()


@ Appl. Gen. Topol. 18, no. 2 (2017), 391-400doi:10.4995/agt.2017.7673
c© AGT, UPV, 2017

Generalized c-distance on cone b-metric spaces

endowed with a graph and fixed point results

Kamal Fallahi a,∗, Mujahid Abbas b and Ghasem Soleimani Rad a,c,∗

a
Department of Mathematics, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran.

(fallahi1361@gmail.com, gh.soleimani2008@gmail.com)
b

Department of Mathematics, Government College University Katchery Road, Lahore 54000,

Pakistan & Department of Mathematics, King Abdulaziz University, P. O. Box 80203, Jeddah

21589, Saudi Arabia. (abbas.mujahid@gmail.com)
c

Young Researchers and Elite club, Central Tehran Branch, Islamic Azad University, Tehran,

Iran. (gha.soleimani.sci@iauctb.ac.ir)

Abstract

The aim of this paper is to present fixed point results of contractive

mappings in the framework of cone b-metric spaces endowed with a

graph and associated with a generalized c-distance. Some corollaries

and an example are presented to support the main result proved herein.

Our results unify, extend and generalize various comparable results in

the literature.

2010 MSC: 46A19; 47H10; 05C20.

Keywords: cone b-metric space; generalized c-distance; fixed point; orbitally
G-continuous mapping.

1. Introduction

The interplay between the notion of a nearness among abstract objects of a
set and fixed point theory is very strong and fruitful. This gives rise to an inter-
esting branch of nonlinear functional analysis called metric fixed point theory.
This theory is studied in the framework of a set equipped with some notion of

∗Corresponding author.

Received 12 May 2017 – Accepted 15 July 2017

http://dx.doi.org/10.4995/agt.2017.7673


K. Fallahi, M. Abbas and G. Soleimani Rad

a distance along with appropriate mappings satisfying certain contraction con-
ditions and has many applications in economics, computer science and other
related disciplines. The concept of a b-metric is one of the important measure
of nearness defined by Bakhtin [2] and Boriceanu [5]. The reader interested in
fixed point results in setup of b-metric spaces is referred to ([1, 6, 12, 14]).

Huang and Zhang [15] defined the concept of cone metric space by replacing
the range of a distance function with an ordered normed space equipped with
an order structure induced by a cone and proved some fixed point results for
contraction type mappings on such spaces ([26]). After that, the concept of b-
metric spaces to cone b-metric spaces or cone metric type spaces are introduced
([11, 18]).

Kada et al. [20] introduced the concept of w-distance on metric spaces and
solved non-convex minimization problems. Cho et al. [7] defined the notion
of a c-distance which is the cone version of a w-distance. Recently, Hussain et
al. [17] defined the concept of wt-distance on b-metric spaces and proved some
fixed point theorems under a wt-distance in partially ordered b-metric spaces
(also, see [21]). Bao et al. [3] defined generalized c-distance in cone b-metric
spaces and obtained some fixed point results in ordered cone b-metric spaces.

The aim of this paper is to prove the existence and uniqueness of fixed
points for contractive mappings defined on cone b-metric spaces endowed with
a graph and associated with a generalized c-distance. Our results generalize
and extend various results in the existing literature. It is worth mentioning
that we have employed the weaker version of continuity of the mapping called
orbitally G-continuity.

2. Preliminaries

Let E be a real Banach space. A subset P of E is called a cone if and only
if:

(i) P is nonempty, closed and P 6= {θ} (where θ is the zero element of E);
(ii) a,b ∈ R,a,b ≥ 0 and x,y ∈ P implies that ax + by ∈ P ;
(iii) P ∩ (−P) = {θ}.

Partial ordering on E is defined with help of a cone P as follows: x � y if and
only if y − x ∈ P . We shall write x ≺ y to indicate that x � y but x 6= y and
x ≪ y stands for y − x ∈ intP , where intP denotes the interior of P . Unless
or otherwise stated, it is assumed that E is a Banach space, P is a cone in E
with intP 6= ∅ and � is partial ordering on E induced by P .

A cone P is normal or semi monotone if

inf{‖ x + y ‖: x,y ∈ P and ‖ x ‖=‖ y ‖= 1} > 0

or equivalently, if there is a number K > 0 such that for all x,y ∈ P , θ �
x � y implies that ‖x‖ ≤ K ‖y‖. The least positive number satisfying the
above inequality is called a normal constant of P . If x = (x1, ...,xn)

T ,y =
(y1, ...,yn)

T ∈ Rn, then x � y means that xi ≤ yi, i = 1, ...,n. In this case, the

c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 392



Generalized c-distance in cone b-metric spaces endowed with a graph

set P = {x = (x1, ...,xn)
T ∈ Rn : xi ≥ 0 for i = 1,2, ...,n} is a normal cone

with K = 1.

Lemma 2.1. Let u,c ∈ E and {xn} a sequence in E. Then we have the
following properties:

(p1) If u � λu where u ∈ P and 0 < λ < 1, then u = θ;
(p2) If c ∈ intP, θ � xn and xn → θ, then there exists n0 such that for all

n > n0 we have xn ≪ c.

Definition 2.2 ([11, 18]). Let X be a nonempty set and s ≥ 1 a given real
number. A mapping d : X × X → E is said to be a cone b-metric on X if for
any x,y,z ∈ X, the following conditions hold:

(d1) θ � d(x,y) and d(x,y) = θ if and only if x = y;
(d2) d(x,y) = d(y,x);
(d3) d(x,z) � s(d(x,y) + d(y,z)).

The pair (X,d) is called a cone b-metric space.

Obviously, for s = 1, the cone b-metric space is a cone metric space. More-
over, if X is any nonempty set, E = R and P = [0,∞), then cone b-metric on
X is a b-metric on X.

Definition 2.3 ([3]). Let (X,d) be a cone b-metric space and s ≥ 1 a given
real number. A mapping q : X × X → E is said to be a generalized c-distance
on X if for any x,y,z ∈ X, the following properties are satisfied:

(q1) θ � q(x,y),
(q2) q(x,z) � s[q(x,y) + q(y,z)],
(q3) If for all n ≥ 1, q(x,yn) � u for some u = ux, then q(x,y) � su, where

{yn} is a sequence in X which converges to y ∈ X;
(q4) for any c ∈ intP , there exists e ∈ E with θ ≪ e such that q(z,x) ≪ e

and q(z,y) ≪ e imply that d(x,y) ≪ c.

If (X,d) is a b-metric space, E = R and P = [0,∞). Then, wt-distance [17]
on a b-metric space X is a generalized c-distance. But the converse does not
hold.

Furthermore, if s = 1, the generalized c-distance is a c-distance defined in
[7]. Also, if in the above definition, we take s = 1, E = R and P = [0,∞),
then we obtain the definition of w-distance [20].

Note that, if q is a generalized c-distance, then q(x,y) = θ is not necessarily
equivalent to x = y. Moreover, q(x,y) = q(y,x) does not necessarily hold for
all x,y ∈ X.

Lemma 2.4. Let (X,d) be a cone b-metric space and q a generalized c-distance
on X. Let {xn} and {yn} be sequences in X, {un} and {vn} two sequences in
P converging to θ. For any x,y,z ∈ X, the following conditions hold:

(qp1) if for all n ∈ N, q(xn,y) � un and q(xn,z) � vn, then y = z. In
particular, if q(x,y) = θ and q(x,z) = θ, then y = z;

(qp2) if for all n ∈ N, q(xn,yn) � un and q(xn,z) � vn, then {yn} converges
to z;

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K. Fallahi, M. Abbas and G. Soleimani Rad

(qp3) if for m,n ∈ N, with m > n, we have q(xn,xm) � un , then {xn} is a
Cauchy sequence in X;

(qp4) if for all n ∈ N, q(y,xn) � un then {xn} is a Cauchy sequence in X.

Proof. Following arguments similar to those given in [7], the Lemma follows.
�

On the other hand, the interplay between the order among abstract objects
of underlying mathematical structure and fixed point theory is very strong and
fruitful. This gives rise to an interesting branch of nonlinear functional analysis
called order oriented fixed point theory. This theory is studied in the framework
of a partially ordered sets along with appropriate mappings satisfying certain
order conditions and has many applications in economics, computer science
and other related disciplines.

Existence of fixed points in partially ordered metric spaces was first investi-
gated in 2004 by Ran and Reurings [25], and then by Nieto and Lopez [23].

Jachymski [19] introduced a new approach in metric fixed point theory by
replacing order structure with a graph structure on a metric space. In this
way, the results obtained in ordered metric spaces are generalized. Employing
the notion of orbits, Nicolae et al. [22] obtained some fixed point results for a
new type of contraction mappings and for G-asymptotic contraction mapping
in metric spaces endowed with a graph. Bojor [4] defined the notion of G-
Reich type mappings and obtained a fixed point theorem for such mappings
in metric spaces endowed with a graph. Cholamjiak [8] proved fixed point
theorems for a Banach type contractive mapping on a complete Tvs-cone metric
spaces endowed with a graph. Also, Hussain et al. [16] proved new fixed point
results for graphic weak ψ-contractive mappings. The following definitions and
notations will be needed in the sequel.

Let (X,d) be a cone b-metric space and ∆ denotes the diagonal of X×X. Let
G be a directed graph such that set V (G) of its vertices is X and E(G) be the
set of edges of a graph G which contains all loops; that is, (x,x) ∈ ∆ ⊂ E(G)
for all x ∈ X. Assume further that graph G has no parallel edges.

Thus one can identify the graph G with the ordered pair (V (G),E(G)). If
x,y ∈ X, then a finite sequence {xi}

N
i=0 consisting of N + 1 vertices is called a

path in G from x to y whenever x0 = x, xN = y and (xi−1,xi) is an edge of G
for i = 1, . . . ,N.

The graph G is called connected if there exists a path in G between any

two vertices of G. The symbols G−1 and G̃ denote the graph which is obtained
from G by reversing the directions of its edges and an undirected graph obtained
from G by ignoring the directions of the edges, respectively. In other words,

V (G−1) = V (G̃) = X, E(G−1) =
{
(x,y) : (y,x) ∈ E(G)

}
and E(G̃) =

E(G) ∪ E(G−1).
We denote by Fix(T) the set of all fixed points of a self mapping T on X

and XT the set of all points x ∈ X such that (x,Tx) is an edge of a graph G.
In other words, XT = {x ∈ X : (x,Tx) ∈ E(G)}.

c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 394



Generalized c-distance in cone b-metric spaces endowed with a graph

Following is the analogue of the concept of Picard operators [24] in cone
b-metric spaces.

Definition 2.5. Let (X,d) be a cone b-metric space. A self mapping T on X
is called a Picard operator if T has a unique fixed point x∗ in X and T

nx → x∗
for any x ∈ X.

Consistent with Jachymski [19, Definition 2.4], we introduce the concept of
orbitally G-continuous for self mapping T on a cone b-metric space (see also
[9]).

Definition 2.6. A mapping T : X → X is called orbitally G-continuous on X if
for any x,y ∈ X and a sequence {bn} of positive integers with (T

bnx,T bn+1x) ∈
E(G) for all n ≥ 1, T bnx → y implies that T(T bnx) → Ty.

Note that a continuous mapping on a cone b-metric space is orbitally G-
continuous for all graphs G but the converse is not true in general.

3. Main results

The following is the main result of this paper.

Theorem 3.1. Let (X,d) be a complete cone b-metric space associated with
the generalized c-distance q and endowed with the graph G and s ≥ 1 be a given
real number. Also, let T : X → X be a orbitally G-continuous mapping such
that the following conditions holds.

(T1) T preserves the edges of G; that is, (x,y) ∈ E(G) implies that (Tx,Ty) ∈
E(G) for all x,y ∈ X;

(T2) there exist nonnegative constants α,β,γ such that

q(Tx,Ty) � αq(x,y) + βq(x,Tx) + γq(y,Ty)

for all x,y ∈ X with (x,y) ∈ E(G), where s(α + β) + γ < 1.

Then T has a fixed point if and only if XT 6= ∅. Moreover, for any x∗
in X with Tx∗ = x∗, we have q(x∗,x∗) = θ. Also, if the subgraph of G with
the vertex set Fix(T) is connected, then the restriction of T to XT is a Picard
operator.

Proof. As Fix(T) ⊆ XT , if T has a fixed point then XT is nonempty. Let x0
be a given point in XT . Define a sequence {xn} in X by xn = Txn−1 = T

nx0.

Clearly, (xn,xn+1) ∈ E(G) for all n ∈ N. Thus

q(xn,xn+1) = q(Txn−1,Txn) � αq(xn−1,xn) + βq(xn−1,Txn−1) + γq(xn,Txn)

�
(
α + β

)
q(xn−1,xn) + γq(xn,xn+1),

which implies that

q(xn,xn+1) �
α + β

1 − γ
q(xn−1,xn)

for all n ∈ N. Continuing this way, we have

q(xn,xn+1) � λ
nq(x0,x1)

c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 395



K. Fallahi, M. Abbas and G. Soleimani Rad

for all n ∈ N, where 0 ≤ λ = α+β
1−γ

< 1
s
. Let m > n. It follows from (q2) and

0 ≤ sλ < 1 that

q(xn,xm) � s[q(xn,xn+1) + q(xn+1,xm)]

� sq(xn,xn+1) + s[sq(xn+1,xn+2) + q(xn+2,xm)]

...

� sq(xn,xn+1) + s
2q(xn+1,xn+2) + · · · + s

m−nq(xm−1,xm)]

� (sλn + s2λn+1 · · · + sm−nλm−1)q(x0,x1)

�
sλn

1 − sλ
q(x0,x1) → θ when n → ∞.

By (iii) of Lemma 2.4, {xn} is a Cauchy sequence in X. Next we assume that
there exists a point x∗ ∈ X such that xn = T

nx0 → x∗ as n → ∞. As x0 ∈ XT ,
(T nx0,T

n+1x0) ∈ E(G) for all n ≥ 0. From orbital G-continuity of T , we get
T n+1x0 → Tx∗ and hence Tx∗ = x∗. Now if Tx∗ = x∗ for any x∗ ∈ X. Then,
from (T2) we have

q(x∗,x∗) = q(Tx∗,Tx∗) � αq(x∗,x∗) + βq(x∗,Tx∗) + γq(x∗,Tx∗)

= (α + β + γ)q(x∗,x∗).

Since 0 ≤ α + β + γ < s(α + β) + γ < 1, therefore q(x∗,x∗) = θ.
Now, if the subgraph of G with the vertex set Fix(T) is connected and

x∗∗ ∈ X is a fixed point of T . Then there exists a path {xi}
N
i=0 in G from

x∗ to x∗∗ such that x1, . . . ,xN−1 ∈ Fix(T); that is, x0 = x∗, xN = x∗∗ and
(xi,xi+1) ∈ E(G) for i = 0, . . . ,N −1. By (T2) and q(xi+1,xi+1) = q(xi,xi) =
θ, we have

q(xi,xi+1) = q(Txi,Txi+1)

� αq(xi,xi+1) + βq(xi,Txi) + γq(xi+1,Txi+1)

= αq(xi,xi+1) + βq(xi,xi) + γq(xi+1,xi+1)

= αq(xi,xi+1).

It follows from (p1) that q(xi,xi+1) = θ. Since q(xi,xi) = θ and q(xi,xi+1) = θ,
by Definition 2.3 we have d(xi,xi+1) = 0; that is, xi = xi−1. Consequently,

x∗ = x0 = x1 = · · · = xN−1 = xN = x∗∗

and hence the fixed point of T is unique and the restriction of T to XT is a
Picard operator. �

Example 3.2. Let X = [0,1], E = C1
R
[0,1] with the norm ‖ϕ‖ = ‖ϕ‖∞ +

‖ϕ′‖∞, and P = {ϕ ∈ E : ϕ(t) ≥ 0 on [0,1]} a non-normal cone . Define the
mapping d : X × X → Y by d(x,y) = |x − y|2 · ϕ(t), where ϕ(t) = 2t ∈ P ⊂ E
with t ∈ [0,1]. Then (X,d) is a cone b-metric space with constant s = 2. Let
the mapping q : X × X → E be given by

q(x,y)(t) = y2 · 2t

c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 396



Generalized c-distance in cone b-metric spaces endowed with a graph

where t ∈ [0,1]. Then q is a generalized c-distance. Define T : X → X by

T(x) =

{
1
2
if x = 1,

x
2

4
if x 6= 1.

Clearly, T is not continuous at x = 1. Now assume that X is endowed with
a graph G = (V (G),E(G)), where V (G) = X and E(G) = {(x,x) : x ∈ X}.
Note that for any x,y ∈ X with (x,y) ∈ E(G), we have x = y. If x,y ∈ X and
{bn} is a sequence of positive integers with (T

bnx,T bn+1x) ∈ E(G) for all n ≥ 1
such that T bnx → y, then {T bnx} is necessarily a constant sequence. Thus, for
some y in X, we have T bnx = y for all n ≥ 1 and hence T(T bnx) → Ty. Take
α = 1

4
, β = 1

5
and γ = 0. Then

1) s(α + β) + γ = 2(1
4
+ 1

5
) = 9

10
< 1;

2) Let x ∈ X with (x,x) ∈ E(G). If x 6= 1, then

q(Tx,Tx)(t) = (
x2

4
)2 · 2t

=
x4

16
· 2t ≤ αq(x,x)(t) + βq(x,Tx)(t) + γq(x,Tx)(t).

If x = 1, then we have

q(T1,T1)(t) = (
1

2
)2 · 2t

=
1

4
· 2t ≤ αq(1,1)(t) + βq(1,T1)(t) + γq(1,T1)(t).

3) Also (0,T0) = (0,0) ∈ E(G), so XT 6= ∅.

Thus, all the conditions of Theorem 3.1 are satisfied. Moreover, x∗ = 0 is a
fixed point of T has a fixed point and q(0,0) = 0.

Remark 3.3. (i) Since we need not to the continuity of mapping, the method of
mentioned theorem generalize, extend and unify all of research papers on fixed
point theorems in cone b-metric spaces associated with a generalized c-distance
and cone metric spaces associated with a c-distances such as: Cho et al. [7],
Bao et al. [3] and Hussain et al. [17] (and also, all references contained in them
about w-distance and c-distance).

(ii) In 2012, Ćirić et al. [10] show that the method of Du [13] for contraction
mappings in cone metric spaces cannot be applied for contraction mappings in
cone metric spaces with a associated c-distance. Also, their notes are hold for
generalized c-distance in cone b-metric spaces. Thus, our results are new and
cannot to derived from the version of wt-distance in b-metric spaces.

If a cone b-metric space X is endowed with the complete graph G0 whose
vertex set coincides with X; that is, V (G0) = X and E(G0) = X × X and we
set G = G0 in Theorem 3.1, then the set XT coincides with the whole set X,
where T is a self mapping on X. Thus, we have the following corollary.

Corollary 3.4. Let (X,d) be a complete cone b-metric space with constant
s ≥ 1 associated with the generalized c-distance q and T : X → X a orbitally

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K. Fallahi, M. Abbas and G. Soleimani Rad

continuous mapping. If there exist nonnegative constants α,β,γ such that

q(Tx,Ty) � αq(x,y) + βq(x,Tx) + γq(y,Ty)

for all x,y ∈ X, where s(α + β) + γ < 1. Then T is a Picard operator.

Suppose that (X,⊑) is a partially ordered set (poset). Let G1 be the graph
such V (G1) = X and E(G1) =

{
(x,y) ∈ X × X : x ⊑ y

}
. Since ⊑ is reflexive,

it follows that E(G1) contain all the loops. If we take G = G1 in Theorem 3.1,
then we obtain the following corollary.

Corollary 3.5. Let (X,d,⊑) be a partially ordered complete cone b-metric
space with constant s ≥ 1 associated with the generalized c-distance q and
endowed with the graph G1. Suppose that T : X → X is a nondecreasing
orbitally G1-continuous mapping. If there exist nonnegative constants α,β,γ

such that

q(Tx,Ty) � αq(x,y) + βq(x,Tx) + γq(y,Ty)

for all x,y ∈ X with x ⊑ y, where s(α + β) + γ < 1. Then T has a fixed
point in X if and only if there exists x0 ∈ X such that x0 ⊑ Tx0. Moreover,
if Tx∗ = x∗ for any x∗ ∈ X, then q(x∗,x∗) = θ. Also, if the subgraph of G1
with the vertex set Fix(T) is connected, then the restriction of T to the set of
all points in x ∈ X satisfying x ⊑ Tx is a Picard operator.

Let X be a poset endowed with the graph G2 given by V (G2) = X and

E(G2) =
{
(x,y) ∈ X × X : x ⊑ y ∨ y ⊑ x

}
.

That is, an ordered pair (x,y) ∈ X × X is an edge of G2 if and only if x and y
are comparable elements of (X,⊑). If we set G = G2 in Theorem 3.1, then we
obtain the following corollary.

Corollary 3.6. Let (X,d,⊑) be a partially ordered complete cone b-metric
space with s ≥ 1 associated with the generalized c-distance q and endowed with
the graph G2. Suppose that T : X → X is a nondecreasing orbitally G2-
continuous mapping which maps comparable elements of X onto comparable

elements. If there exist nonnegative constants α,β,γ such that

q(Tx,Ty) � αq(x,y) + βq(x,Tx) + γq(y,Ty)

for all x,y ∈ X, where x and y are comparable and s(α+β)+γ < 1. Then T has
a fixed point in X if and only if there exists x0 ∈ X such that x0 and Tx0 are
comparable. Moreover, Tx∗ = x∗ for any x∗ in X implies that q(x∗,x∗) = θ.
Also, if every two elements of Fix(T) are comparable, then the restriction of T
to the set of all x ∈ X such x and Tx are comparable is a Picard operator.

Let e ∈ int P with θ ≪ e be a fixed. Recall that two elements x,y ∈ X are
said to be e-closed if d(x,y) ≺ e. Define the e-graph G3 by

V (G3) = X and E(G3) =
{
(x,y) ∈ X × X : d(x,y) ≺ e

}
.

Note that E(G3) contains all loops. Finally, if we set G = G3 in Theorem 3.1,
then we obtain the following result.

c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 398



Generalized c-distance in cone b-metric spaces endowed with a graph

Corollary 3.7. Let (X,d) be a complete cone b-metric space with s ≥ 1 as-
sociated with the generalized c-distance q endowed with the graph G3. Suppose

that T : X → X is an orbitally G3-continuous mapping which maps e-close el-
ements of X onto e-close elements. If there exist nonnegative constants α,β,γ

such that

q(Tx,Ty) � αq(x,y) + βq(x,Tx) + γq(y,Ty)

for all x,y ∈ X, where x and y are e-close elements and s(α+β)+γ < 1. Then
T has a fixed point in X if and only if there exists x0 ∈ X such that x0 and
Tx0 are e-close. Moreover, if Tx∗ = x∗ for any x∗ ∈ X, then q(x∗,x∗) = θ.
Also, if every two elements of Fix(T) are ε-close, then the restriction of T to
the set of all x ∈ X such x and Tx are e-close is a Picard operator.

Note that our results can be easily proved for α,β,γ : X → [0,1) be func-
tions with suitable conditions instead to be constants and for other well-known
contractive conditions. Also, as a new work, it will be interesting to study
common fixed point results for two or more than two mappings with respect
to the generalized c-distance on cone b-metric spaces endowed with the graph
G by considering functions α,β,γ : X → [0,1) with suitable conditions.

Acknowledgements. The authors are grateful to the Editorial Team and

referees for their accurate reading and their helpful suggestions. Also, The

first and the third authors are thankful to the Department of Mathematics of

Payame Noor University for financial support.

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