EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 14, No. 4, 2021, 1148-1160
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Some aspects of b(αn,βn)-hypermetric spaces over Banach
algebras

Akbar Dehghan Nezhad1,∗, Stojan Radenović2

1 School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran
2 Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120
Belgrad 35, Serbia

Abstract. In this paper, we give a definition of a b(αn,βn)-hypermetric spaces over Banach alge-
bras. The purpose of this paper is to prove the concept of extension of fixed point theorems in
b(αn,βn)-hypermetric spaces over Banach algebras.

2020 Mathematics Subject Classifications: 54H25, 47H10, 46B20

Key Words and Phrases: b(αn,βn)-hypermetric spaces, bn-metric space, fixed point.

1. Introduction and preliminaries

Bakhtin (1989), Bourbaki (1974), Czerwik (1993) and Heinonen (2001) generalized
the structure of metric space by weakening the triangle inequality and called it the b-
metric space. In 2017, Kamran et al. [8], introduced the concept of extended b-metric
space by further weakening the triangle inequality. The main purpose of this paper is a
generalization of cone n-metric spaces into b(αn,βn)-hypermetric spaces.

In this section, we recall some definitions, notations and terminologies which will be
used to prove the main results. When good references are available we may not include
the details of all the introduction and proofs (for example, [12] , [11] , [9] , [5] , [13] , [10]
, [1]).

Definition 1. [14] A vector space A over a field K (R or C) is said to be an algebra if it
is closed under multiplication (i.e., for all a, b ∈ A, ab ∈ A) and

(i1) (ab)c = a(bc) for all a, b, c ∈ A,

(i2) a(b+ c) = ab+ ac and (a+ b)c = ab+ bc for all a, b, c ∈ A,

(i3) k(ab) = (ka)b = a(kb) for all a, b ∈ A, for all k ∈ K.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v14i4.4136

Email addresses: dehghannezhad@iust.ac.ir (Akbar Dehghan Nezhad), radens@beotel.net (Stojan
Radenović)

http://www.ejpam.com 1148 © 2021 EJPAM All rights reserved.



A. D.Nezhad, S. Radenović / Eur. J. Pure Appl. Math, 14 (4) (2021), 1148-1160 1149

A Banach space A over a field K (R or C) is said to be a Banach algebra if

(i4) A is an algebra and for all a, b, c ∈ A,

(i5) ∥ ab ∥≤∥ a ∥ · ∥ b ∥ for all a, b ∈ A.

Here we shall always assume that the Banach algebra A is unital, that is it has a
unity element eA such that eAa = aeA = a, for all a ∈ A. Note that the unity element
of a Banach algebra A , if it exists, is unique. A non-zero element b ∈ A is said to be
invertible if its inverse exists i.e. if there exists a non-zero element b−1 ∈ A such that
bb−1 = b−1b = eA, we call b−1 is the inverse of b. One can show that in a Banach algebra
A , with the unity element eA the inverse of an element is unique. Also for all a, b ∈ A ,
we have (ab)−1 = b−1a−1 and (a−1)−1 = a.

Definition 2. [6] A subset P of a unital Banach algebra A is called

(p1) P is non empty, 0A, eA ∈ P, where 0A is the zero element of A.

(p2) If a, b ∈ P and r, s ≥ 0, then ra+ sb ∈ P.

(p3) a, b ∈ P implies ab ∈ K.

(p4) If a,−a ∈ K for some a ∈ A then a = 0A, where 0A is the zero element of A.

A cone P is called a solid cone if int(P) ̸= 0. Each cone P induces a partial ordering
⪯ on A by a ⪯ b if and only if a − b ∈ P. We write a ≺ b if a ⪯ b and a ̸= b. When the
cone is solid a≪ b will stand for a− b ∈ int(P). The cone P is said to be normal if there
exists a number L > 0 such that 0A ⪯ a ⪯ implies ∥ a ∥≤ L ∥ b ∥. The least positive
number L, which satisfies the normality condition is called the normal constant of P.

Remark 1. An ordered ring is a (usually commutative) ring R with a total order ⪯ such
that for all a, b, and c in R:
i) if a ⪯ b, then a+ c ⪯ b+ c
ii) if 0 ⪯ a and 0 ⪯ b, then 0 ⪯ a · b.
We denote R+ a set of non-negative elements of R namely R+ = {g ∈ R : 0 ⪯ g}.

Definition 3. [7] Let X be a non-empty set and A a Banach algebra. A mapping dc :
X ×X −→ A is called a cone metric if it satisfies the following conditions:

(b1) 0A ⪯ dc(x, y), for all x, y ∈ X, dc(x, y) = 0A if and only if x = y,

(b2) dc(x, y) = dc(y, x), for all x, y ∈ X,

(b3) dc(x, y) ⪯ dc(x, z) + dc(z, y) for all x, y, z ∈ X.

In this case, the pair (X, db) is called a cone metric space over Banach algebra.

The concept of a b-metric space is initiated by Bakhtin [2] and thereafter used by
Czerwick [4].



A. D.Nezhad, S. Radenović / Eur. J. Pure Appl. Math, 14 (4) (2021), 1148-1160 1150

Definition 4. [4] Let X be a non-empty set and db : X × X −→ [0,+∞) be a function
satisfying the following conditions:

(b1) db(x, y) = 0 if and only if x = y,

(b2) db(x, y) = db(y, x), for all x, y ∈ X,

(b3) db(x, y) ≤ s(db(x, z) + db(z, y)) for all x, y, z ∈ X, where s ≥ 1.

The function db is called a b-metric and the pair (X, db) is called a b-metric space.

Example 1. [3] Let X = lp[0, 1] be the space of all real functions ϕ(t) with t ∈ [0, 1] such

that
∫ 1
0 |ϕ(t)|p <∞ with 0 < p < 1. Define db : X ×X −→ [0,+∞) as:

db(ϕ, ψ) =)

∫ 1

0
|ϕ(t)− ψ(t)|pdt)

1
p .

Therefore (X, db) is a b-metric space with s = 2
1
p .

Remark 2. [4] The class of b-metric space is larger than the class of metric space. When
s = 1 the concept of b-metric space coincides with the concept of metric space.

In the following we recall the definition of the extended b-metric space.

Definition 5. [8] Let X be a non-empty set and α : X × X −→ [1,+∞). A function
dα : X ×X −→ [0,+∞) is called an extended b-metric if for all x, y, z ∈ X it satisfies the
following conditions:

(b1) dα(x, y) = 0 if and only if x = y,

(b2) dα(x, y) = dα(y, x),

(b3) dα(x, y) ≤ α(x, y)(dα(x, z) + dα(z, y)).

The pair (X, dα) is called extended b-metric space.

For simplicity of notation, R,N denotes the set of real numbers and natural numbers
respectively. R>0 stands for positive reals. Here and subsequently, for n ≥ 2, let Xn de-
notes the n-times Cartesian product X × . . .×X︸ ︷︷ ︸

n−times

. In what follows int(K) and ∂K denote,

respectively, the interior and boundary of K. To simplify, we let (xi)
n
i=1 and (x)n1 stand

for (x1, ..., xn) and (x)ni=1 respectively. Let T be a mapping, for abbreviation, we write Tx
instead of T (x).

2. Main Results

The goal of this section is to describe a few properties and results of the b(αn,βn)-
hypermetric spaces of dimension n.



A. D.Nezhad, S. Radenović / Eur. J. Pure Appl. Math, 14 (4) (2021), 1148-1160 1151

2.1. b(αn,βn)-hypermetric spaces of dimension n

In this section, we will present some fixed point theorems in set-valued metric spaces
over Banach algebra A. Furthermore, we will give examples and application to our main
results. The first result in this work is the following definition.

For n ≥ 2, let Xn denotes the n-times Cartesian product X × . . .×X︸ ︷︷ ︸
n−times

and A be a

Banach algebras. Let P ∗(A) denote the family of all non-empty subsets of A. We begin
with the following definition.

Definition 6. Let X be a non-empty set and αn, βn : Xn −→ A. Let Γ(αn,βn) : X
n −→

P ∗(A) be a mapping (called the b(αn,βn)-hypermetric over Banach algebra A ) satisfiying
for all n-tuples (xi)

n
i=1 X

n in the following conditions:

(G0) 0A ⪯ Γ(αn,βn)(xi)
n
i=1,

(G1) Γ(αn,βn)(xi)
n
i=1 = {0A}, if x1 = . . . = xn,

(G2) Γ(αn,βn)(xi)
n
i=1 ⊇ {0A}, for all x1, ..., xn with xi ̸= xj, for some i, j ∈ {1, ..., n},

(G3) Γ(αn,βn)(xi)
n
i=1 = Γ(αn,βn)(xπi)

n
i=1, for every permutation (π(1), ..., π(n)) of (1, 2, ..., n),

(G4) Γ(αn,βn)((xi)
n−1
i=1 , xn−1) ⊆ Γ(αn,βn)(xi)

n
i=1, for all x1, . . . , xn ∈ X,

(G5) Γ(αn,βn)(xi)
n
i=1 ⊆ αn(xi)

n
i=1 · Γ(αn,βn)(x1, (a)

n
2 ) + βn(xi)

n
i=1 · Γ(αn,βn)(a, (xi)

n
i=2), for

all x1, . . . , xn, a ∈ X.

We denote A+ a set of non-negative elements of A namely A+ = {a ∈ A : 0A ⪯ a}.
Let Ai subsets of X, (i = 1, . . . , n), for any B,B′ ∈ P ∗(A+) and α ∈ A+. We define

Γ(αn,βn)(Ai)
n
i=1 =

⋃{
Γn(xi)

n
i=1 | xi ∈ Ai, i = 1, . . . , n

}
,

B +B′ = {b+ b′ | ∈ B, b′ ∈ B′} and α ·B = {α · b | b ∈ B,α ∈ A+}.

We shall use the following abbreviated notation: The function Γn is called a ordered
b(αn,βn)-hypermetric over Banach algebra A of dimension n, or more specifically a b(αn,βn)-
hypermetric onX over Banach algebraA. The pair (X,Γn) is called an b(αn,βn)-hypermetric
space over Banach algebra A.
For example, we can place A+ = Z0

+ or R0
+, where Z0

+ := N ∪ {0} = {0, 1, 2, . . . } and

R0
+ := [0,+∞). Here, for simplicity we assume that A+ = R0

+. The following useful

properties of a bn-hypermetric are easily derived from the axioms.

Remark 3. If αn(xi)
n
i=1 = βn(xi)

n
i=1 = c for c ≥ 1 and n = 1, then we obtain the

definition of b-metric space (Czerwik [4]). It is clear that for c = 1, this b-metric becomes
a usual metric.



A. D.Nezhad, S. Radenović / Eur. J. Pure Appl. Math, 14 (4) (2021), 1148-1160 1152

Proposition 1. (Example) We assume that A+ = R0
+. Let X = [0, 1] and α2, β2 :

X ×X −→ [1,+∞), with α2(x, y) = 1 + 1
x+y , β2(x, y) = 1 + 2

x+y . Define

Ωα2,β2 : X ×X → P ∗(R0
+)

with,

Ω(α2,β2)(x, y) =


[1, 1

xy ) ; x, y ∈ (0, 1], x ̸= y{
0
}

; x, y ∈ [0, 1], x = y
Ω(α2,β2)(y, x) = [1, 1x) ; y = 0, x ∈ (0, 1]

(1)

and also assume A+B = A ∪B, for all A,B ∈ P ∗(R0
+). Then (X,Ω(α2,β2)) is a b(α2,β2)-

hypermetric space.

Proof. It is sufficient to show that Ω(α2,β2) is satisfied in all properties [(G0)], [(G1)],
[(G2)], . . . , [(G5)] . The proofs of [(G0)], [(G1)], . . . , [(G4)], are immediate from the defini-
tion of Ω(α2,β2). We only need to show that Ω(α2,β2) is satisfied in

Ω(α2,β2)(x, y) ⊆ α2(x, y).Ω(α2,β2)(x, z) + β2(x, y).Ω(α2,β2)(z, y), for all x, y, z ∈ X.

We distinguish the following cases:

(i) Let x, y ∈ (0, 1] For z ∈ (0, 1], we have
Ω(α2,β2)(x, y) ⊆ α2(x, y).Ω(α2,β2)(x, z) + β2(x, y).Ω(α2,β2)(z, y) if and only if

[1, 1
xy ) ⊆ (1+ 1

x+y )[0,
1
xz )+(1+ 2

x+y )[0,
1
zy ) if and only if [1, 1

xy ) ⊆ (1+ 2
x+y )([0,

1
xz )+

[0, 1
zy ))

if and only if [1, 1
xy ) ⊆ (x+y+2

x+y )[0, x+y
xyz ) if and only if z ≤ 2 + x+ y.

If z = 0, then Ω(α2,β2)(x, y) ⊆ α2(x, y).Ω(α2,β2)(x, 0) + β2(x, y).Ω(α2,β2)(0, y)

if and only if [1, 1
xy ) ⊆ (1 + 1

x+y )[0,
1
x) + (1 + 2

x+y )[0,
1
y ) if and only if [1, 1

xy ) ⊆ (1 +
2

x+y )([0,
1
x)+[0, 1y )) if and only if [1, 1

xy ) ⊆ (x+y+2
x+y )[0, x+y

xy ) if and only if 2 ≤ 2+x+y.

(ii) For x ∈ (0, 1] and y = 0, let z ∈ (0, 1],
Ω(α2,β2)(x, 0) ⊆ α2(x, 0).Ω(α2,β2)(x, z) + β2(x, 0).Ω(α2,β2)(z, 0) if and only if

[1, 1x) ⊆ (1+x
x )[0, 1

xz ) + (2+x
x )[0, 1z ) if and only if [1, 1x) ⊆ (2+x

x )([0, 1
xz ) + [0, 1z )) if and

only if [1, 1x) ⊆ (x+2
x )[0, x+1

xz ) if and only if xz ≤ (x+ 1)(x+ 2).

(iii) Let x, y ∈ [0, 1], x = y. Obviously, Ω(α2,β2) is satisfied in the (G5).

Hence (X,Ω(α2,β2)) is a b(α2,β2)-hypermetric space.

Proposition 2. Let (X,Γ(αn,βn)) be a b(αn,βn)-hypermetric space over Banach algebra A.
Then for any x1, ..., xn, a ∈ X it follows that:
(1) If Γ(αn,βn)(xi)

n
i=1 = {0A}, then x1 = ... = xn,

(2) Γ(αn,βn)(xi)
n
i=1 ⊆

∑n
j=2 Γ(αn,βn)((x1)

n−1
1 , xj),

(3) Γ(αn,βn)(xi)
n
i=1 ⊆

∑n
j=1 Γ(αn,βn)(xj , (a)

n
2 ),

(4) Γ(αn,βn)(x1, (x2)
n
2 ) ⊆ (n− 1)Γ(αn,βn)((x1)

n−2
1 , x2).



A. D.Nezhad, S. Radenović / Eur. J. Pure Appl. Math, 14 (4) (2021), 1148-1160 1153

Proposition 3. Let (X,Γ(αn,βn)) be a b(αn,βn)-hypermetric space over Banach algebra A.
Then {0A} ⊆ Γ(αn,βn)(xi)

n
i=1 for every x1, ..., xn ∈ X.

Proof. By the condition (G4) of definition of b(αn,βn)-hypermetric space, we have

{0A} = Γ(αn,βn)(x1)
n
1 ⊆ Γ(αn,βn)(xi)

n
i=1.

Proposition 4. Every b(αn,βn)-hypermetric space (X,Γ(αn,βn)) over Banach algebra A
defines a b(α2,β2)-hypermetric space (X,Γ(α2,β2)) over Banach algebra A as follows:

Γ(α2,β2)(x, y) = Γ(αn,βn)(x, (y)
n
2 ) + Γ(αn,βn)(y, (x)

n
2 ), for all x, y ∈ X,

where α2(x, y) = max{αn(x, (y)
n
2 ), αn(y, (x)

n
2 )} and β2(x, y) = max{βn(x, (y)n2 ), βn(y, (x)n2 )}.

Proof. Note that [(G0)], . . . , [(G4)] trivially hold. We only need to show that Γ(α2,β2)

is satisfied in

Γ(α2,β2)(x, y) ⊆ α2(x, y) · Γ(α2,β2)(x, z) + β2(x, y) · Γ(α2,β2)(z, y), for all x, y, z ∈ X.

The proof is straightforward, by setting α2(x, y) = max{αn(x, (y)
n
2 ), αn(y, (x)

n
2 )} and β2(x, y) =

max{βn(x, (y)n2 ), βn(y, (x)n2 )} and the condition (G5) of definition of b(αn,βn)-hypermetric
space over Banach algebra A.

Proposition 5. Let e be an arbitrary positive real value number, and (X, d) be a metric
space. We define an induced b(α2,β2)-hypermetric over Banach algebra R.

Γe
(α2,β2)

: X ×X → P ∗(R0
+) (2)

Γe
(α2,β2)

(x, y) =


(
d(x, y)− e, d(x, y) + e

)
∪ {0} ; x ̸= y, d(x, y) > e(

d(x, y)− e, d(x, y) + e
)
∩ R0

+ ; x ̸= y, d(x, y) < e
{0} ; x = y ot d(x, y) = e.

(3)
Then (X,Γe

(α2,β2)
) is a b(α2,β2)-hypermetric space over Banach algebra R.

2.2. Quotient b(αn,βn)-hypermetric space over Banach algebra A

Let (X,Γ(αn,βn)) be a b(αn,βn)-hypermetric space over Banach algebra A and X̃ be a

partition of X. For each point p ∈ X, we denote p̃ a point in X̃ containing p, and we
denote the equivalent relation induced by the relation by ∼.

Definition 7. Let (X,Γ(αn,βn)) be a b(αn,βn)-hypermetric space over Banach algebra A.

Let p1, . . . , pn ∈ X, and consider p̃1, . . . , p̃n ∈ X̃. A quotient b(αn,βn)-hypermetric of points

of X̃ induced by Γ(αn,βn) is the function

Γ̃(αn,βn) : X̃
n −→ P ∗(A+) given by Γ̃(αn,βn)(p̃i)

n
i=1 =

⋂
pi∈p̃i Γ(αn,βn)(pi)

n
i=1.



A. D.Nezhad, S. Radenović / Eur. J. Pure Appl. Math, 14 (4) (2021), 1148-1160 1154

Proposition 6. The quotient b(αn,βn)-hypermetric over Banach algebra A induced by

Γ(αn,βn) is well-defined and is a b(αn,βn)-hypermetric on X̃ over Banach algebra A.

Proof. Γ̃(αn,βn) is satisfied in all properties (G0), till (G4).

Γ̃(αn,βn)(p̃i)
n
i=1 ⊆ Γ̃(αn,βn)(p̃1, (q̃)

n
2 ) + Γ(αn,βn)(q̃, (p̃i)

n
i=2) (4)⋂

pi∈P̃i

Γ(αn,βn)(pi)
n
i=1 ⊆

⋂
pi∈P̃i

q∈q̃

(
Γ(αn,βn)(p1, (q)

n
2 ) + Γ(αn,βn)(q, (pi)

n
i=2)

)

⋂
pi∈P̃i

q∈q̃

Γ(αn,βn)(p1, (q)
n
2 ) +

⋂
pi∈P̃i

q∈q̃

Γ(αn,βn)(q, (pi)
n
i=2)

=
⋂

pi∈P̃i

q∈q̃

(
Γ(αn,βn)(p1, (q)

n
2 ) + Γ(αn,βn)(q, (pi)

n
i=2)

)
(5)

Let (X,Γ(αn,βn)) be a b(αn,βn)-hypermetric space of dimension n > 2 over Banach
algebra A. For any arbitrary a in X, define the function Γ(αn−1,βn−1) on Xn−1 by

Γ(αn−1,βn−1)(xi)
n−1
i=1 := Γ(αn,βn)((xi)

n−1
i=1 , a).

Then we have the following result.

Proposition 7. The function Γ(αn−1,βn−1) define a b(αn−1,βn−1)-hypermetric on X over
Banach algebra A.

Proof. We will verify that Γ(αn−1,βn−1) satisfies the five properties of a b(αn−1,βn−1)-
hypermetric over Banach algebra A.

Proposition 8. Let Π : X → Y be an injection from a set X to a set Y . If Γ(αn,βn) :
Y n → P ∗(A+) is a b(αn,βn)-hypermetric on the set Y over Banach algebra A. Then

Γ(αn,βn) : Xn → P ∗(A+), given by the formula Γ(αn,βn)(xi)
n
i=1 = Γ(αn,βn)(Πi)

n
i=1 for all

x1, . . . , xn ∈ X, is a b(αn,βn)-hypermetric on the set X over Banach algebra A.

Proposition 9. Let (X,Γ(αn,βn)) be any b(αn,βn)-hypermetric space over Banach algebra A
and λ ∈ R0

+. Then (X,Γλ
(αn,βn)

) is also a b(αn,βn)-hypermetric space over Banach algebra

A where Γλ
(αn,βn)

(xi)
n
i=1 := {A ∩ {a ∈ A|0A ⪯ a ≺ λ}|A ∈ Γ(αn,βn)(xi)

n
i=1}.

So, on the same X many b(αn,βn)-hypermetric over Banach algebra A can be defined,
as a result of which the same set X is endowed with different metric structures. Another
structure in the next proposition is useful for scaling the b(αn,βn)-hypermetric over Banach
algebra A, so we need the following explanation.

For any non-empty subset B of A+, and λ ∈ A+ we define a set λ · B to be λ · B :={
λ · b | b ∈ B

}
.



A. D.Nezhad, S. Radenović / Eur. J. Pure Appl. Math, 14 (4) (2021), 1148-1160 1155

Proposition 10. Let (X,Γ(αn,βn)) be any b(αn,βn)-hypermetric space over Banach algebra

A. Let Λ be any positive real number. We define Γ̇Λ
(αn,βn)

(xi)
n
i=1 = λ · Γ(αn,βn)(xi)

n
i=1.

Then (X, Γ̇λ
(αn,βn)

)is also a b(αn,βn)-hypermetric space over Banach algebra A.

A sequence {xm} in a b(αn,βn)-hypermetric space (X,Γ(αn,βn)) over Banach algebra A
is said to converge to a point s in X, if for any ϵ ≻ 0A there exists a natural number N
such that for every m1, . . . ,mn−1 ≥ N

Γ(αn,βn)((xmi)
m−1
i=1 , s) ⊆ {a ∈ A|0A ⪯ a ≺ ϵ},

then we shall write

lim
m1,...,mn−1−→+∞

Γ(αn,βn)((xmi)
m−1
i=1 , s) = {0A}.

We shall say that a sequence {xm} has a cluster point x if there exists a subsequence
{xmk

} of {xm} that converges to x.

Proposition 11. Let (X,Γ(αn,βn)) and (X ′,Γ
′

(αn,βn)
) be two b(αn,βn)-hypermetric spaces

over Banach algebra A. Then a function T : X → X ′ is b(αn,βn)-continuous at a point
x ∈ X, if and only if it is b(αn,βn)-sequentially continuous at x; that is, whenever sequence
{xm} is b(αn,βn)-convergent to x one has {T (xm)} is U(αn,βn)-convergent to T (x).

Definition 8. Let (X,Γ(αn,βn)) be a b(αn,βn)-hypermetric spaceover Banach algebra A,
and A ⊆ X. The set A is b(αn,βn)-compact if for every b(αn,βn)-sequence {xm} in A, there
exists a subsequence {xmk

} of {xm} such that b(αn,βn)-convergences to some x0 ∈ A.

Proposition 12. Let (X,Γ(αn,βn)) and (X ′,Γ
′

(αn,βn)
) be two b(αn,βn)-hypermetric spaces

over Banach algebra A and T : X → X ′ a b(αn,βn)-continuous function on X. If X is
b(αn,βn)-compact, then T (X) is b(αn,βn)-compact.

Definition 9. Let (X,Γ(αn,βn)) be a b(αn,βn)-hypermetric space over Banach algebra A.
Then for x0 ∈ X, r ≻ 0A, the b(αn,βn)-hyperball with centre x0 and radius r is

BΓ(αn,βn)
(x0, r) = {y ∈ X : Γ(αn,βn)(x0, (y)

n
2 ) ⊆ {a ∈ A|0A ⪯ a ≺ r}}.

Proposition 13. Let (X,Γ(αn,βn)) be a b(αn,βn)-hypermetric space over Banach algebra
A. Then for x0 ∈ X, r ≻ 0A,
(i) If Γ(αn,βn)(x0, (xi)

n
i=2) ⊆ {a ∈ A|0A ⪯ a ≺ r}, then x2, ..., xn ∈ BΓ(αn,βn)

(x0, r),
(ii) If y ∈ BΓ(αn,βn)

(x0, r), then there exists, δ ≻ 0A such that BΓ(αn,βn)
(y, δ) ⊆ BΓ(αn,βn)

(x0, r).

Proposition 14. The set of all Γ(αn,βn)-balls, Bn = {BΓ(αn,βn)
(x, r) : x ∈ X, r > 0},

forms a basis for a topology T (Γ(αn,βn)) on X.

Definition 10. Let (X,Γ(αn,βn)) be a b(αn,βn)-hypermetric space over Banach algebra A.
The sequence {xn} ⊆ X is b(αn,βn)-convergent to x if it b(αn,βn)-converges to x in the
b(αn,βn)-hypermetric topology over Banach algebra A, T (Γ(αn,βn)).



A. D.Nezhad, S. Radenović / Eur. J. Pure Appl. Math, 14 (4) (2021), 1148-1160 1156

Proposition 15. Let (X,Γ(αn,βn)) be a b(αn,βn)-hypermetric space over Banach algebra
A. Then for a sequence {xm} ⊆ X, and a point x ∈ X the following are equivalent:
(1) {xm} is Γ(αn,βn)-convergent to x,

(2) Γ(αn,βn)((xm)n−1
1 , x) → 0,

(3) Γ(αn,βn)(xm, (x)
n
2 ) → 0.

Definition 11. Let (X,Γ(αn,βn)), (Y,Γ
′

(αm,βm)) be universal hypermetric spaces of dimen-
sion n, m respectively over Banach algebra A. A function T : X −→ Y is b(αn,βn),(αm,βm)-
continuous at point x0 ∈ X, if T−1(B

Γ
′
(αm,βm)

(T (x0), r)) ∈ T (Un), for all r > 0.

We say f is b(αn,βn),(αm,βm)-continuous if it is b(αn,βn),(αm,βm)-continuous at all points
of X; that is, continuous as a function from X with the T (Γ(αn,βn))-topology to Y with

the T (Γ
′

(αm,βm))-topology.

In the sequel, for simplicity we have assume that n = m. Since b(αn,βn)-hypermetric
topologies are metric topologies we have:

Definition 12. Let (X,Γ(αn,βn)) and (Y,Γ
′

(αn,βn)
) be two b(αn,βn)-hypermetric spaces over

Banach algebra A and T : (X,Γ(αn,βn)) → (Y,Γ
′

(αn,βn)
) be a function. The function f is

called b(αn,βn)-continuous at a point a ∈ X if and only if, for given ϵ ≻ 0A, there exists

δ ≻ 0A such that x1, . . . , xn−1 ∈ X and the subset relation Γ(αn,βn)(a, (xi)
n−1
i=1 ) ⊆ {a ∈

A|0A ⪯ a ≺ δ} implies that Γ
′

(αn,βn)
(T (a), (T (xi))

n−1
i=1 ) ⊆ {a ∈ A|0A ⪯ a ≺ ϵ}.

A function f is b(αn,βn)-continuous on X if and only if it is b(αn,βn)-continuous at all
a ∈ X

Proposition 16. Let (X,Γ(αn,βn)), (Y,Γ
′

(αn,βn)
) be b(αn,βn)-hypermetric spaces over Ba-

nach algebra A. A function T : X −→ Y is b(αn,βn)-continuous at point x ∈ X if and only
if it is b(αn,βn)-sequentially continuous at x; that is, whenever {xn} is b(αn,βn)-convergent
to x we have (T (xn)) is b(αn,βn)-convergent to T (x).

Proposition 17. Let (X,Γ(αn,βn)) be a b(αn,βn)-hypermetric space over Banach algebra
A. Then the function Γ(αn,βn)(zi)

n
i=1 is jointly b(αn,βn)-continuous in all n of its variables.

Definition 13. A map T : X −→ Y between b(αn,βn)-hypermetric spaces (X,Γ(αn,βn))

and (Y,Γ
′

(αn,βn)
) over Banach algebra A, is an iso-hypermetry when Γ(αn,βn)(xi)

n
i=1 =

Γ
′

(αn,βn)
(T (xi))

n
i=1 for all x1, . . . , xn ∈ X. If the iso-b(αn,βn)-hypermetry is injective, we

call it iso-b(αn,βn)-hypermetric embedding over Banach algebra A. A bijective iso-b(αn,βn)-
hypermetry is called a b(αn,βn)-hypermetric isomorphism over Banach algebra A.

2.3. Fixed Point Theorem in b(αn,βn)-hypermetric spaces over Banach al-
gebra A

In a b(αn,βn)-hypermetric space over Banach algebra A, the concepts of basic topolog-
ical notions, such as: b(αn,βn)-Cauchy sequence, b(αn,βn)-convergent sequence and b(αn,βn)-
complete b(αn,βn)-hypermetric space over Banach algebraA can be easily adopted as under.



A. D.Nezhad, S. Radenović / Eur. J. Pure Appl. Math, 14 (4) (2021), 1148-1160 1157

We discuss about concept b(αn,βn)-completeness of b(αn,βn)-hypermetric spaces over Banach
algebra A.

Definition 14. Let (X,Γ(αn,βn)) be a b(αn,βn)-hypermetric space over Banach algebra A.
Then a sequence {xm} ⊆ X is called b(αn,βn)-Cauchy if for every ε ≻ 0A, there exists
N ∈ N such that Γ(αn,βn)(xmi)

n
i=1 ≺ ε for all m1,m2, ...,mn ≥ N .

The next proposition follow directly from the definitions.

Proposition 18. In a b(αn,βn)-hypermetric space, (X,Γ(αn,βn)) over Banach algebra A,
the following are equivalent.
(i) The sequence {xm} is b(αn,βn)-Cauchy.
(ii) For every ε ≻ 0A, there exists N ∈ N such that Γ(αn,βn)(xl, (xm)n2 ) ≺ ε, for every
l,m ≥ N .
(iii) {xm} is a Cauchy sequence in the metric space (X, dΓ(αn,βn)

).

Corollary 1. (i) Every b(αn,βn)-convergent sequence in a b(αn,βn)-hypermetric space over
Banach algebra A is b(αn,βn)-Cauchy.
(ii) If a b(αn,βn)-Cauchy sequence in a b(αn,βn)-hypermetric space (X,Γ(αn,βn)) over Banach
algebra A contains a b(αn,βn)-convergent subsequence, then the sequence itself is b(αn,βn)-
convergent.

Definition 15. A b(αn,βn)-hypermetric space (X,Γ(αn,βn)) over Banach algebra A is called
b(αn,βn)-complete if every b(αn,βn)-Cauchy sequence in (X,Γ(αn,βn)) is b(αn,βn)-convergent
in (X,Γ(αn,βn)).

Proposition 19. A b(αn,βn)-hypermetric space (X,Γ(αn,βn)) over Banach algebra A is
b(αn,βn)-complete if and only if (X, dΓ(αn,βn)

) is a complete metric space.

Definition 16. Let (X,Γ(αn,βn)) and (Y,Γ
′

(αn,βn)
) be two b(αn,βn)-hypermetric spaces over

Banach algebra A. A function f : X −→ Y is called a b(αn,βn)-contraction if there exists

a constant k ∈ {a ∈ A|0A ⪯ a ≺ eA} such that Γ
′

(αn,βn)
(f(xi))

n
i=1 ⊆ kΓ(αn,βn)(xi)

n
i=1 for

all x1, . . . , xn ∈ X.

It follows that f is b(αn,βn)-continuous because; Γ(αn,βn)(xi)
n
i=1 ⊆ {a ∈ A|0A ⪯ a ≺ δ}

with k ̸= 0 and δ := ϵk−1 implies Γ
′

(αn,βn)
(f(xi))

n
i=1 ⊆ {a ∈ A|0A ⪯ a ≺ ϵ}.

Theorem 1. Let (X,Γ(αn,βn)) be a b(αn,βn)-complete space and let T : X → X be a
b(αn,βn)-contraction map. Then T has a unique fixed point T (x) = x.

Proof. We consider xm+1 = T (xm), with x0 being any point inX. We have by repeated
use of the (αn, βn)-rectangle inequality and application of contraction property, we obtain

Γ(αn,βn)(xm, (xm+1)
n
2 ) ⊆ kmΓ(αn,βn)(x0, (x1)

n
1 )



A. D.Nezhad, S. Radenović / Eur. J. Pure Appl. Math, 14 (4) (2021), 1148-1160 1158

for all m, s1 ∈ N which m < s1 and k ∈ {a ∈ A|0A ⪯ a ≺ eA}. From the above it follows
that

Γ(αn,βn)(xm, (xs−1)
n
2 ) ⊆ ξ1Γ(αn,βn)(xm, (xm+1)

n
2 )

+ ξ2Γ(αn,βn)(xm+1, (xm+2)
n
2 )

+ ξ3Γ(αn,βn)(xm+2, (xm+3)
n
2 )

+ ... ... ... ... ...
+ ξs1−mΓ(αn,βn)(xs1−1, (xs1)

n
2 )

⊆ ξ(km + km+1 + . . .+ ks1−1)Γ(αn,βn)(x0, (x1)
n
2 )

= ξkm(eA − ks1−m)(eA − k)−1Γ(αn,βn)(x0, (x1)
n
2 ).

(6)

Where ξ1 = αn(xm, (xs1)
n
2 ), ξ2 = βn(xm, (xs1)

n
2 ).αn(xm, (xm+1)

n
2 ), ...

and ξ = max{ξ1, ξ2, ..., ξs1−m} for all xm, ..., xs1 ∈ BΓ(αn,βn)
(x0, r).

Then we have
lim

m, s1 →+∞
Γ(αn,βn)(xm, (xs1)

n
2 ) = {0A} (7)

since
lim

m, s1 →+∞
ξkm(eA − ks1−m)(eA − k)−1Γ(αn,βn)(x0, (x1)

n
2 ) = {0A}. (8)

For m ≤ s1 ≤ s2 ∈ N and (G5) implies that

Γ(αn,βn)(xm, xs1 , (xs2)
n
3 ) ⊆ αn(xm, xs1 , (xs2)

n
3 )Γ(αn,βn)(xm, (xs1)

n
2 )

+ βn(xm, xs1 , (xs2)
n
3 )Γ(αn,βn)(xs1 , (xs2)

n
2 ),

(9)

now taking limit as m, s1, s2 → +∞, we get

Γ(αn,βn)(xm, xs1 , (xs2)
n
3 ) → {0A}.

Now for m ≤ s1 ≤ s2 ≤ . . . ≤ sn−1 ∈ N, we will have

Γ(αn,βn)(xm, (xsi)
n−1
i=1 ) → {0A}; whenever, m, s1, . . . , sn−1 → +∞, (10)

then {xm} is a Cauchy sequence. By completeness of (X,Γ(αn,βn)), there exists a ∈ X
such that {xn} is b(αn,βn)-convergent to a. It follows that the limit xm is a fixed point of

T follows the b(αn,βn)-continuity of T , and

Ta = T lim
m→+∞

xm = lim
m→+∞

Txm = lim
m→+∞

xm+1 = a. (11)

Finally, if a and b are two fixed points, then

{0A} ⊆ Γ(αn,βn)(a, (b)
n
2 ) = Γ(αn,βn)

(
T (a), (T (b))n2

)
⊆ kΓ(αn,βn)(a, (b)

n
2 ).

(12)

We conclude from k ≺ eA that Γn(a, (b)
n
2 ) = {0A}. Consequently a = b and the fixed

point is unique.



REFERENCES 1159

Proposition 20. The equation X l + 1 = (l2 − 1)xl+1 + l2x, for each natural nuber l > 1,
has a unique real solution.

Proof. On can check that if x ∈ R with |x| > 1, then x is not a solution for the above
equation. Now let x = [−1, 1]. Define Ωα2,β2 : X × X → P ∗(R0

+) with, Ωα2,β2(x, y) =
[0, |x−y|] and α2, β2 : X×X −→ [1,+∞), with α2(x, y) = 1+|x|+|y|, β2(x, y) = 2+|x|+|y|.
Then (X,Ω(α2,β2)) is a complete b(α2,β2)-hypermetric space over Banach algebra R . Also,
define the mapping T : X → X by

Tx =
xl + 1

(l2 − 1)xl + l2
.

Now, we study the following cases:
Case I: If x = y. Then

Ω(α2,β2)(T (x), T (y)) = Ω(α2,β2)(T (x), T (x)) = {0} ⊆ 1

l3
Ω(α2,β2)(x, y) = {0}.

Case II: If x ̸= y. Then

Ω(α2,β2)(T (x), T (y)) = Ω(α2,β2)(
xl + 1

(l2 − 1)xl + l2
,

yl + 1

(l2 − 1)yl + l2
) =

[0,
xl + 1

(l2 − 1)xl + l2
− yl + 1

(l2 − 1)yl + l2
] = [0,

|xl − yl|
((l2 − 1)xl + l2)((l2 − 1)yl + l2)

] ⊆

1

l3
|x− y| ⊆ 1

l3
Ω(α2,β2)(x, y),

where we choose k = 1
l3
< 1. Thus, T satisfies all conditions of Theorem 2.32. Therefore,

T has a unique fixed point. Note that the unique fixed point of T is the unique solution
of the equation.

3. Conclusion

The objective of this paper is to study about b(αn,βn)-hypermetric spaces and intro-
duced certain fixed point results of mappings in the setting of b(αn,βn)-hypermetric spaces.
This study is a candidate of a pioneer result and many refined results can be derived in
the near future. The purpose definition is applicable for engineering science.

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