is a let
�-Cauchy sequence. According to the let �-completeness of
(�, �), there exists �∗ ∈ � such that �� →
� �∗ .
�
he following theorem is the main result of this section.
heorem 35. Let (�, �) be a let �-complete quasipseudometric space. Let � : � → ��(�) be a set-valued
map and � : � → R as �(�) = �({�}, ��). If there exists
� ∈ (0, 1) such that for all � ∈ � there exists � ∈ �� satisfying
� ({�} , ��) ≤ � (� (�, �)) ,
� = 0, 1, 2, . . . .
(45)
Claim 1. (�� ) is a let �-Cauchy sequence.
On one hand,
� ({��+1 } , ���+1 ) ≤ � (� (�� , ��+1 )) ,
< inf � (�, ��) .
We get a contradiction, so inf �∈� �(�(�, ��))
inf �∈� �(�, ��) which again contradicts (37).
his completes the proof.
Continuing this process, we can get an iterative sequence (�� )
where ��+1 ∈ ��� ⊆ � and
(42)
then � has a startpoint.
Proof. For any initial �0 ∈ �, there exists �1 ∈ ��0 ⊆ � such
that
� ({�1 } , ��1 ) ≤ � (� (�0 , �1 )) ,
(43)
� ({�2 } , ��2 ) ≤ � (� (�1 , �2 )) .
(44)
and for �1 ∈ �, there is �2 ∈ ��1 ⊆ � such that
Claim 2. �∗ is a startpoint of �.
Observe that the sequence (��� ) = (�({�� }, ��� )) is
decreasing and hence converges to 0. Since � is �(�)-lower
semicontinuous (as supremum of �(�)-lower semicontinuous
functions), we have
0 ≤ � (�∗ ) ≤ lim inf � (�� ) = 0.
�→∞
(51)
Hence, �(�∗ ) = 0; that is, �({�∗ }, ��∗ ) = 0.
his completes the proof.
Example 36. Let � = {1, 1/2, 1/3}. he map � : � × � →
[0, ∞) deined by �(1/�, 1/�) = max{1/� − 1/�, 0} is a �0 quasi-pseudometric on �. Let � : � → 2� be the set
mapping deined by �� = � \ {�} for any � ∈ �. With � = 1/2,
the map � satisies the assumptions of our theorem, so it has
a startpoint, which in the present case is 1/3.
8
Journal of Mathematics
More generally, if we set �� = {1/�, � = 1, 2, . . . , �} and
� as deined above, with � = 1/2, the map � deined by �� =
�\{�} for any � ∈ � satisies the assumptions of our theorem,
so it has a startpoint, which in this case is 1/�.
Corollary 37. Let (�, �) be a right �-complete quasipseudometric space. Let � : � → ��(�) be a set-valued map
and � : � → R deined by �(�) = �(��, {�}). If there exists
� ∈ (0, 1) such that for all � ∈ � there exists � ∈ �� satisfying
� (��, {�}) ≤ � (� (�, �)) ,
(52)
then � has an endpoint.
Corollary 38. Let (�, �) be a bicomplete quasi-pseudometric
space. Let � : � → ��(�) be a set-valued map and
� : � → R deined by �(�) = �� (��, {�}) =
max{�(��, {�}), �({�}, ��)}. If there exists � ∈ (0, 1) such that
for all � ∈ � there exists � ∈ �� satisfying
�� ({�} , ��) ≤ � (min {� (�, �) , � (�, �)}) ,
(53)
then � has a ixed point.
Proof. We give here the main idea of the proof.
Observe that inequality (53) guarantees that the sequence
(�� ) constructed in the proof of heorem 35 is a �� -Cauchy
sequence and hence �� -converges to some �∗ . Using the fact
that � is �(�� )-lower semicontinuous (as supremum of �(�� )continuous functions), we have
0 ≤ � (�∗ ) ≤ lim inf � (�� ) = 0.
�→∞
(54)
Hence, �(�∗ ) = 0; that is, �({�∗ }, ��∗ ) = 0 = �(��∗ , {�∗ }),
and we are done.
Remark 39. All the results given remain true when we replace
accordingly the bicomplete quasi-pseudometric space (�, �)
with a let Smyth sequentially complete/let �-complete or a
right Smyth sequentially complete/right �-complete space.
Conflict of Interests
he author declares that there is no conlict of interests
regarding the publication of this paper.
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