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Topological Vector Space-Valued Cone Metric Spaces and Fixed Point Theorems
Fixed Point Theory and Applications volume 2010, Article number: 170253 (2010)
Abstract
We develop the theory of topological vector space valued cone metric spaces with nonnormal cones. We prove three general fixed point results in these spaces and deduce as corollaries several extensions of theorems about fixed points and common fixed points, known from the theory of (normed-valued) cone metric spaces. Examples are given to distinguish our results from the known ones.
1. Introduction
Ordered normed spaces and cones have applications in applied mathematics, for instance, in using Newton's approximation method [1–4] and in optimization theory [5]. -metric and
-normed spaces were introduced in the mid-20th century ([2], see also [3, 4, 6]) by using an ordered Banach space instead of the set of real numbers, as the codomain for a metric. Huang and Zhang [7] reintroduced such spaces under the name of cone metric spaces but went further, defining convergent and Cauchy sequences in the terms of interior points of the underlying cone. These and other authors (see, e.g., [8–22]) proved some fixed point and common fixed point theorems for contractive-type mappings in cone metric spaces and cone uniform spaces.
In some of the mentioned papers, results were obtained under additional assumptions about the underlying cone, such as normality or even regularity. In the papers [23, 24], the authors tried to generalize this approach by using cones in topological vector spaces (tvs) instead of Banach spaces. However, it should be noted that an old result (see, e.g., [3]) shows that if the underlying cone of an ordered tvs is solid and normal, then such tvs must be an ordered normed space. So, proper generalizations when passing from norm-valued cone metric spaces of [7] to tvs-valued cone metric spaces can be obtained only in the case of nonnormal cones.
In the present paper we develop further the theory of topological vector space valued cone metric spaces (with nonnormal cones). We prove three general fixed point results in these spaces and deduce as corollaries several extensions of theorems about fixed points and common fixed points, known from the theory of (normed-valued) cone metric spaces.
Examples are given to distinguish our results from the known ones.
2. Tvs-Valued Cone Metric Spaces
Let be a real Hausdorff topological vector space (tvs for short) with the zero vector
. A proper nonempty and closed subset
of
is called a (convex) cone if
,
for
and
. We will always assume that the cone P has a nonempty interior int P (such cones are called solid).
Each cone induces a partial order
on
by
.
will stand for
and
, while
will stand for
. The pair
is an ordered topological vector space.
For a pair of elements in
such that
, put

The sets of the form are called order intervals. It is easily verified that order-intervals are convex. A subset
of
is said to be order-convex if
, whenever
and
.
Ordered topological vector space is order-convex if it has a base of neighborhoods of
consisting of order-convex subsets. In this case the cone
is said to be normal. In the case of a normed space, this condition means that the unit ball is order-convex, which is equivalent to the condition that there is a number
such that
and
implies that
. Another equivalent condition is that

It is not hard to conclude from (2.2) that is a nonnormal cone in a normed space
if and only if there exist sequences
such that

Hence, in this case, the Sandwich theorem does not hold.
Note the following properties of bounded sets.
If the cone is solid, then each topologically bounded subset of
is also order-bounded, that is, it is contained in a set of the form
for some
.
If the cone is normal, then each order-bounded subset of
is topologically bounded. Hence, if the cone is both solid and normal, these two properties of subsets of
coincide. Moreover, a proof of the following assertion can be found, for example, in [3].
Theorem 2.1.
If the underlying cone of an ordered tvs is solid and normal, then such tvs must be an ordered normed space.
Example 2.2.
(see [5]) Let with
, and let
. This cone is solid (it has the nonempty interior) but is not normal. Consider, for example,
) and
). Since
and
, it follows that
is a nonnormal cone.
Now consider the space endowed with the strongest locally convex topology
. Then
is also
-solid (it has the nonempty
-interior), but not
-normal. Indeed, if it were normal then, according to Theorem 2.1, the space
would be normed, which is impossible since an infinite-dimensional space with the strongest locally convex topology cannot be metrizable (see, e.g., [25]).
Following [7, 23, 24] we give the following.
Definition 2.3.
Let be a nonempty set and
an ordered tvs. A function
is called a tvs-cone metric and
is called a tvs-cone metric, space if the following conditions hold:
for all
and
if and only if
;
for all
;
for all
.
Let and
be a sequence in
. Then it is said the following.
(i) tvs-cone converges to
if for every
with
there exists a natural number
such that
for all
; we denote it by
or
as
.
(ii) is a tvs-cone Cauchy sequence if for every
with
there exists a natural number
such that
for all
.
(iii) is tvs-cone complete if every tvs-Cauchy sequence is tvs-convergent in
.
Taking into account Theorem 2.1, proper generalizations when passing from norm-valued cone metric spaces of [7] to tvs-cone metric spaces can be obtained only in the case of nonnormal cones.
We will prove now some properties of a real tvs with a solid cone
and a tvs-cone metric space
over it.
Lemma 2.4.
-
(a)
Let
in
, and let
. Then there exists
such that
for each
.
-
(b)
It can happen that
for each
, but
in
.
-
(c)
It can happen that
,
in the tvs-cone metric
, but that
in
. In particular, it can happen that
in
but that
(which is impossible if the cone is normal).
-
(d)
for each
implies that
.
-
(e)
(in the tvs-cone metric) implies that
.
-
(f)
Each tvs-cone metric space is Hausdorff in the sense that for arbitrary distinct points
and
there exist disjoint neighbourhoods in the topology
having the local base formed by the sets of the form
,
.
Proof.
-
(a)
It follows from
that
for
. From
, it follows that
, that is,
.
-
(b)
Consider the sequences
) and
) from Example 2.2. We know that in the ordered Banach space
(2.4)
and that (in the norm of
) but that
in this norm. On the other hand, since
and
, it follows that
. Then also
in the tvs
(the strongest locally convex topology) but
(also considering the interior with respect to
).
We can also consider the tvs-cone metric defined by
,
, and
. Then for the sequence
we have that
in the tvs-cone metric, since
, but
in the tvs
for otherwise it would tend to
in the norm of the space
.
-
(c)
Take the sequence
from (b) and
. Then
, and
in the cone metric
since
and
, but
in
. This means that a tvs-cone metric may be a discontinuous function.
-
(d)
The proof is the same as in the Banach case. For an arbitrary
, it is
for each
, and passing to the limit in
it follows that
, that is,
. Since
is a cone it follows that
.
-
(e)
From
for each
it follows that
(for arbitrary
), which, by (d), means that
.
-
(f)
Suppose, to the contrary, that for the given distinct points
and
there exists a point
. Then
for arbitrary
, implying that
, a contradiction.
The following properties, which can be proved in the same way as in the normed case, will also be needed.
Lemma 2.5.
-
(a)
If
and
, then
.
-
(b)
If
and
, then
.
-
(c)
If
and
, then
.
-
(d)
Let
,
and
be two sequences in
and
, respectively,
, and
for all
. If
, then there exists a natural number
such that
for all
.
3. Fixed Point and Common Fixed Point Results
Theorem 3.1.
Let be a tvs-cone metric space and the mappings
satisfy

for all , where
,
, and
or
. If
and
is a complete subspace of
, then
,
, and
have a unique point of coincidence. Moreover, if
and
are weakly compatible, then
,
, and
have a unique common fixed point.
Recall that a point is called a coincidence point of the pair
and
is its point of coincidence if
. The pair
is said to be weakly compatible if for each
,
implies that
.
Proof.
Let be arbitrary. Using the condition
choose a sequence
such that
and
for all
. Applying contractive condition (3.1) we obtain that

It follows that

that is,

In a similar way one obtains that

Now, from (3.4) and (3.5), by induction, we obtain that

Let

In the case ,

and if ,

Now, for , we have

Similarly, we obtain

Hence, for

where , as
.
Now, using properties (a) and (d) from Lemma 2.5 and only the assumption that the underlying cone is solid, we conclude that is a Cauchy sequence. Since the subspace
is complete, there exist
such that
(
).
We will prove that . Firstly, let us estimate that
. We have that

By the contractive condition (3.1), it holds that

Now it follows from (3.13) that

that is,

Let . Then there exists
such that for
it holds that

and , that is,
for
. Since
was arbitrary, it follows that
, that is,
.
Similarly using that

it can be deduced that . It follows that
is a common point of coincidence for
,
, and
, that is,

Now we prove that the point of coincidence of is unique. Suppose that there is another point
such that

for some . Using the contractive condition we obtain that

Since , it follows that
, that is,
.
Using weak compatibility of the pairs and
and proposition 1.12 from [16], it follows that the mappings
have a unique common fixed point, that is,
.
Corollary 3.2.
Let be a tvs-cone metric space and the mappings
satisfy

for all , where
and
. If
and
is a complete subspace of
, then
, and
have a unique point of coincidence. Moreover, if
and
are weakly compatible, then
, and
have a unique common fixed point.
Putting in this corollary and taking into account that each self-map is weakly compatible with the identity mapping, we obtain the following.
Corollary 3.3.
Let be a complete tvs-cone metric space, and let the mappings
satisfy

for all , where
and
. Then
and
have a unique common fixed point in
. Moreover, any fixed point of
is a fixed point of
, and conversely.
In the case of a cone metric space with a normal cone, this result was proved in [14].
Now put first in Theorem 3.1 and then
. Choosing appropriate values for coefficients, we obtain the following.
Corollary 3.4.
Let be a tvs-cone metric space. Suppose that the mappings
satisfy the contractive condition


or

for all , where
is a constant (
in (3.24) and
in (3.25) and (3.26)). If
and
is a complete subspace of
, then
and
have a unique point of coincidence in
. Moreover, if
and
are weakly compatible, then
and
have a unique common fixed point.
In the case when the space is normed and the cone
is normal, these results were proved in [9].
Similarly one obtains the following.
Corollary 3.5.
Let be a tvs-cone metric space, and let
be such that
. Suppose that

for all , where
and
, and let
imply that
for each
. If
or
is a complete subspace of
, then the mappings
and
have a unique common fixed point in
. Moreover, for any
, the
-
-sequence
with the initial point
converges to the fixed point.
Here, an -
-sequence (also called a Jungck sequence)
is formed in the following way. Let
be arbitrary. Since
, there exists
such that
. Having chosen
,
is chosen such that
.
In the case when the space is normed and under the additional assumption that the cone
is normal, these results were firstly proved in [10].
Corollary 3.6.
Let be a complete tvs-cone metric space. Suppose that the mapping
satisfies the contractive condition


or

for all , where
is a constant (
in (3.28) and
in (3.29) and (3.30)). Then
has a unique fixed point in
, and for any
, the iterative sequence
converges to the fixed point.
In the case when the space is normed and under the additional assumption that the cone
is normal, these results were firstly proved in [7]. The normality condition was removed in [8].
Finally, we give an example of a situation where Theorem 3.1 can be applied, while the results known so far cannot.
Example 3.7 (see [26, Example ]).
Let ,
with the cone
as in Example
and endowed with the strongest locally convex topology
. Let the metric
be defined by
if
and
,
, and
. Further, let
be given by,
,
and
,
. Finally, let
.
Taking ,
, all the conditions of Theorem 3.1 are fulfilled. Indeed, since
, we have only to check that

which is equivalent to

Hence, we can apply Theorem 3.1 and conclude that the mappings have a unique common fixed point (
).
On the other hand, since the space is not an ordered Banach space and its cone is not normal, neither of the mentioned results from [7–10, 14] can be used to obtain such conclusion. Thus, Theorem 3.1 and its corollaries are proper extensions of these results.
Note that an example of similar kind is also given in [24].
The following example shows that the condition " or
" in Theorem 3.1 cannot be omitted.
Example 3.8 (see [26, Example ]).
Let , where
,
,
, and
. Let
be the Euclidean metric in
, and let the tvs-cone metric
(
,
, and
are as in the previous example) be defined in the following way:
, where
is a fixed function, for example,
. Consider the mappings

and let . By a careful computation it is easy to obtain that

for all . We will show that
and
satisfy the following contractive condition: there exist
with
and
,
such that

holds true for all . Obviously,
and
do not have a common fixed point.
Taking (3.34) into account, we have to consider the following cases.
(1)In case , then (3.35) holds for
,
and
.
(2)In case , then (3.35) holds for
,
and
.
(3)In case , then (3.35) holds for
,
and
.
(4)In case , then (3.35) holds for
,
and
.
(5)In case , then (3.35) holds for
,
and
.
4. Quasicontractions in Tvs-Cone Metric Spaces
Definition 4.1.
Let be a tvs-cone metric space, and let
. Then,
is called a quasi-contraction (resp., a
-quasi-contraction) if for some constant
and for all
, there exists

such that

Theorem 4.2.
Let be a complete tvs-cone metric space, and let
be such that
and
is closed. If
is a
-quasi-contraction with
, then
and
have a unique point of coincidence. Moreover, if the pair
is weakly compatible or, at least, occasionally weakly compatible, then
and
have a unique common fixed point.
Recall that the pair of self-maps on
is called occasionally weakly compatible (see [27] or [28]) if there exists
such that
and
.
Proof.
Let us remark that the condition implies that starting with an arbitrary
, we can construct a sequence
of points in
such that
for all
. We will prove that
is a Cauchy sequence. First, we show that

for all . Indeed,

where

The following four cases may occur:
(1)First, .
(2)Second, and so
. In this case, (4.3) follows immediately, because
).
(3)Third, . It follows that (4.3) holds.
(4)Fourth, and so
. Hence, (4.3) holds.
Thus, by putting , we have that
. Now, using (4.3), we have

for all . It follows that

Using properties (a) and (d) from Lemma 2.5, we obtain that is a Cauchy sequence. Therefore, since
is complete and
is closed, there exists
such that

Now we will show that .
By the definition of -quasicontraction, we have that

where . Observe that
and
. Now let
be given. In all of the possible five cases there exists
such that (using (4.9)) one obtains that
:
(1);
(2);
(3); it follows that
;
(4);
(5); it follows that
.
It follows that (
). The uniqueness of limit in a cone metric space implies that
. Thus,
is a coincidence point of the pair
, and
is its point of coincidence. It can be showed in a standard way that this point of coincidence is unique. Using lemma 1.6 of [27] one readily obtains that, in the case when the pair
is occasionally weakly compatible, the point
is the unique common fixed point of
and
.
In the normed case and assuming that the cone is normal (but letting ), this theorem was proved in [11].
Puting in Theorem 4.2 we obtain the following.
Corollary 4.3.
Let be a complete tvs-cone metric space, and let the mapping
be a quasi-contraction with
. Then
has a unique fixed point in
, and for any
, the iterative sequence
converges to the fixed point.
In the case of normed-valued cone metric spaces and under the assumption that the underlying cone is normal (and with
), this result was obtained in [12]. Normality condition was removed in [13].
From Theorem 4.2, as corollaries, among other things, we again recover and extend the results of Huang and Zhang [7] and Rezapour and Hamlbarani [8]. The following three corollaries follow in a similar way.
In the next corollary, we extend the well-known result [29, (9')].
Corollary 4.4.
Let be a complete tvs-cone metric space, and let
be such that
and
is closed. Further, let for some constant
and every
there exists

such that

Then and
have a unique point of coincidence. Moreover, if the pair
is occasionally weakly compatible, then they have a unique common fixed point.
We can also extend the well-known Bianchini's result [29, (5)]
Corollary 4.5.
Let be a complete tvs-cone metric space, and let
be such that
and
is closed. Further, let for some constant
and every
, there exists

such that

Then and
have a unique point of coincidence. Moreover, if the pair
is occasionally weakly compatible, then they have a unique common fixed point.
In the next corollary, we extend the well-known result of Jungck [30, Theorem ].
Corollary 4.6.
Let be a complete tvs-cone metric space, and let
be such that
and
is closed. Further, let for some constant
and every
,

Then and
have a unique point of coincidence. Moreover, if the pair
is occasionally weakly compatible, then they have a unique common fixed point.
Remark 4.7.
Note that in the previous three corollaries it is possible that the parameter takes values from
(and not only in
as in Theorem 4.2). Namely, it is possible to show that the sequence
used in the proof, is a Cauchy sequence because the condition on
is stronger.
Now, we prove the main result of Das and Naik [31] in the frame of tvs-cone metric spaces in which the cone need not be normal.
Theorem 4.8.
Let be a complete tvs-cone metric space. Let
be a self-map on
such that
is continuous, and let
be any self-map on
that commutes with
. Further let
and
satisfy

and let be a
-quasi-contraction. Then
and
have a unique common fixed point.
Proof.
By (4.15), starting with an arbitrary , we can construct a sequence
of points in
such that
,
(as in Theorem 4.2). Now
,
. It can be proved as in Theorem 4.2 that
is a Cauchy sequence and hence convergent to some
. Further, we will show that
. Since

it follows that

because is continuous. Now, we obtain

where

Let be given. Since
and
, choose a natural number
such that for all
we have
and
. Again, we have the following cases:
-
(a)
(4.20)
-
(b)
(4.21)
-
(c)
(4.22)
-
(d)
(4.23)
-
(e)
(4.24)
Therefore, for all
. By property (d) of Lemma 2.4,
, and so
is a common fixed point for
and
. Indeed, putting in the contractivity condition
, we get
. Since
, that is,
, we have that
.
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Acknowledgments
The authors are very grateful to the referees for the valuable comments that enabled them to revise this paper. They are thankful to the Ministry of Science and Technological Development of Serbia.
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Kadelburg, Z., Radenović, S. & Rakočević, V. Topological Vector Space-Valued Cone Metric Spaces and Fixed Point Theorems. Fixed Point Theory Appl 2010, 170253 (2010). https://doi.org/10.1155/2010/170253
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DOI: https://doi.org/10.1155/2010/170253
Keywords
- Normed Space
- Contractive Condition
- Cauchy Sequence
- Topological Vector Space
- Common Fixed Point