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An Order on Subsets of Cone Metric Spaces and Fixed Points of Set-Valued Contractions
Fixed Point Theory and Applications volume 2009, Article number: 723203 (2009)
Abstract
In this paper at first we introduce a new order on the subsets of cone metric spaces then, using this definition, we simplify the proof of fixed point theorems for contractive set-valued maps, omit the assumption of normality, and obtain some generalization of results.
1. Introduction and Preliminary
Cone metric spaces were introduced by Huang and Zhang [1]. They replaced the set of real numbers by an ordered Banach space and obtained some fixed point theorems for mapping satisfying different contractions [1]. The study of fixed point theorems in such spaces followed by some other mathematicians, see [2–8]. Recently Wardowski [9] was introduced the concept of set-valued contractions in cone metric spaces and established some end point and fixed point theorems for such contractions. In this paper at first we will introduce a new order on the subsets of cone metric spaces then, using this definition, we simplify the proof of fixed point theorems for contractive set-valued maps, omit the assumption of normality, and obtain some generalization of results.
Let be a real Banach space. A nonempty convex closed subset
is called a cone in
if it satisfies.
(i) is closed, nonempty, and
,
(ii) and
imply that
(iii) and
imply that
The space can be partially ordered by the cone
; that is,
if and only if
. Also we write
if
, where
denotes the interior of
.
A cone is called normal if there exists a constant
such that
implies
.
In the following we always suppose that is a real Banach space,
is a cone in
and
is the partial ordering with respect to
.
Definition 1.1 (see [1]).
Let be a nonempty set. Assume that the mapping
satisfies
(i) for all
and
iff
(ii) for all
(iii) for all
.
Then is called a cone metric on
, and
is called a cone metric space.
In the following we have some necessary definitions.
(1)Let be a cone metric space. A set
is called closed if for any sequence
convergent to
, we have
(2)A set is called sequentially compact if for any sequence
, there exists a subsequence
of
is convergent to an element of
(3)Denote a collection of all nonempty subsets of
,
a collection of all nonempty closed subsets of
and
a collection of all nonempty sequentially compact subsets of
(4)An element is said to be an endpoint of a set-valued map
if
We denote a set of all endpoints of
by
(5)An element is said to be a fixed point of a set-valued map
if
Denote
(6)A map is called lower semi-continuous, if for any sequence
in
and
such that
as
, we have
(7)A map is called have lower semi-continuous property, and denoted by lsc property if for any sequence
in
and
such that
as
, then there exists
that
for all
(8) called minihedral cone if
exists for all
, and strongly minihedral if every subset of
which is bounded from above has a supremum [10]. Let
a cone metric space, cone
is strongly minihedral and hence, every subset of
has infimum, so for
, we define
Example 1.2.
Let with
. The cone
is normal, minihedral and strongly minihedral with
.
Example 1.3.
Let be a compact set,
and
. The cone
is normal and minihedral but is not strongly minihedral and
.
Example 1.4.
Let be a finite measure space,
countably generated,
,
and
. The cone
is normal, minihedral and strongly minihedral with
.
For more details about above examples, see [11].
Example 1.5.
Let with norm
and
that is not normal cone by [12] and not minihedral by [10].
Example 1.6.
Let and
. This
is strongly minihedral but not minihedral by [10].
Throughout, we will suppose that is strongly minihedral cone in
with nonempty interior and
be a partial ordering with respect to
2. Main Results
Let be a cone metric space and
. For
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F723203/MediaObjects/13663_2009_Article_1171_Equ1_HTML.gif)
At first we prove the closedness of without the assumption of normality.
Lemma 2.1.
Let be a complete cone metric space and
. If the function
for
is lower semi-continuous, then
is closed.
Proof.
Let and
We show that
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F723203/MediaObjects/13663_2009_Article_1171_Equ2_HTML.gif)
so which implies
for some
. Let
with
then, there exists
such that for
,
Now, for
we have,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F723203/MediaObjects/13663_2009_Article_1171_Equ3_HTML.gif)
So is a Cauchy sequence in complete metric space, hence there exist
such that
. Since
is closed, thus
Now by uniqueness of limit we conclude that
Definition 2.2.
Let and
are subsets of
, we write
if and only if there exist
such that for all
,
Also for
, we write
if and only if
and similarly
if and only if
Note that , for every scaler
and
subsets of
.
The following lemma is easily proved.
Lemma 2.3.
Let ,
,
and
.
(1)If and
then
(2)
(3)If then
(4)If then
(5)
(6)If then
The order "" is not antisymmetric, thus this order is not partially order.
Example 2.4.
Let and
. Put
and
so
but
Theorem 2.5.
Let be a complete cone metric space,
, a set-valued map and the function
defined by
,
with lsc property. If there exist real numbers
and
with
such that for all
there exists
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F723203/MediaObjects/13663_2009_Article_1171_Equ4_HTML.gif)
then
Proof.
Let , then there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F723203/MediaObjects/13663_2009_Article_1171_Equ5_HTML.gif)
Let , there exist
such that
and
Continuing this process, we can iteratively choose a sequence
in
such that
,
and
So, for
we have,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F723203/MediaObjects/13663_2009_Article_1171_Equ6_HTML.gif)
Therefore, for every ,
Let
and
be given. Choose
such that
where
Also, choose a
such that
for all
Then
for all
Thus
for all
Namely,
is Cauchy sequence in complete cone metric space, therefore
for some
Now we show that
Let hence there exists
such that
for all
Now
as
so for all
there exists
such that
for all
According to lsc property of , for all
there exists
such that for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F723203/MediaObjects/13663_2009_Article_1171_Equ7_HTML.gif)
So for all
Namely,
thus
for some
and by the closedness of
we have
We notice that implies that for all
there exists
such that
for all
but the inverse is not true.
Example 2.6.
Let with norm
and
that is not normal cone by [12]. Consider
and
so
and
, (see [10]) Define cone metric
with
, for
. Since
namely,
but
. Indeed
in
but
in
Even for
and
in particular
but
.
Example 2.7.
Let with norm
and
that is not normal cone. Define cone metric
with
, for
and set-valued mapping
by
. In this space every Cauchy sequence converges to zero. The function
have lsc property. Also we have
and
. Now for
and for all
take
. Therefore, it satisfies in all of the hypothesis of Theorem 2.5. So
has a fixed point
For sample take
and
Theorem 2.8.
Let be a complete cone metric space,
, a set-valued map, and a function
defined by
,
with lsc property. The following conditions hold:
(i)if there exist real numbers and
with
such that for all
there exists
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F723203/MediaObjects/13663_2009_Article_1171_Equ8_HTML.gif)
then
(ii)if there exist real numbers and
with
such that for all
and
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F723203/MediaObjects/13663_2009_Article_1171_Equ9_HTML.gif)
then
Proof.
-
(i)
It is obvious that
It is enough to show that
for all
However
for some
, it implies
for some
and this is a contradiction.
-
(ii)
By (i), there exists
such that
Then for
and
we have
. Therefore,
This implies that
Corollary 2.9.
Let be a complete cone metric space,
, a set-valued map, and the function
defined by
, for
with lsc property. If there exist real numbers
and
with
such that for all
there exists
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F723203/MediaObjects/13663_2009_Article_1171_Equ10_HTML.gif)
then
To have Theorems??3.1 and ??3.2 in [9], as the corollaries of our theorems we need the following lemma and remarks.
Lemma 2.10.
Let be a cone metric space,
a normal cone with constant one and
, a set-valued map, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F723203/MediaObjects/13663_2009_Article_1171_Equ11_HTML.gif)
Proof.
Put and
we show that
Let then
and so
which implies
For the inverse, let for all . Then
for all
Since , for every
that
there exists
such that
so
for all
Thus
Remark 2.11.
By Proposition ?1.7.59, page 117 in [11], if is an ordered Banach space with positive cone
, then
is a normal cone if and only if there exists an equivalent norm
on
which is monotone. So by renorming the
we can suppose
is a normal cone with constant one.
Remark 2.12.
Let be a cone metric space,
a normal cone with constant one,
, a set-valued map, the function
defined by
,
with lsc property, and
with
. Then
is lower semi-continuous.
Now the Theorems ?3.1 and ?3.2 in [9] is stated as the following corollaries without the assumption of normality, and by Lemma ?2.10 and Remarks ?2.11, ?2.12 we have the same theorems.
Corollary 2.13 (see [9, Theorem ?3.1]).
Let be a complete cone metric space,
, a set-valued map and the function
defined by
,
with lsc property. If there exist real numbers
,
such that for all
there exists
one has
and
then
Corollary 2.14 (see [9, Theorem ?3.2]).
Let be a complete cone metric space,
, a set-valued map and the function
defined by
,
with lsc property. The following hold:
(i)if there exist real numbers ,
such that for all
there exists
one has
and
then
(ii)if there exist real numbers ,
such that for all
and every
one has
and
then
Definition 2.15.
For ,
where
is a set-valued map we define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F723203/MediaObjects/13663_2009_Article_1171_Equ12_HTML.gif)
Note that for
The following theorem is a reform of Theorem 2.5.
Theorem 2.16.
Let be a complete cone metric space,
, a set-valued map, and the function
defined by
,
with lsc property. If there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F723203/MediaObjects/13663_2009_Article_1171_Equ13_HTML.gif)
for all Then
Proof.
For every , then there exist
and
such that
, for all
. Let
, there exist
and
such that
since
. Thus
The remaining is same as the proof of Theorem 2.5.
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Asadi, M., Soleimani, H. & Vaezpour, S.M. An Order on Subsets of Cone Metric Spaces and Fixed Points of Set-Valued Contractions. Fixed Point Theory Appl 2009, 723203 (2009). https://doi.org/10.1155/2009/723203
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DOI: https://doi.org/10.1155/2009/723203