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.