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-Functions on Quasimetric Spaces and Fixed Points for Multivalued Maps
Fixed Point Theory and Applications volume 2011, Article number: 603861 (2011)
Abstract
We discuss several properties of -functions in the sense of Al-Homidan et al.. In particular, we prove that the partial metric induced by any
weighted quasipseudometric space is a
-function and show that both the Sorgenfrey line and the Kofner plane provide significant examples of quasimetric spaces for which the associated supremum metric is a
-function. In this context we also obtain some fixed point results for multivalued maps by using Bianchini-Grandolfi gauge functions.
1. Introduction and Preliminaries
Kada et al. introduced in [1] the concept of -distance on a metric space and extended the Caristi-Kirk fixed point theorem [2], the Ekeland variation principle [3] and the nonconvex minimization theorem [4], for
-distances. Recently, Al-Homidan et al. introduced in [5] the notion of
-function on a quasimetric space and then successfully obtained a Caristi-Kirk-type fixed point theorem,a Takahashi minimization theorem, an equilibrium version of Ekeland-type variational principle, and a version of Nadler's fixed point theorem for a
- function on a complete quasimetric space, generalizing in this way, among others, the main results of [1] because every
-distance is, in fact, a
-function. This interesting approach has been continued by Hussain et al. [6], and by Latif and Al-Mezel [7], respectively. In particular, the authors of [7] have obtained a nice Rakotch-type theorem for
-functions on complete quasimetric spaces.
In Section 2 of this paper, we generalize the basic theory of -functions to
quasipseudometric spaces. Our approach is motivated, in part, by the fact that in many applications to Domain Theory, Complexity Analysis, Computer Science and Asymmetric Functional Analysis,
quasipseudometric spaces (in particular, weightable
quasipseudometric spaces and their equivalent partial metric spaces) rather than quasimetric spaces, play a crucial role (cf. [8–23], etc.). In particular, we prove that for every weighted
quasipseudometric space the induced partial metric is a
-function. We also show that the Sorgenfrey line and the Kofner plane provide interesting examples of quasimetric spaces for which the associated supremum metric is a
-function. Finally, Section 3 is devoted to present a new fixed point theorem for
-functions and multivalued maps on
quasipseudometric spaces, by using Bianchini-Grandolfi gauge functions in the sense of [24]. Our result generalizes and improves, in several ways, well-known fixed point theorems.
Throughout this paper the letter and
will denote the set of positive integer numbers and the set of nonnegative integer numbers, respectively.
Our basic references for quasimetric spaces are [25, 26].
Next we recall several pertinent concepts.
By a quasipseudometric on a set
, we mean a function
such that for all
,
(i),
(ii).
A quasipseudometric
on
that satisfies the stronger condition
(i′)
is called a quasimetric on .
We remark that in the last years several authors used the term "quasimetric" to refer to a quasipseudometric and the term "
quasimetric" to refer to a quasimetric in the above sense.
In the following we will simply write qpm instead of
quasipseudometric if no confusion arises.
A qpm space is a pair
such that
is a set and
is a
qpm on
. If
is a quasimetric on
, the pair
is then called a quasimetric space.
Given a qpm
on a set
, the function
defined by
, is also a
qpm on
, called the conjugate of
, and the function
defined by
is a metric on
, called the supremum metric associated to
.
Thus, every qpm
on
induces, in a natural way, three topologies denoted by
,
and
, respectively, and defined as follows.
(i) is the
topology on
which has as a base the family of
-open balls
, where
, for all
and
.
(ii) is the
topology on
which has as a base the family of
-open balls
, where
, for all
and
.
(iii) is the topology on
induced by the metric
.
Note that if is a quasimetric on
, then
is also a quasimetric, and
and
are
topologies on
.
Note also that a sequence in a
qpm space
is
-convergent (resp.,
-convergent) to
if and only if
(resp.,
.
It is well known (see, for instance, [26, 27]) that there exists many different notions of completeness for quasimetric spaces. In our context we will use the following notion.
A qpm space
is said to be complete if every Cauchy sequence is
-convergent, where a sequence
is called Cauchy if for each
there exists
such that
whenever
.
In this case, we say that is a complete
qpm on
.
2.
-Functions on
qpm-Spaces
We start this section by giving the main concept of this paper, which was introduced in [5] for quasimetric spaces.
Definition 2.1.
A -function on a
qpm space
is a function
satisfying the following conditions:
(Q1), for all
,
(Q2) if , and
is a sequence in
that
-converges to a point
and satisfies
, for all
, then
,
(Q3) for each there exists
such that
and
imply
.
If is a metric space and
satisfies conditions (Q1) and (Q3) above and the following condition:
(Q2′) is lower semicontinuous for all
, then
is called a w-distance on
(cf. [1]).
Clearly is a
-distance on
whenever
is a metric on
.
However, the situation is very different in the quasimetric case. Indeed, it is obvious that if is a
qpm space, then
satisfies conditions (Q1) and (Q2), whereas Example 3.2 of [5] shows that there exists a
qpm space
such that
does not satisfy condition (Q3), and hence it is not a Q-function on
. In this direction, we next present some positive results.
Lemma 2.2.
Let q be a Q-function on a qpm space
. Then, for each
, there exists
such that
and
imply
.
Proof.
By condition (Q3), . Interchanging
and
, it follows that
, so
.
Proposition 2.3.
Let be a
qpm space. If
is a Q-function on
, then
, and hence,
is a metrizable topology on
.
Proof.
Let be a sequence in
which is
-convergent to some
. Then, by Lemma 2.2,
. We conclude that
.
Remark 2.4.
It follows from Proposition 2.3 that many paradigmatic quasimetrizable topological spaces , as the Sorgenfrey line, the Michael line, the Niemytzki plane and the Kofner plane (see [25]), do not admit any compatible quasimetric
which is a
-function on
.
In the sequel, we show that, nevertheless, it is possible to construct an easy but, in several cases, useful -function on any quasimetric space, as well as a suitable
-functions on any weightable
qpm space.
Recall that the discrete metric on a set is the metric
on
defined as
, for all
, and
, for all
with
.
Proposition 2.5.
Let be a quasimetric space. Then, the discrete metric on
is a
-function on
.
Proof.
Since is a metric it obviously satisfies condition (Q1) of Definition 2.1.
Now suppose that is a sequence in
that
-converges to some
, and let
and
such that
, for all
. If
, then
. If
, we deduce that
, for all
. Since
, it follows that
, so
,
and thus
. Hence, condition (Q2) is also satisfied.
Finally, satisfies condition (Q3) taking
for every
Example 2.6.
On the set of real numbers define
as
if
, and
if
. Then,
is a quasimetric on
and the topological space
is the celebrated Sorgenfrey line. Since
is the discrete metric on
, it follows from Proposition 2.5 that
is a
-function on
.
Example 2.7.
The quasimetric on the plane
, constructed in Example 7.7 of [25], verifies that
is the so-called Kofner plane and that
is the discrete metric on
, so, by Proposition 2.5,
is a
-function on
.
Matthews introduced in [14] the notion of a weightable qpm space (under the name of a "weightable quasimetric space"), and its equivalent partial metric space, as a part of the study of denotational semantics of dataflow networks.
A qpm space
is called weightable if there exists a function
such that for all
. In this case, we say that
is a weightable
qpm on
. The function
is said to be a weighting function for
and the triple
is called a weighted
qpm space.
A partial metric on a set is a function
such that, for all
:
(i),
(ii),
(iii),
(iv).
A partial metric space is a pair such that
is a set and
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
.
The precise relationship between partial metric spaces and weightable qpm spaces is provided in the next result.
Theorem 2.8 (Matthews [14]).
-
(a)
Let
be a weightable
qpm space with weighting function. Then, the function
defined by
, for all
,
is a partial metric on
. Furthermore
.
-
(b)
Conversely, let
be a partial metric space. Then, the function
defined by
, for all
is a weightable
qpm on
with weighting function
given by
for all
. Furthermore
.
Remark 2.9.
The domain of words, the interval domain, and the complexity quasimetric space provide distinguished examples of theoretical computer science that admit a structure of a weightable qpm space and, thus, of a partial metric space (see, e.g., [14, 20, 21]).
Proposition 2.10.
Let be a weighted
qpm space. Then, the induced partial metric
is a Q-function on
.
Proof.
We will show that satisfies conditions (Q1), (Q2), and Q(3) of Definition 2.1.
(Q1) Let , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F603861/MediaObjects/13663_2010_Article_1414_Equ1_HTML.gif)
(Q2)Let be a sequence in
which is
-convergent to some
. Let
and
such that
, for all
.
Choose . Then, there exists
such that
, for all
. Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F603861/MediaObjects/13663_2010_Article_1414_Equ2_HTML.gif)
Since is arbitrary, we conclude that
.
(Q3) Given , put
. If
and
, it follows
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F603861/MediaObjects/13663_2010_Article_1414_Equ3_HTML.gif)
3. Fixed Point Results
Given a qpm space
, we denote by
the collection of all nonempty subsets of
, by
the collection of all nonempty
-closed subsets of
, and by
the collection of all nonempty
-closed subsets of
.
Following Al-Homidan et al. [5, Definition 6.1] if is a quasimetric space, we say that a multivalued map
is
-contractive if there exists a
-function
on
and
such that for each
and
there is
satisfying
.
Latif and Al-Mezel (see [7]) generalized this notion as follows.
If is a quasimetric space, we say that a multivalued map
is generalized
-contractive if there exists a
-function
on
such that for each
and
there is
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F603861/MediaObjects/13663_2010_Article_1414_Equ4_HTML.gif)
where is a function such that
for all
.
Then, they proved the following improvement of the celebrated Rakotch fixed point theorem (see [28]).
Theorem 3.1 (Lafit and Al-Mezel [7, Theorem 2.3]).
Let be a complete quasimetric space. Then, for each generalized q-contractive multivalued map
there exists
such that
.
On the other hand, Bianchini and Grandolfi proved in [29] the following fixed point theorem.
Theorem 3.2 (Bianchini and Grandolfi [29]).
Let be a complete metric space and let
be a map such that for each
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F603861/MediaObjects/13663_2010_Article_1414_Equ5_HTML.gif)
where is a nondecreasing function satisfying
, for all
(
denotes the nth iterate of
). Then,
has a unique fixed point.
A function satisfying the conditions of the preceding theorem is called a Bianchini-Grandolfi gauge function (cf [24, 30]).
It is easy to check (see [30, Page 8]) that if is a Bianchini-Grandolfi gauge function, then
, for all
, and hence
.
Our next result generalizes Bianchini-Grandolfi's theorem for Q-functions on complete qpm spaces.
Theorem 3.3.
Let be a complete
qpm space, q a Q-function on
, and
a multivalued map such that for each
and
, there is
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F603861/MediaObjects/13663_2010_Article_1414_Equ6_HTML.gif)
where is a Bianchini-Grandolfi gauge function. Then, there exists
such that
and
.
Proof.
Fix and let
. By hypothesis, there exists
such that
. Following this process, we obtain a sequence
with
and
, for all
. Therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F603861/MediaObjects/13663_2010_Article_1414_Equ7_HTML.gif)
for all .
Now, choose . Let
for which condition (Q3) is satisfied. We will show that there is
such that
whenever
.
Indeed, if , then
and thus
, for all
, so, by condition (Q1),
whenever
.
If ,
, so there is
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F603861/MediaObjects/13663_2010_Article_1414_Equ8_HTML.gif)
Then, for , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F603861/MediaObjects/13663_2010_Article_1414_Equ9_HTML.gif)
In particular, and
whenever
, so, by Lemma 2.2,
whenever
.
We have proved that is a Cauchy sequence in
(in fact, it is a Cauchy sequence in the metric space
. Since
is complete there exists
such that
.
Next, we show that .
To this end, we first prove that . Indeed, choose
. Fix
. Since
whenever
, it follows from condition (Q2) that
whenever
.
Now for each take
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F603861/MediaObjects/13663_2010_Article_1414_Equ10_HTML.gif)
If , it follows that
. Otherwise we obtain
.
Hence, , and by Lemma 2.2,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F603861/MediaObjects/13663_2010_Article_1414_Equ11_HTML.gif)
Therefore, .
It remains to prove that .
Since , we can construct a sequence
in
such that
,
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F603861/MediaObjects/13663_2010_Article_1414_Equ12_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F603861/MediaObjects/13663_2010_Article_1414_IEq434_HTML.gif)
Since , it follows that
, and thus
. So, by Lemma 2.2,
is a Cauchy sequence in
(in fact, it is a Cauchy sequence in
. Let
such that
. Given
, there is
such that
, for all
. By applying condition (Q2), we deduce that
, so
. Since
, it follows from condition (Q1) that
. Therefore,
, for all
, by condition (Q3). We conclude that
, and thus
.
The next example illustrates Theorem 3.3.
Example 3.4.
Let and let
be the
qpm on
given by
. It is well known that
is weightable with weighting function
given by
, for all
. Let
be partial metric induced by
. Then,
is a
-function on
by Proposition 2.10. Note also that, by Theorem 2.8 (a),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F603861/MediaObjects/13663_2010_Article_1414_Equ13_HTML.gif)
for all . Moreover
is clearly complete because
is the Euclidean metric on
and thus
is a compact metric space.
Now define by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F603861/MediaObjects/13663_2010_Article_1414_Equ14_HTML.gif)
for all . Note that
because the nonempty
-closed subsets of
are the intervals of the form
,
.
Let be such that
, for all
, and
, for all
. We wish to show that
is a Bianchini-Grandolfi gauge function.
It is clear that is nondecreasing.
Moreover, , for all
. Indeed, if
we have
whenever
, while for
, we have
so,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F603861/MediaObjects/13663_2010_Article_1414_Equ15_HTML.gif)
and following this process we deduce the known fact that , for all
. We have shown that
is a Bianchini-Grandolfi gauge function.
Finally, for each and
, there exists
such that
. Choose
. Then
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F603861/MediaObjects/13663_2010_Article_1414_Equ16_HTML.gif)
If , then
, and thus
.
We have checked that conditions of Theorem 3.3 are fulfilled, and hence, there is with
. In fact
is the only point of
satisfying
and
(actually
. The following consequence of Theorem 3.3, which is also illustrated by Example 3.4, improves and generalizes in several directions the Banach Contraction Principle for partial metric spaces obtained in Theorem 5.3 of [14].
Corollary 3.5.
Let be a partial metric space such that the induced weightable
qpm
is complete and let
be a multivalued map such that for each
and
, there is
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F603861/MediaObjects/13663_2010_Article_1414_Equ17_HTML.gif)
where is a Bianchini-Grandolfi gauge function. Then, there exists
such that
and
.
Proof.
Since (see Theorem 2.8), we deduce from Proposition 2.10 that
is a
-function for the complete (weightable)
qpm space
. The conclusion follows from Theorem 3.3.
Observe that if is a nondecreasing function such that
, for all
, then the function
given by
, is a Bianchini-Grandolfi gauge function (compare [31, Proposition 8]). Therefore, the following variant of Theorem 3.1, which improves Corollary 2.4 of [7], is now a consequence of Theorem 3.3.
Corollary 3.6.
Let be a complete
qpm space. Then, for each generalized q-contractive multivalued map
with q nondecreasing, there exists
such that
and
.
Remark 3.7.
The proof of Theorem 3.3 shows that the condition that is complete can be replaced by the more general condition that every Cauchy sequence in the metric space
is
-convergent.
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Acknowledgments
The authors thank one of the reviewers for suggesting the inclusion of a concrete example to which Theorem 3.3 applies. They acknowledge the support of the Spanish Ministry of Science and Innovation, Grant no. MTM2009-12872-C02-01.
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Marín, J., Romaguera, S. & Tirado, P. -Functions on Quasimetric Spaces and Fixed Points for Multivalued Maps.
Fixed Point Theory Appl 2011, 603861 (2011). https://doi.org/10.1155/2011/603861
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DOI: https://doi.org/10.1155/2011/603861