Startpoints and Contractions in Qua

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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 Ya´e Ulrich Gaba; gabayae2@gmail.com

Received 7 May 2014; Accepted 18 June 2014; Published 2 July 2014

Academic Editor: Bruce A. Watson

Copyright © 2014 Ya´e 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 ofstartpointandendpointfor 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 Cl´emence 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 ofCauchy 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,completenessis a key element, since most of the constructed sequences will be assumed to have aCauchy typeproperty.

It is the aim of this paper to continue the study of quasi-pseudometric 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 Sections4and5, the notions ofstartpointand

endpointfor set-valued mappings are introduced and

difer-ent 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 aquasi-pseudometricon�if

(i)�(�, �) = 0for 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�−1deined by�−1(�, �) = �(�, �)whenever�, � ∈ �is also a quasi-pseudometric on�, called theconjugateof�. In the literature,�−1is 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�. Volume 2014, Article ID 709253, 8 pages

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Let(�, �)be a quasi-pseudometric space. For� ∈ �and

� > 0,

��(�, �) = {� ∈ � : � (�, �) < �} (1)

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,

��(�, �) = {� ∈ � : � (�, �) ≤ �} (2)

denotes the closed�-ball at�. We will say that a subset� ⊂ �

isjoin-closedif 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�-convergenceorlet-convergenceand denoted by�_��→�

�, is deined in the following way:

�� → � ⇐⇒ � (�, �� ) �→ 0. (3)

Similarly, the convergence of a sequence (�_�)to�with respect to�(�−1), called�−1-convergenceorright-convergence

and denoted by�_� �

Finally, in a quasi-pseudometric space(�, �), we will say that a sequence(�_�) ��-convergesto�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

Dually, we deine, in the same way,right�-Cauchyand

right�-Cauchysequences.

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 completeif any let�-Cauchy

sequence is��-convergent.

he dual notions ofright-completenessare easily derived from the above deinition.

Deinition 7. A�₀-quasi-pseudometric space(�, �)is called

bicompleteprovided that the metric��on�is complete.

As usual, a subset�of a quasi-pseudometric space(�, �) will be calledboundedprovided that there exists a positive real constant�such that�(�, �) < �whenever�, � ∈ �. coincides with that of a bounded set in a metric space.

Let (�, �) be a quasi-pseudometric space. We set

P₀(�) := 2�\ {0}_where2�_{denotes the power set of}�_{. For}

hen �is an extended quasi-pseudometric onP₀(�)_.

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� ∈ �,

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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� := � −

We begin by recalling the following.

Deinition 9. A function� : [0, ∞) → [0, ∞)is called a(�)

-comparison function if

(�₁)�is nondecreasing;

(�₂)∑∞_�=1��(�) < ∞for all� > 0, where��is the�th iterate of�.

We will denote byΓthe set of such functions. Note that for any� ∈ Γ, �(�) < �for any� > 0.

We then introduce the following deinitions.

Deinition 10. Let(�, �)be a quasi-pseudometric type space.

A function� : � → �is called�-sequentially continuousor

let-sequentially continuousif for any�-convergent sequence

(��)with��→ �� , the sequence(��) �-converges to��; that

is,��_��→ �� .

Similarly, we deine a�−1-sequentially continuousor

right-sequentially continuous function.

Deinition 11. Let(�, �)be a quasi-pseudometric space, and

let� : � → �and� : � × � → [0, ∞)be mappings. We say that�is�-admissible if

� (�, �) ≥ 1 �⇒ � (��, ��) ≥ 1, (13)

whenever�, � ∈ �.

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� ∈ Γsuch that

� (�, �) � (��, ��) ≤ � (� (�, �)) , (14)

whenever�, � ∈ �.

We now state the irst ixed point theorem.

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.

From assumption (i), we derive

� (�0, ��0) = � (�0, �1) ≥ 1

�⇒ � (��0, ��1) = � (�1, �2) ≥ 1.

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Recursively, we get

� (��, ��+1) ≥ 1 ∀� = 0, 1, 2, . . . . (16)

Since�is(�, �)-contractive, we can write

� (��, ��+1) = � (��−1, ��)

herefore, for any� ≥ 1, using the triangle inequality, we get

� (��, ��+�) ≤ � (��, ��+1) + � (��+1, ��+2) and � �-sequentially continuous, there exists�∗ such that

�� → �� ∗ and��+1 �→ �� ∗. Since�is Hausdorf, we have

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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.

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�, � ∈ �;

(iii)�is��-sequentially continuous.

Proof. Following the proof ofheorem 13, it is clear that the

sequence(�_�)in�deined by�_�+1= ��_�for all� = 0, 1, 2, . . .

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) ≥ 1and�(��₀, �₀) ≥ 1.

We conclude this section by the following results which are in fact consequences ofheorem 13.

heorem 17. Let (�, �) be a Hausdorf let�-complete �₀

-quasi-pseudometric space. Suppose that � : � → � is

an (�, �)-contractive mapping which satisies the following conditions:

Proof. Following the proof ofheorem 13, we know that the

sequence(�_�)deined by�_�+1= ��_�for all� = 0, 1, 2, . . . � -converges to some�∗and satisies�(�_�, �_�+1) ≥ 1for� ≥ 1. From condition (iii), we know that there exists a subsequence

(��(�))of(��)such that�(��(�), �∗) ≥ 1for all�. Since�is an

(�, �)-contractive mapping, we get

� (��(�)+1, ��∗) = � (��(�), ��∗)

Corollary 18. Let (�, �) be a Hausdorf right �-complete

�0-quasi-pseudometric space. Suppose that � : � → �

is an(�, �)-contractive mapping which satisies the following

conditions:

space. Suppose that� : � → � is an (�, �)-contractive

mapping which satisies the following conditions:

(i)�is�-admissible and the function�is symmetric;

(iii)if(�_�)is a sequence in�such that�(�_�, �_�) ≥ 1for all

�, � ∈Nand�_� �

�

�→ �, then there exists a subsequence

(��(�))of(��)such that�(�, ��(�)) ≥ 1for all�.

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 corre-sponding 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

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}

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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

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 exam-ples. 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. Nevertheless, a simple computation shows that�({0}, �0) =

0, and hence 0is 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,

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

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, andlim 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 start-point property and the approximate endstart-point property. Just observe that�({�}, ��) = sup_�∈��(�, �)and�(��, {�}) = sup_�∈��(�, �). Conversely, suppose �has the approximate mix-point property. hen (as supremum of�(��)-continuous mappings); we have that

��is�(��)-closed.

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Assume by the way of contradiction that�(�_�) = ∞for

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

� =inf

�∈Ninf{lim inf� → ∞ (��,�− � (��,�)) : (��,�, ��,�) ∈ ��,

��,�= � (��,�, ��,�) �→ � (��)as� �→ ∞} .

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Now we show that � > 0 (notice � ≥ 0). Arguing by contradiction, we assume� = 0; then by the deinition of�, there exists a sequence�_�such that�_� → �₀and lim_{� → ∞}(�_�−

�(��)) = 0. hen lim� → ∞�(��) = �0. But since�is upper

semicontinuous and�₀> 0, then

�0=_{� → ∞}lim� (��) ≤ � (�0) < �0. (32)

0. It follows from the Cantor intersection theorem that

⋂�∈��= {�0}.

� (��, ��) ≤ � (� (�, �)) , �� ℎ �, � ∈ �, (34)

for each� > 0, andlim inf_{� → ∞}(� − �(�)) > 0. If�has the

approximate mix-point property then�has a ixed point.

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.

heorem 31. Let(�, �) be a bicomplete quasi-pseudometric

� (��, ��) ≤ �� (�, �) , �� ℎ �, � ∈ �, (35)

where0 ≤ � < 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�(�) = ��inheorem 29.

We then deduce the following result for single-valued maps.

heorem 32. Let (�, �)be a bicomplete quasi-pseudometric

space. Let� : � → �be a map that satisies

� (��, ��) ≤ � (� (�, �)) , �� ℎ �, � ∈ �, (36)

for each� > 0, andlim inf_{� → ∞}(� − �(�)) > 0. hen�has the

approximate startpoint property.

Proof. By the way of contradiction, suppose that

inf_�∈��(�, ��) > 0. hen

Now, on the contrary, suppose again that

inf

�∈�� (� (�, ��)) =�∈�inf� (�, ��) . (38)

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may assume that lim_{� → ∞}�(�(�_�, �(�_�)))exists. hen from

� (��, ��) ≤ � (� (�, �)) , �� ℎ �, � ∈ �, (40)

for each� > 0, andlim inf_{� → ∞}(� − �(�)) > 0. hen�has the

approximate endpoint property.

We inish this section by the following ixed point result.

� (��, ��) ≤ � (� (�, �)) , for each�, � ∈ �, (41)

for each� > 0, andlim 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

he following theorem is the main result of this section.

heorem 35. Let (�, �) be a let �-complete

quasi-pseudometric space. Let � : � → ��(�)be a set-valued

map and� : � → Ras�(�) = �({�}, ��). If there exists

� ∈ (0, 1)such that for all� ∈ �there exists� ∈ ��satisfying

� ({�} , ��) ≤ � (� (�, �)) , (42)

then�has a startpoint.

Proof. For any initial�₀∈ �, there exists�₁ ∈ ��₀⊆ �such

that

� ({�1} , ��1) ≤ � (� (�0, �1)) , (43)

and for�₁∈ �, there is�₂∈ ��₁⊆ �such that

� ({�2} , ��2) ≤ � (� (�1, �2)) . (44)

Continuing this process, we can get an iterative sequence(�_�) where�_�+1∈ ��_�⊆ �and

By the two above inequalities, we have

� (��+1, ��+2) ≤ �� (��, ��+1) � = 0, 1, 2, . . . ,

� ({��+1} , ��+1) ≤ �� ({��} , ��) � = 0, 1, 2, . . . .

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We then deduce by iteration that

� (��, ��+1) ≤ �� (�0, �1) � = 0, 1, 2, . . . ,

�-Cauchy sequence. According to the let�-completeness of

(�, �), there exists�∗∈ �such that�_��→ �� ∗.

Claim 2.�∗is a startpoint of�.

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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 is1/�.

Corollary 37. Let (�, �) be a right �-complete

quasi-pseudometric space. Let� : � → ��(�)be a set-valued map

and� : � → Rdeined by�(�) = �(��, {�}). If there exists

� ∈ (0, 1)such that for all� ∈ �there exists� ∈ ��satisfying

� (��, {�}) ≤ � (� (�, �)) , (52)

then�has an endpoint.

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 ofheorem 35is 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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