Int. J. Anal. Appl. (2023), 21:46

Fixed Point Theorems in Cone Metric Spaces via c−Distance Over Topological
Module

Shallu Sharma∗, Pooja Saproo, Iqbal Kour, Naresh Digra

Department of Mathematics, University of Jammu, Jammu, India

∗Corresponding author: shallujamwal09@gmail.com

Abstract. In 2011, Wang and Guo introduced c-distance in cone metric spaces. The idea of cone

metric spaces over topological modules was presented by Branga and Olaru in 2020. Combining these

two ideas, we introduce cone metric spaces with c−distance over topological module and establish a
fixed point theorem.

1. Introduction

Cone metric spaces were first introduced by Huang and Zhang [9]. For detailed study of cone

metric spaces, refer [5,6,12,14–16]. Other authors have also established fixed point theorems in cone

metric spaces (for instance, [1–3, 10, 11]). Wang and Guo [18] presented cone metric spaces with

c-distance and proved some fixed point theorems. Cone metric spaces over topological module were

introduced by Branga and Olaru [4]. In this paper, we present a new concept namely “cone metric

spaces with c-distance over topological module" and prove a fixed point theorem.

2. Preliminaries

Definition 2.1. [8] Let (G, +) be a group with partial order relation ≤ . Then G is said to be a
partially ordered group if translation in G is order preserving:

x ≤ y ⇒ z + x + w ≤ z + y + w,∀ x,y,w,z ∈ G

Definition 2.2. [17] Consider a ring (R, +, .) and 1 be an identity of (R, +, .) such that 1 6= 0 and
≤ is a partial order on R. Then R is called a partially ordered ring if:

Received: Mar. 23, 2023.

2020 Mathematics Subject Classification. 37C25; 47H10; 46H25; 47L07.

Key words and phrases. cone metric spaces with c-distance; topological module; fixed point theorems.

https://doi.org/10.28924/2291-8639-21-2023-46
ISSN: 2291-8639

© 2023 the author(s).

https://doi.org/10.28924/2291-8639-21-2023-46


2 Int. J. Anal. Appl. (2023), 21:46

(1) (R, +, .) is a partially ordered group;

(2) z ≥ 0 and w ≥ 0 implies z.w ≥ 0 for all z,w ∈ R.

Remark 2.1. Throughout the paper, R+ = {r ∈ R : r ≥ 0}, U(R) and U+(R) are the notations for
the positive cone of R, the set of invertible elements of R and U(R) ∩R+, respectively.

Definition 2.3. [20] Consider an abelian group (G, +). Then G is called a topological group if G is

endowed with a topology G such that the conditions mentioned below are satisfied:

(1) For (g1,g2) ∈ G×G, the map (g1,g2) → g1 + g2 is continuous where g1 + g2 ∈ G and G×G
is endowed with the product topology;

(2) For g ∈ G, the map g →−g is continuous, where −g ∈ G.

(G, +,G) or (G,G) is the notation for the topological group.

Definition 2.4. [20] A ring (R, +, .) is called a topological ring if R considered with the topology

R such that (R, +,R) is a topological group and for (r1, r2) ∈ R × R, the map (r1, r2) → r1.r2 is
continuous, where r1.r2 ∈ R and R×R is endowed with the product topology.

(R, +, .,R) is called a Hausdorff topological ring [20] if the topology R is Hausdorff. Also,
(R, +, .,R) or (R,R) is the notation for the topological ring.

Definition 2.5. [20] Consider a topological ring (R,R). A left R−module (E, +, .) is called a
topological R−module if a topology E is defined on E such that (E, +) is a topological abelian group
and the condition mentioned below is satisfied:

For (r,x) ∈ R×E, the map (r,x) → r.x is continuous, where r.x ∈ E.
Further, (E, +, .,E) or (E,E) is the notation used for the topological left R−module.

Definition 2.6. [4] A set P ⊂ E, where (E, +, .,E) is a topological module, is called a cone if :

(1) P is closed, nonempty and P 6= {0E};
(2) s.x + t.y ∈ P, whenever x,y ∈ P and s,t ∈ R+;
(3) x ∈ P and −x ∈ P implies x = 0E.

Remark 2.2. Throughout the paper, P◦ and P̄ are be the notations for the interior and closure of P ,

respectively. Furthermore, the cone P is said to be solid if P◦ is non-empty. Also, a partial ordering ≤P
with respect to P is defined by y −x ∈ P if and only if x ≤P y and x <P y indicates x ≤P y,x 6= y.
Moreover, the partial ordering x � y indicates y −x ∈ P◦.

Next, we shall consider the following hypotheses:

(Hyp I) [4] Consider a Hausdorff topological ring (R, +,�,R) such that:
(1) U+(R) is non-empty.
(2) 0R is a limit point of U+(R).



Int. J. Anal. Appl. (2023), 21:46 3

(3) ≤R indicates the partial ordering on R.
(Hyp II) [4] (E, +, .,E) is a topological left R−module.
(Hyp III) [4] P is a solid cone contained in E.

Proposition 2.1. [4] Consider a topological left R−module (E, +, .,E) and P be a cone contained
in E such that the hypothesis HypI,HypII and HypIII are satisfied. Then:

(1) P◦ + P◦ ⊆ P◦;
(2) β �P◦ ⊆ P◦, where β is an element of U+(R);
(3) If v ≤P u and β ∈ R+, then β �v ≤P β �u;
(4) If y ≤P x and x � z, then y � z;
(5) If y � x and x ≤P z, then y � z;
(6) If y � x and x � w, then y � w;
(7) If 0E ≤P y � v,∀v ∈ P◦, then v = 0;
(8) If 0E � v and {bn} is a sequence in E such that bn → 0E, then there is a natural number m0

such that bn � u for m ≥ m0.

From now onwards, (R, +, .,R) denotes a Hausdorff topological ring.

Definition 2.7. [4] A directed set is a partially ordered set (Λ,≤) such that the condition mentioned
below is fulfilled:

If λ1,λ2 in Λ there is λ3 in Λ so that λ1 ≤ λ3 and λ2 ≤ λ3.

Definition 2.8. [4] A sequence {xλ}λ∈Λ in R is a family of elements in R that is indexed by a directed
set.

Definition 2.9. [4] A family {xλ}λ∈Λ contained in R is convergent to an element x ∈ R if for each
neighborhood W of x there exists λ0 ∈ Λ such that xλ is an element of W for every λ ∈ Λ, where
λ ≥ λ0.

Definition 2.10. [4] A sequence {xλ}λ∈Λ is called a Cauchy sequence if for each neighborhood W of
0R there exists λ0 ∈ Λ such that xλ1 −xλ2 ∈ W for each λ0 ≤ λ1 and λ0 ≤ λ2.

Theorem 2.1. [4] Every convergent sequence {xλ}λ∈Λ in R is a Cauchy sequence.

To exemplify the summability of the family of elements of a topological ring, consider H(Λ) to be
the set of all finite sets contained in Λ directed by the inclusion ⊆ .

Definition 2.11. [4] An element t of R is sum of family {xλ}λ∈Λ contained in R if the sequence
{tI}I∈H(Λ) is convergent to t, where for each I ∈H(Λ),

tI =
∑
λ∈I

xλ.

The family {xλ}λ∈I is said to be summable if {xλ}λ∈I has a sum t in R.



4 Int. J. Anal. Appl. (2023), 21:46

Definition 2.12. [4] A family {xλ}λ∈I contained in R is said to satisfy Cauchy condition if for every
neighborhood W of 0R there exists IW in H(Λ) such that

∑
λ∈J xλ ∈ W, for each J ∈ H(Λ) disjoint

with IW .

Theorem 2.2. [4] A sequence {tI}I∈H(Λ) is a Cauchy sequence if and only if the family {xλ}λ∈Λ
contained in R satisfies Cauchy condition.

Theorem 2.3. [4] Let {xλ}λ∈Λ be a summable family in R. Then for each neighborhood W of 0R,
there exists J in H(Λ) so that xλ ∈ W for each λ ∈ Λ \J.

Definition 2.13. [4] Let (R, +, .,R) be a topological ring. Then R is said to be complete if the
topological additive group of ring (R, +,R) is complete.

Definition 2.14. [4] Let Xbe a non-empty set, (E, +, .,E) be a topological left R−module and the
map d : X ×X → E satisfies:

(1) d(x,y) ≥P 0E ∀x,y ∈ X.
(2) d(x,y) = 0E if and only if x = y ∀x,y ∈ X.
(3) d(x,y) = d(y,x) ∀x,y ∈ X.
(4) d(x,y) ≤P d(x,z) + d(z,y) ∀x,y ∈ X.

Then d is said to be a cone metric on X and the pair (X,d) is said to be a cone metric space over

topological left R−module E.

Definition 2.15. [4] Let (X,d) be a cone metric space over the topological left R−module E, x be
an element of X and {xn} be a sequence in X. Then

(1) {xn} is said to be convergent to x if for each u � 0, there is a natural number N such that
d(xn,x) � u, ∀ n > N.

(2) {xn} contained in X is said to be a Cauchy sequence if for each u � 0, there is a natural
natural number N such that d(xn,xm) � u, ∀ m,n > N.

3. Fixed point theorem via c-distance

In this section we first introduce a new notion namely c−distance in cone metric spaces over
topological module. Next we discuss some results regarding the same. Further, a fixed point theorem

in cone metric spaces via c−distance over topological module has been established.

Definition 3.1. Let (X,d) be a cone metric space over a topological left R−module E. Then a map
p : X ×X → E is said to be a c−distance on X if it satisfies the following conditions:

(p1) p(x,y) ≥P 0E ∀ x,y ∈ X;
(p2) p(x,z) ≤P p(x,y) + p(y,z) ∀ x,y,z ∈ X;
(p3) For every y ∈ X and n ≥ 1, p(y,xn) ≤P u for some u = uy ∈ P, then p(y,x) ≤P u whenever

{xn} in X is convergent to a point x ∈ X;



Int. J. Anal. Appl. (2023), 21:46 5

(p4) For each u � 0, there exists v � 0 such that p(z,x) � v and p(z,y) � v implies that
d(x,y) � u where u,v ∈ E.

Lemma 3.1. Let (X,d) be a cone metric space over a topological left R-module E,p be a c−distance
on X. Consider the sequences {xn} and {yn} in X. Next, let {an} be a sequence in P convergent to
0E and x,y,z ∈ X. Then the following hold:

(i) If p(xn,yn) ≤P an and p(xn,z) ≤P an for every natural number n, then {yn} is convergent to
z.

(ii) If p(xn,y) ≤P an and p(xn,z) ≤P an for every natural number n, then y = z.
(iii) If p(xn,xm) ≤P an for all m > n, where m,n are natural numbers, then {xn} is a Cauchy

sequence in X.

(iv) If p(y,xn) ≤P an, then {xn} is a Cauchy sequence in X.

Proof. (i) Let u � 0. Then there is v � 0 such that p(u
′
,v
′
) � v and p(u

′
,z) � v implies

that d(v
′
,z) � u. Let m0 be any natural number such that an � v and bn � v for each

n ≥ m0. Next for each n ≥ m0,p(xn,yn) ≤P an � v and p(xn,z) ≤P bn � v. Therefore,
d(yn,z) � u. This proves that {yn} is convergent to z.

(ii) Proof clearly follows from (i).

(iii) Let u � 0. Proceeding as proof of (i) let v � 0 and m0 be any natural number. Next, for
n,m ≥ m0 + 1,p(xm0,xn) ≤ um0 � v and p(xm0,xm) ≤ um0 � v. Therefore, d(xn,xm) � v.
This proves that {xn} is a Cauchy sequence.

(iv) Proof clearly follows from (iii).

�

Theorem 3.1. Let (X,d) be a cone metric space over a topological left R−module E and p be a
c−distance on X. Suppose that the hypothesis HypI,HypII and HypIII are satisfied. Define

S = {r ∈ R+|{rn} is a summable family}.

and the map T : X → X satisfies:

p(Tx,Ty) ≤P rp(x,y), ∀ x,y ∈ X.

Then T has a fixed point x∗ in X and for each x ∈ X,{Tnx} is convergent to the fixed point. If
ζ = Tζ, then p(ζ,ζ) = 0. Also, T has a unique fixed point.

Proof. Fix x0 ∈ X. Let x1 = Tx0,x2 = Tx1 = T 2x0, . . . ,xn+1 = Txn = Tn+1x0. Then
p(xn,xn+1) = p(Txn−1,Txn) ≤P rp(xn−1,xn) ≤P r2p(xn−2,xn−1) ≤P . . . ≤P rnp(x0,x1).



6 Int. J. Anal. Appl. (2023), 21:46

On the basis of above inequality, for all q ≥ 1, we have

p(xn,xn+q) ≤P p(xn,xn+1) + . . . + p(xn+q−1,xn+q)

≤P rnp(x0,x1) + . . . + rn+q−1p(x0,x1)

≤P rn(1R + r + . . . + rq−1)p(x0,x1)

≤P rn(
+∞∑
i=0

r i )p(x0,x1).

Using Theorem 2.3, we have rn
R→ 0R as n →∞. Also {rn} is a summable family, right multiplication

is continuous and using Proposition 2.1 (8) we see that for each u � 0, there is a natural number
N such that p(xn,xn+q) � u ∀ n ≥ N and q ≥ 1. Then {xn} is a Cauchy sequence in X. By the
completeness of X, there is x∗ in X such that xn is convergent to x∗ as n tends to ∞. By (p3) we
see that

p(xn,x
∗) � u. (0.3.1)

Also,

p(xn,Tx
∗) = p(Txn−1,Tx

∗)

≤P rp(xn−1,x∗)

≤P rn(
+∞∑
i=0

r i )p(x0,x1)

� u. (0.3.2)

Now, p(xn,x∗) � u and p(xn,Tx∗) � u. Let v � 0. Then by (p4), we have d(x∗,T (x∗)) � v. Next
using Proposition 2.1 (7) we have

x∗ = Tx∗.

Hence x∗ is a fixed point of T. Next suppose that y∗ is a fixed point of T. Then for u ∈ P◦, we
have q(x∗,y∗) = q(Tx∗,Ty∗) ≤P rq(x∗,y∗) ≤P r2q(x∗,y∗) ≤p . . . ≤P rnq(x∗,y∗) � u. Hence
q(x∗,y∗) = 0. Also, q(x∗,x∗) = 0. Using Lemma 3.1 (ii), we have x∗ = y∗. Hence T has a unique

fixed point. �

Corollary 3.1. Let (X,d) be a cone metric space over a topological left R−module E and p be a
c−distance on X. Define

S = {r ∈ R+|{rn} is a summable family}.

and the map T : X → X satisfies:

p(Tnx,Tny) ≤P rp(x,y), ∀ x,y ∈ X.

Then T has a fixed point x∗ in X. If ζ = Tζ, then p(ζ,ζ) = 0.



Int. J. Anal. Appl. (2023), 21:46 7

Proof. Using Theorem 3.1, we see that x∗ is a unique fixed point of Tn. Also, Tn(Tx∗) = T (Tnx∗) =

T (x∗). Therefore, T (x∗) is a fixed point of Tn. So, x∗ = Tx∗. This shows that x∗ is a fixed point of

T. Also, the fixed point of T is also a fixed point of Tn shows that T has a unique fixed point.

Next suppose that ζ = Tζ. We see that the fixed point of T is also a fixed point of Tn. From this,

for u � 0, we have
p(ζ,ζ) = p(Tζ,Tζ) = p(Tnζ,Tnζ) ≤P rp(ζ,ζ) ≤P r2q(ζ,ζ) ≤P . . . ≤P rnp(ζ,ζ) � u.
Hence p(ζ,ζ) = 0. �

Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publi-

cation of this paper.

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	1. Introduction
	2. Preliminaries
	3. Fixed point theorem via c-distance
	References