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A note on cone b-metric and its related results: generalizations or equivalence?

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.

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 U τ be the base at θ, consisting of all absolutely convex neighborhood of θ. Let

L={: is a Minkowski functional of U for U U τ }.

Then is a family of seminorms on Y. For each L, let

V()= { y Y : ( y ) < 1 } ,

and let

U L = { U : U = r 1 V ( 1 ) r 2 V ( 2 ) r n V ( n ) , r k > 0 , k L , 1 k n , n N } .

Then U L is a base at θ, and the topology Γ L generated by U L is the weakest topology for Y such that all seminorms in are continuous and τ= Γ L . Moreover, given any neighborhood O θ of θ, there exists U U L such that θU O θ (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 θ E . A nonempty subset K of E is called a cone if λKK for λ0. A cone K is said to be pointed if K(K)={ θ E }. For a given cone KE, we can define a partial ordering (or K ) with respect to K by

xyyxK.

xy will stand for xy and xy, while xy will stand for yxintK, where intK denotes the interior of K.

Let E be a t.v.s. and K a convex cone with intK in E. Then it is obvious that

intK+intKintK+KintK

and

λintKintKfor all λ>0.

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, eintK and a partial ordering with respect to K.

The nonlinear scalarization function [1, 4, 5] ξ e :YR is defined as follows:

ξ e (y)=inf{rR:yreK}for all yY.

Lemma 1.1 (See, e.g., [1, 4, 5])

For each rR and yY, the following statements are satisfied:

  1. (i)

    ξ e (y)ryreK,

  2. (ii)

    ξ e (y)>ryreK,

  3. (iii)

    ξ e (y)ryreintK,

  4. (iv)

    ξ e (y)<ryreintK,

  5. (v)

    ξ e () is positively homogeneous and continuous on Y,

  6. (vi)

    if y 1 y 2 +K (i.e. y 2 y 1 ), then ξ 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 2 Y.

Remark 1.2

  1. (a)

    Clearly, ξ e (θ)=0.

  2. (b)

    It is worth mentioning that the reverse statement of (vi) in Lemma 1.1 (i.e., ξ e ( y 2 ) ξ e ( y 1 ) y 2 y 1 ) does not hold in general. For example, let Y= R 2 , K= R + 2 ={(x,y) R 2 :x,y0}, and let e=(1,1). Then K is a proper, closed, convex and pointed cone in Y with intK={(x,y) R 2 :x,y>0} and eintK. For r=1, it is easy to see that y 1 =(8,15)reintK, and y 2 =(0,0)reintK. By applying (iii) and (iv) of Lemma 1.1, we have ξ e ( y 2 )<1 ξ e ( y 1 ), while y 1 y 2 +K.

1.1 TVS-cone metric spaces

Definition 1.3 (See [1])

Let X be a nonempty set. Suppose that a vector-valued function p:X×XY satisfies:

  1. (C1)

    θp(x,y) for all x,yX and p(x,y)=θ if and only if x=y,

  2. (C2)

    p(x,y)=p(y,x) for all x,yX,

  3. (C3)

    p(x,y)p(x,z)+p(z,y) for all x,y,zX.

Then, the function p is called a TVS-cone metric on X. Furthermore, the pair (X,p) is called a TVS-cone metric space (in short, TVS-CMS).

Lemma 1.4 (See [1])

Let (X,p) be a TVS-CMS. Then, d p :X×X[0,) defined by d p = ξ e 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 TVS-CMS. 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×X[0,) defined by d p = ξ e p is also a metric.

Definition 1.6 (See [1])

Let (X,p) be a TVS-CMS, xX and { x n } n N a sequence in X.

  1. (i)

    { x n } n N TVS-cone converges to xX whenever for every θcY, there is a natural number M such that p( x n ,x)c for all nM and denoted by cone- lim n x n =x (or x n cone x as n),

  2. (ii)

    { x n } n N TVS-cone Cauchy sequence in (X,p) whenever for every θcY, there is a natural number M such that p( x n , x m )c for all n,mM,

  3. (iii)

    (X,p) is TVS-cone complete if every sequence TVS-cone Cauchy sequence in X is a TVS-cone convergent.

Theorem 1.7 (See [1, 6])

Let (X,p) be a TVS-CMS, xX, and let { x n } n N be a sequence in X. Set d p = ξ e p. Then the following statements hold:

  1. (i)

    { x n } n N converges to x in TVS-CMS (X,p) if and only if d p ( x n ,x)0 as n,

  2. (ii)

    { x n } n N is a Cauchy sequence in TVS-CMS (X,p) if and only if { x n } n N is a Cauchy sequence in (X, d p ),

  3. (iii)

    (X,p) is a complete TVS-CMS 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 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 (X,p) be a complete TVS-CMS and 0γ<1. If T:XX satisfies the contractive condition

p(Tx,Ty)γp(x,y) for all x,yX,

then T has a unique fixed point in X. Moreover, for each xX, the iterative sequence { T n x } n = 1 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 ρ1 such that for all x,yV:θxyxρy. 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.

Definition 1.10 (See [710])

Let X be a nonempty set, and let s1 be a given real number. A function d:X×X[0,) is called a b-metric if the following conditions are satisfied:

  1. (1)

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

  2. (2)

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

  3. (3)

    d(x,z)s[d(x,y)+d(y,z)] for all x,y,zX.

A pair (X,d) 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 s1 be a given real number. A vector-valued function p:X×XY is said to be TVS-cone b-metric if the following conditions are satisfied:

  1. (BM1)

    θp(x,y) for all x,yX and p(x,y)=θ if and only if x=y;

  2. (BM2)

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

  3. (BM3)

    p(x,z)s[p(x,y)+p(y,z)] for all x,y,zX.

The pair (X,p) 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 [1113]. 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 (X,p) be a TVS-cone b-metric space. Then, d p :X×X[0,) defined by d p = ξ e p is a b-metric.

Proof Clearly, d p (x,y)= d p (y,x) for all x,yX. By Lemma 1.1, we have d p (x,y)0 for all x,yX. If x=y, then, by (BM1), d p (x,y)= ξ e (θ)=0. Conversely, if d p (x,y)=0, then by Lemma 1.1 p(x,y)K(K)={θ}, which implies that x=y. Since s1, by applying (v), (vi) and (vii) of Lemma 1.1, we have

ξ e ( p ( x , z ) ) s ( ξ e ( p ( x , y ) ) + ξ e ( p ( y , z ) ) )

or

d p (x,z)s [ d p ( x , y ) + d p ( y , z ) ] for all x,y,zX.

So we prove that d p is a b-metric. □

The following consequence of Theorem 2.1 is evident.

Corollary 2.2 Let (X,p) be a cone b-metric space. Then, d p :X×X[0,) defined by d p = ξ e p is a b-metric.

Following the idea of Du [1], we can define the following.

Definition 2.3 Let (X,p) be a TVS-cone b-metric space, let xX, and let { x n } n N be a sequence in X.

  1. (i)

    { x n } n N TVS-cone converges to xX whenever for every θcY, there is a natural number M such that p( x n ,x)c for all nM and denoted by cone- lim n x n =x (or x n cone x as n),

  2. (ii)

    { x n } n N TVS-cone Cauchy sequence in (X,p) whenever for every θcY, there is a natural number M such that p( x n , x m )c for all n,mM,

  3. (iii)

    (X,p) 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 (X,p) be a TVS-cone b-metric space, let xX, let and { x n } n = 1 be a sequence in X. Set d p = ξ e p. Then the following statements hold:

  1. (i)

    { x n } n N converges to x in TVS-cone b-metric space (X,p) if and only if d p ( x n ,x)0 as n,

  2. (ii)

    { x n } n N is a Cauchy sequence in TVS-cone b-metric space (X,p) if and only if { x n } n N is a Cauchy sequence in (X, d p ),

  3. (iii)

    (X,p) is a complete TVS-cone b-metric space if and only if (X, d p ) 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 (X,p) be a complete TVS-cone b-metric space with s1 and 0γ<1. If T:XX satisfies the contractive condition

p(Tx,Ty)γp(x,y) for all x,yX,

then T has a unique fixed point in X. Moreover, for each xX, the iterative sequence { T n x } n N converges to the unique fixed point of T.

Proof Set d p = ξ e p. Due to Theorem 2.4, we conclude that (X, d p ) is a complete b-metric space. On the other hand, from Lemma 1.1, we derive that

p(Tx,Ty)γp(x,y) d p (Tx,Ty)γ d p (x,y)for all x,yX.

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 (X,p) be a complete TVS-cone b-metric space with s1, and let T:XX satisfy the contractive condition

p(Tx,Ty) λ 1 p(x,Tx)+ λ 2 p(y,Ty)+ λ 3 p(x,Ty)+ λ 4 p(y,Tx) for all x,yX,

where λ i [0,1), i=1,2,3,4, and λ 1 + λ 2 +s( λ 3 + λ 4 )<min{1, 2 s }. Then T has a unique fixed point in X. Moreover, for each xX, the iterative sequence { T n x } n 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 = ξ e p. Due to Theorem 2.4, we conclude that (X, d p ) is a complete b-metric space. On the other hand, from Lemma 1.1, we derive that

p(Tx,Ty) λ 1 p(x,Tx)+ λ 2 p(y,Ty)+ λ 3 p(x,Ty)+ λ 4 p(y,Tx)

implies that

d p (Tx,Ty) λ 1 d p (x,Tx)+ λ 2 d p (y,Ty)+ λ 3 d p (x,Ty)+ λ 4 d(y,Tx)for all x,yX.

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 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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Correspondence to Erdal Karapınar.

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The authors declare that they have no competing interests.

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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

  • TVS-cone metric space
  • b-metric
  • TVS-cone b-metric
  • nonlinear scalarization function
  • fixed point