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A note on cone bmetric and its related results: generalizations or equivalence?
Fixed Point Theory and Applicationsvolume 2013, Article number: 210 (2013)
Abstract
Very recently, a notion of cone bmetric was introduced as a generalization of bmetric, and some related fixed point results were obtained. In this paper, we investigate the answer to the question whether the given results generalize the existing ones or are equivalent to them.
MSC:46N40, 47H10, 54H25, 46T99.
1 Introduction and preliminaries
Topological vector spacevalued metric space (or TVScone metric space) introduced by Du [1] as a generalization of the Banachvalued/cone metric space [2]. Recently, Du [1] noted that fixed point theorems in generalized cone metric spaces and in usual metric spaces are equivalent. In particular, the author proved that the celebrated fixed point theorems of Banach, Kannan, Chatterjea, etc. in both TVScone metric can be derived easily from the usual metric space setup, by a simple manipulation, namely, using a scalarization function. Very recently, a number of publications, dealing with the cone bmetric space structure and fixed point theorems on such spaces, appeared. In this paper, we show that fixed point theorems in cone bmetric and usual bmetric spaces are equivalent. This paper can be considered as a continuation of the report [1].
A topological vector space (t.v.s. for short) is a vector space with a topology such that the vector space operations (addition and scalar multiplication) are continuous. A topological vector space is locally convex if its origin has a basis of neighborhoods that are convex. Let Y be a locally convex Hausdorff t.v.s. with its zero vector θ, let τ denote the topology of Y, and let ${\mathcal{U}}_{\tau}$ be the base at θ, consisting of all absolutely convex neighborhood of θ. Let
Then ℒ is a family of seminorms on Y. For each $\ell \in \mathcal{L}$, let
and let
Then ${\mathcal{U}}_{\mathcal{L}}$ is a base at θ, and the topology ${\mathrm{\Gamma}}_{\mathcal{L}}$ generated by ${\mathcal{U}}_{\mathcal{L}}$ is the weakest topology for Y such that all seminorms in ℒ are continuous and $\tau ={\mathrm{\Gamma}}_{\mathcal{L}}$. Moreover, given any neighborhood ${\mathcal{O}}_{\theta}$ of θ, there exists $U\in {\mathcal{U}}_{\mathcal{L}}$ such that $\theta \in U\subset {\mathcal{O}}_{\theta}$ (see, e.g., [[3], Theorem 12.4 in II.12, p.113]).
Throughout this paper, we follow all notations considered in [1]. Let E be a t.v.s. with its zero vector ${\theta}_{E}$. A nonempty subset K of E is called a cone if $\lambda K\subseteq K$ for $\lambda \ge 0$. A cone K is said to be pointed if $K\cap (K)=\{{\theta}_{E}\}$. For a given cone $K\subseteq E$, we can define a partial ordering ≾ (or ${\precsim}_{K}$) with respect to K by
$x\prec y$ will stand for $x\precsim y$ and $x\ne y$, while $x\ll y$ will stand for $yx\in intK$, where intK denotes the interior of K.
Let E be a t.v.s. and K a convex cone with $intK\ne \mathrm{\varnothing}$ in E. Then it is obvious that
and
In the following, unless otherwise specified, we always assume that Y is a locally convex Hausdorff t.v.s. with its zero vector θ, K a proper, closed and convex pointed cone in Y with $intK\ne \mathrm{\varnothing}$, $e\in intK$ and ≾ a partial ordering with respect to K.
The nonlinear scalarization function [1, 4, 5]${\xi}_{e}:Y\to \mathbb{R}$ is defined as follows:
Lemma 1.1 (See, e.g., [1, 4, 5])
For each $r\in \mathbb{R}$ and $y\in Y$, the following statements are satisfied:

(i)
${\xi}_{e}(y)\le r\iff y\in reK$,

(ii)
${\xi}_{e}(y)>r\iff y\notin reK$,

(iii)
${\xi}_{e}(y)\ge r\iff y\notin reintK$,

(iv)
${\xi}_{e}(y)<r\iff y\in reintK$,

(v)
${\xi}_{e}(\cdot )$ is positively homogeneous and continuous on Y,

(vi)
if ${y}_{1}\in {y}_{2}+K$ (i.e. ${y}_{2}\precsim {y}_{1}$), then ${\xi}_{e}({y}_{2})\le {\xi}_{e}({y}_{1})$,

(vii)
${\xi}_{e}({y}_{1}+{y}_{2})\le {\xi}_{e}({y}_{1})+{\xi}_{e}({y}_{2})$ for all ${y}_{1},{y}_{2}\in Y$.
Remark 1.2

(a)
Clearly, ${\xi}_{e}(\theta )=0$.

(b)
It is worth mentioning that the reverse statement of (vi) in Lemma 1.1 (i.e., ${\xi}_{e}({y}_{2})\le {\xi}_{e}({y}_{1})\u27f9{y}_{2}\precsim {y}_{1}$) does not hold in general. For example, let $Y={\mathbb{R}}^{2}$, $K={\mathbb{R}}_{+}^{2}=\{(x,y)\in {\mathbb{R}}^{2}:x,y\ge 0\}$, and let $e=(1,1)$. Then K is a proper, closed, convex and pointed cone in Y with $intK=\{(x,y)\in {\mathbb{R}}^{2}:x,y>0\}\ne \mathrm{\varnothing}$ and $e\in intK$. For $r=1$, it is easy to see that ${y}_{1}=(8,15)\notin reintK$, and ${y}_{2}=(0,0)\in reintK$. By applying (iii) and (iv) of Lemma 1.1, we have ${\xi}_{e}({y}_{2})<1\le {\xi}_{e}({y}_{1})$, while ${y}_{1}\notin {y}_{2}+K$.
1.1 TVScone metric spaces
Definition 1.3 (See [1])
Let X be a nonempty set. Suppose that a vectorvalued function $p:X\times X\to Y$ satisfies:

(C1)
$\theta \precsim p(x,y)$ for all $x,y\in X$ and $p(x,y)=\theta $ if and only if $x=y$,

(C2)
$p(x,y)=p(y,x)$ for all $x,y\in X$,

(C3)
$p(x,y)\precsim p(x,z)+p(z,y)$ for all $x,y,z\in X$.
Then, the function p is called a TVScone metric on X. Furthermore, the pair $(X,p)$ is called a TVScone metric space (in short, TVSCMS).
Lemma 1.4 (See [1])
Let $(X,p)$ be a TVSCMS. Then, ${d}_{p}:X\times X\to [0,\mathrm{\infty})$ defined by ${d}_{p}={\xi}_{e}\circ p$ is a metric.
Remark 1.5 We notice that a cone metric space (in short, CMS), introduced by Huang and Zhang [2], is a special case of TVSCMS. Indeed, the authors [2] considered E as a real Banach space instead of TVS in Definition 1.3. Further, for a CMS $(X,p)$, the function ${d}_{p}:X\times X\to [0,\mathrm{\infty})$ defined by ${d}_{p}={\xi}_{e}\circ p$ is also a metric.
Definition 1.6 (See [1])
Let $(X,p)$ be a TVSCMS, $x\in X$ and ${\{{x}_{n}\}}_{n\in \mathbb{N}}$ a sequence in X.

(i)
${\{{x}_{n}\}}_{n\in \mathbb{N}}$ TVScone converges to $x\in X$ whenever for every $\theta \ll c\in Y$, there is a natural number M such that $p({x}_{n},x)\ll c$ for all $n\ge M$ and denoted by $\text{cone}{lim}_{n\to \mathrm{\infty}}{x}_{n}=x$ (or ${x}_{n}\stackrel{\mathrm{cone}}{\u27f6}x$ as $n\to \mathrm{\infty}$),

(ii)
${\{{x}_{n}\}}_{n\in \mathbb{N}}$ TVScone Cauchy sequence in $(X,p)$ whenever for every $\theta \ll c\in Y$, there is a natural number M such that $p({x}_{n},{x}_{m})\ll c$ for all $n,m\ge M$,

(iii)
$(X,p)$ is TVScone complete if every sequence TVScone Cauchy sequence in X is a TVScone convergent.
Let $(X,p)$ be a TVSCMS, $x\in X$, and let ${\{{x}_{n}\}}_{n\in \mathbb{N}}$ be a sequence in X. Set ${d}_{p}={\xi}_{e}\circ p$. Then the following statements hold:

(i)
${\{{x}_{n}\}}_{n\in \mathbb{N}}$ converges to x in TVSCMS $(X,p)$ if and only if ${d}_{p}({x}_{n},x)\to 0$ as $n\to \mathrm{\infty}$,

(ii)
${\{{x}_{n}\}}_{n\in \mathbb{N}}$ is a Cauchy sequence in TVSCMS $(X,p)$ if and only if ${\{{x}_{n}\}}_{n\in \mathbb{N}}$ is a Cauchy sequence in $(X,{d}_{p})$,

(iii)
$(X,p)$ is a complete TVSCMS if and only if $(X,{d}_{p})$ is a complete metric space.
Remark 1.8 From Theorem 1.7, we conclude that for every complete TVScone metric space, there exists a correspondent isomorphic complete usual metric space. Notice that the cone should have a nonempty interior.
Proposition 1.9 (See [1])
Let $(X,p)$ be a complete TVSCMS and $0\le \gamma <1$. If $T:X\to X$ satisfies the contractive condition
then T has a unique fixed point in X. Moreover, for each $x\in X$, the iterative sequence ${\{{T}^{n}x\}}_{n=1}^{\mathrm{\infty}}$ converges to the unique fixed point of T.
In particular, if K is a cone of a real Banach space V, then it is called normal if there is a number $\rho \ge 1$ such that for all $x,y\in V:\theta \precsim x\precsim y\u27f9\parallel x\parallel \le \rho \parallel y\parallel $. The least positive integer ρ, satisfying this inequality, is called the normal constant of K.
1.2 bMetric spaces
The notion of a bmetric space was considered by Bakhtin [7] and Czerwik [8] as a generalization of metric space.
Let X be a nonempty set, and let $s\ge 1$ be a given real number. A function $d:X\times X\to [0,\mathrm{\infty})$ is called a bmetric if the following conditions are satisfied:

(1)
$d(x,y)=0$ if and only if $x=y$;

(2)
$d(x,y)=d(y,x)$;

(3)
$d(x,z)\le s[d(x,y)+d(y,z)]$ for all $x,y,z\in X$.
A pair $(X,d)$ is called a bmetric space.
In this paper, we first introduce the concept of TVScone bmetric space which generalize the concept of bmetric space and cone bmetric space.
Definition 1.11 Let X be a nonempty set and $s\ge 1$ be a given real number. A vectorvalued function $p:X\times X\to Y$ is said to be TVScone bmetric if the following conditions are satisfied:

(BM1)
$\theta \precsim p(x,y)$ for all $x,y\in X$ and $p(x,y)=\theta $ if and only if $x=y$;

(BM2)
$p(x,y)=p(y,x)$;

(BM3)
$p(x,z)\precsim s[p(x,y)+p(y,z)]$ for all $x,y,z\in X$.
The pair $(X,p)$ is called a TVScone bmetric space.
If we replace Y by a real Banach space in Definition 1.11, we get the cone bmetric space in the sense of [11–13]. It is evident that Definition 1.10 coincides with Definition 1.11 if we replace Y by a set of nonnegative real numbers.
2 Main results
The following theorem is one of main results in this paper. Although it is the mimic of the proof of Lemma 1.4, we give the proof for the sake of completeness and for the readers’ convenience.
Theorem 2.1 Let $(X,p)$ be a TVScone bmetric space. Then, ${d}_{p}:X\times X\to [0,\mathrm{\infty})$ defined by ${d}_{p}={\xi}_{e}\circ p$ is a bmetric.
Proof Clearly, ${d}_{p}(x,y)={d}_{p}(y,x)$ for all $x,y\in X$. By Lemma 1.1, we have ${d}_{p}(x,y)\ge 0$ for all $x,y\in X$. If $x=y$, then, by (BM1), ${d}_{p}(x,y)={\xi}_{e}(\theta )=0$. Conversely, if ${d}_{p}(x,y)=0$, then by Lemma 1.1 $p(x,y)\in K\cap (K)=\{\theta \}$, which implies that $x=y$. Since $s\ge 1$, by applying (v), (vi) and (vii) of Lemma 1.1, we have
or
So we prove that ${d}_{p}$ is a bmetric. □
The following consequence of Theorem 2.1 is evident.
Corollary 2.2 Let $(X,p)$ be a cone bmetric space. Then, ${d}_{p}:X\times X\to [0,\mathrm{\infty})$ defined by ${d}_{p}={\xi}_{e}\circ p$ is a bmetric.
Following the idea of Du [1], we can define the following.
Definition 2.3 Let $(X,p)$ be a TVScone bmetric space, let $x\in X$, and let ${\{{x}_{n}\}}_{n\in \mathbb{N}}$ be a sequence in X.

(i)
${\{{x}_{n}\}}_{n\in \mathbb{N}}$ TVScone converges to $x\in X$ whenever for every $\theta \ll c\in Y$, there is a natural number M such that $p({x}_{n},x)\ll c$ for all $n\ge M$ and denoted by $\text{cone}{lim}_{n\to \mathrm{\infty}}{x}_{n}=x$ (or ${x}_{n}\stackrel{\mathrm{cone}}{\u27f6}x$ as $n\to \mathrm{\infty}$),

(ii)
${\{{x}_{n}\}}_{n\in \mathbb{N}}$ TVScone Cauchy sequence in $(X,p)$ whenever for every $\theta \ll c\in Y$, there is a natural number M such that $p({x}_{n},{x}_{m})\ll c$ for all $n,m\ge M$,

(iii)
$(X,p)$ is TVScone complete if every sequence TVScone Cauchy sequence in X is a TVScone convergent.
Using a similar argument as in the proof of [[2], Theorem 2.2], we can prove the following result.
Theorem 2.4 Let $(X,p)$ be a TVScone bmetric space, let $x\in X$, let and ${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$ be a sequence in X. Set ${d}_{p}={\xi}_{e}\circ p$. Then the following statements hold:

(i)
${\{{x}_{n}\}}_{n\in \mathbb{N}}$ converges to x in TVScone bmetric space $(X,p)$ if and only if ${d}_{p}({x}_{n},x)\to 0$ as $n\to \mathrm{\infty}$,

(ii)
${\{{x}_{n}\}}_{n\in \mathbb{N}}$ is a Cauchy sequence in TVScone bmetric space $(X,p)$ if and only if ${\{{x}_{n}\}}_{n\in \mathbb{N}}$ is a Cauchy sequence in $(X,{d}_{p})$,

(iii)
$(X,p)$ is a complete TVScone bmetric space if and only if $(X,{d}_{p})$ is a complete bmetric space.
Remark 2.5 From Theorem 2.4, we conclude that for every complete TVScone bmetric space there exists a correspondent isomorphic complete usual (associated) bmetric space.
Theorem 2.6 Let $(X,p)$ be a complete TVScone bmetric space with $s\ge 1$ and $0\le \gamma <1$. If $T:X\to X$ satisfies the contractive condition
then T has a unique fixed point in X. Moreover, for each $x\in X$, the iterative sequence ${\{{T}^{n}x\}}_{n\in \mathbb{N}}$ converges to the unique fixed point of T.
Proof Set ${d}_{p}={\xi}_{e}\circ p$. Due to Theorem 2.4, we conclude that $(X,{d}_{p})$ is a complete bmetric space. On the other hand, from Lemma 1.1, we derive that
We conclude the results from the characterization of the Banach contraction mapping principle in the context of bmetric space (see, e.g., [[14], Theorem 2]). The proof is completed. □
Theorem 2.7 Let $(X,p)$ be a complete TVScone bmetric space with $s\ge 1$, and let $T:X\to X$ satisfy the contractive condition
where ${\lambda}_{i}\in [0,1)$, $i=1,2,3,4$, and ${\lambda}_{1}+{\lambda}_{2}+s({\lambda}_{3}+{\lambda}_{4})<min\{1,\frac{2}{s}\}$. Then T has a unique fixed point in X. Moreover, for each $x\in X$, the iterative sequence ${\{{T}^{n}x\}}_{n\in \mathbb{N}}$ converges to the unique fixed point of T.
The idea of the proof is the same with the proof of Theorem 2.6. For the sake of completeness, we put it here.
Proof Set ${d}_{p}={\xi}_{e}\circ p$. Due to Theorem 2.4, we conclude that $(X,{d}_{p})$ is a complete bmetric space. On the other hand, from Lemma 1.1, we derive that
implies that
We conclude the result from [[14], Corollary 4.1] with taking $S=T$. The proof is completed. □
3 Conclusion
In this paper, we just show that two fixed point theorems in the setting of cone bmetric spaces can be easily derived from the existing result in the context of bmetric space. Hence, the notion of ‘cone bmetric’ is not a real generalization of neither bmetric nor metric. By using the techniques above, one can easily prove the equivalence of other fixed point results (published, unpublished/that will be published) in the context of cone bmetric space. Regarding the published papers on the equivalence of cone metric and usual (associated) metric in the literature, it is natural to conclude that some other techniques can also be developed for the equivalence of the mentioned notions.
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Acknowledgements
The authors would like to express their sincere thanks to the anonymous referees for their valuable comments and useful suggestions in improving the paper. The first author was supported partially by grant No. NSC 1012115M017001 of the National Science Council of the Republic of China.
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Both authors contributed equally and significantly in writing this paper. Both authors read and approved the final manuscript.
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Keywords
 TVScone metric space
 bmetric
 TVScone bmetric
 nonlinear scalarization function
 fixed point