Open Access Research

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

Mehdi Asadi1*, Billy E Rhoades2 and Hossein Soleimani3

Author Affiliations

1 Department of Mathematics, Zanjan Branch, Islamic Azad University, Zanjan, Iran

2 Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, USA

3 Department of Mathematics, Malayer Branch, Islamic Azad University, Malayer, Iran

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Fixed Point Theory and Applications 2012, 2012:87  doi:10.1186/1687-1812-2012-87


The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2012/1/87


Received:9 May 2011
Accepted:21 May 2012
Published:21 May 2012

© 2012 Asadi et al; licensee Springer.

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.

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

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M1">View MathML</a>

x < y will stand for x ≼ y and x y 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 <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M2">View MathML</a> is defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M3">View MathML</a>

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) ξe(y) ≤ r y re − K.

(ii) ξe(y) > r y re − K.

(iii) ξe(y) ≥ r y re − K°.

(iv) ξe(y) < r y re − K°.

(v) ξe(.) is positively homogeneous and continuous on Y .

(vi) y1 y2 + K ξe(y2) ≤ ξe(y1)

(vii) ξe(y1 + y2) ≤ ξe(y1) + ξe(y2) for all y1, y2 Y.

Definition 1.2 [1]Let X be a nonempty set. A vector-valued function d : X × X Y is said to be a TVS-cone 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 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 × 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 TVS-cone metric space.

In the following, some conclusions are listed.

Lemma 1.3 [3]Let (X, D) be a cone metric space. Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M4">View MathML</a>

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) = ξe(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 <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M5">View MathML</a>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 {xn} ⊆ 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

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M6">View MathML</a>

Put vn := unn then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M7">View MathML</a> and D(xn, x) ≼ vn. Now if xn x in (X, d) then d(xn, x) → 0 and so vn → 0 too, therefore for all c ≻≻ 0 there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M8">View MathML</a>such that vn ≺≺ c for all n ≥ N. This implies that D(xn, x) ≺≺ c for all n ≥ N. Namely xn x in (X, D).

Conversely, for every real ε > 0, choose c E with c ≻≻ 0 and ||c|| < ε. Then there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M9">View MathML</a> such that D(xn, x) ≺≺ c for all n ≥ N. This means that for all ε > 0 there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M10">View MathML</a>such that d(xn, x) ||c|| < ε for all n ≥ N. Therefore d(xn, x) → 0 as n so xn x in (X, d).

Theorem 2.2 If d1(x, y) = inf {||u||: D(x, y) ≼ u} and d2(x, y) = ξe(D(x, y)) where D is a cone metric on X. Then d1 is equivalent with d2.

Proof. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M11">View MathML</a>then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M12">View MathML</a>so by Theorem 2.1 in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M13">View MathML</a>so

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M14">View MathML</a>

and or εe − D(xn, x) ∈ for all n ≥ N. So D(xn, x) ∈ e - for n ≥ N. Now by [[1], Lemma 1.1 (iv)] ξe(D(xn, x)) < ε for all n ≥ N. Namely d2(xn, x) < ε for all n ≥ N therefore <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M15">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M16">View MathML</a>.

Conversely, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M17">View MathML</a> hence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M18">View MathML</a> so <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M19">View MathML</a>, therefore

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M20">View MathML</a>

So D(xn, x) ∈ εe−K° for n ≥ N by [[1], Lemma 1.1 (iv)]. Hence, D(xn, x) = εe−k for some k , so D(xn, x) ≺≺ εe for n ≥ N this implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M21">View MathML</a>and again by Theorem 2.1 <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M22">View MathML</a>. □

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

Example 2.3 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M23">View MathML</a>with ||a|| = 1 and for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M24">View MathML</a>define

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M25">View MathML</a>

Then D is a cone metric on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M26">View MathML</a> and its equivalent metric d is

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M27">View MathML</a>

which is a discrete metric.

Example 2.4 Let a, b ≥ 0 and consider the cone metric <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M28">View MathML</a>with

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M29">View MathML</a>

where d1, d2 are metrics on . Then its equivalent metric is

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M31">View MathML</a>

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

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M32">View MathML</a>

Example 2.5 For q > 0, b > 1, E = lq, P = {{xn}n≥1 : xn 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

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M33">View MathML</a>

Then its equivalent metric on × is

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/87/mathml/M34">View MathML</a>

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.

References

  1. Du, WS: A note on cone metric fixed point theory and its equivalence. Nonlinear Anal. 72, 2259–2261 (2010). Publisher Full Text OpenURL

  2. Huang, LG, Zhang, X: Cone metric spaces and fixed point theorems of contractive mapping. J Math Anal Appl. 322(2), 1468–1476 (2007)

  3. Feng, Y, Mao, W: Equivalence of cone metric spaces and metric spaces. Fixed Point Theory. 11(2), 259–264 (2010)

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