Some notes on the paper "The equivalence of cone metric spaces and metric spaces"

  • Mehdi Asadi1Email author,

    Affiliated with

    • Billy E Rhoades2 and

      Affiliated with

      • Hossein Soleimani3

        Affiliated with

        Fixed Point Theory and Applications20122012:87

        DOI: 10.1186/1687-1812-2012-87

        Received: 9 May 2011

        Accepted: 21 May 2012

        Published: 21 May 2012


        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 + KK and λKK for each λ 0. A convex cone K is said to be pointed if K ∩ - K = {θ}. For a given cone KE, we can define a partial ordering ≼ with respect to K by
        x < y will stand for x ≼ y and xy while x ≺≺ y stands for y − x, where 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 ≠ ∅, e and ≼ a partial ordering with respect to K. The nonlinear scalarization function is defined by

        for all yY.

        We will use P instead of K when E is a real Banach spaces.

        Lemma 1.1 [1]For each rR and yY, the following statements are satisfied:
        1. (i)

          ξ e (y) ≤ ryre − K.

        2. (ii)

          ξ e (y) > ryre − K.

        3. (iii)

          ξ e (y) ≥ ryre − K°.

        4. (iv)

          ξ e (y) < ryre − K°.

        5. (v)

          ξ e (.) is positively homogeneous and continuous on Y .

        6. (vi)

          y 1y 2 + Kξ e (y 2) ≤ ξ e (y 1)

        7. (vii)

          ξ e (y 1 + y 2) ≤ ξ e (y 1) + ξ e (y 2) for all y 1, y 2Y.

        Definition 1.2 [1]Let X be a nonempty set. A vector-valued function d : X × XY is said to be a TVS-cone metric, if the following conditions hold:
        1. (C1)

          θd(x, y) for all x, yX and d(x, y) = θ iff x = y

        2. (C2)

          d(x, y) = d(y, x) for all x, yX

        3. (C3)

          d(x, y) ≼ (x, z) + d(z, y) for all x, y, zX.


        The pair (X, d) is then called a TVS-cone metric space.

        Huang and Zhang [2] discuss the case in which Y is a real Banach space and call a vector-valued function d : X × XY 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 TVS-cone 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 TVS-cone 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 × XE 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 xX 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 cE 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) ∈ for all n ≥ N. So D(x n , x) ∈ e - 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, 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):= α|x − 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 × XE which is the same cone metric as [[4], Example 1.3] by
        Then its equivalent metric on × is



        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.

        Authors’ Affiliations

        Department of Mathematics, Zanjan Branch, Islamic Azad University
        Department of Mathematics, Indiana University
        Department of Mathematics, Malayer Branch, Islamic Azad University


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        4. Haghi RH, Rezapour Sh: Fixed points of multifunctions on regular cone metric spaces. Expositiones Mathematicae 2010,28(1):71–77.MathSciNetView ArticleMATH


        © Asadi et al; licensee Springer. 2012