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.
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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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F493298/MediaObjects/13663_2009_Article_1292_Equ1_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F493298/MediaObjects/13663_2009_Article_1292_Equ2_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F493298/MediaObjects/13663_2009_Article_1292_Equ3_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F493298/MediaObjects/13663_2009_Article_1292_Equ4_HTML.gif)
for all
Now, for each define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F493298/MediaObjects/13663_2009_Article_1292_Equ5_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F493298/MediaObjects/13663_2009_Article_1292_Equ6_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F493298/MediaObjects/13663_2009_Article_1292_Equ7_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F493298/MediaObjects/13663_2009_Article_1292_Equ8_HTML.gif)
and for each we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F493298/MediaObjects/13663_2009_Article_1292_Equ9_HTML.gif)
Therefore is a Caristi
-mapping on
without fixed point, a contradiction. This concludes the proof.
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Acknowledgments
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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Romaguera, S. A Kirk Type Characterization of Completeness for Partial Metric Spaces. Fixed Point Theory Appl 2010, 493298 (2009). https://doi.org/10.1155/2010/493298
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DOI: https://doi.org/10.1155/2010/493298