Abstract
In the setting of quasimetric spaces, we prove some new results on the existence of fixed points for contractive type maps with respect to -function. Our results either improve or generalize many known results in the literature.
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1. Introduction and Preliminaries
Let be a metric space with metric
. We use
to denote the collection of all nonempty subsets of
for the collection of all nonempty closed subsets of
for the collection of all nonempty closed bounded subsets of
and
for the Hausdorff metric on
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F178306/MediaObjects/13663_2010_Article_1382_Equ1_HTML.gif)
where is the distance from the point
to the subset
For a multivalued map , we say
is contraction [1] if there exists a constant
, such that for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F178306/MediaObjects/13663_2010_Article_1382_Equ2_HTML.gif)
T is weakly contractive [2] if there exist constants
, such that for any
, there is
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F178306/MediaObjects/13663_2010_Article_1382_Equ3_HTML.gif)
where .
A point is called a fixed point of a multivalued map
if
. We denote
A sequence in
is called an
of
at
if
for all integer
. A real valued function
on
is called lower semicontinuous if for any sequence
with
implies that
Using the Hausdorff metric, Nadler Jr. [1] has established a multivalued version of the well-known Banach contraction principle in the setting of metric spaces as follows.
Theorem 1.1.
Let be a complete metric space, then each contraction map
has a fixed point.
Without using the Hausdorff metric, Feng and Liu [2] generalized Nadler's contraction principle as follows.
Theorem 1.2.
Let be a complete metric space and let
be a weakly contractive map, then
has a fixed point in
provided the real valued function
on
is a lower semicontinuous.
In [3], Kada et al. introduced the concept of -distance in the setting of metric spaces as follows.
A function is called a
-distance on
if it satisfies the following:
(w1) for all
(w2) is lower semicontinuous in its second variable;
(w3)for any there exists
, such that
and
imply
Note that in general for ,
and not either of the implications
necessarily holds. Clearly, the metric
is a
-distance on
. Many other examples and properties of
-distances are given in [3].
In [4], Suzuki and Takahashi improved Nadler contraction principle (Theorem 1.1) as follows.
Theorem 1.3.
Let be a complete metric space and let
. If there exist a
-distance
on
and a constant
, such that for each
and
, there is
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F178306/MediaObjects/13663_2010_Article_1382_Equ4_HTML.gif)
then has a fixed point.
Recently, Latif and Albar [5] generalized Theorem 1.2 with respect to -distance (see, Theorem
in [5]), and Latif [6] proved a fixed point result with respect to
-distance ( see, Theorem
in [6]) which contains Theorem 1.3 as a special case.
A nonempty set together with a quasimetric
(i.e., not necessarily symmetric) is called a quasimetric space. In the setting of a quasimetric spaces, Al-Homidan et al. [7] introduced the concept of a
-function on quasimetric spaces which generalizes the notion of a
-distance.
A function is called a
-function on
if it satisfies the following conditions:
(Q1) for all
(Q2)If is a sequence in
such that
and for
,
for some
, then
(Q3)for any there exists
, such that
and
imply
Note that every -distance is a
-function, but the converse is not true in general [7]. Now, we state some useful properties of
-function as given in [7].
Lemma 1.4.
Let be a complete quasimetric space and let
be a
-function on
. Let
and
be sequences in
. Let
and
be sequences in
converging to
, then the following hold for any
:
(i)if and
for all
then
in particular, if
and
, then
(ii)if and
for all
then
converges to
;
(iii)if for any
with
then
is a Cauchy sequence;
(iv)if for any
then
is a Cauchy sequence.
Using the concept -function, Al-Homidan et al. [7] recently studied an equilibrium version of the Ekeland-type variational principle. They also generalized Nadler's fixed point theorem (Theorem 1.1) in the setting of quasimetric spaces as follows.
Theorem 1.5.
Let be a complete quasimetric space and let
. If there exist
-function
on
and a constant
, such that for each
and
, there is
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F178306/MediaObjects/13663_2010_Article_1382_Equ5_HTML.gif)
then has a fixed point.
In the sequel, we consider as a quasimetric space with quasimetric
.
Considering a multivalued map , we say
-
(c)
is weakly
-contractive if there exist
- function
on
and constants
,
, such that for any
, there is
satisfying
(1.6)
where and
-
(d)
is generalized
-contractive if there exists a
- function
on
, such that for each
and
, there is
satisfying
(1.7)
where is a function of
to
, such that
for all
Clearly, t he class of weakly- contractive maps contains the class of weakly contractive maps, and the class of generalized
-contractive maps contains the classes of generalized
-contraction maps [6],
-contractive maps [4], and
-contractive maps [7].
In this paper, we prove some new fixed point results in the setting of quasimetric spaces for weakly -contractive and generalized
-contractive multivalued maps. Consequently, our results either improve or generalize many known results including the above stated fixed point results.
2. The Results
First, we prove a fixed point theorem for weakly -contractive maps in the setting of quasimetric spaces.
Theorem 2.1.
Let be a complete quasimetric space and let
be a weakly
- contractive map. If a real valued function
on
is lower semicontinuous, then there exists
, such that
Further, if
then
is
a
fixed
point of
.
Proof.
Let Since
is weakly contractive, there is
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F178306/MediaObjects/13663_2010_Article_1382_Equ8_HTML.gif)
where Continuing this process, we can get an orbit
of
at
satisfying
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F178306/MediaObjects/13663_2010_Article_1382_Equ9_HTML.gif)
Since and
thus we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F178306/MediaObjects/13663_2010_Article_1382_Equ10_HTML.gif)
If we put , then also we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F178306/MediaObjects/13663_2010_Article_1382_Equ11_HTML.gif)
Thus, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F178306/MediaObjects/13663_2010_Article_1382_Equ12_HTML.gif)
and since , hence the sequence
which is decreasing, converges to 0. Now, we show that
is a Cauchy sequence. Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F178306/MediaObjects/13663_2010_Article_1382_Equ13_HTML.gif)
Now, for any integer with
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F178306/MediaObjects/13663_2010_Article_1382_Equ14_HTML.gif)
and thus by Lemma 1.4, is a Cauchy sequence. Due to the completeness of
, there exists some
, such that
Now, since
is lower semicontinuous, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F178306/MediaObjects/13663_2010_Article_1382_Equ15_HTML.gif)
and thus, It follows that there exists a sequence
in
, such that
Now, if
then by Lemma 1.4,
. Since
is closed,
we
get
Now, we prove the following useful lemma.
Lemma 2.2.
Let be a complete quasimetric space and let
be a generalized
-contractive map, then there exists an orbit
of
at
, such that the sequence of nonnegative numbers
is decreasing to zero and
is a Cauchy
sequence.
Proof.
Let be an arbitrary but fixed element of
and let
. Since
is generalized as a
-contractive, there is
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F178306/MediaObjects/13663_2010_Article_1382_Equ16_HTML.gif)
Continuing this process, we get a sequence in
, such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F178306/MediaObjects/13663_2010_Article_1382_Equ17_HTML.gif)
Thus, for all , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F178306/MediaObjects/13663_2010_Article_1382_Equ18_HTML.gif)
Write . Suppose that
, then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F178306/MediaObjects/13663_2010_Article_1382_Equ19_HTML.gif)
Now, taking limits as on both sides, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F178306/MediaObjects/13663_2010_Article_1382_Equ20_HTML.gif)
which is not possible, and hence the sequence of nonnegative numbers which is decreasing, converges to 0. Finally, we show that
is a Cauchy sequence. Let
. There exists real number
such that
. Then for sufficiently large
,
, and thus for sufficiently large
, we have
Consequently, we obtain
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F178306/MediaObjects/13663_2010_Article_1382_Equ21_HTML.gif)
Now, for any integers ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F178306/MediaObjects/13663_2010_Article_1382_Equ22_HTML.gif)
and thus by Lemma 1.4, is a Cauchy sequence.
Applying Lemma 2.2, we prove a fixed point result for generalized -contractive maps.
Theorem 2.3.
Let be a complete quasimetric space then each generalized q -contractive map
has a fixed point.
Proof.
It follows from Lemma 2.2 that there exists a Cauchy sequence in
such that the decreasing sequence
converges to 0. Due to the completeness of
, there exists some
such that
Let
be arbitrary fixed positive integer then for all positive integers
with
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F178306/MediaObjects/13663_2010_Article_1382_Equ23_HTML.gif)
Let , then
. Now, note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F178306/MediaObjects/13663_2010_Article_1382_Equ24_HTML.gif)
Since was arbitrary fixed, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F178306/MediaObjects/13663_2010_Article_1382_Equ25_HTML.gif)
Note that converges to
. Now, since
and
is a generalized
-contractive map, then there is
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F178306/MediaObjects/13663_2010_Article_1382_Equ26_HTML.gif)
And for large , we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F178306/MediaObjects/13663_2010_Article_1382_Equ27_HTML.gif)
thus, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F178306/MediaObjects/13663_2010_Article_1382_Equ28_HTML.gif)
Thus, it follows from Lemma 1.4 that . Since
is closed, we get
Corollary 2.4.
Let be a complete quasimetric space and
a
-function on
. Let
be a multivalued map, such that for any
and
, there is
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F178306/MediaObjects/13663_2010_Article_1382_Equ29_HTML.gif)
where is a monotonic increasing function from
to
, then
has a fixed point.
Finally, we conclude with the following remarks concerning our results related to the known fixed point results.
Remark 2.5.
(1)Theorem 2.1 generalizes Theorem 1.2 according to Feng and Liu [2] and Latif and Albar [5, Theorem ].
(2)Theorem 2.3 generalizes Theorem 1.3 according to Suzuki and Takahashi [4] and Theorem 1.5 according to Al-Homidan et al. [7] and contains Latif's Theorem in [6].
(3)Theorem 2.3 also generalizes Theorem in [8] in several ways.
(4)Corollary 2.4 improves and generalizes Theorem in [9].
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Acknowledgments
The authors thank the referees for their kind comments. The authors also thank King Abdulaziz University and the Deanship of Scientific Research for the research Grant no. 3-35/429.
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Latif, A., Al-Mezel, S.A. Fixed Point Results in Quasimetric Spaces. Fixed Point Theory Appl 2011, 178306 (2011). https://doi.org/10.1155/2011/178306
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DOI: https://doi.org/10.1155/2011/178306