© J. Marín et al. 2011
Received: 14 December 2010
Accepted: 31 January 2011
Published: 28 February 2011
We discuss several properties of -functions in the sense of Al-Homidan et al.. In particular, we prove that the partial metric induced by any weighted quasipseudometric space is a -function and show that both the Sorgenfrey line and the Kofner plane provide significant examples of quasimetric spaces for which the associated supremum metric is a -function. In this context we also obtain some fixed point results for multivalued maps by using Bianchini-Grandolfi gauge functions.
1. Introduction and Preliminaries
Kada et al. introduced in  the concept of -distance on a metric space and extended the Caristi-Kirk fixed point theorem , the Ekeland variation principle  and the nonconvex minimization theorem , for -distances. Recently, Al-Homidan et al. introduced in  the notion of -function on a quasimetric space and then successfully obtained a Caristi-Kirk-type fixed point theorem,a Takahashi minimization theorem, an equilibrium version of Ekeland-type variational principle, and a version of Nadler's fixed point theorem for a - function on a complete quasimetric space, generalizing in this way, among others, the main results of  because every -distance is, in fact, a -function. This interesting approach has been continued by Hussain et al. , and by Latif and Al-Mezel , respectively. In particular, the authors of  have obtained a nice Rakotch-type theorem for -functions on complete quasimetric spaces.
In Section 2 of this paper, we generalize the basic theory of -functions to quasipseudometric spaces. Our approach is motivated, in part, by the fact that in many applications to Domain Theory, Complexity Analysis, Computer Science and Asymmetric Functional Analysis, quasipseudometric spaces (in particular, weightable quasipseudometric spaces and their equivalent partial metric spaces) rather than quasimetric spaces, play a crucial role (cf. [8–23], etc.). In particular, we prove that for every weighted quasipseudometric space the induced partial metric is a -function. We also show that the Sorgenfrey line and the Kofner plane provide interesting examples of quasimetric spaces for which the associated supremum metric is a -function. Finally, Section 3 is devoted to present a new fixed point theorem for -functions and multivalued maps on quasipseudometric spaces, by using Bianchini-Grandolfi gauge functions in the sense of . Our result generalizes and improves, in several ways, well-known fixed point theorems.
Next we recall several pertinent concepts.
We start this section by giving the main concept of this paper, which was introduced in  for quasimetric spaces.
(Q2′) is lower semicontinuous for all , then is called a w-distance on (cf. ).
However, the situation is very different in the quasimetric case. Indeed, it is obvious that if is a qpm space, then satisfies conditions (Q1) and (Q2), whereas Example 3.2 of  shows that there exists a qpm space such that does not satisfy condition (Q3), and hence it is not a Q-function on . In this direction, we next present some positive results.
It follows from Proposition 2.3 that many paradigmatic quasimetrizable topological spaces , as the Sorgenfrey line, the Michael line, the Niemytzki plane and the Kofner plane (see ), do not admit any compatible quasimetric which is a -function on .
In the sequel, we show that, nevertheless, it is possible to construct an easy but, in several cases, useful -function on any quasimetric space, as well as a suitable -functions on any weightable qpm space.
Now suppose that is a sequence in that -converges to some , and let and such that , for all . If , then . If , we deduce that , for all . Since , it follows that , so , and thus . Hence, condition (Q2) is also satisfied.
On the set of real numbers define as if , and if . Then, is a quasimetric on and the topological space is the celebrated Sorgenfrey line. Since is the discrete metric on , it follows from Proposition 2.5 that is a -function on .
The quasimetric on the plane , constructed in Example 7.7 of , verifies that is the so-called Kofner plane and that is the discrete metric on , so, by Proposition 2.5, is a -function on .
Matthews introduced in  the notion of a weightable qpm space (under the name of a "weightable quasimetric space"), and its equivalent partial metric space, as a part of the study of denotational semantics of dataflow networks.
A qpm space is called weightable if there exists a function such that for all . In this case, we say that is a weightable qpm on . The function is said to be a weighting function for and the triple is called a weighted qpm space.
The domain of words, the interval domain, and the complexity quasimetric space provide distinguished examples of theoretical computer science that admit a structure of a weightable qpm space and, thus, of a partial metric space (see, e.g., [14, 20, 21]).
3. Fixed Point Results
Following Al-Homidan et al. [5, Definition 6.1] if is a quasimetric space, we say that a multivalued map is -contractive if there exists a -function on and such that for each and there is satisfying .
Latif and Al-Mezel (see ) generalized this notion as follows.
Then, they proved the following improvement of the celebrated Rakotch fixed point theorem (see ).
Theorem 3.1 (Lafit and Al-Mezel [7, Theorem 2.3]).
On the other hand, Bianchini and Grandolfi proved in  the following fixed point theorem.
Theorem 3.2 (Bianchini and Grandolfi ).
It is easy to check (see [30, Page 8]) that if is a Bianchini-Grandolfi gauge function, then , for all , and hence .
Since , it follows that , and thus . So, by Lemma 2.2, is a Cauchy sequence in (in fact, it is a Cauchy sequence in . Let such that . Given , there is such that , for all . By applying condition (Q2), we deduce that , so . Since , it follows from condition (Q1) that . Therefore, , for all , by condition (Q3). We conclude that , and thus .
The next example illustrates Theorem 3.3.
We have checked that conditions of Theorem 3.3 are fulfilled, and hence, there is with . In fact is the only point of satisfying and (actually . The following consequence of Theorem 3.3, which is also illustrated by Example 3.4, improves and generalizes in several directions the Banach Contraction Principle for partial metric spaces obtained in Theorem 5.3 of .
Observe that if is a nondecreasing function such that , for all , then the function given by , is a Bianchini-Grandolfi gauge function (compare [31, Proposition 8]). Therefore, the following variant of Theorem 3.1, which improves Corollary 2.4 of , is now a consequence of Theorem 3.3.
The authors thank one of the reviewers for suggesting the inclusion of a concrete example to which Theorem 3.3 applies. They acknowledge the support of the Spanish Ministry of Science and Innovation, Grant no. MTM2009-12872-C02-01.
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