Open Access

Topological Vector Space-Valued Cone Metric Spaces and Fixed Point Theorems

  • Zoran Kadelburg1,
  • Stojan Radenović2Email author and
  • Vladimir Rakočević3
Fixed Point Theory and Applications20102010:170253

DOI: 10.1155/2010/170253

Received: 18 December 2009

Accepted: 19 July 2010

Published: 2 August 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 [14] 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., [822]) 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
(2.1)

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
(2.2)
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
(2.3)

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.
  1. (a)

    Let in , and let . Then there exists such that for each .

     
  2. (b)

    It can happen that for each , but in .

     
  3. (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).

     
  4. (d)

    for each implies that .

     
  5. (e)

    (in the tvs-cone metric) implies that .

     
  6. (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.
  1. (a)

    It follows from that for . From , it follows that , that is, .

     
  2. (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 .
  1. (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.

     
  2. (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 .

     
  3. (e)

    From for each it follows that (for arbitrary ), which, by (d), means that .

     
  4. (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.
  1. (a)

    If and , then .

     
  2. (b)

    If and , then .

     
  3. (c)

    If and , then .

     
  4. (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
(3.1)

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
(3.2)
It follows that
(3.3)
that is,
(3.4)
In a similar way one obtains that
(3.5)
Now, from (3.4) and (3.5), by induction, we obtain that
(3.6)
Let
(3.7)
In the case ,
(3.8)
and if ,
(3.9)
Now, for , we have
(3.10)
Similarly, we obtain
(3.11)
Hence, for
(3.12)

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
(3.13)
By the contractive condition (3.1), it holds that
(3.14)
Now it follows from (3.13) that
(3.15)
that is,
(3.16)
Let . Then there exists such that for it holds that
(3.17)

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

Similarly using that
(3.18)
it can be deduced that . It follows that is a common point of coincidence for , , and , that is,
(3.19)
Now we prove that the point of coincidence of is unique. Suppose that there is another point such that
(3.20)
for some . Using the contractive condition we obtain that
(3.21)

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
(3.22)

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
(3.23)

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
(3.24)
(3.25)
or
(3.26)

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
(3.27)

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
(3.28)
(3.29)
or
(3.30)

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
(3.31)
which is equivalent to
(3.32)

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 [710, 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
(3.33)
and let . By a careful computation it is easy to obtain that
(3.34)
for all . We will show that and satisfy the following contractive condition: there exist with and , such that
(3.35)

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
(4.1)
such that
(4.2)

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
(4.3)
for all . Indeed,
(4.4)
where
(4.5)

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
(4.6)
for all . It follows that
(4.7)
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
(4.8)

Now we will show that .

By the definition of -quasicontraction, we have that
(4.9)

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
(4.10)
such that
(4.11)

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
(4.12)
such that
(4.13)

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 ,
(4.14)

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
(4.15)

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
(4.16)
it follows that
(4.17)
because is continuous. Now, we obtain
(4.18)
where
(4.19)
Let be given. Since and , choose a natural number such that for all we have and . Again, we have the following cases:
  1. (a)
    (4.20)
     
  1. (b)
    (4.21)
     
  1. (c)
    (4.22)
     
  1. (d)
    (4.23)
     
  1. (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 .

Declarations

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.

Authors’ Affiliations

(1)
Faculty of Mathematics, University of Belgrade
(2)
Faculty of Mechanical Engineering, University of Belgrade
(3)
Department of Mathematics, Faculty of Sciences and Mathematics, University of Niš

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