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

        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 + 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
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-87/MediaObjects/13663_2011_186_Equa_HTML.gif
        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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-87/MediaObjects/13663_2011_186_IEq1_HTML.gif is defined by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-87/MediaObjects/13663_2011_186_Equb_HTML.gif

        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
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-87/MediaObjects/13663_2011_186_Equc_HTML.gif

        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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-87/MediaObjects/13663_2011_186_IEq2_HTML.gif 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
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-87/MediaObjects/13663_2011_186_Equd_HTML.gif

        Put v n := u nn then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-87/MediaObjects/13663_2011_186_IEq3_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-87/MediaObjects/13663_2011_186_IEq4_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-87/MediaObjects/13663_2011_186_IEq5_HTML.gif such that D(x n , x) ≺≺ c for all n ≥ N. This means that for all ε > 0 there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-87/MediaObjects/13663_2011_186_IEq6_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-87/MediaObjects/13663_2011_186_IEq7_HTML.gif then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-87/MediaObjects/13663_2011_186_IEq8_HTML.gif so by Theorem 2.1 in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-87/MediaObjects/13663_2011_186_IEq9_HTML.gif so
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-87/MediaObjects/13663_2011_186_Eque_HTML.gif

        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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-87/MediaObjects/13663_2011_186_IEq10_HTML.gif or http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-87/MediaObjects/13663_2011_186_IEq11_HTML.gif .

        Conversely, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-87/MediaObjects/13663_2011_186_IEq12_HTML.gif hence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-87/MediaObjects/13663_2011_186_IEq13_HTML.gif so http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-87/MediaObjects/13663_2011_186_IEq14_HTML.gif , therefore
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-87/MediaObjects/13663_2011_186_Equf_HTML.gif

        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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-87/MediaObjects/13663_2011_186_IEq15_HTML.gif and again by Theorem 2.1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-87/MediaObjects/13663_2011_186_IEq16_HTML.gif . □

        In the following examples, we use the metric of Lemma 1.3.

        Example 2.3 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-87/MediaObjects/13663_2011_186_IEq17_HTML.gif with ||a|| = 1 and for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-87/MediaObjects/13663_2011_186_IEq18_HTML.gif define
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-87/MediaObjects/13663_2011_186_Equg_HTML.gif
        Then D is a cone metric on http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-87/MediaObjects/13663_2011_186_IEq19_HTML.gif and its equivalent metric d is
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-87/MediaObjects/13663_2011_186_Equh_HTML.gif

        which is a discrete metric.

        Example 2.4 Let a, b ≥ 0 and consider the cone metric http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-87/MediaObjects/13663_2011_186_IEq20_HTML.gif with
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-87/MediaObjects/13663_2011_186_Equi_HTML.gif
        where d 1, d 2 are metrics on http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-87/MediaObjects/13663_2011_186_IEq21_HTML.gif . Then its equivalent metric is
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-87/MediaObjects/13663_2011_186_Equj_HTML.gif
        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
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-87/MediaObjects/13663_2011_186_Equk_HTML.gif
        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
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-87/MediaObjects/13663_2011_186_Equl_HTML.gif
        Then its equivalent metric on × is
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-87/MediaObjects/13663_2011_186_Equm_HTML.gif

        Declarations

        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.

        Authors’ Affiliations

        (1)
        Department of Mathematics, Zanjan Branch, Islamic Azad University
        (2)
        Department of Mathematics, Indiana University
        (3)
        Department of Mathematics, Malayer Branch, Islamic Azad University

        References

        1. Du WS: A note on cone metric fixed point theory and its equivalence. Nonlinear Anal 2010, 72:2259–2261.MathSciNetView ArticleMATH
        2. Huang LG, Zhang X: Cone metric spaces and fixed point theorems of contractive mapping. J Math Anal Appl 2007,322(2):1468–1476.View Article
        3. Feng Y, Mao W: Equivalence of cone metric spaces and metric spaces. Fixed Point Theory 2010,11(2):259–264.MathSciNetMATH
        4. Haghi RH, Rezapour Sh: Fixed points of multifunctions on regular cone metric spaces. Expositiones Mathematicae 2010,28(1):71–77.MathSciNetView ArticleMATH

        Copyright

        © Asadi et al; licensee Springer. 2012