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Some Fixed-Point Theorems for Multivalued Monotone Mappings in Ordered Uniform Space
Fixed Point Theory and Applications volume 2011, Article number: 186237 (2011)
Abstract
We use the order relation on uniform spaces defined by Altun and Imdad (2009) to prove some new fixed-point and coupled fixed-point theorems for multivalued monotone mappings in ordered uniform spaces.
1. Introduction
There exists considerable literature of fixed-point theory dealing with results on fixed or common fixed-points in uniform space (e.g., between [1–14]). But the majority of these results are proved for contractive or contractive type mapping (notice from the cited references). Also some fixed-point and coupled fixed-point theorems in partially ordered metric spaces are given in [15–20]. Recently, Aamri and El Moutawakil [2] have introduced the concept of -distance function on uniform spaces and utilize it to improve some well-known results of the existing literature involving both
-contractive or
-expansive mappings. Lately, Altun and Imdad [21] have introduced a partial ordering on uniform spaces utilizing
-distance function and have used the same to prove a fixed-point theorem for single-valued nondecreasing mappings on ordered uniform spaces. In this paper, we use the partial ordering on uniform spaces which is defined by [21], so we prove some fixed-point theorems of multivalued monotone mappings and some coupled fixed-point theorems of multivalued mappings which are given for ordered metric spaces in [22] on ordered uniform spaces.
Now, we recall some relevant definitions and properties from the foundation of uniform spaces. We call a pair to be a uniform space which consists of a nonempty set
together with an uniformity
wherein the latter begins with a special kind of filter on
whose all elements contain the diagonal
. If
and
,
then
and
are said to be
-close. Also a sequence
in
, is said to be a Cauchy sequence with regard to uniformity
if for any
, there exists
such that
and
are
-close for
. An uniformity
defines a unique topology
on
for which the neighborhoods of
are the sets
when
runs over
.
A uniform space is said to be Hausdorff if and only if the intersection of all the
reduces to diagonal
of
, that is,
for
implies
. Notice that Hausdorffness of the topology induced by the uniformity guarantees the uniqueness of limit of a sequence in uniform spaces. An element of uniformity
is said to be symmetrical if
. Since each
contains a symmetrical
and if
then
and
are both
and
-close and then one may assume that each
is symmetrical. When topological concepts are mentioned in the context of a uniform space
, they are naturally interpreted with respect to the topological space
.
2. Preliminaries
We will require the following definitions and lemmas in the sequel.
Definition 2.1 (see [2]).
Let be a uniform space. A function
is said to be an
-distance if
for any , there exists
, such that
and
for some
imply
,
, for all
.
The following lemma embodies some useful properties of -distance.
Let be a Hausdorff uniform space and
be an
-distance on
. Let
and
be arbitrary sequences in
and
,
be sequences in
converging to 0. Then, for
, the following holds:
(a)if and
for all
, then
. In particular, if
and
, then
,
(b)if and
for all
, then
converges to
,
(c)if for all
, then
is a Cauchy sequence in
.
Let be a uniform space equipped with
-distance
. A sequence in
is
-Cauchy if it satisfies the usual metric condition. There are several concepts of completeness in this setting.
Let be a uniform space and
be an
-distance on
. Then
(i) said to be
-complete if for every
-Cauchy sequence
there exists
with
,
(ii) is said to be
-Cauchy complete if for every
-Cauchy sequence
there exists
with
with respect to
,
(iii) is
-continuous if
implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F186237/MediaObjects/13663_2010_Article_1384_Equ1_HTML.gif)
(iv) is
-continuous if
with respect to
implies
with respect to
.
Remark 2.4 (see [2]).
Let be a Hausdorff uniform space and let
be a
-Cauchy sequence. Suppose that
is
-complete, then there exists
such that
. Then Lemma 2.2(b) gives that
with respect to the topology
which shows that
-completeness implies
-Cauchy completeness.
Lemma 2.5 (see [15]).
Let be a Hausdorff uniform space,
be
-distance on
and
. Define the relation "
" on
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F186237/MediaObjects/13663_2010_Article_1384_Equ2_HTML.gif)
Then "" is a (partial) order on
induced by
.
3. The Fixed-Point Theorems of Multivalued Mappings
Theorem 3.1.
Let a Hausdorff uniform space and
is an
-distance on
,
be a function which is bounded below and "
" the order introduced by
. Let
be also a
-Cauchy complete space,
be a multivalued mapping,
and
. Suppose that:
(i) is upper semicontinuous, that is,
and
with
and
, implies
,
(ii),
(iii)for each ,
.
Then has a fixed-point
and there exists a sequence
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F186237/MediaObjects/13663_2010_Article_1384_Equ3_HTML.gif)
such that . Moreover if
is lower semicontinuous, then
for all
.
Proof.
By the condition (ii), take . From (iii), there exist
and
. Again from (iii), there exist
. Thus
.
Continuing this procedure we get a sequence satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F186237/MediaObjects/13663_2010_Article_1384_Equ4_HTML.gif)
So by the definition of "", we have
, that is, the sequence
is a nonincreasing sequence in
. Since
is bounded from below,
is convergent and hence it is Cauchy, that is, for all
, there exists
such that for all
we have
. Since
, we have
or
. Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F186237/MediaObjects/13663_2010_Article_1384_Equ5_HTML.gif)
which shows that (in view of Lemma 2.2(c)) that is
-Cauchy sequence. By the
-Cauchy completeness of
,
converges to
. Since
is upper semicontinuous,
.
Moreover, when is lower semicontinuous, for each
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F186237/MediaObjects/13663_2010_Article_1384_Equ6_HTML.gif)
So , for all
.
Similarly, we can prove the following.
Theorem 3.2.
Let a Hausdorff uniform space and
an
-distance on
,
be a function which is bounded above and "
" the order introduced by
. Let
be also a
-Cauchy complete space,
be a multivalued mapping,
and
. Suppose that
(i) is upper semicontinuous, that is,
and
with
and
, implies
,
(ii),
(iii)for each ,
.
Then has a fixed-point
and there exists a sequence
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F186237/MediaObjects/13663_2010_Article_1384_Equ7_HTML.gif)
such that . Moreover, if
is upper semicontinuous, then
for all
.
Corollary 3.3.
Let a Hausdorff uniform space and
is an
-distance on
,
be a function which is bounded below and "
" the order introduced by
. Let
be also a
-Cauchy complete space,
be a multivalued mapping and
. Suppose that:
(i) is upper semicontinuous, that is,
and
with
and
, implies
,
(ii) satisfies the monotonic condition: for any
,
with
and any
, there exists
such that
,
(iii)there exists an such that
.
Then has a fixed-point
and there exists a sequence
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F186237/MediaObjects/13663_2010_Article_1384_Equ8_HTML.gif)
such that . Moreover if
is lower semicontinuous, then
for all
.
Proof.
By (iii), . For
, take
and
. By the monotonicity of
, there exists
such that
. So
, and
. The conclusion follows from Theorem 3.1.
Corollary 3.4.
Let a Hausdorff uniform space and
is an
-distance on
,
be a function which is bounded above and "
" the order introduced by
. Let
be also a
-Cauchy complete space,
be a multivalued mapping and
. Suppose that:
(i) is upper semicontinuous,
(ii) satisfies the monotonic condition; for any
with
and any
, there exists
such that
,
(iii)there exists an such that
.
Then has a fixed-point
and there exists a sequence
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F186237/MediaObjects/13663_2010_Article_1384_Equ9_HTML.gif)
such that . Moreover if
is upper semicontinuous, then
for all
.
Corollary 3.5.
Let a Hausdorff uniform space and
is an
-distance on
,
be a function which is bounded below and "
" the order introduced by
. Let
be also a
-Cauchy complete space,
be a map and
. Suppose that:
(i) is
-continuous,
(ii),
(iii)for each ,
.
Then has a fixed-point
and the sequence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F186237/MediaObjects/13663_2010_Article_1384_Equ10_HTML.gif)
converges to . Moreover if
is lower semicontinuous, then
for all
.
Corollary 3.6.
Let be a Hausdorff uniform space,
is an
-distance on
,
be a function which is bounded above, and "
" the order introduced by
. Let
be also a
-Cauchy complete space,
be a map and
. Suppose that:
(i) is
-continuous,
(ii),
(iii)for each ,
.
Then has a fixed-point
. And the sequence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F186237/MediaObjects/13663_2010_Article_1384_Equ11_HTML.gif)
converges to . Moreover, if
is upper semicontinuous, then
for all
.
Corollary 3.7.
Let be a Hausdorff uniform space,
is an
-distance on
,
be a function which is bounded below, and "
" the order introduced by
. Let
be also a
-Cauchy complete space,
be a map and
. Suppose that:
(i) is
-continuous,
(ii) is monotone increasing, that is, for
we have
,
(iii)there exists an , with
.
Then has a fixed-point
and the sequence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F186237/MediaObjects/13663_2010_Article_1384_Equ12_HTML.gif)
converges to . Moreover if
is lower semicontinuous, then
for all
.
Example 3.8.
Let and
. Define
as
for all
,
,
ve
. Since definition of
,
and this show that the uniform space
is a Hausdorff uniform space. On the other hand,
,
and
for
and thus
is an
-distance as it is a metric on
. Next define
,
,
. Since
, therefore
. But as
therefore
and
. Again similarly
and
which show that this ordering is partial and hence
is a partially ordered uniform space. Define
as
,
and
, then by a routine calculation one can verify that all the conditions of Corollary 3.7 are satisfied and
has a fixed-point. Notice that
which shows that
is neither
-contractive nor
expansive, therefore the results of [2] are not applicable in the context of this example. Thus, this example demonstrates the utility of our result.
Corollary 3.9.
Let be a Hausdorff uniform space,
is an
-distance on
,
be a function which is bounded above and "
" the order introduced by
. Let
be also a
-Cauchy complete space and
be a map. Suppose that
(i) is
-continuous,
(ii) is monotone increasing, that is, for
we have
,
(iii)there exists an with
.
Then has a fixed-point
. And the sequence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F186237/MediaObjects/13663_2010_Article_1384_Equ13_HTML.gif)
converges to . Moreover if
is upper semicontinuous, then
for all
.
Theorem 3.10.
Let be a Hausdorff uniform space,
is an
-distance on
,
be a continuous function bounded below and "
" the order introduced by
. Let
be also a
-Cauchy complete space,
be a multivalued mapping and
. Suppose that
(i) satisfies the monotonic condition: for each
and each
there exists
such that
,
(ii) is compact for each
,
(iii).
Then has a fixed-point
.
Proof.
We will prove that has a maximum element. Let
be a totally ordered subset in
, where
is a directed set. For
and
, one has
, which implies that
for
. Since
is bounded below,
is a convergence net in
. From
, we get that
is a
-cauchy net in
. By the
-Cauchy completeness of
, let
converge to
in
.
For given . So
for all
.
For , by the condition (i), for each
, there exists a
such that
. By the compactness of
, there exists a convergence subnet
of
. Suppose that
converges to
. Take
such that
implies
.
We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F186237/MediaObjects/13663_2010_Article_1384_Equ14_HTML.gif)
So for all
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F186237/MediaObjects/13663_2010_Article_1384_Equ15_HTML.gif)
So and this gives that
. Hence we have proven that
has an upper bound in
.
By Zorn's Lemma, there exists a maximum element in
. By the definition of
, there exists a
such that
. By the condition (i), there exists a
such that
. Hence
. Since
is the maximum element in
, it follows that
and
. So
is a fixed-point of
.
Theorem 3.11.
Let be a Hausdorff uniform space,
is an
-distance on
,
be a continuous function bounded above and "
" the order introduced by
. Let
be also a
-Cauchy complete space,
be a multivalued mapping and
. Suppose that
(i) satisfies the following condition; for each
and
, there exists
such that
,
(ii) is compact for each
,
(iii).
Then has a fixed-point.
Corollary 3.12.
Let be a Hausdorff uniform space,
is an
-distance on
,
be a continuous function bounded below and "
" the order introduced by
. Let
be also a
-Cauchy complete space and
be a map. Suppose that;
(i) is monotone increasing, that is for
,
,
(ii)there is an such that
.
Then has a fixed-point.
Corollary 3.13.
Let be a Hausdorff uniform space,
is an
-distance on
,
be a continuous function bounded above and "
" the order introduced by
. Let
be also a
-Cauchy complete space and
be a map. Suppose that;
(i) is monotone increasing, that is, for
,
;
(ii)there is an such that
.
Then has a fixed-point.
4. The Coupled Fixed-Point Theorems of Multivalued Mappings
Definition 4.1.
An element is called a coupled fixed-point of the multivalued mapping
if
,
.
Theorem 4.2.
Let be a Hausdorff uniform space,
is an
-distance on
,
be a function bounded below and "
" be the order in
introduced by
. Let
be also a
-Cauchy complete space,
be a multivalued mapping,
,
, and
,
and
. Suppose that:
(i) is upper semicontinuous, that is,
,
and
, with
,
and
implies
,
(ii),
(iii)for each , there is
such that
and
.
Then has a coupled fixed-point
, that is,
and
. And there exist two sequences
and
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F186237/MediaObjects/13663_2010_Article_1384_Equ16_HTML.gif)
such that and
.
Proof.
By the condition (ii), take . From (iii), there exist
such that
,
and
,
. Again from (iii), there exist
such that
,
and
,
.
Continuing this procedure we get two sequences and
satisfying
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F186237/MediaObjects/13663_2010_Article_1384_Equ17_HTML.gif)
So
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F186237/MediaObjects/13663_2010_Article_1384_Equ18_HTML.gif)
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F186237/MediaObjects/13663_2010_Article_1384_Equ19_HTML.gif)
From this we get that and
are convergent sequences. By the definition of "
" as in the proof of Theorem 3.1, it is easy to prove that
and
are
-Cauchy sequences. Since
is
-Cauchy complete, let
converge to
and
converge to
. Since
is upper semicontinuous,
and
. Hence
is a coupled fixed-point of
.
Corollary 4.3.
Let be a Hausdorff uniform space,
is an
-distance on
,
be a function bounded below, and "
" be the order in
introduced by
. Let
be also a
-Cauchy complete space,
be a mapping and
and
and
. Suppose that;
(i) is
-continuous,
(ii),
(iii)for each ,
and
.
Then has a coupled fixed-point
, that is,
and
. And there exist two sequences
and
with
,
,
such that
and
.
Corollary 4.4.
Let be a Hausdorff uniform space,
is an
-distance on
,
be a function bounded below, and "
" be the order in
introduced by
. Let
be also a
-Cauchy complete space,
be a mapping and
and
and
. Suppose that;
(i) is
-continuous,
(ii),
(iii) is mixed monotone, that is for each
and
,
.
Then has a coupled fixed-point
. And there exist two sequences
and
with
,
,
such that
and
.
Theorem 4.5.
Let be a Hausdorff uniform space,
is an
-distance on
,
be a continuous function, and "
" be the order in
introduced by
. Let
be also a
-Cauchy complete space,
be a multivalued mapping,
,
, and
,
and
. Suppose that;
(i) is mixed monotone, that is, for
,
and
,
, there exist
,
such that
,
,
(ii),
(iii) is compact for each
.
Then has a coupled fixed-point.
Proof.
By (ii), there exists with
,
and
. Let
,
,
and
. Then
. Define the order relation "
" in
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F186237/MediaObjects/13663_2010_Article_1384_Equ20_HTML.gif)
It is easy to prove that becomes an ordered space.
We will prove that has a maximum element. Let
be a totally ordered subset in
, where
is a directed set. For
and
, one has
. So
and
, which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F186237/MediaObjects/13663_2010_Article_1384_Equ21_HTML.gif)
for .
Since and
are convergence nets in
. From
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F186237/MediaObjects/13663_2010_Article_1384_Equ22_HTML.gif)
we get that and
are
-Cauchy nets in
. By the
-Cauchy completeness of
, let
convergence to
and
convergence to
in
. For given
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F186237/MediaObjects/13663_2010_Article_1384_Equ23_HTML.gif)
So and
for all
.
For , by the condition (i), for each
with
and
with
, there exist
and
such that
and
. By the compactness of
and
, there exist convergence subnets
of
and
of
. Suppose that
converges to
and
converges to
). Take
, such that
implies
. We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F186237/MediaObjects/13663_2010_Article_1384_Equ24_HTML.gif)
So and
for all
. And
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F186237/MediaObjects/13663_2010_Article_1384_Equ25_HTML.gif)
So and
, this gives that
. Hence we have proven that
has an upper bound in
.
By Zorn's lemma, there exists a maximum element in
. By the definition of
, there exist
,
, such that
,
and
,
. By the condition (i) there exist
,
such that
and
. Hence
and
. Since
is maximum element in
, it follows that
, and it follows that
and
. So
is a coupled fixed-point of
.
Corollary 4.6.
Let be a Hausdorff uniform space,
is an
-distance on
,
be a continuous function, and "
" be the order in
introduced by
. Let
be also a
-Cauchy complete space and
be a mapping. Suppose that;
(i) is mixed monotone, that is for
,
and
,
(ii)there exist such that
and
.
Then has a coupled fixed-point.
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Turkoglu, D., Binbasioglu, D. Some Fixed-Point Theorems for Multivalued Monotone Mappings in Ordered Uniform Space. Fixed Point Theory Appl 2011, 186237 (2011). https://doi.org/10.1155/2011/186237
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DOI: https://doi.org/10.1155/2011/186237