On Equivalent Cone Metric Spaces
We explore the necessary and sufficient conditions for the two cone metrics to be topologically equivalent.
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Institute of Mathematics, NAS of Ukraine
2013
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508460067586048 |
|---|---|
| author | Aytar, S. Ölmez, Ö. Айтар, С. Олмез, О. |
| author_facet | Aytar, S. Ölmez, Ö. Айтар, С. Олмез, О. |
| author_sort | Aytar, S. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:26:06Z |
| description | We explore the necessary and sufficient conditions for the two cone metrics to be topologically equivalent. |
| first_indexed | 2026-03-24T02:25:33Z |
| format | Article |
| fulltext |
К О Р О Т К I П О В I Д О М Л Е Н Н Я
UDC 517.91
Ö. Ölmez, S. Aytar (Süleyman Demirel Univ., Isparta, Turkey)
ON EQUIVALENT CONE METRIC SPACES*
ПРО ЕКВIВАЛЕНТНI КОНIЧНI МЕТРИЧНI ПРОСТОРИ
We explore the necessary and sufficient conditions for the two cone metrics to be topologically equivalent.
Дослiджено необхiднi та достатнi умови для топологiчної еквiвалентностi двох конiчних метрик.
1. Introduction. The concept of cone metric space was first introduced by Huang and Zhang [7]
in 2007. They also obtained some fixed point theorems for mappings satisfying certain contractive
conditions. Afterwards, many authors generalized fixed point theorems from metric spaces to cone
metric spaces (see 1, 3, 8, 9, 10, 12). In 2010, Du [4] obtained an ordinary metric corresponding to
a cone metric using the following nonlinear scalarization function: Let E be a Banach space and P
be a cone in E. The nonlinear scalarization function ξe : E → R is defined as follows:
ξe(y) = inf{r ∈ R : y ∈ re− P} for all y ∈ E.
If (X,D) is a cone metric space, Du [4] showed that ρD := ξe ◦ D is an ordinary metric on
X. Abdeljawad [2] proved that for every complete cone metric space there exists a correspondent
complete usual metric space such that the spaces are topologically equivalent.
In this paper, we introduce the concept of equivalent cone metrics on the same cone. We present
the relations between the notions of convergence and equivalence in cone metric spaces. We obtain
the necessary and sufficient conditions for two cone metrics to be equivalent. We also present an
alternative definition for the equivalence of cone metrics, which is called the Lipschitz equivalence.
Finally, we compare these two definitions.
2. Preliminaries. Let E be a real Banach space. A nonempty convex closed subset P ⊂ E is
called a cone in E if it satisfies the following conditions:
(i) P is closed, nonempty and P 6= {0} ,
(ii) a, b ∈ R, a, b ≥ 0 and x, y ∈ P imply that ax+ by ∈ P ,
(iii) x ∈ P and −x ∈ P imply that x = 0 [7].
Given a cone P ⊂ E, we define a partial ordering ≤ with respect to P by x ≤ y if and only
if y − x ∈ P . We shall write x < y to indicate that x ≤ y but x 6= y, while x � y will stand for
y − x ∈ intP, intP denotes the interior of P [7].
In the sequel, one also has to note that by using the properties of the cone and the definition of
the interior that intP + P ⊆ intP [11].
Let X be a nonempty set. Suppose the mapping D : X ×X → E satisfies
(d1) 0 < D(x, y) for all x, y ∈ X, and D(x, y) = 0 if and only if x = y,
(d2) D(x, y) = D(y, x) for all x, y ∈ X ,
(d3) D(x, y) ≤ D(x, z) +D(z, y) for all x, y, z ∈ X .
* This paper was supported by grant SDU-BAP-2330-YL-10 from the Suleyman Demirel University, Isparta, Turkey.
c© Ö. ÖLMEZ, S. AYTAR, 2013
1712 ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 12
ON EQUIVALENT CONE METRIC SPACES 1713
Then D is called a cone metric on X, and (X,D) is called a cone metric space. It is obvious that
a cone metric space is a generalization of an ordinary metric space [7].
Example 2.1. Let P =
{
{xn} ∈ l1 : xn ≥ 0, for all n
}
, (X, d) be any metric space andD : X×
×X → l1 defined by D(x, y) =
{
min{1, d(x, y)}
n2
}
. Then (X,D) is a cone metric space.
Let (X,D) be a cone metric space, x ∈ X and {xn} be a sequence in X. Then {xn} is said to be
convergent to x provided that, for every c ∈ E with 0� c there is a positive integer N = N(c) such
that D(xn, x)� c for all n ≥ N. We denote this by D − limn→∞xn = x or xn
D→ x as n→∞.
Let (X,D) be a cone metric space and A ⊆ X.
(i) A point a ∈ A is called an interior point of A if there exists a point c with 0 � c such that
BD (a, c) ⊆ A, where BD (a, c) := {y ∈ X : D(a, y)� c} is called the D-ball of a.
(ii) A subset A ⊆ X is called D-open if each element of A is an interior point of A. The family
β = {BD (x, e) : x ∈ X, 0� e} is a subbasis for a topology on X. We denote this cone topology
by τc. The topology τc is Hausdorff and first countable [2, 3, 6].
Theorem 2.1 [2, 4]. Let (X,D) be a cone metric space, x ∈ X and {xn} be a sequence in X.
Define ρD := ξe ◦D. Then the following statements hold:
(i) {xn} converges to x in the cone metric space (X,D) if and only if ρD(xn, x) → 0 as
n→∞,
(ii) {xn} is a Cauchy sequence in the cone metric space (X,D) if and only if {xn} is a Cauchy
sequence in (X, ρD),
(iii) (X,D) is a complete cone metric space if and only if (X, ρD) is a complete metric space.
3. Main results.
Definition 3.1. Let D1 and D2 be cone metrics on a set X . If each D1-open subset of X is
D2-open and each D2-open subset of X is D1-open, then D1 and D2 are said to be equivalent.
Now we give an important result which characterizes the concept of equivalence of two cone
metrics. Its proof is similar to the ordinary case (see [5], Proposition 1.32).
Proposition 3.1. Let D1 and D2 be cone metrics on the same set X. Then a necessary and
sufficient condition for D1 to be equivalent to D2 is that, given any point x ∈ X, each D1-ball at x
contains some D2-ball at x, and each D2-ball at x contains some D1-ball at x.
Proof. Necessity. Assume that D1 is equivalent to D2. Let x ∈ X. Consider an arbitrary D1-ball
BD1(x, c) at x. Since BD1(x, c) is D1-open, by our assumption it must be D2-open as well. Hence
at the point x ∈ BD1(x, c) there is some D2-ball BD2(x, e) with BD2(x, e) ⊂ BD1(x, c). Similarly,
for each D2-ball BD2(x, c
′), there is some D1-ball BD1(x, e
′) with BD1(x, e
′) ⊂ BD2(x, c
′).
Sufficiency. Assume the given conditions hold. We will show that each D1-open set is D2-open
and each D2-open set is D1-open. Let M be a D1-open subset of X. Let x ∈ M. Since M is
D1-open, there is some 0 � c with BD1(x, c) ⊂ M. By assumption, there is some 0 � e with
BD2(x, e) ⊂ BD1(x, c). Then BD2(x, e) ⊂M. Since x is an arbitrary point of M, we conclude that
M is D2-open.
Let N be a D2-open subset of X. Let x ∈ N. Since N is D2-open, there is some 0 � c′
with BD2(x, c
′) ⊂ N. By assumption, there is some 0 � e′ with BD1(x, e
′) ⊂ BD2(x, c
′). Then
BD1(x, e
′) ⊂ N. Since x is an arbitrary point of N, we conclude that N is D1-open.
Proposition 3.1 is proved.
Now each D-ball at a point x in a cone metric space (X,D) is a D-neighborhood of x, and each
D-neighborhood of x contains some D-ball at x. Hence we have the following criterion.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 12
1714 Ö. ÖLMEZ, S. AYTAR
Theorem 3.1. Let {xn} be a sequence in a cone metric space (X,D), and let x ∈ X. Then
a necessary and sufficient condition for {xn} to converge to x in (X,D) is that for each D-
neighborhood V of x there exists some N ∈ N with xn ∈ V whenever n ≥ N.
Corollary 3.1. In the notation of Theorem 3.1, let D1 be a cone metric on X that is equivalent
to D2. Then {xn} converges to x in (X,D1) if and only if it converges to x in (X,D2).
Now we define two cone metrics which are not equivalent. Let P =
{
(x, y) ∈ R2 : x, y ≥ 0
}
⊂
⊂ R2 and D1, D2 : X ×X → R2 such that
D1(x, y) = (dA(x, y), αdA(x, y)),
D2(x, y) = (dE(x, y), αdE(x, y)),
where α > 0 is a constant and the metrics dA and dE denote the discrete and Euclidean metrics
on R, respectively. By Corollary 3.1, in order to show these cone metrics to be nonequivalent, it is
enough to prove that convergence in these cone metric spaces do not require each other.
Definition 3.2. Let D1 and D2 be two cone metrics on a set X . We say that D1 and D2 are
Lipschitz equivalent on X if there exist two positive constants t1 and t2 such that
t1D2(x, y) ≤ D1(x, y) ≤ t2D2(x, y) for all x, y ∈ X.
The next theorem says that Lipschitz equivalence implies the equivalence in the sense of Defini-
tion 3.1.
Theorem 3.2. Let D1 and D2 be two cone metrics on a set X . If the cone metrics D1 and D2
are Lipschitz equivalent, then they are equivalent in the sense of Definition 3.1.
Proof. Let t1D2(x, y) ≤ D1(x, y) ≤ t2D2(x, y). It suffices to show that BD1(x, t1c) ⊂
⊂ BD2(x, c) and BD2
(
x,
c
t2
)
⊂ BD1(x, c) for all x ∈ X and c ∈ intP. Take
y ∈ BD1(x, t1c) ⇒ D1(x, y) � t1c ⇒ t1c−D1(x, y) ∈ intP. (3.1)
On the other hand, we get
t1D2(x, y) ≤ D1(x, y) ⇒ D1(x, y)− t1D2(x, y) ∈ P. (3.2)
Combining the inequalities (3.1) and (3.2), we get t1c− t1D2(x, y) ∈ intP +P . Since the inclusion
intP + P ⊆ intP holds, we obtain
t1c− t1D2(x, y) ∈ intP. (3.3)
Since λintP ⊂ intP for all λ > 0, we have c −D2(x, y) ∈ intP from the expression (3.3). Then
D2(x, y)� c.
Now we prove that the inclusion BD2
(
x,
c
t2
)
⊂ BD1(x, c) is valid. Clearly,
y ∈ BD2
(
x,
c
t2
)
⇒ D2(x, y) �
c
t2
⇒ c
t2
−D2(x, y) ∈ intP. (3.4)
Since λ intP ⊂ intP for all λ > 0, using the expression (3.4) we have
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 12
ON EQUIVALENT CONE METRIC SPACES 1715
c− t2D2(x, y) ∈ intP. (3.5)
On the other hand, we get
D1(x, y) ≤ t2D2(x, y)⇒ t2D2(x, y)−D1(x, y) ∈ P. (3.6)
Combining the inequalities (3.5) and (3.6) we get c−D1(x, y) ∈ intP+P . Since intP+P ⊆ intP,
we have c−D1(x, y) ∈ intP, i.e., D1(x, y)� c.
Theorem 3.2 is proved.
The converse of Theorem 3.2 is not true in general as can be seen in the example below.
Example 3.1. Define D1, D2 : X × X → l1 as D1(x, y) =
{
d(x, y)
n2
}
and D2(x, y) =
=
{
min{1, d(x, y)}
n2
}
. Let X = R, d : X × X → R such that d(x, y) = |x− y| and P =
= {{xn} ∈ l1 : xn ≥ 0 for all n}. It is easily to show that the cone metrics D1 and D2 are
equivalent but they are not Lipschitz equivalent. A cone metric space is a first-countable topological
space [3]. That is why, in order to show that these cone metrics are equivalent it suffices to show that
xk
D1→ x0 iff xk
D2→ x0 as k →∞.
Definition 3.3. Let D1 and D2 be two cone metrics on a set X . We say that D1 and D2 are
strong Lipschitz equivalent on X if there exist positive constants t1 and t2 such that
t1D2(x, y)� D1(x, y)� t2D2(x, y)
for all x, y ∈ X .
Theorem 3.3. Let D1 and D2 be two cone metrics on a set X . If the cone metrics D1 and D2
are strong Lipschitz equivalent, then these cone metrics are equivalent in the sense of Definition 3.1.
This theorem can be proved using the similar arguments in the proof of Theorem 3.2.
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Received 30.03.12,
after revision — 13.06.13
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 12
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| spelling | umjimathkievua-article-25482020-03-18T19:26:06Z On Equivalent Cone Metric Spaces Про еквівалентні конічні метричні простори Aytar, S. Ölmez, Ö. Айтар, С. Олмез, О. We explore the necessary and sufficient conditions for the two cone metrics to be topologically equivalent. Досліджено необхідні та достатні умови для топологічної еквівалентності двох конічних метрик. Institute of Mathematics, NAS of Ukraine 2013-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2548 Ukrains’kyi Matematychnyi Zhurnal; Vol. 65 No. 12 (2013); 1712–1715 Український математичний журнал; Том 65 № 12 (2013); 1712–1715 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2548/1856 https://umj.imath.kiev.ua/index.php/umj/article/view/2548/1857 Copyright (c) 2013 Aytar S.; Ölmez Ö. |
| spellingShingle | Aytar, S. Ölmez, Ö. Айтар, С. Олмез, О. On Equivalent Cone Metric Spaces |
| title | On Equivalent Cone Metric Spaces |
| title_alt | Про еквівалентні конічні метричні простори |
| title_full | On Equivalent Cone Metric Spaces |
| title_fullStr | On Equivalent Cone Metric Spaces |
| title_full_unstemmed | On Equivalent Cone Metric Spaces |
| title_short | On Equivalent Cone Metric Spaces |
| title_sort | on equivalent cone metric spaces |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2548 |
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