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

Open Access
Research

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.

Keywords

Banach Space Vector Space Differential Geometry Scalarization Function Convex Cone 

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 y - x K . Open image in new window
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 ξ e : Y Open image in new window is defined by
ξ e ( y ) = inf { r : y r e - K } Open image in new window

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:

(i) ξ e (y) ≤ ryre − K.

(ii) ξ e (y) > ryre − K.

(iii) ξ e (y) ≥ ryre − K°.

(iv) ξ e (y) < ryre − K°.

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

(vi) y1y2 + Kξ e (y2) ≤ ξ e (y1)

(vii) ξ e (y1 + y2) ≤ ξ e (y1) + ξ e (y2) for all y1, y2Y.

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:

(C1) θd(x, y) for all x, yX and d(x, y) = θ iff x = y

(C2) d(x, y) = d(y, x) for all x, yX

(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
d ( x , y ) = inf { u P | D ( x , y ) u } | | u | | , x , y X Open image in new window

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 d2 : X × X → [0, ∞) defined by d2(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 d : X × X + Open image in new window 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
n , m u n m s u c h t h a t | | u n m | | < d ( x n , x ) + 1 m , D ( x n , x ) u n m . Open image in new window

Put v n := u nn then | | v n | | < d ( x n , x ) + 1 n Open image in new window 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 N Open image in new window 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 N Open image in new window such that D(x n , x) ≺≺ c for all n ≥ N. This means that for all ε > 0 there exists N Open image in new window 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 d1(x, y) = inf {||u||: D(x, y) ≼ u} and d2(x, y) = ξ e (D(x, y)) where D is a cone metric on X. Then d1 is equivalent with d2.

Proof. Let x n d 1 x Open image in new windowthen d 1 ( x n , x ) 0 Open image in new windowso by Theorem 2.1 in x n D x Open image in new window so
ε > 0 , e 0 N n ( n N D ( x n , x ) ε e ) , Open image in new window

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 d2(x n , x) < ε for all n ≥ N therefore d 2 ( x n , x ) 0 Open image in new window or x n d 2 x Open image in new window.

Conversely, x n d 2 x Open image in new window hence d 2 ( x n , x ) 0 Open image in new window so ξ e ( D ( x n , x ) ) 0 Open image in new window, therefore
ε > 0 N n ( n N ξ e ( D ( x n , x ) ) < ε ) . Open image in new window

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 x n D x Open image in new window and again by Theorem 2.1 x n d 1 x Open image in new window. □

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

Example 2.3 Let 0 a P n Open image in new window with ||a|| = 1 and for every x , y n Open image in new windowdefine
D ( x , y ) = a , x y ; 0 , x = y . Open image in new window
Then D is a cone metric on n Open image in new window and its equivalent metric d is
d ( x , y ) = 1 , x y ; 0 , x = y , Open image in new window

which is a discrete metric.

Example 2.4 Let a, b ≥ 0 and consider the cone metric D : × 2 Open image in new windowwith
D ( x , y ) = ( a d 1 ( x , y ) , b d 2 ( x , y ) ) Open image in new window
where d1, d2 are metrics on Open image in new window . Then its equivalent metric is
d ( x , y ) = a 2 + b 2 | | ( d 1 ( x , y ) , d 2 ( x , y ) ) | | . Open image in new window
In particular if d1(x, y):= |x − y| and d2(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
d ( x , y ) = 1 + α 2 | x - y | . Open image in new window
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
D ( x , y ) = ρ ( x , y ) b n 1 q n 1 . Open image in new window
Then its equivalent metric on × is
d ( x , y ) = ρ ( x , y ) b n 1 q n 1 l q = n = 1 ρ ( x , y ) b n 1 q = ρ ( x , y ) b - 1 1 q . Open image in new window

Notes

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.

References

  1. 1.
    Du WS: A note on cone metric fixed point theory and its equivalence. Nonlinear Anal 2010, 72: 2259–2261. 10.1016/j.na.2009.10.026MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Huang LG, Zhang X: Cone metric spaces and fixed point theorems of contractive mapping. J Math Anal Appl 2007, 322(2):1468–1476.CrossRefGoogle Scholar
  3. 3.
    Feng Y, Mao W: Equivalence of cone metric spaces and metric spaces. Fixed Point Theory 2010, 11(2):259–264.MATHMathSciNetGoogle Scholar
  4. 4.
    Haghi RH, Rezapour Sh: Fixed points of multifunctions on regular cone metric spaces. Expositiones Mathematicae 2010, 28(1):71–77. 10.1016/j.exmath.2009.04.001MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Asadi et al; licensee Springer. 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  • Mehdi Asadi
    • 1
  • Billy E Rhoades
    • 2
  • Hossein Soleimani
    • 3
  1. 1.Department of Mathematics, Zanjan BranchIslamic Azad UniversityZanjanIran
  2. 2.Department of MathematicsIndiana UniversityBloomingtonUSA
  3. 3.Department of Mathematics, Malayer BranchIslamic Azad UniversityMalayerIran

Personalised recommendations