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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 269246, 6 pages
http://dx.doi.org/10.1155/2013/269246
Research Article

Generalized Contractive Set-Valued Maps on Complete Preordered Quasi-Metric Spaces

Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Camí de Vera s/n, 46022 València, Spain

Received 12 October 2012; Accepted 8 December 2012

Academic Editor: Gerald Beer

Copyright © 2013 J. Marín et al. This 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.

Abstract

By using a suitable modification of the notion of a -distance we obtain some fixed point results for generalized contractive set-valued maps on complete preordered quasi-metric spaces. We also show that several distinguished examples of non-metrizable quasi-metric spaces and of cones of asymmetric normed spaces admit -distances of this type. Our results extend and generalize some well-known fixed point theorems.

1. Introduction and Preliminaries

Throughout this paper the letters , , , and will denote the set of real numbers, the set of non-negative real numbers, the set of positive integer numbers and the set of non-negative integer numbers, respectively. Our basic references for quasi-metric spaces are [1, 2] and for asymmetric normed space it is [3].

A quasi-pseudometric on a set is a function such that for all (i) ; (ii) .

If satisfies conditions (i) and (ii) above but we allow , then is said to be an extended quasi-pseudometric on .

Following the modern terminology, a quasi-pseudometric on satisfying () if and only if is called a quasi-metric on .

If the quasi-metric satisfies the stronger condition () if and only if , we say that is a quasi-metric on .

A quasi-metric space is a pair such that is a nonempty set and is a quasi-metric on .

Each extended quasi-pseudometric on a set induces a topology on which has as a base the family of open balls , where for all and .

The closure with respect to of a subset of will be denoted by .

Note that if is quasi-metric then is a topology, and if a quasi-metric then is a topology on .

Given a quasi-metric on , the function defined by for all , is also a quasi-metric on , and the function defined by for all , is a metric on .

There exist several different notions of Cauchyness and quasi-metric completeness in the literature (see, e.g., [2]). In our context will be useful the following general notion.

Definition 1. A quasi-metric on a set will be called complete if every Cauchy sequence in converges with respect to the topology (i.e., there exists such that ), where the sequence is said to be Cauchy if for each there exists such that whenever . If is complete we will say that the quasi-metric space is complete.

Kada et al. introduced in [4] the notion of -distance for metric spaces and obtained, among other results, -distance versions of the celebrated Ekeland variational principle [5] and the nonconvex minimization theorem [6]. In [7] Park extended this concept to quasi-metric spaces in order to generalize and unify different versions of Ekeland's variational principle. Park’s approach was continued by Al-Homidan et al. [8], and recently by Latif and Al-Mezel [9], and Marín et al. [10, 11], among others. Thus in [8] were obtained extensions and generalizations of Caristi-Kirk’s type fixed point theorem [12] as well as a Takahashi type minimization theorem and generalizations of Ekeland’s variational principle and of Nadler’s fixed point theorem [13], respectively, while in [911] were proved several fixed point theorems for single and set-valued maps on quasi-metric spaces by using -functions in the sense of [8] and -distances.

Definition 2 (see [7, 8]). A -distance for a quasi-metric space is a function satisfying the following three conditions:(W1) for all ;(W2) is lower semicontinuous on for all ;(W3) for each there exists such that and , imply .

Note that every quasi-metric on satisfies conditions (W1) and (W2) above.

If is a metric on , then Definition 2 provides the notion of a -distance for the metric space as defined in [4]. In particular, every metric on is a -distance for .

Unfortunately, the situation is quite different when is a quasi-metric. In fact, it was shown in [10] that if a quasi-metric on is also a -distance for , then the topology induced by is metrizable. Hence, many distinguished examples of nonmetrizable quasi-metrizable topological spaces do not admit any compatible quasi-metric which is also a -distance.

Motivated by this fact, in Section 2 we will show that the use of (pre)ordered quasi-metric spaces, with a suitable adaptation of the notion of -distance to this setting, allows us to generate several interesting examples of preordered quasi-metric spaces for which the quasi-metric is a -distance in this new sense. In Section 3 we will prove a fixed point theorem for set-valued maps on complete preordered quasi-metric spaces by means of the modified notion of -distance, that generalizes and extends several well-known fixed point theorems and allows us to deduce fixed point results involving the lower Hausdorff distance of a complete preordered quasi-metric space. We illustrate these results with some examples.

2. Preordered Quasi-Metric Spaces, -Distances, and Examples

We start this section by recalling some pertinent concepts.

A preorder on a (nonempty) set is a reflexive and transitive (binary) relation on . If, in addition, is antisymmetric (i.e., condition and , implies ), is called a partial order or, simply, an order on . The usual order on is denoted by .

Let be a preorder on . Given the set will be denoted by . A sequence in is said to be nondecreasing if for all .

Remark 3. Given a (nonempty) set , the (trivial) relation given by if and only if is obviously a preorder on

According to [14], a (pre)ordered quasi-metric space is a triple such that is a (pre)order on and is a quasi-metric on .

Observe that if is a quasi-metric space, then the relation on defined by if and only if is a partial order on called the specialization order of . So is an ordered quasi-metric space.

Definition 4. A -distance for a preordered quasi-metric space is a function satisfying conditions (W1) and (W2) of Definition 2, and: () for each there exists such that , , and , imply .

Example 5. Let be a -distance for a quasi-metric space . Then is obviously a -distance for the preordered quasi-metric space .

Example 6. Let be a quasi-metric space. Consider the ordered quasi-metric space . Of course, satisfies conditions (W1) and (W2). Moreover, it trivially satisfies condition () of Definition 4. Hence is a -distance for .

Example 7. Let and let be the quasi-metric on given by if , and if . Then induces the Sorgenfrey topology on . We show that is is a -distance for the ordered quasi-metric . Indeed, since is a quasi-metric, we only need to show condition (3) of Definition 4. To this end, choose . Put , and let and with . Therefore and . Since , we have . We conclude that is a -distance for .

Our next example should be compared with Example 3.1 of [8]. Recall [3, 15] that an asymmetric norm on a real vector space is a function such that for each and (i) if and only if ; (ii) ; (iii) .

Then, the pair is called an aysmmetric normed space. Asymmetric norms are called quasi-norms in [16, 17], and so forth.

Example 8. Let be a normed lattice. Denote by the positive cone of , that is, , and define as for all . Then is an aysmmetric norm on (see, e.g., [17, Theorem 3.1]), and thus the function defined by for all is a quasi-metric on , so is an ordered quasi-metric space. Hence is also an ordered quasi-metric space, where denotes the restriction of to .
We will show that the function defined by for all , is a -distance for . Indeed, first note that condition (W1) is trivially satisfied. Now fix and let be a sequence in such that for some . Since for all , we deduce that is lower semicontinuous for , and thus condition (W2) is satisfied. Finally, choose and put . Suppose and with . Therefore
Consequently condition () is also satisfied, so is a -distance for .

Definition 9. A preordered quasi-metric space is called complete if for each nondecreasing Cauchy sequence the following two conditions hold:() there exists satisfying ;() each satisfying verifies that for all .

Next we give some examples of complete preordered quasi-metric spaces.

Example 10. Let be any complete quasi-metric space. Then is obviously a complete preordered quasi-metric space.

Example 11. Let be a partial order on a set . Then, for every complete quasi-metric on such that if and only if , we have that is a complete ordered quasi-metric space (note that in this case the partial order coincides with the specialization order ). Indeed, let be a nondecreasing Cauchy sequence and let be such that . Choose any . Then, for each arbitrary there is such that . Since , we have , so by the triangle inequality, . Since is arbitrary we deduce that . Hence .

Example 12. Let be the ordered quasi-metric space of Example 7. If is a Cauchy sequence in that is also nondecreasing, then it is clear that only for . Therefore is a complete ordered quasi-metric space.

Example 13. Let and let be the complete quasi-metric on given by for all . Then is not a complete preordered quasi-metric space in our sense because any (nondecreasing Cauchy) sequence in satisfies , so condition () of Definition 9 does not hold. However, since if and only if , it follows from Example 11 that is a complete ordered quasi-metric space.

3. Fixed Point Results

Answering a question posed by Reich [18], Mizoguchi and Takahashi [19] (see also [20, 21]) obtained a set-valued generalization-improvement of the Rakotch fixed point theorem [22, Corollary of Theorem 2]. Recently, Latif and Al-Mezel [9, Theorem 2.3] extended Mizoguchi-Takahashi’s theorem to the framework of complete quasi-metric spaces by using -distances (actually they states their result in a slightly more general form by using -functions in the sense of [8], instead of -distances). Here we obtain a fixed point theorem for complete preordered quasi-metric spaces from which [9, Theorem 2.3] can be deduced as a special case. Several other consequences are deduced and some illustrative examples are given.

We first introduce the notions of contractiveness that we will use in the rest of the paper.

If is a quasi-metric space, we denote by the set of all nonempty subsets of and by the set of all nonempty -closed subsets of .

Definition 14. Let be a preordered quasi-metric space and let be a set-valued map such that for all . We say that is -contractive if there exist a -distance for and a constant , such that for each , with , and there is satisfying .

Definition 15. Let be a preordered quasi-metric space and let be a set-valued map such that for all . We say that is generalized -contractive if there exist a -distance for and a function with for all , and such that for each , with , and there is satisfying .

Theorem 16. Let be a complete preordered quasi-metric space and be a generalized -contractive set-valued map. Then has a fixed point.

Proof. Since is generalized -contractive, there is a -distance for and a function with for all , and such that for each , with , and there is satisfying
Fix . Since there exists such that . Taking and , we deduce the existence of an such that and
Repeating the above argument, there is such that and
Hence, following this process we construct a sequence in such that for every ,(a),(b), and(c).
Next we show that is a Cauchy sequence in the quasi-metric space .
To this end, first suppose that there is such that . Thus whenever , by conditions (c) and (W1). Then, from conditions (b) and (3) we deduce that is a Cauchy sequence in .
Now suppose that for all . Put , . Then is a strictly decreasing sequence of non-negative real numbers. Let be such that . Then
Hence there exist and such that for all . By condition (c) we deduce that for all . Now choose . Then, there is for which condition (3) follows. Since by (7) and (W1) there is such that whenever , we deduce from (3) that whenever . Therefore is a nondecreasing Cauchy sequence in .
Since is a complete preordered quasi-metric space, there exists such that and for all .
Next we show that .
Indeed, choose . Then, there is such that whenever . Given there is, by condition (W2), an such that
Thus
Therefore .
Finally, since for all , we can find a sequence in such that and for all . Hence . We deduce from (3) that . So because . This concludes the proof.

Corollary 17. Let be a complete preordered quasi-metric space and be a -contractive set-valued map. Then has a fixed point.

Corollary 18. Let be a complete preordered quasi-metric space for which d is a -distance and let be a self-map. If there is a function with for all , and such that for each , with , one has then has a fixed point.

Proof. Since is a topology, then for all . The result is now an immediate consequence of Theorem 16.

Remark 19. Putting and taking into account Example 5, we deduce that [9, Theorem 2.3] and [8, Theorem 6.1] are, for -distances, special cases of Theorem 16 and Corollary 17 respectively, whereas Corollary 18 provides a quasi-metric generalization of Rakotch’s fixed point theorem.

Next we give an easy example where Corollary 17, and hence Theorem 16, can be applied to the involved complete ordered quasi-metric space , but not to the complete ordered metric space .

Example 20. Let be the complete ordered quasi-metric space of Example 12 and let defined by for all . Since is a -distance for (see Example 7), and for each with , we have then all conditions of Corollary 17, and thus of Theorem 16, are satisfied. However, for with , we have so Theorem 16 cannot be applied to the complete ordered metric space and the self-map .

In the sequel we will apply Corollary 17 to deduce a fixed point result for set-valued maps on complete preordered quasi-metric spaces involving the (lower) Hausdorff distance.

Let be a quasi-metric space. For each let

Then , and will be called the lower Hausdorff distance of , the upper Hausdorff distance of and the Hausdorff distance of , respectively (compare e.g., [2326]).

It is interesting to note that , and are extended quasi-pseudometrics on , but not quasi-metrics, in general.

Corollary 21. Let be a complete preordered quasi-metric space for which is a -distance and let be a set-valued map such that for all . If there is such that for each , with , then has a fixed point.

Proof. Take . Then is a -contractive set-valued map for the -distance and the constant . By Corollary 17, has a fixed point.

Remark 22. Observe that for the ordered quasi-metric space , any set-valued map such that for all , satisfies that every is a fixed point of . Indeed, condition implies , so , that is, . Note also that the contraction condition (15) is, in this case, equivalent to the following:
We finish the paper with two examples that illustrate Corollary 21 and Remark 22, respectively.

Example 23. Let be the set of all continuous functions from into itself and let be the quasi-metric on defined as (compare [27, Example 4]):
Let be the usual pointwise partial order on , that is, if and only if for all . By standard arguments we deduce that is a complete ordered quasi-metric space: Indeed, given a nondecreasing Cauchy sequence in , then only for the function defined by for all .
Moreover is a -distance for because given we take , and then for , and , we obtain
Now construct the set-valued map given by
Note that . Indeed, suppose that there is . Then, there is a subsequence of such that for all . Since for all , we can assume, without loss of generality, that for all . Consequently, we have for each and each ,
Since , we deduce that for all , which contradicts that . We conclude that .
Moreover for all because for all and thus .
Finally, let , with , and . Then, there is such that for all . Taking for all , we have , , and
Hence
By Corollary 21, has a fixed point. In fact, the function defined by for all , satisfies .

Example 24. Consider the Banach lattice , where denotes the vector space of all infinite sequences of real numbers such that , denotes the usual order on and for all .
Now denote by the positive cone of and by the quasi-metric on defined by for all (compare Example 8). Then is a complete quasi-metric space by [28, Theorem 2].
Let be nondecreasing and such that for all . Define as for all . Then (compare Remark 22). In fact for all .
Finally note that given with and , we have that and , so , and hence
Since , we deduce that condition (16) of Remark 22 is also satisfied.

Acknowledgments

The authors thank the referees for some useful suggestions and corrections. This research is supported by the Ministry of Economy and Competitiveness of Spain, Grant MTM2012-37894-C02-01. S. Romaguera and P. Tirado also acknowledge the support of Universitat Politècnica de València, Grant PAID-06-12.

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