A Kirk Type Characterization of Completeness for Partial Metric Spaces

Abstract

We extend the celebrated result of W. A. Kirk that a metric space is complete if and only if every Caristi self-mapping for has a fixed point, to partial metric spaces.

1. Introduction and Preliminaries

Caristi proved in [1] that if is a selfmapping of a complete metric space such that there is a lower semicontinuous function satisfying

(1.1)

for all then has a fixed point.

This classical result suggests the following notion. A selfmapping of a metric space for which there is a function satisfying the conditions of Caristi's theorem is called a Caristi mapping for

There exists an extensive and well-known literature on Caristi's fixed point theorem and related results (see, e.g., [2–10], etc.).

In particular, Kirk proved in [7] that a metric space is complete if and only if every Caristi mapping for has a fixed point. (For other characterizations of metric completeness in terms of fixed point theory see [11–14], etc., and also [15, 16] for recent contributions in this direction.)

In this paper we extend Kirk's characterization to a kind of complete partial metric spaces.

Let us recall that partial metric spaces were introduced by Matthews in [17] as a part of the study of denotational semantics of dataflow networks. In fact, it is widely recognized that partial metric spaces play an important role in constructing models in the theory of computation (see [18–25], etc.).

A partial metric [17] on a set is a function such that for all : (i) ; (ii) ; (iii) ; (iv)

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

Next we give some pertinent concepts and facts on completeness for partial metric spaces.

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) ([17, Definition ])

Note that is a Cauchy sequence in if and only if it is a Cauchy sequence in the metric space (see, e.g., [17, 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 ([17, Definition ]).

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 order to give an appropriate notion of a Caristi mapping in the framework of partial metric spaces, we naturally propose the following two alternatives.

(i)A selfmapping of a partial metric space is called a -Caristi mapping on if there is a function which is lower semicontinuous for and satisfies , for all

(ii)A selfmapping of a partial metric space is called a -Caristi mapping on if there is a function which is lower semicontinuous for and satisfies , for all

It is clear that every -Caristi mapping is -Caristi but the converse is not true, in general.

In a first attempt to generalize Kirk's characterization of metric completeness to the partial metric framework, one can conjecture that a partial metric space is complete if and only if every -Caristi mapping on has a fixed point.

The following easy example shows that this conjecture is false.

Example 1.1.

On the set of natural numbers construct the partial metric given by

(1.2)

Note that is not complete, because the metric induces the discrete topology on , and is a Cauchy sequence in . However, there is no -Caristi mappings on as we show in the next.

Indeed, let and suppose that there is a lower semicontinuous function from into such that for all If we have , which means that for any so by lower semicontinuity of which contradicts condition Therefore which again contradicts condition We conclude that is not a -Caristi mapping on

Unfortunately, the existence of fixed point for each -Caristi mapping on a partial metric space neither characterizes completeness of as follows from our discussion in the next section.

2. The Main Result

In this section we characterize those partial metric spaces for which every -Caristi mapping has a fixed point in the style of Kirk's characterization of metric completeness. This will be done by means of the notion of a 0-complete partial metric space which is introduced as follows.

Definition 2.1.

A sequence in a partial metric space is called 0-Cauchy if We say that is 0-complete if every 0-Cauchy sequence in converges, with respect to to a point such that

Note that every 0-Cauchy sequence in is Cauchy in and that every complete partial metric space is 0-complete.

On the other hand, the partial metric space where denotes the set of rational numbers and the partial metric is given by provides a paradigmatic example of a 0-complete partial metric space which is not complete.

In the proof of the "only if" part of our main result we will use ideas from [11, 26], whereas the following auxiliary result will be used in the proof of the "if" part.

Lemma 2.2.

Let be a partial metric space. Then, for each the function given by is lower semicontinuous for

Proof.

Assume that then

(2.1)

This yields because

Theorem 2.3.

A partial metric space is 0-complete if and only if every -Caristi mapping on has a fixed point.

Proof.

Suppose that is 0-complete and let be a -Caristi mapping on then, there is a which is lower semicontinuous function for and satisfies

(2.2)

for all

Now, for each define

(2.3)

Observe that because Moreover is closed in the metric space since is lower semicontinuous for .

Fix Take such that Clearly . Hence, for each we have

(2.4)

Following this process we construct a sequence in such that its associated sequence of closed subsets in satisfies

(i) for all

(ii) for all

Since and, by (i) and (ii), for all it follows that so is a 0-Cauchy sequence in and by our hypothesis, there exists such that and thus Therefore

Finally, we show that To this end, we first note that

(2.5)

for all Consequently so by (ii), for all Since and it follows that Hence since so

Conversely, suppose that there is a 0-Cauchy sequence of distinct points in which is not convergent in Construct a subsequence of such that for all

Put and define by if and for all

Observe that is closed in

Now define by if and for all

Note that for all and that for all

From this fact and the preceding lemma we deduce that is lower semicontinuous for

Moreover, for each we have

(2.6)

and for each we have

(2.7)

Therefore is a Caristi -mapping on without fixed point, a contradiction. This concludes the proof.