sign in sign up
<<
Home , Convex cone, Symmetric set

View metadata, citation and similar papers at core.ac.uk brought to you by CORE

provided by Elsevier - Publisher Connector

Applied Letters 25 (2012) 429–433

Contents lists available at SciVerse ScienceDirect

Applied Mathematics Letters

journal homepage: www.elsevier.com/locate/aml

On an equivalence of topological valued cone metric spaces and metric spaces

Hüseyin Çakallı a,1, Ayşe Sönmez b,∗, Çiğdem Genç c a Maltepe University, Department of Mathematics, Faculty of Science and Letters, Marmara Eğitim Köyü, TR 34857, Maltepe, İstanbul, Turkey b Department of Mathematics, Gebze Institute of Technology, Cayirova Campus 41400 Gebze-Kocaeli, Turkey c Department of Mathematics, Maltepe University, Marmara Eğitim Köyü, TR 34857, Maltepe, İstanbul, Turkey article info a b s t r a c t

Article history: Scalarization method is an important tool in the study of vector optimization as Received 15 November 2010 corresponding solutions of vector optimization problems can be found by solving Received in revised form 12 September optimization problems. Recently this has been applied by Du (2010) [14] to investigate 2011 the equivalence of vectorial versions of fixed point theorems of contractive mappings in Accepted 19 September 2011 generalized cone metric spaces and scalar versions of fixed point theorems in general metric spaces in usual sense. In this paper, we find out that the topology induced by Keywords: valued cone metric coincides with the topology induced by the TVS-cone metric space Generalized TVS-cone metric space metric obtained via a nonlinear scalarization function, i.e any topological vector space Cone Banach spaces valued cone metric space is metrizable, prove a completion theorem, and also obtain some Set-valued operators more results in topological vector space valued cone normed spaces. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The concept of metric, and any concept related to metric play a very important role not only in pure mathematics but also in other branches of science involving mathematics especially in computer science, information science, and biological science.

In 1980, Rzepecki [1] introduced a generalized metric dE on a set X in a way that dE : X × X −→ K, replacing the set of real numbers with a E in the metric function where K is a normal cone in E with a partial order ≤. Seven years later, Lin [2] considered the notion of K-metric spaces by replacing the set of non-negative real numbers with a cone K in the metric function. Twenty years after Lin’s work, Long-Guang and Xian [3] announced the notion of a cone metric space by replacing real numbers with an ordering Banach space, which is the same as either the definition of Rzepecki or of Lin. Their discussion of some properties of convergence of sequences and proofs of the fixed point theorems of contractive mappings for cone metric spaces drew many authors to publish papers on fixed point theorems of contractive mappings for cone metric spaces, and on theoretical properties of cone metric spaces. Rezapour [4], Turkoglu and Abuloha [5], Sonmez [6] and some authors discuss the case in the sense of Long-Guang and Xian [3] that Y is a real Banach space and called a vector valued function p : X × X −→ Y is a cone metric if p satisfies (CM1), (CM2), and (CM3) of Definition 2.1 in Section 2. Clearly the lemmas, the theorems, and the corollaries given in above works related to topological properties are obvious consequences of Theorem 3.2. in Section 3. However

∗ Corresponding author. Tel.: +90 262 6051389. E-mail addresses: [email protected], [email protected] (H. Çakallı), [email protected], [email protected] (A. Sönmez), [email protected] (Ç. Genç). 1 Tel.: +90 216 6261050x1206; fax: +90 216 6261113.

0893-9659/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2011.09.029 430 H. Çakallı et al. / Applied Mathematics Letters 25 (2012) 429–433 results related to non-topological properties, namely, metric properties, are not direct consequences of that theorem. Such a property is a completion concept which involves Cauchy sequences, and so is a non-topological property, which is involved in Theorem 3.3 in Section 3. Karapinar [7], Turkoglu et al. [8], and some authors discuss the case in the sense of that Y is a real Banach space and called a vector valued function ||| · ||| : X −→ Y is a cone if ||| · ||| satisfies (CN1), (CN2), and (CN3) of Definition 3.1 in Section 3. Clearly the lemmas, the theorems, and the corollaries related to topological properties for Banach space valued cone normed spaces are obvious consequences of Theorem 3.4. in Section 3. However results related to non-topological properties, namely, norm properties, are not direct consequences of the theorem. The result that the collection of open subsets of a cone metric space is a first countable Hausdorff topology suggests we ask if the topology of a cone metric space coincides with an appreciate metric defined on X. There are some proofs of metrizability of Banach space valued, and topological vector space valued cone metric spaces in the literature proved independently of each other (see [9,10]), some of which are interesting. The equivalent metric in this paper is different from the others, much more simple and short (see also [11]). Our aim in this paper is to prove that any TVS-cone metric space can be metrizable, i.e. the topology induced by the TVS- cone metric p coincides with the topology induced by an appreciate metric, that any TVS-cone metric space has a completion, and obtain some results in topological vector space valued cone normed spaces.

2. Preliminaries

Let Y be a topological vector space (t.v.s. for short) with its zero vector θE . A nonempty subset K of Y is called a if K + K ⊆ K and λK ⊆ K for λ ≥ 0. A convex cone K is said to be pointed if K ∩ (−K) = {θE }. For a given cone K ⊆ Y , we can define a partial ordering -K with respect to K by

x -K y ⇔ y − x ∈ K. x ≺K y will stand for x -K y and x ̸= y, while x ≪ y stands for y − x ∈ int K, where int K denotes the interior of K. In the following, unless otherwise specified, we always suppose that Y is a locally convex Hausdorff t.v.s. with its zero vector θ, K a proper, closed and convex pointed cone in Y with int K ̸= ∅, e ∈ int K and -K a partial ordering with respect to K. Beg et al. [12,13] replaced the set of an ordered Banach space by locally convex Hausdorff t.v.s. in the definition of cone metric and generalized cone metric space. Finally, Du [14] showed that the Banach contraction principles in general metric spaces and in topological vector space valued cone metric spaces are equivalent. Now, we first recall the concept of a topological vector space valued cone metric space.

Definition 2.1. Let X be a nonempty set and let a vector-valued function p : X × X → Y , be a function satisfying the following:

(CM1) θ -K p(x, y) for all x, y ∈ X and p(x, y) = θ if and if only x = y; (CM2) p(x, y) = p(y, x) for all x, y ∈ X;

(CM3) p(x, z) -K p(x, y) + p(y, z) for all x, y, z ∈ X. Then the function p is called a topological vector space valued cone metric (TVS-cone metric for short) on X, and (X, p) is said to be a topological vector space valued cone metric space.

The well-known nonlinear scalarization function ξe : Y → R is defined as follows:

ξe(y) = inf{r ∈ R : y ∈ re − K}, for all y ∈ Y .

Lemma 2.1 ([15]). For each r ∈ R and y ∈ Y , the following statements are satisfied:

(i) ξe(y) ≤ r ⇔ y ∈ re − K; (ii) ξe(y) > r ⇔ y ̸∈ re − K; (iii) ξe(y) ≥ r ⇔ y ̸∈ re − int K; (iv) ξe(y) < r ⇔ y ∈ re − int K; (v) ξe(.) is positively homogeneous and continuous on Y ; (vi) If y1 ∈ y2 + K, then ξe(y2) ≤ ξe(y1); (vii) ξe(y1 + y2) ≤ ξe(y1) + ξe(y2) for all y1, y2 ∈ Y.

3. Results

A TVS-cone metric space is a generalization of a cone metric space in the sense that the ordered Banach space in the definition is replaced by an ordered t.v.s. Some properties of a cone metric space are also satisfied in TVS-cone metric space. H. Çakallı et al. / Applied Mathematics Letters 25 (2012) 429–433 431

In the setting, we have to deal with neighborhoods while in the former we deal with norm when we are to use the partial ordering. However, due to the change in the settings, the proofs are not always be analogous to those in Banach space valued cone metric spaces. In this section, we prove that not only cone metric spaces are metrizable, but also TVS-cone metric spaces are, and present a completion theorem. 1 ∈ ≥ First we note that for any fixed point in intK, there exists a positive m such that n c int K for n m. It is easy to see that for any open neighborhood U of zero there is a c0 ∈ int K such that c0 ∈ U. We also note that for each c ∈ int K, there exists an open neighborhood U of zero such that c − x ∈ int K whenever x ∈ U. We will need the results in the following lemma in the sequel.

Lemma 3.1. Let K be a cone in Y . Then the following conditions are satisfied: (i) λint K ⊆ int K for any positive λ; (ii) int K + int K ⊆ int K; (iii) K + int K ⊆ int K; (iv) if a ≪ b and b ≪ c, then a ≪ c for any a, b, c ∈ Y; (v) if a -K b and b ≪ c, then a ≪ c for any a, b, c ∈ Y. Proof. The proofs of assertions are easily obtained by using main properties of a cone, so are omitted.  To obtain one of our main results that any TVS-cone metric space is a topological space, we prove the following lemma.

Lemma 3.2. Let (X, p) be a TVS-cone metric space. Then, for each c1 ∈ int K and c2 ∈ int K, there exists c ∈ int K such that c ≪ c1 and c ≪ c2.

Proof. Let c1 ∈ int K, and c2 ∈ int K. Since c1 ∈ int K, there exists an open, convex, and symmetric neighborhood U of zero such that U ⊂ −c1 + int K. Similarly there exists an open, convex, and symmetric neighborhood V of zero such that V ⊂ −c2 + int K. Write W = U ∩ V . Thus W is an open, convex, and symmetric neighborhood of zero. Therefore ⊂ − + ⊂ − + c1 W c1 int K and W c2 int K. Since for this fixed c1, the sequence ( n ) converges to 0, there exists positive integer c1 ∈ = c1 − ∈ − + ∈ − + ∈ m such that m W . c m , say. Since W is a , then c W . Therefore c c1 int K and c c2 int K. Consequently, we find c ≪ c1 and c ≪ c2.  Theorem 3.1. Every TVS-cone metric space (X, p) is a topological space.

Proof. For c ∈ int K, let B(x, c) = {y ∈ X : d(x, y) ≪ c} and β = {B(x, c) : x ∈ X, c ∈ int K}. Then, τc = {U ⊂ X : ∀x ∈ U, ∃B ∈ β, x ∈ B ⊂ U} ∪ {∅}is a topology for X.

(τ1) ∅, X ∈ τc . (τ2) Let U, V ∈ τc and let x ∈ U ∩ V then x ∈ U and x ∈ V , find c1 ∈ int K, c2 ∈ int K such that x ∈ B(x, c1) ⊂ U and x ∈ B(x, c2) ⊂ V . By Lemma 3.2, find c ∈ int K such that c ≪ c1 and c ≪ c2. Then, x ∈ B(x, c) ⊂ B(x, c1) ∩ B(x, c2) ⊂ U ∩ V . Hence V ∩ V ∈ τc . ∈ ∈ ∈ ∪ ∃ ∈ ∈ ∈ (τ3) Let Uα τc for each α ∆ and let x α∈∆ Uα then α0 ∆ such that x Uα0 . Hence find c int K such that ∈ ⊂ ⊂ ∪ ∪ ∈ x B(x, c) Uα0 α∈∆ Uα. That is α∈∆ Uα τc .  Now we give a theorem which enables us to obtain that the topology induced by a certain metric which is coarser than the topology induced by the cone metric p. := ◦ ∈ = Lemma 3.3. Let (X, p) be a TVS-cone metric space, dp ξe p, and x X. Then (X, dp) is a metric space, and Bdp (x, r) Bp(x, re). ∈ = Proof. It is known that dp is a metric by Theorem 2.1 in [14]. Now assume that y Bdp (x, r), then dp(x, y) ξe(p(x, y)) < r. By the (iv) of Lemma 2.1, if we have p(x, y) ∈ re − int K, then p(x, y) ≪ re. Finally we have y ∈ Bp(x, re). Similarly y ∈ Bp(x, re), then p(x, y) ∈ re − int K. By the (iv) of Lemma 2.1, we have dp(x, y) = ξe(p(x, y)) < r. Finally we have ∈ y Bdp (x, r).   e :  Lemma 3.4. The class Bp(x, n ) n is a positive integer is a local base at the point x for each x in the TVS-cone metric space (X, p). ··· 1 ≪ 1 ≪ 1 ≪ 1 −→ Proof. It is easy to see that n+2 e n+1 e n e e for each positive integer n. We also see that n e 0 as n tends to ∞ where the convergence is in the order obtained by the cone K. Otherwise we can construct a sequence of positive e e c (nk) such that c − ̸∈ int K for each positive integer k and for a fixed element c in the int K. Hence c − ∈ (int K) for each nk nk positive integer k where (int K)c denotes the complement of int K. This implies that c ∈ (int K)c . This is a contradiction. This completes the proof of the lemma.  Theorem 3.2. Let (X, p) be a TVS-cone metric space. Then there exists a metric d on X which induces the same topology on X as the TVS-cone metric topology induced by p. := ◦ Proof. Consider the metric defined by dp ξe p. Let τdp , and τp denote the topology induced by the metric dp, and the ⊂ topology induced by the cone metric p, respectively. It follows from Lemma 3.3 that τdp τp, and Lemma 3.5 enables us to ⊂ obtain τp τdp . Thus the topology of (X, p) coincides with the topology of (X, dp).  Now we give the following completion theorem for TVS-cone metric spaces. 432 H. Çakallı et al. / Applied Mathematics Letters 25 (2012) 429–433

Theorem 3.3. Let (X, p) be a TVS-cone metric space. For (X, p) there exists a complete TVS-cone metric space (Xˆ , dˆ) which has a subspace W that is isometric with X and is dense in X.ˆ This space Xˆ is unique except for isometries, that is, if X˜ is any complete TVS-cone metric space having a dense subspace W˜ isometric with X, then X˜ and Xˆ are isometric. ˆ ˆ Proof. Consider the metric defined by dp = ξ ◦ p. So (X, dp) has a completion (X, dp). By using an improvement of Theorem 2.2 in [14], one can obtain that a sequence is a Cauchy in (X, p) if and only if it is Cauchy in (X, dp) and a sequence ˆ is convergent to a point x in (X, p) if and only if it is convergent to x in (X, dp). Noticing that dp = ξ ◦ pˆ, we get a completion ˆ ˆ ˆ (X, dp) of (X, dp). It is easy to see that (X, pˆ) is a completion of (X, p). This completes the proof of the theorem.  If we take E as a Banach space, then we obtain the result proved in [16,6] as a special case.

Corollary 3.1. A cone metric space (X, p) is complete if and only if (X, dp) is complete. Now, we give a definition of TVS-cone normed space.

Definition 3.1. Let X be a vector space. A vector valued function |||· ||| : X → Y is said to be a TVS-cone normed space, if the following conditions hold: (CN1) |||x||| = 0 ⇔ x = 0 (CN2) |||αx||| = |α||||x|||

(CN3) |||x + y||| -K |||x||| + |||y|||. The pair (X, ||| · |||) is then called a TVS-cone normed space.

We note that in [5,17,7,8] the case that Y is a real Banach space and called a vector-valued function ||| · ||| : X → Y is a cone norm if ||| · ||| satisfies (CN1), (CN2) and (CN3). Clearly, a cone normed space in the sense of above mentioned works is a special case of a TVS-cone normed space. It is clear that p(x, y) = |||x − y||| is a TVS-cone metric and so dp := ξe ◦ p is a metric on X.

Lemma 3.5. Let (X, ||| · |||) be a TVS-cone normed space. Then ‖·‖: X → [0, ∞) defined by ‖ · ‖ := ξe ◦ ||| · ||| is a norm.

Proof. By (CN1) and Lemma 3.2, ‖x‖ = 0 if and only if x = 0. Since ξe(.) is positively homogeneous and (CN2) ‖αx‖ = ξe(|||αx|||) = ξe(|α||||x|||) = |α|ξe(|||x|||) = |α||||x|||. Applying (vi) and (vii) of Lemma 3.2 and (CN3), we have

ξe(|||x + y|||) -K ξe(|||x||| + |||y|||) -K ξe(|||x|||) + ξe(|||y|||) for all x, y, z ∈ X.

That is, ‖·‖ satisfies the triangle inequality. So we have obtained that ‖·‖ is a norm. 

Theorem 3.4. Any TVS-cone normed space is normable, i.e. the topology of the cone normed space coincides with the topology of the norm defined in Lemma 3.5.

Proof. Write ‖ · ‖ := ξe ◦ |||· |||, and d(x, y) := ‖x − y‖ = ξe ◦ |||x − y|||. Then it follows from Lemmas 3.3–3.5 that the topology of the cone normed space (X, ||| · |||) coincides with the topology of the normed space (X, ‖·‖). 

4. Conclusion

We would like to emphasize that the present work contains not only the proofs of metrizability of a topological vector space valued cone metric space and normability of a topological vector space valued cone normed space, but also some useful theorems involving Cauchy sequences. Using the theorems obtained in the paper, we suggest to investigate further results related to convergence of a sequence, and continuity of a function both in a topological vector space valued cone metric space and a topological vector space valued cone normed space. We also suggest to investigate the case for fuzzy cone metric spaces and fuzzy cone normed spaces. However due to the change in settings, the definitions and methods of proofs will not always be analogous to those of the present work.

Acknowledgments

The authors would like to thank the referees for a careful reading and some constructive comments that have improved the presentation of the results.

References

[1] B. Rzepecki, On fixed point theorems of Maia type, Publications de l’Institut Mathematique 28 (42) (1980) 179–186. MR 83a:54073. [2] S.D. Lin, A common fixed point theorem in abstract spaces, Indian J. Pure Appl. Math. 18 (8) (1987) 685–690. MR 88h:54062. H. Çakallı et al. / Applied Mathematics Letters 25 (2012) 429–433 433

[3] H. Long-Guang, Z. Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468–1476. MR 2008d:47111. [4] Sh. Rezapour, A review on topological properties of cone metric spaces, Conference, Analysis, Topology and Applications Serbia, 2008. [5] D. Turkoglu, M. Abuloha, Cone metric spaces and fixed point theorems in diametrically contractive mappings, Acta Math. Sin. (Engl. Ser.) 26 (3) (2010) 489–496. [6] A. Sonmez, On paracompactness in cone metric spaces, Appl. Math. Lett. 23 (2010) 494–497. [7] E. Karapinar, Fixed point theorems in cone Banach spaces, Fixed Point Theory Appl. 2009 (2009) doi:10.1155/2009/609281. Article ID 609281, 2010m:47081. [8] D. Turkoglu, M. Abuloha, T. Abdeljawad, Some theorems and examples of cone Banach spaces, J. Comput. Anal. Appl. 12 (4) (2010) 739–753. [9] Z. Kadelburg, S. Radenovic, V. Rakocevic, A note on equivalence of some metric and cone metric fixed point results, Appl. Math. Lett., in press (doi:10. 1016/j.aml.2010.10.030). [10] A. Amini-Harandi, M. Fakhar, Fixed point theory in cone metric spaces obtained via the scalarization method, Comput. Math. Appl 59 (2010) 3529–3534. [11] Hüseyin Çakallı, Ayşe Sönmez, Çiğdem Genç, Metrizability of topological vector space valued cone metric spaces, July 2010, http://arxiv.org/abs/1007. 3123v2. [12] I. Beg, M. Abbas, T. Nazir, Generalized cone metric space, J. Nonlinear Sci. Appl. 3 (1) (2010) 21–31. [13] I. Beg, A. Azam, M. Arshad, Fixed point in topological vector space-valued cone metric spaces, Fixed Point Theory Appl. 2010 (2010) doi:10.1155/2010/ 604084. Article ID 604084. [14] W.-S. Du, A note on cone metric fixed point theory and its equivalence, Nonlinear Anal. 72 (2010) 2259–2261. 2010k:47119. [15] A. Gpfert, Chr. Tammer, H. Riahi, C. Zlinescu, Variational Methods in Partially Ordered Spaces, Springer-Verlag, New York, 2003. [16] T. Abdeljawad, Completion of cone metric spaces, Hacet. J. Math. Stat. 39 (1) (2010) 67–74. [17] A. Sonmez, H. Cakalli, Cone normed space and weighted means, Math. Comput. Modelling 52 (2010) 1660–1666.