Some Equivalences between Cone b -Metric Spaces and b -Metric Spaces

Abstract

We introduce a b -metric on the cone b -metric space and then prove some equivalences between them. As applications, we show that fixed point theorems on cone b -metric spaces can be obtained from fixed point theorems on b -metric spaces.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Poom Kumam 1 and Nguyen Van Dung 2 and Vo Thi Le Hang 3

Academic Editor:Hassen Aydi

1, Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bang Mod, Thrung Khru, Bangkok 10140, Thailand
2, Faculty of Mathematics and Information Technology Teacher Education, Dong Thap University, Cao Lanh City, Dong Thap Province 871200, Vietnam
3, Journal of Science, Dong Thap University, Cao Lanh City, Dong Thap Province 871200, Vietnam

Received 20 June 2013; Revised 24 August 2013; Accepted 27 August 2013

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction and Preliminaries

The fixed point theory in b -metric spaces was investigated by Bakhtin [1], Czerwik [2], Akkouchi [3], Olatinwo and Imoru [4], and P...curar [5]. A b -metric space was also called a metric-type space in [6]. The fixed point theory in metric-type spaces was investigated in [6, 7]. Recently, Hussain and Shah introduced the notion of a cone b -metric as a generalization of a b -metric in [8]. Some fixed point theorems on cone b -metric spaces were stated in [8-10].

Note that the relation between a cone b -metric and a b -metric is likely the relation between a cone metric [11] and a metric. Some authors have proved that fixed point theorems on cone metric spaces are, essentially, fixed point theorems on metric space; see [12-16] for example. Very recently, Du used the method in [12] to introduce a b -metric on a cone b -metric space and stated some relations between fixed point theorems on cone b -metric spaces and on b -metric spaces [17].

In this paper, we use the method in [13] to introduce another b -metric on the cone b -metric space and then prove some equivalences between them. As applications, we show that fixed point theorems on cone b -metric spaces can be obtained from fixed point theorems on b -metric spaces.

Now, we recall some definitions and lemmas.

Definition 1 (see [1]).

Let X be a nonempty set and d:X×X[arrow right][0,+∞) . Then, d is called a b -metric on X if

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

(2) d(x,y)=d(y,x) for all x,y∈X ;

(3) there exists s...5;1 such that d(x,z)...4;s[d(x,y)+d(y,z)] for all x,y,z∈X .

The pair (X,d) is called a b-metric space . A sequence {_xn } is called convergent to x in X , written _{limn[arrow right]∞xn} =x , if _{limn[arrow right]∞} d(_xn ,x)=0 . A sequence {_xn } is called a Cauchy sequence if _{limn,m[arrow right]∞} d(_xn ,_xm )=0 . The b -metric space (X,d) is called complete if every Cauchy sequence in X is a convergent sequence.

Remark 2.

On a b -metric space (X,d) , we consider a topology induced by its convergence. For results concerning b -metric spaces, readers are invited to consult papers [1, 2].

Remark 3.

Let (X,d) be a b -metric space. For each r>0 and x∈X , we set [figure omitted; refer to PDF] In [3], Akkouchi claimed that the topology ...AF;(d) on X associated with d is given by setting U∈...AF;(d) if and only if, for each x∈U , there exists some r>0 such that B(x,r)⊂U and the convergence of {_xn}n in the b -metric space (X,d) and that in the topological space (X,...AF;(d)) are equivalent. Unfortunately, this claim is not true in general; see Example 13. Note that; on a b -metric space, we always consider the topology induced by its convergence. Most of concepts and results obtained for metric spaces can be extended to the case of b -metric spaces. For results concerning b -metric spaces, readers are invited to consult papers [1, 2].

In what follows, let E be a real Banach space, P a subset of E , θ the zero element of E , and intP the interior of P . We define a partially ordering ...4; with respect to P by x...4;y if and only if y-x∈P . We also write x<y to indicate that x...4;y and x...0;y and write x...a;y to indicate that y-x∈intP . Let ||·|| denote the norm on E .

Definition 4 (see [11]).

P is called a cone if and only if

(1) P is closed and nonempty and P...0;{θ} ;

(2) a,b∈...; a,b...5;0; x,y∈P imply that ax+by∈P ;

(3) P∩(-P)={θ} .

The cone P is called normal if there exists K...5;1 such that, for all x,y∈E , we have θ...4;x...4;y implies ||x||...4;K||y|| . The least positive number K satisfying the above is called the normal constant of P .

Definition 5 (see [11, Definition 1]).

Let X be a nonempty set and d:X×X[arrow right]E satisfy

(1) θ...4;d(x,y) for all x,y∈X and d(x,y)=θ if and only if x=y ;

(2) d(x,y)=d(y,x) for all x,y∈X ;

(3) d(x,y)...4;d(x,z)+d(z,y) for all x,y,z∈X .

Then d is called a cone metric on X , and (X,d) is called a cone metric space .

Definition 6 (see [8, Definition 2.1]).

Let X be a nonempty set and d:X×X[arrow right]P satisfy

(1) θ...4;d(x,y) for all x,y∈X and d(x,y)=θ if and only if x=y ;

(2) d(x,y)=d(y,x) for all x,y∈X ;

(3) d(x,y)...4;s[d(x,z)+d(z,y)] for some s...5;1 and all x,y,z∈X .

Then d is called a cone b-metric with coefficient s on X and (X,d) is called a cone b-metric space with coefficient s .

Definition 7 (see [8, Definition 2.4]).

Let (X,d) be a cone b -metric space and {_xn } a sequence in X .

(1) {_xn } is called convergent to x , written _{limn[arrow right]∞xn} =x , if for each c∈E with θ...a;c , there exists _n0 such that d(_xn ,x)...a;c for all n...5;_n0 .

(2) {_xn } is called a Cauchy sequence if for each c∈E with θ...a;c there exists _n0 such that d(_xn ,_xm )...a;c for all n,m...5;_n0 .

(3) (X,d) is called complete if every Cauchy sequence in X is a convergent sequence.

Lemma 8 (see [8, Proposition 2.5]).

Let (X,d) be a cone b -metric space, P a normal cone with normal constant K , x∈X , and {_xn } a sequence in X . Then one has the following.

(1) _{limn[arrow right]∞xn} =x if and only if _{limn[arrow right]∞} d(_xn ,x)=θ .

(2) The limit point of a convergent sequence is unique.

(3) Every convergent sequence is a Cauchy sequence.

(4) {_xn } is a Cauchy sequence if _{limn,m[arrow right]∞} d(_xn ,_xm )=θ .

Lemma 9 (see [8, Remark 2.6]).

Let (X,d) be a cone b -metric space over an ordered real Banach space E with a cone P . Then one has the following.

(1) If a...4;b and b...a;c , then a...a;c .

(2) If a...a;b and b...a;c , then a...a;c .

(3) If θ...4;u...a;c for all c∈intP , then u=θ .

(4) If c∈intP , θ...4;_an for all n∈... and _{limn[arrow right]∞an} =θ , then there exists _n0 such that _an ...a;c for all n...5;_n0 .

(5) If θ...a;c , θ...4;d(_xn ,x)...4;_bn for all n∈... and _{limn[arrow right]∞bn} =θ , then d(_xn ,x)...a;c eventually.

(6) If θ...4;_an ...4;_bn for all n∈... and _{limn[arrow right]∞an} =a , _{limn[arrow right]∞bn} =b , then a...4;b .

(7) If a∈P , 0...4;λ<1 , and a...4;λ·a , then a=θ .

(8) For each α>0 , one has α·intP⊂intP .

(9) For each δ>0 and x∈intP , there exists 0<γ<1 such that ||γ·x||<δ .

(10) For each θ...a;_c1 and _c2 ∈P , there exists θ...a;d such that _c1 ...a;d and _c2 ...a;d .

(11) For each θ...a;_c1 and θ...a;_c2 , there exists θ...a;e such that e...a;_c1 and e...a;_c2 .

Remark 10 (see [10, Remark 1.3]).

Every cone metric space is a cone b -metric space. Moreover, cone b -metric spaces generalize cone metric spaces, b -metric spaces, and metric spaces.

Example 11 (see [10, Example 2.2]).

Let [figure omitted; refer to PDF] and d(x,y)(t)=|x-y^|2et for all x,y∈X and t∈[0,1] . Then (X,d) is a cone b -metric space with coefficient s=2 , but it is not a cone metric space.

Example 12 (see [10, Example 2.3]).

Let X be the set of Lebesgue measurable functions on [0,1] such that ^∫01 |u(x)^|2 dx<+∞ , E=_C... [0,1] , P={[straight phi]∈E:[straight phi]...5;0} . Define d:X×X[arrow right]E as [figure omitted; refer to PDF] for all u,v∈X and t∈[0,1] . Then (X,d) is a cone b -metric space with coefficient s=2 , but it is not a cone metric space.

2. Main Results

The following example shows that the family of all balls B(x,r) does not form a base for any topology on a b -metric space (X,d) .

Example 13.

Let X={0,1,1/2,...,1/n,...} and [figure omitted; refer to PDF] Then we have the following.

(1) d is a b -metric on X with coefficient s=8/3 .

(2) 0∈B(1,2) but B(0,r)⊄B(1,2) for all r>0 .

Proof.

(1) For all x,y∈X , we have d(x,y)...5;0 , d(x,y)=0 if and only if x=y and d(x,y)=d(y,x) .

If d(x,y)=d(0,1)=1 , then [figure omitted; refer to PDF] If d(x,y)=d(0,1/2n)=1/2n , then [figure omitted; refer to PDF] If d(x,y)=d(1/2k,1/2n)=|1/2k-1/2n| , then [figure omitted; refer to PDF] If d(x,y)=d(1/2k,1/(2n+1))=4 with 1/(2n+1)...0;1 , then [figure omitted; refer to PDF] If d(x,y)=d(1/(2k+1),1/(2n+1))=4 with 1/(2k+1)...0;1 and 1/(2n+1)...0;1 , then [figure omitted; refer to PDF] If d(x,y)=d(1/2k,1)=4 , then [figure omitted; refer to PDF] If d(x,y)=d(1/(2k+1),1)=4 , then [figure omitted; refer to PDF] If d(x,y)=d(1/(2k+1),0)=4 , then [figure omitted; refer to PDF]

By the previous calculations, we get d(x,y)...4;(8/3)[d(x,z)+d(z,y)] for all x,y,z∈X . This proves that d is a b -metric on X with s=8/3 .

(2) We have B(1,2)={x∈X:d(x,1)<2}={1,0} . Then 0∈B(1,2) .

For each r>0 , since d(0,1/2n)=1/2n , we have 1/2n∈B(0,r) for n being large enough. Note that d(1,1/2n)=4 , so 1/2n∉B(1,2) for all n∈... . This proves that B(0,r)⊄B(1,2) .

We introduce a b -metric on the cone b -metric space and then prove some equivalences between them as follows.

Theorem 14.

Let (X,d) be a cone b -metric space with coefficient s and [figure omitted; refer to PDF] for all x,y∈X . Then one has the following.

(1) D is a b -metric on X .

(2) _{limn[arrow right]∞xn} =x in the cone b -metric space (X,d) if and only if _{limn[arrow right]∞xn} =x in the b -metric space (X,D) .

(3) {_xn } is a Cauchy sequence in the cone b -metric space (X,d) if and only if {_xn } is a Cauchy sequence in the b -metric space (X,D) .

(4) The cone b -metric space (X,d) is complete if and only if the b -metric space (X,D) is complete.

Proof.

(1) For all x,y∈X , it is obvious that D(x,y)...5;0 and D(x,y)=D(y,x) .

If x=y , then D(x,y)=inf{||u||:u∈P, u...5;θ}=0 .

If D(x,y)=inf{||u||:u∈P, u...5;(1/s)d(x,y)}=0 , then, for each n∈... , there exists _un ∈P such that _un ...5;(1/s)d(x,y) and ||_un ||<1/n . Then _{limn[arrow right]∞un} =θ , and by Lemma 9(6), we have d(x,y)...4;θ . It implies that d(x,y)∈P∩(-P) . Therefore, d(x,y)=θ ; that is, x=y .

For each x,y,z∈X , we have [figure omitted; refer to PDF] Since _u2 ,_u3 ∈P and _u2 ...5;(1/s)d(x,y) , _u3 ...5;(1/s)d(y,z) , we have [figure omitted; refer to PDF] Then we have [figure omitted; refer to PDF] It implies that [figure omitted; refer to PDF] Now, we have [figure omitted; refer to PDF] By the previously metioned, D is a b -metric on X .

(2) Necessity. Let _{limn[arrow right]∞xn} =x in the cone b -metric space (X,d) . For each [straight epsilon]>0 , by Lemma 9(8), if θ...a;c , then θ...a;s·[straight epsilon]·(c/||c||) . Then, for each c∈E with θ...a;c , there exists _n0 such that d(_xn ,x)...a;s·[straight epsilon]·(c/||c||) for all n...5;_n0 . Using Lemma 9(8) again, we get (1/s)d(_xn ,x)...a;[straight epsilon]·(c/||c||) . It implies that [figure omitted; refer to PDF] for all n...5;_n0 . This proves that _{limn[arrow right]∞} D(_xn ,x)=0 ; that is, _{limn[arrow right]∞xn} =x in the b -metric space (X,D) .

Sufficiency. Let _{limn[arrow right]∞xn} =x in the b -metric space (X,D) . For each θ...a;c , there exists [straight epsilon]>0 such that c+B(0,[straight epsilon])⊂P . For this [straight epsilon] , there exists _n0 such that [figure omitted; refer to PDF] Then, there exist v∈P and d(_xn ,x)...4;v such that ||v||...4;[straight epsilon]/2 . So -v∈B(0,[straight epsilon]) , and we have c-v∈intP . Therefore, d(_xn ,x)...4;v...a;c for all n...5;_n0 . By Lemma 9(1), we get d(_xn ,x)...a;c for all n...5;_n0 . This proves that _{limn[arrow right]∞xn} =x in the cone b -metric space (X,d) .

(3) Necessity. Let {_xn } be a Cauchy sequence in the cone b -metric space (X,d) . For each [straight epsilon]>0 , by Lemma 9(6), if θ...a;c , then θ...a;s·[straight epsilon]·(c/||c||) . Then for each c∈E with θ...a;c , there exists _n0 such that d(_xn ,_xm )...a;s·[straight epsilon]·(c/||c||) for all n,m...5;_n0 . Using Lemma 9(6) again, we get (1/s)d(_xn ,_xm )...a;[straight epsilon]·(c/||c||) . It implies that [figure omitted; refer to PDF] for all n,m...5;_n0 . This proves that {_xn } is a Cauchy sequence in the b -metric space (X,D) .

Sufficiency. Let {_xn } be a Cauchy sequence in the b -metric space (X,D) . Then _{limn,m[arrow right]∞} D(_xn ,_xm )=0 . For each θ...a;c , there exists [straight epsilon]>0 such that c+B(0,[straight epsilon])⊂P . For this [straight epsilon] , there exists _n0 such that [figure omitted; refer to PDF] for all n,m...5;_n0 . Then, there exists v∈P , d(_xn ,_xm )...4;v such that ||v||...4;[straight epsilon]/2 . So -v∈B(0,[straight epsilon]) , and we have c-v∈intP . Therefore, d(_xn ,_xm )...4;v...a;c for all n,m...5;_n0 . By Lemma 9(1), we get d(_xn ,_xm )...a;c for all n,m...5;_n0 . This proves that {_xn } is a Cauchy sequence in the cone b -metric space (X,d) .

(4) It is a direct consequence of (2) and (3).

By choosing s=1 in Theorem 14, we get the following results.

Corollary 15 (see [13, Lemma 2.1]).

Let (X,d) be a cone metric space. Then [figure omitted; refer to PDF] for all x,y∈X is a metric on X .

Corollary 16 (see [10, Theorem 2.2]).

Let (X,d) be a cone metric space and [figure omitted; refer to PDF] for all x,y∈X . Then the metric space (X,D) is complete if and only if the cone metric space (X,d) is complete.

The following examples show that Corollaries 15 and 16 are not applicable to cone b -metric spaces in general.

Example 17.

Let (X,d) be a cone b -metric space as in Example 11. We have [figure omitted; refer to PDF] It implies that [figure omitted; refer to PDF] Then D is not a metric on X . This proves that Corollaries 15 and 16 are not applicable to given cone b -metric space (X,d) .

Example 18.

Let (X,d) be a cone b -metric space as in Example 12. We have [figure omitted; refer to PDF] For u(s)=0 , v(s)=1 , and w(s)=2 for all s∈[0,1] , we have [figure omitted; refer to PDF] Then D is not a metric on X . This proves that Corollaries 15 and 16 are not applicable to given cone b -metric space (X,d) .

Next, by using Theorem 14, we show that some contraction conditions on cone b -metric spaces can be obtained from certain contraction conditions on b -metric spaces.

Corollary 19.

Let (X,d) be a cone b -metric space with coefficient s , let T:X[arrow right]X be a map, and let D be defined as in Theorem 14. Then the following statements hold.

(1) If d(Tx,Ty)...4;kd(x,y) for some k∈[0,1) and all x,y∈X , then [figure omitted; refer to PDF] for all x,y∈X .

(2) If d(Tx,Ty)...4;_λ1 d(x,Tx)+_λ2 d(y,Ty)+_λ3 d(x,Ty)+_λ4 d(y,Tx) for some _λ1 ,_λ2 ,_λ3 ,_λ4 ∈[0,1) with _λ1 +_λ2 +s(_λ3 +_λ4 )<min{1,2/s} and all x,y∈X , then [figure omitted; refer to PDF] for all x,y∈X .

Proof.

(1) For each x,y∈X and v∈P with v...5;(1/s)d(x,y) , it follows from Lemma 9(8) that [figure omitted; refer to PDF] Thus, {kv:v∈P, v...5;(1/s)d(x,y)}⊂{u:u∈P, u...5;(1/s)d(Tx,Ty)} . Then we have [figure omitted; refer to PDF] It implies that D(Tx,Ty)...4;kD(x,y) .

(2) Let x,y∈X and _v1 ,_v2 ,_v3 ,_v4 ∈P satisfy [figure omitted; refer to PDF]

From Lemma 9(8), we have [figure omitted; refer to PDF] It implies that [figure omitted; refer to PDF] Then we have [figure omitted; refer to PDF] This proves that D(Tx,Ty)...4;_λ1 D(x,Tx)+_λ2 D(y,Ty)+_λ3 D(x,Ty)+_λ4 D(y,Tx) .

Now, we show that main results in [9] are consequences of preceding results on b -metric spaces.

Corollary 20.

Let (X,d) be a complete cone b -metric space with coefficient s , and let T:X[arrow right]X be a map. Then the following statements hold.

(1) (see [9, Theorem 2.1]) If d(Tx,Ty)...4;kd(x,y) for all x,y∈X , then T has a unique fixed point.

(2) (see [9, Theorem 2.3]) If d(Tx,Ty)...4;_λ1 d(x,Tx)+_λ2 d(y,Ty)+_λ3 d(x,Ty)+_λ4 d(y,Tx) for some _λ1 ,_λ2 ,_λ3 ,_λ4 ∈[0,1) with _λ1 +_λ2 +s(_λ3 +_λ4 )<min{1,2/s} and all x,y∈X , then T has a unique fixed point.

Proof.

Let D be defined as in Theorem 14. It follows from Theorem 14(4) that (X,D) is a complete b -metric space.

(1) By Corollary 19 (1), we see that T satisfies all assumptions of [5, Theorem 2]. Then T has a unique fixed point.

(2) By Corollary 19 (2), we see that T satisfies all assumptions in [6, Theorem 3.7], where K=s , f=T , g is the identity, and _a1 =0, _a2 =_λ1 , _a3 =_λ2 , and _a4 =_λ3 , _a5 =_λ4 . Note that condition (3.10) in [6, Theorem 3.7] was used to prove (3.16) and K(_a2 +_a3 +_a4 +_a5 )<2 at line 3, page 7 in the proof of [6, Theorem 3.7]. These claims also hold if _a1 =0 and _λ1 +_λ2 +s(_λ3 +_λ4 )<min{1,2/s} . Then T has a unique fixed point.

Remark 21.

By similar arguments as in Corollaries 19 and 20, we may get fixed point theorems on cone b -metric spaces in [8, 10] from preceding ones on b -metric spaces in [3, 5].

Acknowledgments

The authors are thankful for an anonymous referee for his useful comments on this paper. This research was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission under the Computational Science and Engineering Research Cluster (CSEC Grant no. NRU56000508).

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