#### Abstract

We introduce the concept of *startpoint* 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.

*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 modified. In the same reference, which uses an approach based on uniformities, the* bicompletion* of a -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 a* Cauchy 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 Section 3 a suitable notion of -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 of* startpoint* and* endpoint* 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. *Let be a nonempty set. A function is called a* quasi-pseudometric* on if (i) for all ,(ii) for all .Moreover, if , then is said to be a -*quasi-pseudometric.* The latter condition is referred to as the -condition.

*Remark 2. *(i) Let be a quasi-pseudometric on ; then the map defined by whenever is also a quasi-pseudometric on , called the* conjugate* of . In the literature, is also denoted by or .

(ii) It is easy to verify that the function defined by , that is, , defines a metric on whenever is a -quasi-pseudometric on .

Let be a quasi-pseudometric space. For and , denotes the open -ball at . The collection of all such balls yields a base for the topology induced by on . Hence, for any , we will, respectively, denote by and the interior and the closure of the set with respect to the topology .

Similarly, for and ,
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 . The topology is finer than the topologies and .

*Definition 3. *Let be a quasi-pseudometric space. The convergence of a sequence to with respect to , called -*convergence* or* left-convergence* and denoted by , is defined in the following way:

Similarly, the convergence of a sequence to with respect to , called -*convergence* or* right-convergence* and denoted by , is defined in the following way:

Finally, in a quasi-pseudometric space , we will say that a sequence -*converges* to if it is both left and right convergent to , and we denote it by or when there is no confusion. Hence,

*Definition 4. *A sequence in a quasi-pseudometric is called(a) *left **-Cauchy* if for every , there exist and such that
(b) *left **-Cauchy* if for every , there exists such that
(c) -*Cauchy* if for every , there exists such that

Dually, we define, in the same way,* right **-Cauchy* and* right **-Cauchy* sequences.

*Remark 5. *Consider the following:

(i) -Cauchy left -Cauchy left -Cauchy. The same implications hold for the corresponding right notions. None of the above implications is reversible.

(ii) A sequence is left -Cauchy with respect to if and only if it is right -Cauchy with respect to .

(iii) A sequence is left -Cauchy with respect to if and only if it is right -Cauchy with respect to .

(iv) A sequence is -Cauchy if and only if it is both left and right -Cauchy.

*Definition 6. *A quasi-pseudometric space is called(i)*left-**-complete* provided that any left -Cauchy sequence is -convergent,(ii)*left Smyth sequentially complete* if any left -Cauchy sequence is -convergent.The dual notions of* right-completeness* are easily derived from the above definition.

*Definition 7. *A -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 . This is equivalent to saying that there exist , such that .

We also define the* diameter * of by . Hence, is bounded if and only if . It is not difficult to see that this definition coincides with that of a bounded set in a metric space.

Let be a quasi-pseudometric space. We set where denotes the power set of . For and , we define and we define by

Then is an extended quasi-pseudometric on . Moreover, we know from [9] that the restriction of to is an extended -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 fixed , the mapping is -upper semicontinuous -usc in short and -lower semicontinuous -lsc in short. For every fixed , the mapping is -lsc and -usc.*

*Proof. *To prove that is -usc and -lsc, we have to show that the set is -open and is -open, for every , properties that are easy to check.

Indeed, for such that , let . If is such that , then
showing that .

Similarly, for with take . If satisfies , then
so that . Consequently, .

#### 3. Some First Results

We begin by recalling the following.

*Definition 9. *A function is called a -comparison function if () is nondecreasing;() for all , where is the th iterate of .We will denote by the set of such functions. Note that for any , for any .

We then introduce the following definitions.

*Definition 10. *Let be a quasi-pseudometric type space. A function is called -*sequentially continuous* or* left-sequentially continuous* if for any -convergent sequence with , the sequence -converges to ; that is, .

Similarly, we define a -*sequentially continuous* or* right-sequentially continuous function*.

*Definition 11. *Let be a quasi-pseudometric space, and let and be mappings. We say that is -admissible if
whenever .

*Definition 12. *Let be a quasi-pseudometric space and let be a mapping. We say that is an -contractive mapping if there exist two functions and such that
whenever .

We now state the first fixed point theorem.

Theorem 13. *Let be a Hausdorff left -complete -quasi-pseudometric space. Suppose that is an -contractive mapping which satisfies the following conditions: *(i)* is -admissible;*(ii)*there exists such that ;*(iii)* is -sequentially continuous.**Then has a fixed point.*

*Proof. *By (ii), there exists such that . Let us define the sequence in by for all . Without loss of generality, we can always assume that for all , since if for some , the proof is complete.

From assumption (i), we derive Recursively, we get Since is -contractive, we can write for all . Inductively, we obtain Therefore, for any , using the triangle inequality, we get Letting , we derive that . Hence, is a left -Cauchy sequence. Since is left -complete and -sequentially continuous, there exists such that and . Since is Hausdorff, we have that .

Corollary 14. *Let be a Hausdorff right -complete -quasi-pseudometric space. Suppose that is an -contractive mapping which satisfies the following conditions: *(i)* is -admissible;*(ii)*there exists such that ;*(iii)* is -sequentially continuous.**Then has a fixed point.*

Corollary 15. *Let be a bicomplete quasi-pseudometric space. Suppose that is an -contractive mapping which satisfies the following conditions: *(i)* is -admissible and the function is symmetric; that is, for any ;*(ii)*there exists such that ;*(iii)* is -sequentially continuous.**Then has a fixed point.*

*Proof. *Following the proof of Theorem 13, it is clear that the sequence in defined by for all is -Cauchy. Since is complete and sequentially continuous, there exists such that and . Since is Hausdorff, 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 for which and .

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

Theorem 17. *Let be a Hausdorff left -complete -quasi-pseudometric space. Suppose that is an -contractive mapping which satisfies the following conditions:*(i)* is -admissible;*(ii)*there exists such that ;*(iii)*if is a sequence in such that for all and , then there exists a subsequence of such that for all .**Then has a fixed point.*

*Proof. *Following the proof of Theorem 13, we know that the sequence defined by for all -converges to some and satisfies for . From condition (iii), we know that there exists a subsequence of such that for all . Since is an -contractive mapping, we get
Letting , we obtain . Since is Hausdorff, we have that .

This completes the proof.

Corollary 18. *Let be a Hausdorff right -complete -quasi-pseudometric space. Suppose that is an -contractive mapping which satisfies the following conditions:*(i)* is -admissible;*(ii)*there exists such that ;*(iii)*if is a sequence in such that for all and , then there exists a subsequence of such that for all .**Then has a fixed point.*

Corollary 19. *Let be a bicomplete quasi-pseudometric space. Suppose that is an -contractive mapping which satisfies the following conditions:*(i)* is -admissible and the function is symmetric;*(ii)*there exists such that ;*(iii)*if is a sequence in such that for all and , then there exists a subsequence of such that for all .**Then 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 be a -quasi-pseudometric space.

*Definition 20. *Let be a set-valued map. An element is said to be (i)a fixed point of if ,(ii)a startpoint of if ,(iii)an endpoint of if ,(iv)an -startpoint of for some if ,(v)an -endpoint of for some if .

*Remark 21. *It is therefore obvious that if is both a startpoint of and an endpoint of , then is a fixed point of . In fact, is a singleton. But a fixed point need not be a startpoint nor an endpoint.

Indeed, consider the -quasi-pseudometric space , where and is defined by , , and for . We define on the set-valued map by . Obviously, is a fixed point, but .

Lemma 22. *Let be a -quasi-pseudometric space and let be a set-valued map. An element is a startpoint of if and only if it is an -startpoint of for every .*

Lemma 23. *Let be a -quasi-pseudometric space and let be a set-valued map. An element is an endpoint of if and only if it is an -endpoint of for every .*

*Definition 24. *Let be a -quasi-pseudometric space. We say that a set-valued map has the* approximate startpoint property* (resp.,* approximate endpoint property*) if

*Definition 25. *Let be a -quasi-pseudometric space. We say that a set-valued map has the* approximate mix-point property* if
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 definition.

*Definition 26. *Let be a single-valued map on a -quasi-pseudometric space . Then has the* approximate startpoint property* (resp.,* approximate endpoint property*) if and only if

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. *Let . The map defined by , , , and for all is a -quasi-pseudometric on . Let be the set mapping defined by for any . By definition, does not have any fixed point. Nevertheless, a simple computation shows that , and hence is a startpoint and it is the only one. Also there is no endpoint. Again, with a direct computation, we have , showing that has the approximate startpoint property, but , showing that does not have the approximate endpoint property.

*Example 28. *Let . The map defined by is a -quasi-pseudometric on . Let be the set-valued mapping defined by for any . By definition, does not have any fixed point.

For a fixed ,
Similarly for a fixed ,
Hence, does not have any startpoint nor endpoint (which also implies that does not have any fixed point).

But for a given , there exists such that . We also know from definition that , so, is an -startpoint of . More generally, for a given , there exists such that is an -startpoint of . Moreover, for any , is an -startpoint of .

Similarly, we can show that admits an -endpoint.

We can now state our first result.

Theorem 29. *Let be a bicomplete quasi-pseudometric space. Let be a set-valued map that satisfies
**
where is upper semicontinuous, for each , and . Then there exists a unique 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 and . Conversely, suppose has the approximate mix-point property. Then
for each . Also it is clear that for each , . The map is -lower semicontinuous (as supremum of -continuous mappings); we have that is -closed.

Next we prove that, for each is bounded.

Assume by the way of contradiction that for each . Then there exist such that . From (26), we obtain that
whenever .

Therefore,
whenever . Hence,
This contradicts our assumption. Now we show that . On the contrary, assume (note that the sequence is nonincreasing and bounded below and then has a limit). Let
Now we show that (notice ). Arguing by contradiction, we assume ; then by the definition of , there exists a sequence such that and . Then . But since is upper semicontinuous and , then
This contradiction shows that . For each , let be a sequence such that , as . Then from (29) we get
Hence, . This contradiction shows that . It follows from the Cantor intersection theorem that .

Thus, . For uniqueness, if is an arbitrary startpoint of , then , and so .

This completes the proof.

Corollary 30. *Let be a bicomplete quasi-pseudometric space. Let be a set-valued map that satisfies
**
where is upper semicontinuous, for each , and . If has the approximate mix-point property then has a fixed point.*

*Proof. *From Theorem 29, we conclude that there exists which is both a startpoint and an endpoint; that is, . The -condition therefore guarantees the desired result.

Theorem 31. *Let be a bicomplete quasi-pseudometric space. Let be a set-valued map that satisfies
**
where . Then there exists a unique which is both a startpoint and an endpoint of if and only if has the approximate mix-point property.*

*Proof. *Take in Theorem 29.

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

Theorem 32. *Let be a bicomplete quasi-pseudometric space. Let be a map that satisfies
**
where is upper semicontinuous, for each , and . Then has the approximate startpoint property.*

*Proof. *By the way of contradiction, suppose that . Then
Since , then .

Now, on the contrary, suppose again that
Let be a sequence such that . By passing to subsequences if necessary, we may assume that exists. Then from (37) we have

We get a contradiction, so which again contradicts (37).

This completes the proof.

Corollary 33. *Let be a bicomplete quasi-pseudometric space. Let be a map that satisfies
**
where is upper semicontinuous, for each , and . Then has the approximate endpoint property.*

We finish this section by the following fixed point result.

Corollary 34. *Let be a bicomplete quasi-pseudometric space. Let be a map that satisfies
**
where is upper semicontinuous, for each , and . Then has a fixed point.*

*Proof. *From Theorem 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

The following theorem is the main result of this section.

Theorem 35. *Let be a left -complete quasi-pseudometric space. Let be a set-valued map and as . If there exists such that for all there exists satisfying
**
then has a startpoint.*

*Proof. *For any initial , there exists such that
and for , there is such that
Continuing this process, we can get an iterative sequence where and
*Claim **1. * is a left -Cauchy sequence.

On one hand,
On the other hand, implies
By the two above inequalities, we have
We then deduce by iteration that
Then for , ,
and since as we conclude that is a left -Cauchy sequence. According to the left -completeness of , there exists such that .*Claim **2. * is a startpoint of .

Observe that the sequence is decreasing and hence converges to . Since is -lower semicontinuous (as supremum of -lower semicontinuous functions), we have
Hence, ; that is, .

This completes the proof.

*Example 36. *Let . The map defined by is a -quasi-pseudometric on . Let be the set mapping defined by for any . With , the map satisfies the assumptions of our theorem, so it has a startpoint, which in the present case is .

More generally, if we set and as defined above, with , the map defined by for any satisfies the assumptions of our theorem, so it has a startpoint, which in this case is .

Corollary 37. *Let be a right -complete quasi-pseudometric space. Let be a set-valued map and defined by . If there exists such that for all there exists satisfying
**
then has an endpoint.*

Corollary 38. *Let be a bicomplete quasi-pseudometric space. Let be a set-valued map and defined by . If there exists such that for all there exists satisfying
**
then has a fixed point.*

*Proof. *We give here the main idea of the proof.

Observe that inequality (53) guarantees that the sequence constructed in the proof of Theorem 35 is a -Cauchy sequence and hence -converges to some . Using the fact that is -lower semicontinuous (as supremum of -continuous functions), we have
Hence, ; that is, , and we are done.

*Remark 39. *All the results given remain true when we replace accordingly the bicomplete quasi-pseudometric space with a left Smyth sequentially complete/left -complete or a right Smyth sequentially complete/right -complete space.

#### Conflict of Interests

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