A Kirk Type Characterization of Completeness for Partial Metric Spaces
© Salvador Romaguera. 2010
Received: 1 October 2009
Accepted: 25 November 2009
Published: 7 December 2009
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  that if is a selfmapping of a complete metric space such that there is a lower semicontinuous function satisfying
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
In particular, Kirk proved in  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  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  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.
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.
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.
Let be a partial metric space. Then, for each the function given by is lower semicontinuous for
This yields because
A partial metric space is 0-complete if and only if every -Caristi mapping on has a fixed point.
Now, for each define
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
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
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
Therefore is a Caristi -mapping on without fixed point, a contradiction. This concludes the proof.
The author is very grateful to the referee for his/her useful suggestions. This work was partially supported by the Spanish Ministry of Science and Innovation, and FEDER, Grant MTM2009-12872-C02-01.
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