Topological Vector Space-Valued Cone Metric Spaces and Fixed Point Theorems
© Zoran Kadelburg et al. 2010
Received: 18 December 2009
Accepted: 19 July 2010
Published: 2 August 2010
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.
Ordered normed spaces and cones have applications in applied mathematics, for instance, in using Newton's approximation method [1–4] and in optimization theory . -metric and -normed spaces were introduced in the mid-20th century (, 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  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., ) 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  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).
Hence, in this case, the Sandwich theorem does not hold.
Note the following properties of bounded sets.
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 .
If the underlying cone of an ordered tvs is solid and normal, then such tvs must be an ordered normed space.
(see ) 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., ).
Taking into account Theorem 2.1, proper generalizations when passing from norm-valued cone metric spaces of  to tvs-cone metric spaces can be obtained only in the case of nonnormal cones.
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 , .
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 ).
The following properties, which can be proved in the same way as in the normed case, will also be needed.
3. Fixed Point and Common Fixed Point Results
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 ( ).
Using weak compatibility of the pairs and and proposition 1.12 from , it follows that the mappings have a unique common fixed point, that is, .
In the case of a cone metric space with a normal cone, this result was proved in .
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 .
Similarly one obtains the following.
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.
In the case when the space is normed and under the additional assumption that the cone is normal, these results were firstly proved in .
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 ]).
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 .
Example 3.8 (see [26, Example ]).
Taking (3.34) into account, we have to consider the following cases.
4. Quasicontractions in Tvs-Cone Metric Spaces
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.
The following four cases may occur:
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  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 .
From Theorem 4.2, as corollaries, among other things, we again recover and extend the results of Huang and Zhang  and Rezapour and Hamlbarani . The following three corollaries follow in a similar way.
In the next corollary, we extend the well-known result [29, (9')].
We can also extend the well-known Bianchini's result [29, (5)]
In the next corollary, we extend the well-known result of Jungck [30, Theorem ].
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  in the frame of tvs-cone metric spaces in which the cone need not be normal.
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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