Abstract
Very recently, a notion of cone b-metric was introduced as a generalization of b-metric, 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.
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1 Introduction and preliminaries
Topological vector space-valued metric space (or TVS-cone metric space) introduced by Du [1] as a generalization of the Banach-valued/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 TVS-cone metric can be derived easily from the usual metric space set-up, by a simple manipulation, namely, using a scalarization function. Very recently, a number of publications, dealing with the cone b-metric space structure and fixed point theorems on such spaces, appeared. In this paper, we show that fixed point theorems in cone b-metric and usual b-metric 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 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 , let
and let
Then is a base at θ, and the topology generated by is the weakest topology for Y such that all seminorms in ℒ are continuous and . Moreover, given any neighborhood of θ, there exists such that (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 . A nonempty subset K of E is called a cone if for . A cone K is said to be pointed if . For a given cone , we can define a partial ordering ≾ (or ) with respect to K by
will stand for and , while will stand for , where intK denotes the interior of K.
Let E be a t.v.s. and K a convex cone with 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 , and ≾ a partial ordering with respect to K.
The nonlinear scalarization function [1, 4, 5] is defined as follows:
Lemma 1.1 (See, e.g., [1, 4, 5])
For each and , the following statements are satisfied:
-
(i)
,
-
(ii)
,
-
(iii)
,
-
(iv)
,
-
(v)
is positively homogeneous and continuous on Y,
-
(vi)
if (i.e. ), then ,
-
(vii)
for all .
Remark 1.2
-
(a)
Clearly, .
-
(b)
It is worth mentioning that the reverse statement of (vi) in Lemma 1.1 (i.e., ) does not hold in general. For example, let , , and let . Then K is a proper, closed, convex and pointed cone in Y with and . For , it is easy to see that , and . By applying (iii) and (iv) of Lemma 1.1, we have , while .
1.1 TVS-cone metric spaces
Definition 1.3 (See [1])
Let X be a nonempty set. Suppose that a vector-valued function satisfies:
-
(C1)
for all and if and only if ,
-
(C2)
for all ,
-
(C3)
for all .
Then, the function p is called a TVS-cone metric on X. Furthermore, the pair is called a TVS-cone metric space (in short, TVS-CMS).
Lemma 1.4 (See [1])
Let be a TVS-CMS. Then, defined by 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 TVS-CMS. Indeed, the authors [2] considered E as a real Banach space instead of TVS in Definition 1.3. Further, for a CMS , the function defined by is also a metric.
Definition 1.6 (See [1])
Let be a TVS-CMS, and a sequence in X.
-
(i)
TVS-cone converges to whenever for every , there is a natural number M such that for all and denoted by (or as ),
-
(ii)
TVS-cone Cauchy sequence in whenever for every , there is a natural number M such that for all ,
-
(iii)
is TVS-cone complete if every sequence TVS-cone Cauchy sequence in X is a TVS-cone convergent.
Let be a TVS-CMS, , and let be a sequence in X. Set . Then the following statements hold:
-
(i)
converges to x in TVS-CMS if and only if as ,
-
(ii)
is a Cauchy sequence in TVS-CMS if and only if is a Cauchy sequence in ,
-
(iii)
is a complete TVS-CMS if and only if is a complete metric space.
Remark 1.8 From Theorem 1.7, we conclude that for every complete TVS-cone 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 be a complete TVS-CMS and . If satisfies the contractive condition
then T has a unique fixed point in X. Moreover, for each , the iterative sequence 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 such that for all . The least positive integer ρ, satisfying this inequality, is called the normal constant of K.
1.2 b-Metric spaces
The notion of a b-metric space was considered by Bakhtin [7] and Czerwik [8] as a generalization of metric space.
Let X be a nonempty set, and let be a given real number. A function is called a b-metric if the following conditions are satisfied:
-
(1)
if and only if ;
-
(2)
;
-
(3)
for all .
A pair is called a b-metric space.
In this paper, we first introduce the concept of TVS-cone b-metric space which generalize the concept of b-metric space and cone b-metric space.
Definition 1.11 Let X be a non-empty set and be a given real number. A vector-valued function is said to be TVS-cone b-metric if the following conditions are satisfied:
-
(BM1)
for all and if and only if ;
-
(BM2)
;
-
(BM3)
for all .
The pair is called a TVS-cone b-metric space.
If we replace Y by a real Banach space in Definition 1.11, we get the cone b-metric 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 non-negative 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 be a TVS-cone b-metric space. Then, defined by is a b-metric.
Proof Clearly, for all . By Lemma 1.1, we have for all . If , then, by (BM1), . Conversely, if , then by Lemma 1.1 , which implies that . Since , by applying (v), (vi) and (vii) of Lemma 1.1, we have
or
So we prove that is a b-metric. □
The following consequence of Theorem 2.1 is evident.
Corollary 2.2 Let be a cone b-metric space. Then, defined by is a b-metric.
Following the idea of Du [1], we can define the following.
Definition 2.3 Let be a TVS-cone b-metric space, let , and let be a sequence in X.
-
(i)
TVS-cone converges to whenever for every , there is a natural number M such that for all and denoted by (or as ),
-
(ii)
TVS-cone Cauchy sequence in whenever for every , there is a natural number M such that for all ,
-
(iii)
is TVS-cone complete if every sequence TVS-cone Cauchy sequence in X is a TVS-cone convergent.
Using a similar argument as in the proof of [[2], Theorem 2.2], we can prove the following result.
Theorem 2.4 Let be a TVS-cone b-metric space, let , let and be a sequence in X. Set . Then the following statements hold:
-
(i)
converges to x in TVS-cone b-metric space if and only if as ,
-
(ii)
is a Cauchy sequence in TVS-cone b-metric space if and only if is a Cauchy sequence in ,
-
(iii)
is a complete TVS-cone b-metric space if and only if is a complete b-metric space.
Remark 2.5 From Theorem 2.4, we conclude that for every complete TVS-cone b-metric space there exists a correspondent isomorphic complete usual (associated) b-metric space.
Theorem 2.6 Let be a complete TVS-cone b-metric space with and . If satisfies the contractive condition
then T has a unique fixed point in X. Moreover, for each , the iterative sequence converges to the unique fixed point of T.
Proof Set . Due to Theorem 2.4, we conclude that is a complete b-metric 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 b-metric space (see, e.g., [[14], Theorem 2]). The proof is completed. □
Theorem 2.7 Let be a complete TVS-cone b-metric space with , and let satisfy the contractive condition
where , , and . Then T has a unique fixed point in X. Moreover, for each , the iterative sequence 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 . Due to Theorem 2.4, we conclude that is a complete b-metric 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 . The proof is completed. □
3 Conclusion
In this paper, we just show that two fixed point theorems in the setting of cone b-metric spaces can be easily derived from the existing result in the context of b-metric space. Hence, the notion of ‘cone b-metric’ is not a real generalization of neither b-metric 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 b-metric 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 101-2115-M-017-001 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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Du, WS., Karapınar, E. A note on cone b-metric and its related results: generalizations or equivalence?. Fixed Point Theory Appl 2013, 210 (2013). https://doi.org/10.1186/1687-1812-2013-210
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DOI: https://doi.org/10.1186/1687-1812-2013-210