AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 573740 10.1155/2013/573740 573740 Research Article Some Equivalences between Cone b-Metric Spaces and b-Metric Spaces Kumam Poom 1 Dung Nguyen Van 2 Hang Vo Thi Le 3 Aydi Hassen 1 Department of Mathematics Faculty of Science King Mongkut’s University of Technology Thonburi (KMUTT) Bang Mod Thrung Khru Bangkok 10140 Thailand kmutt.ac.th 2 Faculty of Mathematics and Information Technology Teacher Education Dong Thap University Cao Lanh City Dong Thap Province 871200 Vietnam dthu.edu.vn 3 Journal of Science Dong Thap University Cao Lanh City Dong Thap Province 871200 Vietnam dthu.edu.vn 2013 3 10 2013 2013 20 06 2013 24 08 2013 27 08 2013 2013 Copyright © 2013 Poom Kumam et al. 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.

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.

1. Introduction and Preliminaries

The fixed point theory in b-metric spaces was investigated by Bakhtin , Czerwik , Akkouchi , Olatinwo and Imoru , and Pǎcurar . A b-metric space was also called a metric-type space in . 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 . Some fixed point theorems on cone b-metric spaces were stated in .

Note that the relation between a cone b-metric and a b-metric is likely the relation between a cone metric  and a metric. Some authors have proved that fixed point theorems on cone metric spaces are, essentially, fixed point theorems on metric space; see  for example. Very recently, Du used the method in  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 .

In this paper, we use the method in  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 [<xref ref-type="bibr" rid="B2">1</xref>]).

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

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

d(x,y)=d(y,x) for all x,yX;

there exists s1 such that d(x,z)s[d(x,y)+d(y,z)] for all x,y,zX.

The pair (X,d) is called a b-metric space. A sequence {xn} is called convergent to x in X, written limnxn=x, if limnd(xn,x)=0. A sequence {xn} is called a Cauchy sequence if limn,md(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 xX, we set (1)B(x,r)={yX:d(x,y)<r}. In , Akkouchi claimed that the topology 𝒯(d) on X associated with d is given by setting U𝒯(d) if and only if, for each xU, 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,𝒯(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 with respect to P by xy if and only if y-xP. We also write x<y to indicate that xy and xy and write xy to indicate that y-xintP. Let · denote the norm on E.

Definition 4 (see [<xref ref-type="bibr" rid="B9">11</xref>]).

P is called a cone if and only if

P is closed and nonempty and P{θ};

a,b;a,b0;  x,yP imply that ax+byP;

P(-P)={θ}.

The cone P is called normal if there exists K1 such that, for all x,yE, we have θxy implies xKy. The least positive number K satisfying the above is called the normal constant of P.

Definition 5 (see [<xref ref-type="bibr" rid="B9">11</xref>, Definition 1]).

Let X be a nonempty set and d:X×XE satisfy

θd(x,y) for all x,yX and d(x,y)=θ if and only if x=y;

d(x,y)=d(y,x) for all x,yX;

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

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

Definition 6 (see [<xref ref-type="bibr" rid="B11">8</xref>, Definition 2.1]).

Let X be a nonempty set and d:X×XP satisfy

θd(x,y) for all x,yX and d(x,y)=θ if and only if x=y;

d(x,y)=d(y,x) for all x,yX;

d(x,y)s[d(x,z)+d(z,y)] for some s1 and all x,y,zX.

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 [<xref ref-type="bibr" rid="B11">8</xref>, Definition 2.4]).

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

{xn} is called convergent to x, written limnxn=x, if for each cE with θc, there exists n0 such that d(xn,x)c for all nn0.

{xn} is called a Cauchy sequence if for each cE with θc there exists n0 such that d(xn,xm)c for all n,mn0.

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

Lemma 8 (see [<xref ref-type="bibr" rid="B11">8</xref>, Proposition 2.5]).

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

limnxn=x if and only if  limnd(xn,x)=θ.

The limit point of a convergent sequence is unique.

Every convergent sequence is a Cauchy sequence.

{xn} is a Cauchy sequence if  limn,md(xn,xm)=θ.

Lemma 9 (see [<xref ref-type="bibr" rid="B11">8</xref>, 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.

If ab and bc, then ac.

If ab and bc, then ac.

If θuc for all cintP, then u=θ.

If cintP, θan for all n and limnan=θ, then there exists n0 such that anc for all nn0.

If θc, θd(xn,x)bn for all n and limnbn=θ, then d(xn,x)c eventually.

If θanbn for all n and limnan=a, limnbn=b, then ab.

If aP, 0λ<1, and aλ·a, then a=θ.

For each α>0, one has α·intPintP.

For each δ>0 and xintP, there exists 0<γ<1 such that γ·x<δ.

For each θc1 and c2P, there exists θd such that c1d and c2d.

For each θc1 and θc2, there exists θe such that ec1 and ec2.

Remark 10 (see [<xref ref-type="bibr" rid="B17">10</xref>, 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 [<xref ref-type="bibr" rid="B17">10</xref>, Example 2.2]).

Let (2)E=C1[0,1],P={φE:φ0},X=[1,+), and d(x,y)(t)=|x-y|2et for all x,yX 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 [<xref ref-type="bibr" rid="B17">10</xref>, Example 2.3]).

Let X be the set of Lebesgue measurable functions on [0,1] such that 01|u(x)|2dx<+, E=C[0,1], P={φE:φ0}. Define d:X×XE as (3)d(u(t),v(t))=et01|u(s)-v(s)|2ds, for all u,vX 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 (4)d(x,y)={0if  x=y1if  xy{0,1}|x-y|if  xy{0,12n,12m}4otherwise. Then we have the following.

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

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

Proof.

(1) For all x,yX, we have d(x,y)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 (5)d(x,z)+d(z,y)={d(0,12n)+d(12n,1)=12n+4if  z=12nd(0,12n+1)+d(12n+1,1)=4+4if  z=12n+1. If d(x,y)=d(0,1/2n)=1/2n, then (6)d(x,z)+d(z,y)={d(0,12m)+d(12m,12n)=12m+|12m-12n|ifz=12md(0,12m+1)+d(12m+1,12n)=4+4if  z=12m+11d(0,1)+d(1,12n)=1+4if  z=1. If d(x,y)=d(1/2k,1/2n)=|1/2k-1/2n|, then (7)d(x,z)+d(z,y)={d(12k,12m)+d(12m,12n)=|12k-12m|+|12m-12n|if  z=12md(12k,12m+1)+d(12m+1,12n)=4+4if  z=12m+1d(12k,0)+d(0,12n)=12k+12nif  z=0. If d(x,y)=d(1/2k,1/(2n+1))=4 with 1/(2n+1)1, then (8)d(x,z)+d(z,y)={d(12k,0)+d(0,12n+1)=12k+4if  z=0d(12k,12m)+d(12m,12n+1)=|12k-12m|+4if  z=12md(12k,12m+1)+d(12m+1,12n+1)=4+4if  z=12m+1. If d(x,y)=d(1/(2k+1),1/(2n+1))=4 with 1/(2k+1)1 and 1/(2n+1)1, then (9)d(x,z)+d(z,y)={d(12k+1,0)+d(0,12n+1)=4+4if  z=0d(12k+1,12m)+d(12m,12n+1)=4+4if  z=12md(12k+1,12m+1)+d(12m+1,12n+1)=4+4if  z=12m+1. If d(x,y)=d(1/2k,1)=4, then (10)d(x,z)+d(z,y)={d(12k,0)+d(0,1)=12k+1if  z=0d(12k,12m)+d(12m,1)=|12k-12m|+4if  z=12md(12k,12m+1)+d(12m+1,1)=4+4if  z=12m+11. If d(x,y)=d(1/(2k+1),1)=4, then (11)d(x,z)+d(z,y)={d(12k+1,0)+d(0,1)=4+1if  z=0d(12k+1,12m)+d(12m,1)=4+4if  z=12md(12k+1,12m+1)+d(12m+1,1)=4+4if  z=12m+11. If d(x,y)=d(1/(2k+1),0)=4, then (12)d(x,z)+d(z,y)={d(12k+1,1)+d(1,0)=4+1if  z=1d(12k+1,12m)+d(12m,0)=4+12mif  z=12md(12k+1,12m+1)+d(12m+1,0)=4+4if  z=12m+11.

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

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

For each r>0, since d(0,1/2n)=1/2n, we have 1/2nB(0,r) for nbeing large enough. Note that d(1,1/2n)=4, so1/2nB(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 (13)D(x,y)=inf{u:uP,u1sd(x,y)}, for all x,yX. Then one has the following.

D is a b-metric on X.

limnxn=x in the cone b-metric space (X,d) if and only if limnxn=x in the b-metric space (X,D).

{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).

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,yX, it is obvious that D(x,y)0 and D(x,y)=D(y,x).

If x=y, then D(x,y)=inf{u:uP,uθ}=0.

If D(x,y)=inf{u:uP,u(1/s)d(x,y)}=0, then, for each n, there exists unP such that un(1/s)d(x,y) and un<1/n. Then limnun=θ, and by Lemma 9(6), we have d(x,y)θ. It implies that d(x,y)P(-P). Therefore, d(x,y)=θ; that is, x=y.

For each x,y,zX, we have (14)D(x,z)=inf{u1:u1P,u11sd(x,z)},D(x,y)=inf{u2:u2P,u21sd(x,y)},D(y,z)=inf{u3:u3P,u31sd(y,z)}. Since u2,u3P and u2(1/s)d(x,y),u3(1/s)d(y,z), we have (15)s(u2+u3)d(x,y)+d(y,z)1sd(x,z). Then we have (16){u1P:u11sd(x,z)}{1sd(x,y)s(u2+u3)P:u21sd(x,y),u31sd(y,z)}. It implies that (17)inf{1sd(y,z)s(u2+u3):u2,u3P,u21sd(x,y),u31sd(y,z)}inf{u1:u1P,u11sd(x,z)}. Now, we have (18)D(x,z)=inf{u1:u1P,u11sd(x,z)}inf{u21ss(u2+u3):u2,u3P,u21sd(x,y),u31sd(y,z)}=sinf{1sd(y,z)u2+u3:u2,u3P,u21sd(x,y),u31sd(y,z)}sinf{1su2+u3:u2,u3P,u21sd(x,y),u31sd(y,z)}=sinf{u2:u2P,u21sd(x,y)}+sinf{u3:u3P,u31sd(y,z)}=s[D(x,y)+D(y,z)]. By the previously metioned, D is a b-metric on X.

(2) Necessity. Let limnxn=x in the cone b-metric space (X,d). For each ε>0, by Lemma 9(8), if θc, then θs·ε·(c/c). Then, for each cE with θc, there exists n0 such that d(xn,x)s·ε·(c/c) for all nn0. Using Lemma 9(8) again, we get (1/s)d(xn,x)ε·(c/c). It implies that (19)D(xn,x)=inf{u:uP,u1sd(xn,x)}ε·cc=ε, for all nn0. This proves that limnD(xn,x)=0; that is, limnxn=x in the b-metric space (X,D).

Sufficiency. Let limnxn=x in the b-metric space (X,D). For each θc, there exists ε>0 such that c+B(0,ε)P. For this ε, there exists n0 such that (20)D(xn,x)=inf{u:uP,u1sd(xn,x)}ε4. Then, there exist vPandd(xn,x)v such that vε/2. So -vB(0,ε), and we have c-vintP. Therefore, d(xn,x)vc for all nn0. By Lemma 9(1), we get d(xn,x)c for all nn0. This proves that limnxn=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 ε>0, by Lemma 9(6), if θc, then θs·ε·(c/c). Then for each cE with θc, there exists n0 such that d(xn,xm)s·ε·(c/c) for all n,mn0. Using Lemma 9(6) again, we get (1/s)d(xn,xm)ε·(c/c). It implies that (21)D(xn,xm)=inf{u:uP,u1sd(xn,xm)}ε·cc=ε, for all n,mn0. 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,mD(xn,xm)=0. For each θc, there exists ε>0 such that c+B(0,ε)P. For this ε, there exists n0 such that (22)D(xn,xm)=inf{u:uP,u1sd(xn,xm)}ε4, for all n,mn0. Then, there exists vP, d(xn,xm)v such that vε/2. So -vB(0,ε), and we have c-vintP. Therefore, d(xn,xm)vc for all n,mn0. By Lemma 9(1), we get d(xn,xm)c for all n,mn0. 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 [<xref ref-type="bibr" rid="B6">13</xref>, Lemma 2.1]).

Let (X,d) be a cone metric space. Then (23)D(x,y)=inf{u:uP,ud(x,y)}, for all x,yX is a metric on X.

Corollary 16 (see [<xref ref-type="bibr" rid="B17">10</xref>, Theorem 2.2]).

Let (X,d) be a cone metric space and (24)D(x,y)=inf{u:uP,ud(x,y)}, for all x,yX. 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 (25)D(x,y)=inf{u:uP,ud(x,y)}=d(x,y)=sup{|x-y|2et:t[0,1]}  =e|x-y|2. It implies that (26)D(0,2)=4e>D(0,1)+D(1,2)=e+e=2e. 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 (27)D(u,v)=inf{z:zP,zd(u,v)}=d(u,v)=sup{et01|u(s)-v(s)|2ds:t[0,1]}=e01|u(s)-v(s)|2ds. For u(s)=0, v(s)=1, and w(s)=2 for all s[0,1], we have (28)D(u,w)=4e>D(u,v)+D(v,w)=e+e=2e. 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:XX be a map, and let D be defined as in Theorem 14. Then the following statements hold.

If d(Tx,Ty)kd(x,y) for some k[0,1) and all x,yX, then (29)D(Tx,Ty)kD(x,y), for all x,yX.

If d(Tx,Ty)λ1d(x,Tx)+λ2d(y,Ty)+λ3d(x,Ty)+λ4d(y,Tx) for some λ1,λ2,λ3,λ4[0,1) with λ1+λ2+s(λ3+λ4)<min{1,2/s} and all x,yX, then (30)D(Tx,Ty)λ1D(x,Tx)+λ2D(y,Ty)+λ3D(x,Ty)+λ4D(y,Tx), for all x,yX.

Proof.

(1) For each x,yX and vP with v(1/s)d(x,y), it follows from Lemma 9(8) that (31)kvk1sd(x,y)1sd(Tx,Ty). Thus, {kv:vP,v(1/s)d(x,y)}{u:uP,u(1/s)d(Tx,Ty)}. Then we have (32)D(Tx,Ty)=inf{u:uP,u1sd(Tx,Ty)}inf{kv:vP,v1sd(x,y)}=kinf{v:vP,v1sd(x,y)}=kD(x,y). It implies that D(Tx,Ty)kD(x,y).

(2) Let x,yX and v1,v2,v3,v4P satisfy (33)v11sd(x,Tx),v21sd(y,Ty),v31sd(x,Ty),v41sd(y,Tx).

From Lemma 9(8), we have (34)λ1v1+λ2v2+λ3v3+λ4v41s[λ1d(x,Tx)+λ2d(y,Ty)+λ3d(x,Ty)+λ4d(y,Tx)]1sd(Tx,Ty). It implies that (35){v:vP,v1sd(Tx,Ty)}{1sλ1v1+λ2v2+λ3v3+λ4v4:v1,v2,v3,v4P,v11sd(x,Tx),v21sd(y,Ty),v31sd(x,Ty),v41sd(y,Tx)}. Then we have (36)D(Tx,Ty)=inf{v:vP,v1sd(Tx,Ty)}inf{1sλ1v1+λ2v2+λ3v3+λ4v4:v1,v2,v3,v4P,v11sd(x,Tx),v21sd(y,Ty),v31sd(x,Ty),v41sd(y,Tx)}inf{1sλ1v1+λ2v2+λ3v3+λ4v4:v1,v2,v3,v4P,v11sd(x,Tx),v21sd(y,Ty),v31sd(x,Ty),v41sd(y,Tx)}=inf{λ1v1:v1P,v11sd(x,Tx)}+inf{λ2v2:v2P,v21sd(y,Ty)}+inf{λ3v3:v3P,v31sd(x,Ty)}+inf{λ4v4:v4P,v41sd(y,Tx)}=λ1inf{v1:v1P,v11sd(x,Tx)}+λ2inf{v2:v2P,v21sd(y,Ty)}+λ3·inf{v3:v3P,v31sd(x,Ty)}+λ4·inf{v4:v4P,v41sd(y,Tx)}=λ1D(x,Tx)+λ2D(y,Ty)+λ3D(x,Ty)+λ4D(y,Tx). This proves that D(Tx,Ty)λ1D(x,Tx)+λ2D(y,Ty)+λ3D(x,Ty)+λ4D(y,Tx).

Now, we show that main results in  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:XX be a map. Then the following statements hold.

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

(see [9, Theorem 2.3]) If d(Tx,Ty)λ1d(x,Tx)+λ2d(y,Ty)+λ3d(x,Ty)+λ4d(y,Tx) for some λ1,λ2,λ3,λ4[0,1) with λ1+λ2+s(λ3+λ4)<min{1,2/s} and all x,yX, 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.

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

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).

Bakhtin I. A. The contraction mapping principle in quasimetric spaces Functional Analysis 1989 30 26 37 Czerwik S. Contraction mappings in b-metric spaces Acta Mathematica et Informatica Universitatis Ostraviensis 1993 1 1 5 11 MR1250922 Akkouchi M. A common fixed point theorem for expansive mappings under strict implicit conditions on b-metric spaces Acta Universitatis Palackianae Olomucensis, Facultas Rerum Naturalium. Mathematica 2011 50 1 5 15 MR2920694 ZBL1246.54035 Olatinwo M. O. Imoru C. O. A generalization of some results on multi-valued weakly Picard mappings in b-metric space Fasciculi Mathematici 2008 40 45 56 MR2499257 Păcurar M. A fixed point result for φ-contractions on b-metric spaces without the boundedness assumption Fasciculi Mathematici 2010 43 127 137 MR2666110 Jovanović M. Kadelburg Z. Radenović S. Common fixed point results in metric-type spaces Fixed Point Theory and Applications 2010 2010 15 10.1155/2010/978121 978121 MR2747253 ZBL1207.54058 Hussain N. Dorić D. Kadelburg Z. Radenović S. Suzuki-type fixed point results in metric type spaces Fixed Point Theory and Applications 2012 2012, article 126 10.1186/1687-1812-2012-126 MR2969325 Hussain N. Shah M. H. KKM mappings in cone b-metric spaces Computers & Mathematics with Applications 2011 62 4 1677 1684 10.1016/j.camwa.2011.06.004 MR2834154 Huang H. Xu S. Fixed point theorems of contractive mappings in cone b-metric spaces and applications Fixed Point Theory and Applications 2013 2013, article 112 14 10.1186/1687-1812-2013-112 MR3071995 Shi L. Xu S. Common fixed point theorems for two weakly compatible self-mappings in cone b-metric spaces Fixed Point Theory and Applications 2013 2013, article 120 10.1186/1687-1812-2013-120 MR3066467 Huang L. G. Zhang X. Cone metric spaces and fixed point theorems of contractive mappings Journal of Mathematical Analysis and Applications 2007 332 2 1468 1476 10.1016/j.jmaa.2005.03.087 MR2324351 ZBL1118.54022 Du W. S. A note on cone metric fixed point theory and its equivalence Nonlinear Analysis: Theory, Methods & Applications 2010 72 5 2259 2261 10.1016/j.na.2009.10.026 MR2577793 ZBL1205.54040 Feng Y. Mao W. The equivalence of cone metric spaces and metric spaces Fixed Point Theory 2010 11 2 259 264 MR2743780 ZBL1221.54055 Harandi A. A. Fakhar M. Fixed point theory in cone metric spaces obtained via the scalarization method Computers & Mathematics with Applications 2010 59 11 3529 3534 10.1016/j.camwa.2010.03.046 MR2646324 ZBL1197.54055 Kaekhao A. Sintunavarat W. Kumam P. Common fixed point theorems of c-distance on cone metric spaces Journal of Nonlinear Analysis and Application 2012 2012 11 jnaa-00137 10.5899/2012/jnaa-00137 Khani M. Pourmahdian M. On the metrizability of cone metric spaces Topology and Its Applications 2011 158 2 190 193 10.1016/j.topol.2010.10.016 MR2739889 ZBL1206.54026 Du W. S. Karapinar E. A note on cone b-metric and its related results: generalizations or equivalence? Fixed Point Theory and Applications 2013 2013, article 210 6 10.1186/1687-1812-2013-210