Abstract
In this article, we shall show that the metrics defined by Feng and Mao, and Du are equivalent. We also provide some examples for one of the metrics.
1 Introduction and preliminary
Let E be a topological vector space (t.v.s.) with zero vector θ. A nonempty subset K of E is called a convex cone if K + K ⊆ K and λK ⊆ K for each λ ≥ 0. A convex cone K is said to be pointed if K ∩  K = {θ}. For a given cone K ⊆ E, we can define a partial ordering ≼ with respect to K by
x < y will stand for x ≼ y and x ≠ y while x ≺≺ y stands for y − x ∈ K°, where K° denotes the interior of K. In the following, we shall always assume that Y is a locally convex Hausdorff t.v.s. with zero vector θ, K is a proper, closed, and convex pointed cone in Y with K° ≠ ∅, e ∈ K° and ≼ a partial ordering with respect to K. The nonlinear scalarization function is defined by
for all y ∈ Y.
We will use P instead of K when E is a real Banach spaces.
Lemma 1.1 [1]For each r ∈ R and y ∈ Y, the following statements are satisfied:
(i) ξ_{e}(y) ≤ r ⇔ y ∈ re − K.
(ii) ξ_{e}(y) > r ⇔ y ∉ re − K.
(iii) ξ_{e}(y) ≥ r ⇔ y ∉ re − K°.
(iv) ξ_{e}(y) < r ⇔ y ∈ re − K°.
(v) ξ_{e}(.) is positively homogeneous and continuous on Y .
(vi) y_{1 }∈ y_{2 }+ K ⇒ ξ_{e}(y_{2}) ≤ ξ_{e}(y_{1})
(vii) ξ_{e}(y_{1 }+ y_{2}) ≤ ξ_{e}(y_{1}) + ξ_{e}(y_{2}) for all y_{1}, y_{2 }∈ Y.
Definition 1.2 [1]Let X be a nonempty set. A vectorvalued function d : X × X → Y is said to be a TVScone metric, if the following conditions hold:
(C1) θ ≼ d(x, y) for all x, y ∈ X and d(x, y) = θ iff x = y
(C2) d(x, y) = d(y, x) for all x, y ∈ X
(C3)d(x, y) ≼ (x, z) + d(z, y) for all x, y, z ∈ X.
The pair (X, d) is then called a TVScone metric space.
Huang and Zhang [2] discuss the case in which Y is a real Banach space and call a vectorvalued function d : X × X → Y a cone metric if d satisfies (C1)(C3). Clearly, a cone metric space, in the sense of Huang and Zhang, is a special case of a TVScone metric space.
In the following, some conclusions are listed.
Lemma 1.3 [3]Let (X, D) be a cone metric space. Then
is a metric on X.
Theorem 1.4 [3]The metric space (X, d) is complete if and only if the cone metric space (X, D) is complete .
Theorem 1.5 [1]Let (X, D) be a TVScone metric space. Then d_{2 }: X × X → [0, ∞) defined by d_{2}(x, y) = ξ_{e}(D(x, y)) is a metric.
2 Main results
We first show that the metrics introduced the Lemma 1.3 and the Theorem 1.5 are equivalent. Then, we provide some examples involving the metric defined in Lemma 1.3.
Theorem 2.1 For every cone metric D : X × X → E there exists a metric which is equivalent to D on X.
Proof. Define d(x, y) = inf {u: D(x, y) ≼ u}. By the Lemma 1.3 d is a metric. We shall now show that each sequence {x_{n}} ⊆ X which converges to a point x ∈ X in the (X, d) metric also converges to x in the (X, D) metric, and conversely. We have
Put v_{n }:= u_{nn }then and D(x_{n}, x) ≼ v_{n}. Now if x_{n }→ x in (X, d) then d(x_{n}, x) → 0 and so v_{n }→ 0 too, therefore for all c ≻≻ 0 there exists such that v_{n }≺≺ c for all n ≥ N. This implies that D(x_{n}, x) ≺≺ c for all n ≥ N. Namely x_{n }→ x in (X, D).
Conversely, for every real ε > 0, choose c ∈ E with c ≻≻ 0 and c < ε. Then there exists such that D(x_{n}, x) ≺≺ c for all n ≥ N. This means that for all ε > 0 there exists such that d(x_{n}, x) ≤ c < ε for all n ≥ N. Therefore d(x_{n}, x) → 0 as n → ∞ so x_{n }→ x in (X, d).
□
Theorem 2.2 If d_{1}(x, y) = inf {u: D(x, y) ≼ u} and d_{2}(x, y) = ξ_{e}(D(x, y)) where D is a cone metric on X. Then d_{1 }is equivalent with d_{2}.
Proof. Let then so by Theorem 2.1 in so
and or εe − D(x_{n}, x) ∈ K° for all n ≥ N. So D(x_{n}, x) ∈ e  K° for n ≥ N. Now by [[1], Lemma 1.1 (iv)] ξ_{e}(D(x_{n}, x)) < ε for all n ≥ N. Namely d_{2}(x_{n}, x) < ε for all n ≥ N therefore or .
Conversely, hence so , therefore
So D(x_{n}, x) ∈ εe−K° for n ≥ N by [[1], Lemma 1.1 (iv)]. Hence, D(x_{n}, x) = εe−k for some k ∈ K°, so D(x_{n}, x) ≺≺ εe for n ≥ N this implies that and again by Theorem 2.1 . □
In the following examples, we use the metric of Lemma 1.3.
Example 2.3 Let with a = 1 and for every define
Then D is a cone metric on and its equivalent metric d is
which is a discrete metric.
Example 2.4 Let a, b ≥ 0 and consider the cone metric with
where d_{1}, d_{2 }are metrics on . Then its equivalent metric is
In particular if d_{1}(x, y):= x − y and d_{2}(x, y):= αx − y, where α ≥ 0 then D is the same famous cone metric which has been introduced in [[2], Example 1] and its equivalent metric is
Example 2.5 For q > 0, b > 1, E = l^{q}, P = {{x_{n}}_{n}_{≥1 }: x_{n }≥ 0, for all n} and (X, ρ) a metric space, define D : X × X → E which is the same cone metric as [[4], Example 1.3] by
Then its equivalent metric on × is
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
All authors have read and approved the final manuscript.
Acknowledgements
This research was supported by the Zanjan Branch, Islamic Azad University, Zanjan, Iran. Mehdi Asadi would like to acknowledge this support. The first and third authors would like proudly to dedicate this paper to Professor Billy E. Rhoades in recognition of his the valuable works in mathematics. The authors would also like to thank Professor S. Mansour Vaezpour for his helpful advise which led them to present this article. They also express their deep gratitude to the referee for his/her valuable comments and suggestions.
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