New Results in the Startpoint Theory for Quasipseudometric Spaces

Abstract

We give two generalizations of Theorem 35 proved by Gaba (2014). More precisely, we change the structure of the contractive condition; namely, we introduce a function instead of a simple constant .

Dedicated to Professor Guy A. Degla for his mentorship

1. Introduction and Preliminaries

In [1], we introduced the concept of startpoint and endpoint for set-valued mappings defined on quasipseudometric spaces. As mentioned there, the purpose of this theory is to study fixed point like related properties. In the present, we give more results from the theory. More precisely, we generalize Theorem 35 of [1] by changing the structure of the contractive condition; namely, we introduce a function instead of a simple constant (as it appears in the original statement). This new condition is interesting in the sense that it allows us to have a condition involving a functional of the variables and not just the variables themselves. For the convenience of the reader, we will recall some necessary definitions but for a detailed exposé of the definition and examples, the interested reader is referred to [1].

Definition 1. Let be a nonempty set. A function is called quasipseudometric on if (i)  ;(ii)  .
Moreover, if , then is said to be a -quasipseudometric. The latter condition is referred to as the -condition.

Remark 2. (i) Let be quasipseudometric on ; then the map defined by whenever is also a quasipseudometric on , called the conjugate of . In the literature, is also denoted as or .
(ii) It is easy to verify that the function defined by , that is, , defines a metric on whenever is a -quasipseudometric on .

The quasipseudemetric induces a topology on .

Definition 3. Let be a quasipseudometric space. The -convergence of a sequence to a point , also called left-convergence and denoted by , is defined in the following way:
Similarly, we define the -convergence of a sequence to a point or right convergence and denote it by , in the following way:
Finally, in a quasipseudometric space , we will say that a sequence ()  -converges to if it is both left and right convergent to , and we denote it as or   when there is no confusion. Hence

Definition 4. A sequence in quasipseudometric 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 right -Cauchy and right -Cauchy sequences.

Definition 5. A quasipseudometric 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.

Definition 6. A -quasipseudometric space is called bicomplete provided that the metric on is complete.

As usual, a subset of a quasipseudometric space will be called bounded provided that there exists a positive real constant such that whenever .

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

Then is an extended quasipseudometric on . Moreover, we know from [2] that the restriction of to is an extended -quasipseudometric. We will denote by the collection of all nonempty bounded and -closed subsets of .

Definition 7 (compare [1]). 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 .

We complete this section by recalling the following lemma.

Lemma 8 (compare [1]). Let be a quasipseudometric space. For every fixed , the mapping is upper semicontinuous in short and lower semicontinuous (-lsc in short). For every fixed , the mapping is -lsc and -usc.

2. Main Results

We commence this section with the main result of this paper.

Theorem 9. Let be a left -complete quasipseudometric space. Let be a set-valued map and define as . Let be a function such that for each . Moreover, assume that for any there exists satisfying and then has a startpoint.

Proof. First observe that, since for any , it follows that for any , and hence for any and .
For any initial , there exists (for all actually) such that
From (9) we get where is defined by
Now choosing , we have that and from (9) we get
Continuing this process, we obtain a sequence where , with
For simplicity, denote and for all . So from (17) we can write for all . Hence is a strictly decreasing sequence and hence there exists such that
From (16), it is easy to see that
Thus the sequence is bounded and so there is such that and hence a subsequence of such that . From (17) we have and thus
This together with the fact for each implies that . Then from (19) and (20) we derive that .
Claim 1. is a left -Cauchy sequence.
Now let and such that . This choice of is always possible since . Then there is such that for all . So from (18) we have for all . Then by induction we get for all . Combining this and inequality (20) we get for all . Hence 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 converges to . Since is -lower semicontinuous (as supremum of -lower semicontinuous functions), we have
Hence ; that is, .
This completes the proof.

We give below an example to illustrate the theorem.

Example 10. Let and be the mapping defined by . Then is a -quasipseudometric on . Observe that any left -Cauchy sequence in is -convergent to . Indeed, if is a left -Cauchy sequence, for every , there exists such that
This entails that
Hence ; that is, . Therefore is left -complete. Let be such that
Let be defined by
An explicit computation of gives
Moreover, for each , and we have
Of course inequality (29) also holds in the case of and . Therefore, all assumptions of Theorem 9 are satisfied and the endpoint of is .

Remark 11. In fact, every sequence in -converges to .

Corollary 12. Let be a right -complete quasipseudometric space. Let be a set-valued map and define as . Let be a function such that for each . Moreover, assume that for any there exists satisfying and then has an endpoint.

Corollary 13. Let be a bicomplete quasipseudometric space. Let be a set-valued map and defined by . If there exists such that for all there exists satisfying where and , then has a fixed point.

Proof. We give here the main idea of the proof. Observe that inequality (31) guarantees that the sequence constructed in the proof of Theorem 9 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.

The following theorem is the second generalization that we propose.

Theorem 14. Let be a left -complete quasipseudometric space. Let be a set-valued map and define as . Let be a nondecreasing function. Let be a function such that for each and for each . Moreover, assume that for any there exists satisfying and then has a startpoint.

Proof. Observe that because for all , for any . Let be arbitrary. Then we can choose such that
Define the function by . Hence (35) and (36) together sum to
Now we choose such that which lead to
Continuing this process, we get an iterative sequence where and, denoting and for all , we can write that for all . Hence
If for some , then we trivially have and the conclusion is immediate. So without loss of generality, we can assume that for all and, from (41), we have for all .
Observe also that if, for some , , we are led to a contradiction. Indeed from (40) and using the fact that the function is nondecreasing, we have
Hence for all . Thus there exist and such that and . From (41), we get and hence (since for all ). Moreover, since this forces to be ; that is, .
Furthermore, setting and letting be a positive number such that , there is such that for all . Hence from (41) and (44), we get for all . So for all . Using a similar argument as the one used in the proof of Theorem 9, we conclude that is a left -Cauchy sequence and that its limit point is a starpoint of .

Example 15. Let and be the mapping defined by . Then is a -quasipseudometric on . We know that is left -complete. Let be such that
Let be defined by
Let be defined by
An explicit computation of gives
Moreover, for each , there exists and we have
Therefore, all assumptions of Theorem 14 are satisfied and the endpoint of is .

Corollary 16. Let be a right -complete quasipseudometric space. Let be a set-valued map and define as . Let be a nondecreasing function. Let be a function such that for each and for each . Moreover, assume that for any there exists satisfying , and for each . Moreover, assume that for any there exists satisfying and then has an endpoint.

Corollary 17. Let be a bicomplete quasipseudometric space. Let be a set-valued map and define as . Let , be a nondecreasing function. Let be a function such that and for each . Moreover, assume that for any there exists satisfying where and . Then has a fixed point.

Remark 18. All the results given remain true when we replace accordingly the bicomplete quasipseudometric space by 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.