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 [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 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 [<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,y∈X;

there exists s≥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 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
(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, 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 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 [<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∈ℝ;a,b≥0;x,y∈P imply that ax+by∈P;

P∩(-P)={θ}.

The cone P 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 [<xref ref-type="bibr" rid="B9">11</xref>, Definition 1]).

Let X be a nonempty set and d:X×X→E satisfy

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

d(x,y)=d(y,x) for all x,y∈X;

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

Let X be a nonempty set and d:X×X→P satisfy

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

d(x,y)=d(y,x) for all x,y∈X;

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 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 limn→∞xn=x, if for each c∈E with θ≪c, there exists n0 such that d(xn,x)≪c for all n≥n0.

{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.

(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, x∈X, and {xn} a sequence in X. Then one has the following.

limn→∞xn=x if and only if limn→∞d(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,m→∞d(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 a≤b and b≪c, then a≪c.

If a≪b and b≪c, then a≪c.

If θ≤u≪c for all c∈intP, then u=θ.

If c∈intP, θ≤an for all n∈ℕ and limn→∞an=θ, then there exists n0 such that an≪c for all n≥n0.

If θ≪c, θ≤d(xn,x)≤bn for all n∈ℕ and limn→∞bn=θ, then d(xn,x)≪c eventually.

If θ≤an≤bn for all n∈ℕ and limn→∞an=a, limn→∞bn=b, then a≤b.

If a∈P, 0≤λ<1, and a≤λ·a, then a=θ.

For each α>0, one has α·intP⊂intP.

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

For each θ≪c1 and c2∈P, there exists θ≪d such that c1≪d and c2≪d.

For each θ≪c1 and θ≪c2, there exists θ≪e such that e≪c1 and e≪c2.

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=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 [<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=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 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)={0ifx=y1ifx≠y∈{0,1}|x-y|ifx≠y∈{0,12n,12m}4otherwise.
Then we have the following.

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

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)≥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+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 get d(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 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 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 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∥:u∈P,u≥1sd(x,y)},
for all x,y∈X. Then one has the following.

D is a b-metric on X.

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).

{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,y∈X, it is obvious that D(x,y)≥0 and D(x,y)=D(y,x).

If x=y, then D(x,y)=inf{∥u∥:u∈P,u≥θ}=0.

If D(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 each x,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 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∥:u∈P,u≥d(x,y)},
for all x,y∈X 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∥: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 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∥: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 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→X 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,y∈X, then
(29)D(Tx,Ty)≤kD(x,y),
for all x,y∈X.

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
(30)D(Tx,Ty)≤λ1D(x,Tx)+λ2D(y,Ty)+λ3D(x,Ty)+λ4D(y,Tx),
for all x,y∈X.

Proof.

(1) For each x,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) Let x,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 Lemma 9(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.

(see [9, Theorem 2.1]) If d(Tx,Ty)≤kd(x,y) for all x,y∈X, 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,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.

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).

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