Fixed Point for Partial Metric Spaces

Abstract

We consider the partial metric on a set , define -fixed point for maps, and obtain some sufficient and necessary conditions on that, also we obtain some sufficient and necessary theorems on -fixed point.

1. Introduction

The partial metric spaces were introduced in [1] as a part of the study of denotational semantics of dataflow networks. In particular, he established the precise rela- tionship between partial metric spaces and the so-called weightable quasimetric spaces and proved a partial metric generalization of Banach contraction mapping theorem.

A partial metric [1] on a set is a function such that, for all (1); (2); (3); (4).

A partial metric space is a pair , where is a partial metric on . Each partial metric on induces a topology on which has as a base the family of open balls , where for all and .

If is a partial metric on , then the function given by is a metric on .

A sequence in a partial metric space is called a Cauchy sequence if there exists (and is finite) [1, Definition 5.2].

Note that is a Cauchy sequence in if and only if it is a Cauchy sequence in the metric space [1, page 194].

A partial metric space is said to be complete if every Cauchy sequence in converges, with respect to , to a point such that [1, Definition 5.3].

It is well known and easy to see that a partial metric space is complete if and only if the metric space is complete.

In [2], S. J. O’Neill proposed one significant change to Matthews definition of the partial metrics, and that was to extend their range from to . In the following, partial metrics in the O’Neill sense will be called dualistic partial metrics and a pair such that is a nonempty set and is a dualistic partial metric on will be called a dualistic partial metric space.

2. -Fixed Point

Our basic references for quasi-metric spaces are [3, 4]. In our context, by a quasi-metric on a set we mean a nonnegative real-valued function d on such that, for all :(i), (ii).

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

Each quasi-metric on generates a -topology on which has as a base the family of open -balls , where for all and .

If is a quasi-metric on , then the function defined on by is a metric on .

Lemma 2.1. If is a dualistic partial metric space, then the function defined by , is a quasi-metric on such that .

Definition 2.2. Let be a dualistic partial metric space and let be a map. Then is -fixed point for if

Theorem 2.3. Let be a dualistic partial metric space and let be a map, , and . If as for some , then has an -fixed point.

Proof. Since as , , Then . Therefore is an -fixed point of .

Theorem 2.4. Let be a dualistic partial metric space and let be a map also for all , then has an fixed point in partial metric. Moreover, if are -fixed points of , then .

Proof. Suppose , then Therefore as . From Theorem 2.3, has an -fixed point and Since Then .

Theorem 2.5. Let be a mapping of a dualistic partial metric space into itself such that for all . Then has an -fixed point, for all .

Proof. Fix , then it is clear that, for each , also We deduce that Hence
Therefore, for ,
Similarly, we obtain that Then Therefore, has an -fixed point.

Example 2.6. Let , and let p be the dualistic metric on given by for all .

Let be the mapping from into itself defined by , for all . It is immediate to see that for all . However, has no fixed point, of course. But by the Theorem 2.5, for every , has an -fixed point. That is, there exists such that since

Theorem 2.7. Let be a mapping of a dualistic partial metric space into itself such that where .
If is an -fixed point for , then is an -fixed point for .

Proof. We have therefore Since , Since is an -fixed point for , then . So is an -fixed point for .

Theorem 2.8. Let be a dualistic partial metric, and let be a mapping and . If and . then has -fixed point. Moreover, if are -fixed points of , then .

Proof. We have Therefore also so and for every , we have Thus, since ,    as . Now, by Theorem 2.3, has an -fixed point and since Then .

Corollary 2.9. Let be a dualistic partial metric, and let be a mapping and . If and , then has an -fixed point.

References

1. S. G. Matthews, “Partial metric topology,” in Proceedings of the 11th Summer Conference on General Topology and Applications, vol. 728, pp. 183–197, The New York Academy of Sciences, Gorham, Me, USA, Augusts 1995. View at: Google Scholar
2. S. J. O'Neill, “Partial metrics, valuations and domain theory,” in Proceedings of the 11th Summer Conference on General Topology and Applications, vol. 806, pp. 304–315, The New York Academy of Sciences, Gorham, Me, USA, Augusts 1995. View at: Google Scholar
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4. H. P. A. Kunzi, “Nonsymmetric distances and their associated topologies: about the origins of basic ideas in the area of asymmetric topology,” in Handbook of the History of General Topology, vol. 3, pp. 853–968, Kluwer Academic, Dordrecht, The Netherlands, 2001. View at: Google Scholar | Zentralblatt MATH

Copyright

Copyright © 2011 S. A. M. Mohsenalhosseini 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.