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Best Approximations Theorem for a Couple in Cone Banach Space
Fixed Point Theory and Applications volume 2010, Article number: 784578 (2010)
Abstract
The notion of coupled fixed point is introduced by Bhaskar and Lakshmikantham, (2006). In this manuscript, some result of Mitrović, (2010) extended to the class of cone Banach spaces.
1. Introduction and Preliminaries
Banach, valued metric space was considered by Rzepecki [1], Lin [2], and lately by Huang and Zhang [3]. Basically, for nonempty set , the definition of metric
is replaced by a new metric, namely, by an ordered Banach space
:
. Such metric spaces are called cone metric spaces (in short CMSs). In 1980, by using this idea Rzepecki [1] generalized the fixed point theorems of Maia type. Seven years later, Lin [2] extends some results of Khan and Imdad [4] by considering this new metric space construction. In 2007, Huang and Zhang [3] discussed some properties of convergence of sequences and proved the fixed point theorems of contractive mapping for cone metric spaces: any mapping
of a complete cone metric space
into itself that satisfies, for some
, the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F784578/MediaObjects/13663_2010_Article_1343_Equ1_HTML.gif)
for all , has a unique fixed point. Recently, many results on fixed point theorems have been extended to cone metric spaces (see, e.g., [3, 5–11]). In [3], the authors extends to cone metric spaces over regular cones. In this manuscript, some results of some result of Mitrović in [12] are extended to the class of cone metric spaces.
Throughout this paper stands for real Banach space. Let
always be closed subset of
.
is called cone if the following conditions are satisfied:
,
for all
and nonnegative real numbers
,
and
.
For a given cone , one can define a partial ordering (denoted by
or
) with respect to
by
if and only if
. The notation
indicates that
and
while
will show
, where
denotes the interior of
. It can be easily shown that
and
where
. Throughout this manuscript
.
The cone is called
normal if there is a number such that for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F784578/MediaObjects/13663_2010_Article_1343_Equ2_HTML.gif)
regular if every increasing sequence which is bounded from above is convergent. That is, if is a sequence such that
for some
, then there is
such that
.
In , the least positive integer
satisfying (1.2) is called the normal constant of
. Note that, in [3, 5], normal constant
is considered a positive real number, (
), although it is proved that there is no normal cone for
in (see e.g., Lemma
, [5]).
Lemma 1.1 (see e.g., [13]).
One has the following.
(i)Every regular cone is normal.
(ii)For each , there is a normal cone with normal constant
.
(iii)The cone is regular if every decreasing sequence which is bounded from below is convergent.
Definition 1.2 (see [14]).
is called minihedral cone if
exists for all
; and strongly minihedral if every subset of
which is bounded from above has a supremum.
Example 1.3.
Let with the supremum norm and
Since the sequence
is monotonically decreasing, but not uniformly convergent to
, thus,
is not strongly minihedral.
Definition 1.4.
Let be nonempty set. Suppose that the mapping
satisfies the following:
for all
,
if and only if
,
, for all
.
for all
Then is called cone metric on
, and the pair
is called a cone metric space (CMS).
Example 1.5.
Let and
and
. Define
by
, where
are positive constants. Then
is a CMS. Note that the cone
is normal with the normal constant
It is quite natural to consider Cone Normed Spaces (CNSs).
Definition 1.6 (see e.g., [9, 15, 16]).
Let be a vector space over
. Suppose that the mapping
satisfies the following:
for all
,
if and only if
,
, for all
.
for all
.
Then is called cone norm on
, and the pair
is called a cone normed space (CNS).
Note that each CNS is CMS. Indeed, .
Definition 1.7.
Let be a CNS,
and
a sequence in
. Then one has the following.
(i)converges to
whenever for every
with
there is a natural number
, such that
for all
. It is denoted by
or
.
(ii) is a Cauchy sequence whenever for every
with
there is a natural number
, such that
for all
.
(iii) is a complete cone normed space if every Cauchy sequence is convergent.
Complete cone-normed spaces will be called cone Banach spaces.
Lemma 1.8.
Let be a CNS, let
be a normal cone with normal constant
, and let
be a sequence in
. Then, one has the following:
(i)the sequence converges to
if and only if
, as
,
(ii)the sequence is Cauchy if and only if
as
,
(iii)the sequence converges to
and the sequence
converges to
and then
.
The proof is direct by applying Lemmas 1, 4, and 5 in [3] to the cone metric space , where
, for all
.
Lemma 1.9 (see, e.g., [6, 7]).
Let be a CNS over a cone
in
. Then (1)
and
. (2) If
then there exists
such that
implies
. (3) For any given
and
there exists
such that
. (4) If
are sequences in
such that
,
and
for all
then
.
Definition 1.10.
Let be a CNS and let
be the closed unit interval. A continuous mapping
is said to be a convex structure on
if for all
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F784578/MediaObjects/13663_2010_Article_1343_Equ3_HTML.gif)
holds for all . A CNS
together with a convex structure is said to be convex CNS. A subset
is convex, if
holds for all
and
.
Definition 1.11.
Let be a CNS, and
and
the nonempty convex subsets of
. A mapping
is said to be almost quasiconvex with respect to
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F784578/MediaObjects/13663_2010_Article_1343_Equ4_HTML.gif)
where for all
and
.
2. Couple Fixed Theorems on Cone Metric Spaces
Let be a CMS and
. Then the mapping
such that
forms a cone metric on
. A sequence
is said to be a double sequence of
. A sequence
is convergent to
if, for every
, there exists a natural number
such that
for all
.
Lemma 2.1.
Let and
. Then,
if and only if
and
.
Proof.
Suppose . Thus, for any
, there exist
such that
for all
. Hence,
and
for all
, that is,
and
.
Conversely, assume and
. Thus, for any
, there exist
such that
for all
, and also
for all
. Hence,
for all
, where
.
Definition 2.2.
Let be a CMS. A function
is said to be sequentially continuous if
implies that
. Analogously, a function
is sequentially continuous if
implies that
.
Lemma 2.3 (see [6]).
Let be a CNS. Then
is continuous if and only if
is sequentially continuous.
Definition 2.4 (see [10, 17, 18]).
Let be partially ordered set and
.
is said to have mixed monotone property if
is monotone nondecreasing in
and is monotone nonincreasing in
, that is, for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F784578/MediaObjects/13663_2010_Article_1343_Equ5_HTML.gif)
Note that this definition reduces the notion of mixed monotone function on where
represents usual total order
in
.
Definition 2.5 (see [10, 17, 18]).
An element is said to be a couple fixed point of the mapping
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F784578/MediaObjects/13663_2010_Article_1343_Equ6_HTML.gif)
Throughout this paper, let be partially ordered set and let
be a cone metric on
such that
is a complete CMS over the normal cone
with the normal constant
. Further, the product spaces
satisfy the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F784578/MediaObjects/13663_2010_Article_1343_Equ7_HTML.gif)
Definition 2.6 (see [3]).
Let be a CMS and
.
is said to be sequentially compact if for any sequence
in
there is a subsequence
of
such that
is convergent in
.
Remark 2.7 (see [19]).
Every cone metric space is a topological space which is denoted by
. Moreover, a subset
is sequentially compact if and only if
is compact.
Definition 2.8.
Let be a nonempty subset of a CNS
. A set-valued map
is called KKM map if for every finite subset
of
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F784578/MediaObjects/13663_2010_Article_1343_Equ8_HTML.gif)
where denotes the convex hull.
Lemma 2.9.
Let be a topological vector space, let
be a nonempty subset of
and let
be called KKM map with closed values. If
is compact for at least one
then
.
Theorem 2.10.
Let be a CNS over strongly minidhedral cone
, and let
be a nonempty convex compact subset of
. If
is continuous mapping and
is continuous almost quasiconvex mapping with respect to
, then there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F784578/MediaObjects/13663_2010_Article_1343_Equ9_HTML.gif)
Proof.
Let by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F784578/MediaObjects/13663_2010_Article_1343_Equ10_HTML.gif)
for each . Since
, then
. Regarding that the mappings
and
are continuous,
is closed for each
. Since
is compact, then
is compact for each
. Thus,
is a KKM map.
Let ,
where
and
are finite subsets of
. Then, there exists
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F784578/MediaObjects/13663_2010_Article_1343_Equ11_HTML.gif)
From the first expression in (2.7), one can get that there exist such that
and
. Set
and
then
,
and
,
. Regarding that
is almost quasiconvex with respect to
yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F784578/MediaObjects/13663_2010_Article_1343_Equ12_HTML.gif)
where and
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F784578/MediaObjects/13663_2010_Article_1343_Equ13_HTML.gif)
Taking (2.7) into account, one can get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F784578/MediaObjects/13663_2010_Article_1343_Equ14_HTML.gif)
for all which is a contradiction. Hence
is a KKM mapping. It follows that there exists
such that
for all
. Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F784578/MediaObjects/13663_2010_Article_1343_Equ15_HTML.gif)
Theorem 2.11.
Let be a CNS over strongly minidhedral cone
, and let
be a nonempty convex compact subset of
. If
is continuous mapping and
is continuous almost quasiconvex mapping with respect to
such that
, then
and
have a coupled coincidence point.
Proof.
Due to Theorem 2.10, there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F784578/MediaObjects/13663_2010_Article_1343_Equ16_HTML.gif)
Since ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F784578/MediaObjects/13663_2010_Article_1343_Equ17_HTML.gif)
then .
Thus, and
.
If we take as an identity,
, in Theorem 2.11, then we get the following result.
Theorem 2.12.
Let be a CNS over strongly minidhedral cone
, and let
be a nonempty convex compact subset of
. If
is continuous mapping, then
has a coupled fixed point.
Theorem 2.13.
Let be a CNS over strongly minidhedral cone
, and let
be a nonempty convex compact subset of
. If
is continuous mapping, then either
has a coupled fixed point or there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F784578/MediaObjects/13663_2010_Article_1343_Equ18_HTML.gif)
for all .
Proof.
If has a coupled fixed point, then we are done. Suppose that
has no coupled fixed points. Due to Theorem 2.10, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F784578/MediaObjects/13663_2010_Article_1343_Equ19_HTML.gif)
Take which implies (2.14). It is sufficient to show that
. The inequality (2.14) implies that either
or
.
Consider the first case: . Suppose
. Since
is convex, then there exists
such that
. Thus
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F784578/MediaObjects/13663_2010_Article_1343_Equ20_HTML.gif)
This is a contradiction. Analogously one can get the contradiction from the case . Thus,
.
Theorem 2.14.
Let be a CNS over strongly minidhedral cone
, and let
be a nonempty convex compact subset of
. Suppose that
is continuous mapping. Then
has a coupled fixed point if one of the following conditions is satisfied for all
such that
:
(i)there exists a such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F784578/MediaObjects/13663_2010_Article_1343_Equ21_HTML.gif)
(ii)there exists an such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F784578/MediaObjects/13663_2010_Article_1343_Equ22_HTML.gif)
(iii)
Proof.
It is clear that (iii)(ii)
(i). To finalize proof, it is sufficient to show that
is satisfied. Suppose that
holds but
has no coupled fixed point. Take Theorem 2.13 into account; then there exist
such that (2.14) holds which contradicts
.
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Karapınar, E., Türkoğlu, D. Best Approximations Theorem for a Couple in Cone Banach Space. Fixed Point Theory Appl 2010, 784578 (2010). https://doi.org/10.1155/2010/784578
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DOI: https://doi.org/10.1155/2010/784578