# Some Equivalences between Coneb-Metric Spaces and b-Metric Spaces

Authors: Poom Kumam; Nguyen Van Dung; Vo Thi Le Hang
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

We introduce ab-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.


---

## Body

## 1. Introduction and Preliminaries

The fixed point theory inb-metric spaces was investigated by Bakhtin [1], Czerwik [2], Akkouchi [3], Olatinwo and Imoru [4], and Pǎcurar [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 coneb-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]).
LetX be a nonempty set and d:X×X→[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 existss≥1 such that d(x,z)≤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→∞xn=x, if limn→∞d(xn,x)=0. A sequence {xn} is called a Cauchy sequence if limn,m→∞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 ab-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
(1)B(x,r)={y∈X:d(x,y)<r}.
In [3], Akkouchi claimed that the topology 𝒯(d) on X associated with d is given by setting U∈𝒯(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,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, letE 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 x≤y if and only if y-x∈P. We also write x<y to indicate that x≤y and x≠y and write x≪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≠{θ};(2)
a,b∈ℝ;a,b≥0;x,y∈P imply that ax+by∈P;(3)
P∩(-P)={θ}.
The coneP is called normal if there exists K≥1 such that, for all x,y∈E, we have θ≤x≤y implies ∥x∥≤K∥y∥. The least positive number K satisfying the above is called the normal constant of P.Definition 5 (see [11, Definition 1]).
LetX be a nonempty set and d:X×X→E satisfy (1)
θ≤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)≤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]).
LetX be a nonempty set and d:X×X→P satisfy (1)
θ≤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)≤s[d(x,z)+d(z,y)] for some s≥1 and all x,y,z∈X.Then d is called a coneb-metric with coefficient s on X and (X,d) is called a coneb-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→∞xn=x, if for each c∈E with θ≪c, there exists n0 such that d(xn,x)≪c for all n≥n0.(2)
{xn} is called a Cauchy sequence if for each c∈E with θ≪c there exists n0 such that d(xn,xm)≪c for all n,m≥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→∞xn=x if and only if  limn→∞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→∞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)
Ifa≤b and b≪c, then a≪c.(2)
Ifa≪b and b≪c, then a≪c.(3)
Ifθ≤u≪c for all c∈intP, then u=θ.(4)
Ifc∈intP, θ≤an for all n∈ℕ and limn→∞an=θ, then there exists n0 such that an≪c for all n≥n0.(5)
Ifθ≪c, θ≤d(xn,x)≤bn for all n∈ℕ and limn→∞bn=θ, then d(xn,x)≪c eventually.(6)
Ifθ≤an≤bn for all n∈ℕ and limn→∞an=a, limn→∞bn=b, then a≤b.(7)
Ifa∈P, 0≤λ<1, and a≤λ·a, then a=θ.(8)
For eachα>0, one has α·intP⊂intP.(9)
For eachδ>0 and x∈intP, there exists 0<γ<1 such that ∥γ·x∥<δ.(10)
For eachθ≪c1 and c2∈P, there exists θ≪d such that c1≪d and c2≪d.(11)
For eachθ≪c1 and θ≪c2, there exists θ≪e such that e≪c1 and e≪c2.Remark 10 (see [10, Remark 1.3]).
Every cone metric space is a coneb-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(2)E=Cℝ1[0,1],P={φ∈E:φ≥0},X=[1,+∞),
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]).
LetX be the set of Lebesgue measurable functions on [0,1] such that ∫01|u(x)|2dx<+∞, E=Cℝ[0,1], P={φ∈E:φ≥0}. Define d:X×X→E as
(3)d(u(t),v(t))=et∫01|u(s)-v(s)|2ds,
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 ballsB(x,r) does not form a base for any topology on a b-metric space (X,d).Example 13.
LetX={0,1,1/2,…,1/n,…} and
(4)d(x,y)={0ifx=y1ifx≠y∈{0,1}|x-y|ifx≠y∈{0,12n,12m}4otherwise.
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 allx,y∈X, we have d(x,y)≥0, d(x,y)=0 if and only if x=y and d(x,y)=d(y,x).
Ifd(x,y)=d(0,1)=1, then
(5)d(x,z)+d(z,y)={d(0,12n)+d(12n,1)=12n+4ifz=12nd(0,12n+1)+d(12n+1,1)=4+4ifz=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+4ifz=12m+1≠1d(0,1)+d(1,12n)=1+4ifz=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|ifz=12md(12k,12m+1)+d(12m+1,12n)=4+4ifz=12m+1d(12k,0)+d(0,12n)=12k+12nifz=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+4ifz=0d(12k,12m)+d(12m,12n+1)=|12k-12m|+4ifz=12md(12k,12m+1)+d(12m+1,12n+1)=4+4ifz=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+4ifz=0d(12k+1,12m)+d(12m,12n+1)=4+4ifz=12md(12k+1,12m+1)+d(12m+1,12n+1)=4+4ifz=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+1ifz=0d(12k,12m)+d(12m,1)=|12k-12m|+4ifz=12md(12k,12m+1)+d(12m+1,1)=4+4ifz=12m+1≠1.
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+1ifz=0d(12k+1,12m)+d(12m,1)=4+4ifz=12md(12k+1,12m+1)+d(12m+1,1)=4+4ifz=12m+1≠1.
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+1ifz=1d(12k+1,12m)+d(12m,0)=4+12mifz=12md(12k+1,12m+1)+d(12m+1,0)=4+4ifz=12m+1≠1.
By the previous calculations, we getd(x,y)≤(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 haveB(1,2)={x∈X:d(x,1)<2}={1,0}. Then 0∈B(1,2).
For eachr>0, since d(0,1/2n)=1/2n, we have 1/2n∈B(0,r) for nbeing large enough. Note that d(1,1/2n)=4, so1/2n∉B(1,2) for all n∈ℕ. This proves that B(0,r)⊄B(1,2).We introduce ab-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∥:u∈P,u≥1sd(x,y)},
for all x,y∈X. Then one has the following.(1)
D is a b-metric on X.(2)
limn→∞xn=x in the cone b-metric space (X,d) if and only if limn→∞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 coneb-metric space (X,d) is complete if and only if the b-metric space (X,D) is complete.Proof.
(1) For allx,y∈X, it is obvious that D(x,y)≥0 and D(x,y)=D(y,x).
Ifx=y, then D(x,y)=inf{∥u∥:u∈P,u≥θ}=0.
IfD(x,y)=inf{∥u∥:u∈P,u≥(1/s)d(x,y)}=0, then, for each n∈ℕ, there exists un∈P such that un≥(1/s)d(x,y) and ∥un∥<1/n. Then limn→∞un=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 eachx,y,z∈X, we have
(14)D(x,z)=inf{∥u1∥:u1∈P,u1≥1sd(x,z)},D(x,y)=inf{∥u2∥:u2∈P,u2≥1sd(x,y)},D(y,z)=inf{∥u3∥:u3∈P,u3≥1sd(y,z)}.
Since u2,u3∈P 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){u1∈P:u1≥1sd(x,z)}⊃{1sd(x,y)s(u2+u3)∈P:u2≥1sd(x,y),u31s≥d(y,z)}.
It implies that
(17)inf{1sd(y,z)∥s(u2+u3)∥:u2,u3∈P,u2≥1sd(x,y),u3≥1sd(y,z)}≥inf{∥u1∥:u1∈P,u1≥1sd(x,z)}.
Now, we have
(18)D(x,z)=inf{∥u1∥:u1∈P,u1≥1sd(x,z)}≤inf{u21s∥s(u2+u3)∥:u2,u3∈P,u2≥1sd(x,y),u3≥1sd(y,z)}=sinf{1sd(y,z)∥u2+u3∥:u2,u3∈P,u2≥1sd(x,y),u3≥1sd(y,z)}≤sinf{1s∥u2∥+∥u3∥:u2,u3∈P,u2≥1sd(x,y),u3≥1sd(y,z)}=sinf{∥u2∥:u2∈P,u2≥1sd(x,y)}+sinf{∥u3∥:u3∈P,u3≥1sd(y,z)}=s[D(x,y)+D(y,z)].
By the previously metioned, D is a b-metric on X.
(2)Necessity. Let limn→∞xn=x in the cone b-metric space (X,d). For each ε>0, by Lemma 9(8), if θ≪c, then θ≪s·ε·(c/∥c∥). Then, for each c∈E with θ≪c, there exists n0 such that d(xn,x)≪s·ε·(c/∥c∥) for all n≥n0. Using Lemma 9(8) again, we get (1/s)d(xn,x)≪ε·(c/∥c∥). It implies that
(19)D(xn,x)=inf{∥u∥:u∈P,u≥1sd(xn,x)}≤ε·∥c∥c∥∥=ε,
for all n≥n0. This proves that limn→∞D(xn,x)=0; that is, limn→∞xn=x in the b-metric space (X,D).
Sufficiency. Let limn→∞xn=x in the b-metric space (X,D). For each θ≪c, 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∥:u∈P,u≥1sd(xn,x)}≤ε4.
Then, there exist v∈Pandd(xn,x)≤v such that ∥v∥≤ε/2. So -v∈B(0,ε), and we have c-v∈intP. Therefore, d(xn,x)≤v≪c for all n≥n0. By Lemma 9(1), we get d(xn,x)≪c for all n≥n0. This proves that limn→∞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 ε>0, by Lemma 9(6), if θ≪c, then θ≪s·ε·(c/∥c∥). Then for each c∈E with θ≪c, there exists n0 such that d(xn,xm)≪s·ε·(c/∥c∥) for all n,m≥n0. Using Lemma 9(6) again, we get (1/s)d(xn,xm)≪ε·(c/∥c∥). It implies that
(21)D(xn,xm)=inf{∥u∥:u∈P,u≥1sd(xn,xm)}≤ε·∥c∥c∥∥=ε,
for all n,m≥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→∞D(xn,xm)=0. For each θ≪c, 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∥:u∈P,u≥1sd(xn,xm)}≤ε4,
for all n,m≥n0. Then, there exists v∈P, d(xn,xm)≤v such that ∥v∥≤ε/2. So -v∈B(0,ε), and we have c-v∈intP. Therefore, d(xn,xm)≤v≪c for all n,m≥n0. By Lemma 9(1), we get d(xn,xm)≪c for all n,m≥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 choosings=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
(23)D(x,y)=inf{∥u∥:u∈P,u≥d(x,y)},
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
(24)D(x,y)=inf{∥u∥:u∈P,u≥d(x,y)},
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 Corollaries15 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∥:u∈P,u≥d(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∥:z∈P,z≥d(u,v)}=∥d(u,v)∥=sup{et∫01|u(s)-v(s)|2ds:t∈[0,1]}=e∫01|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 Theorem14, 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→X be a map, and let D be defined as in Theorem 14. Then the following statements hold. (1)
Ifd(Tx,Ty)≤kd(x,y) for some k∈[0,1) and all x,y∈X, then
(29)D(Tx,Ty)≤kD(x,y),
for all x,y∈X.(2)
Ifd(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,y∈X, then
(30)D(Tx,Ty)≤λ1D(x,Tx)+λ2D(y,Ty)+λ3D(x,Ty)+λ4D(y,Tx),
for all x,y∈X.Proof.
(1) For eachx,y∈X and v∈P with v≥(1/s)d(x,y), it follows from Lemma 9(8) that
(31)kv≥k1sd(x,y)≥1sd(Tx,Ty).
Thus, {kv:v∈P,v≥(1/s)d(x,y)}⊂{u:u∈P,u≥(1/s)d(Tx,Ty)}. Then we have
(32)D(Tx,Ty)=inf{∥u∥:u∈P,u≥1sd(Tx,Ty)}≤inf{∥kv∥:v∈P,v≥1sd(x,y)}=kinf{∥v∥:v∈P,v≥1sd(x,y)}=kD(x,y).
It implies that D(Tx,Ty)≤kD(x,y).
(2) Letx,y∈X and v1,v2,v3,v4∈P satisfy
(33)v1≥1sd(x,Tx),v2≥1sd(y,Ty),v3≥1sd(x,Ty),v4≥1sd(y,Tx).
From Lemma9(8), we have
(34)λ1v1+λ2v2+λ3v3+λ4v4≥1s[λ1d(x,Tx)+λ2d(y,Ty)+λ3d(x,Ty)+λ4d(y,Tx)]≥1sd(Tx,Ty).
It implies that
(35){v:v∈P,v≥1sd(Tx,Ty)}⊃{1sλ1v1+λ2v2+λ3v3+λ4v4:v1,v2,v3,v4∈P,v1≥1sd(x,Tx),v2≥1sd(y,Ty),v3≥1sd(x,Ty),v4≥1sd(y,Tx)}.
Then we have
(36)D(Tx,Ty)=inf{∥v∥:v∈P,v≥1sd(Tx,Ty)}≤inf{1s∥λ1v1+λ2v2+λ3v3+λ4v4∥:v1,v2,v3,v4∈P,v1≥1sd(x,Tx),v2≥1sd(y,Ty),v3≥1sd(x,Ty),v4≥1sd(y,Tx)}≤inf{1sλ1∥v1∥+λ2∥v2∥+λ3∥v3∥+λ4∥v4∥:v1,v2,v3,v4∈P,v1≥1sd(x,Tx),v2≥1sd(y,Ty),v3≥1sd(x,Ty),v4≥1sd(y,Tx)}=inf{λ1∥v1∥:v1∈P,v1≥1sd(x,Tx)}+inf{λ2∥v2∥:v2∈P,v2≥1sd(y,Ty)}+inf{λ3∥v3∥:v3∈P,v3≥1sd(x,Ty)}+inf{λ4∥v4∥:v4∈P,v4≥1sd(y,Tx)}=λ1inf{∥v1∥:v1∈P,v1≥1sd(x,Tx)}+λ2inf{∥v2∥:v2∈P,v2≥1sd(y,Ty)}+λ3·inf{∥v3∥:v3∈P,v3≥1sd(x,Ty)}+λ4·inf{∥v4∥:v4∈P,v4≥1sd(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 [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→X be a map. Then the following statements hold. (1)
(see [9, Theorem 2.1]) If d(Tx,Ty)≤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)≤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,y∈X, then T has a unique fixed point.Proof.
LetD be defined as in Theorem 14. It follows from Theorem 14(4) that (X,D) is a complete b-metric space.(1)
By Corollary19 (1), we see that T satisfies all assumptions of [5, Theorem 2]. Then T has a unique fixed point.(2)
By Corollary19 (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 Corollaries19 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].



---

*Source: 573740-2013-10-03.xml*