Abstract:
Introduction/purpose: The aim of this paper is to present the concept of b(αn,βn) -hypermetric spaces. Methods: Conventional theoretical methods of functional analysis. Results: This study presents the initial results on the topic of b(αn,βn)-hypermetric spaces. In the first part, we generalize an n-dimensional (n ≥ 2) hypermetric distance over an arbitrary non-empty set X. The b(αn,βn)-hyperdistance function is defined in any way we like, the only constraint being the simultaneous satisfaction of the three properties, viz, non-negativity and positive-definiteness, symmetry and (αn, βn)-triangle inequality. In the second part, we discuss the concept of (αn, βn)-completeness, with respect to this b(αn,βn)-hypermetric, and the fixed point theorem which plays an important role in applied mathematics in a variety of fields. Conclusion: With proper generalisations, it is possible to formulate well-known results of classical metric spaces to the case of b(αn,βn)-hypermetric spaces.
Keywords: b(αn,βn)-hypermetric spaces, G-metric, fixed point.
Pезюме:
Введение/цель: Целью данной статьи является представление концепции b(αn,βn)-гиперметрических пространств. Методы: В статье применены конвенциональные теоретические методы функционального анализа. Результаты: В статье представлены инициальные результаты в области b(αn,βn)-гиперметрических пространств. В первой части обобщается n-мерное (n ≥ 2) гиперметрическое расстояние на произвольном непустом множестве X. Функцию b(αn,βn)-гиперрастояния можно определить произвольно при наличии трех свойств: не отрицательность, положительная определенность, симметрия и (αn,βn)-неравенство треугольника. Во второй части статьи рассматривается концепция (αn,βn)-полноты по отношению к b(αn,βn)-гиперметрике и теореме о неподвижной точке, которая играет важную роль в прикладной математике в нескольких областях. Выводы: С помощью соответствующих обобщений можно сформулировать известные результаты классических метрических пространств в случае b(αn,βn)-гиперметрических пространств.
Ключевые слова: b(αn,βn)-гиперметрические пространства, G-метрика, неподвижные точки.
Abstract:
Увод/циљ: Циљ овог рада јесте да се представи концепт b(αn,βn)-хиперметричких простора. Методе: Примењене су конвенционалне теоретске методе функционалне анализе. Резултати: У раду су представљени иницијални резултати који се односе на b(αn,βn)-хиперметричке просторе. У првом делу генерализује се n-димензионално (n ≥ 2) хиперметричко растојање на произвољном непразном скупу X. Функција b(αn,βn)-хиперрастојања може се дефинисати на произвољан начин докле год су задовољене три особине: ненегативност, позитивна дефинитност, симетрија и (αn, βn)-неједнакост троугла. У другом делу рада разматрани су концепт (αn, βn)-комплетности у односу на b(αn,βn)-хиперметрику и теорема фиксне тачке, која има значајну улогу у примењеној математици на више поља. Закључак: Одговарајућим генерализацијама могуће је формулисати познате резултате класичних метричких простора на случај b(αn,βn)-хиперметричких простора.
Keywords: b(αn,βn)-хиперметрички простори, G-метрика, фиксне тачке.
Original scientific papers
A different approach to b(αn,βn)-hypermetric spaces
Иной подход к b(αn,βn)-гиперметрическим пространствам
Другачији приступ према b(αn,βn)-хиперметричким просторима
Akbar Dehghan Nezhad dehghannezhad@iust.ac.ir
Nikola Mirkov nmirkov@vin.bg.ac.rs
Vesna Todorčević vesna.todorcevic@fon.bg.ac.rs
Stojan Radenović radens@beotel.rs
Received: 08 December 2021
Revised document received: 04 January 2022
Accepted: 05 January 2022
In human effort to describe the surrounding world, the concept of distance has long been fundamental. Our intuitive understanding of distance as an exact value may however differ from its mathematical definition and its properties. If one is to include the measurement error, encountered in real life attempt to measure the distance between two objects, the distance will be defined as an interval. This is, for example, where we may come across a set-valued distance function. This approach will in fact be our main motivation for presenting a generalized concept of the distance as a set-valued function in this paper.
The notion of 2-metric spaces, as a possible generalization of metric spaces, was introduced by Gähler (Gähler, 1963). The 2-metric d(x, y, z) is a function of three variables, and Gähler geometrically interpreted it as an area of triangle with vertices at x , y and z, respectively.
B. C. Dhage, in his PhD thesis (1992), introduced the notion of D-metric (Dhage et al, 2000) spaces that generalize metric spaces. However, most of the claims concerning the fundamental topological properties of D-metric spaces are incorrect, as shown in 2003 by Mustafa and Sims (Mustafa & Sims, 2003). This led them to introduce the notion of G-metric spaces (Mustafa & Sims, 2006), as a generalization of the metric spaces. In this type of spaces, a non-negative real number is assigned to every triplet of elements.
The G-metric spaces were generalized to universal metrics by Dehghan Nezhad et al, in a series of papers (Dehghan Nezhad & Aral, 2011; Dehghan Nezhad & Khajuee, 2013; Dehghan Nezhad et al, 2017; Dehghan Nezhad et al, 2021; Dehghan Nezhad & Mazaheri, 2010). The interpretation of the perimeter of a triangle is applied, but this time on G-metric spaces. Since then, many authors have obtained fixed point results for Gmetric spaces.
In an attempt to generalize the notion of a G-metric space to more than three variables, Khan first introduced the notion of a K-metric, and later the notion of a generalized n-metric space( for any n ≥ 2) (Khan, 2012, 2014), in 1975. He also proved the common fixed point theorem for such spaces.
Bakhtin (Bakhtin, 1989) and Czerwik (Czerwik, 1993) generalized the structure of metric space by weakening the triangle inequality and called it the b-metric space. In 2017, Kamran et al. (Kamran et al, 2017) introduced the concept of extended b-metric space by further weakening the triangle inequality. For more details also see (Agarwal et al, 2015; Debnath et al, 2021; Kirk & Shahzad, 2014; Todorčević, 2019). Also, for a broader perspective on extended b-metric spaces, dislocated b-metric spaces, rectangular b-metric spaces, b-metric like spaces, and applications see (Younis et al, 2021a,b,c; Younis & Singh, 2021).
The main purpose of this paper is a generalization of universal metric spaces into b(αn,βn) -hypermetric spaces of the n-dimension.
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}.
The concept of a b-metric space is initiated by Bakhtin (Bakhtin, 1989) and later used by Czerwick (Czerwik, 1993).
DEFINITION 1. (Czerwik, 1993) 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, z ∈ 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 bmetric space.
EXAMPLE 1. (Berinde, 1993) Let X = lp[0, 1] be the space of all real functions ϕ(t) with t ∈ [0, 1] such that 
with 0 < p < 1. Define db : X × X −→ [0, +∞) as:
Therefore, (X, db) is a b-metric space with 
.
REMARK 2. (Czerwik, 1993) The class of the b-metric space is larger than the class of the metric space. When s = 1, the concept of the b-metric space coincides with the concept of the metric space.
In the following we recall the definition of the extended b-metric space.
DEFINITION 2. (Kamran et al, 2017) Let X be a non-empty set and r : X × X −→ [1, +∞). A function dr : X × X −→ [0, +∞) is called an extended b-metric if for all x, y, z ∈ X it satisfies the following conditions:
(b1) dr(x, y) = 0 if and only if x = y,
(b2) dr(x, y) = dr(y, x),
(b3) dr(x, y) ≤ r(x, y)(dr(x, z) + dr(z, y)).
The pair (X, dr) is called the extended b-metric space.
The goal of this section is to describe a few properties and the results of the b(αn,βn)-hypermetric spaces of the dimension n.
Now we first recall and introduce some notation. For n ≥ 2, let Xn denote the n-times Cartesian product 
and R be an ordered ring. Let P∗(R) denote the family of all non-empty subsets of R. We begin with the following definition.
DEFINITION 3. Let X be a non-empty set and αn, βn : Xn −→ [1, +∞). Let 
be a function that satisfies the following conditions:
Let Ai be the subsets of X,(i = 1, . . . , n), for any D, D′ ∈ P*(R+) and α ∈ R+. We define
We shall use the following abbreviated notation: The function 
is called an ordered b(αn,βn)-hypermetric ring of the dimension n, or more specifically a b(αn,βn)-hypermetric on X. The pair 
is called an b(αn,βn)-hypermetric space.
For example, we can place 
, where 
= {0, 1, 2, . . . } and 
. In the sequel, for simplicity we assume that 
. The following useful properties of a bn-hypermetric are easily derived from the axioms.
REMARK 3. If αn(x1, x2, . . . , xn) = βn(x1, x2, . . . , xn) = c for c ≥ 1 and n = 1, then we obtain the definition of a b-metric space (Czerwik, 1993). It is clear that for c = 1, this b-metric becomes a usual metric.
PROPOSITION 1. (Example) Let X = [0, 1] and α2, β2 : X × X −→ [1, +∞), with 
. Define
with,
and also assume A + B = A ∪ B, for all 
. Then 
is a b(α2,β2) -hypermetric space.
Proof. It is sufficient to show that 
is satisfied in all properties (U1),(U2), . . . ,(U5) . The proofs of (U1), . . . ,(U4), follow immediately from the definition of 
. We only need to show that 
is satisfied in
We distinguish the following cases:
Hence 
is a b(α2,β2) -hypermetric space.
PROPOSITION 2. Let 
be a 
-hypermetric space, then for any x1, ..., xn, a ∈ X it follows that:
PROPOSITION 3. Let 
be a 
-hypermetric space, then 
or all x1, ..., xn ∈ X.
Proof. By condition (U4) of the definition of a b(αn,βn)-hypermetric space, we have
PROPOSITION 4. Every b(αn,βn)-hypermetric space 
defines a b(α2,β2)-hypermetric space 
as follows:
Proof. Note that (U1), . . . ,(U4) trivially hold. We only need to show that 
is satisfied in
By setting
and
This completes the proof.
PROPOSITION 5. Let e be an arbitrary positive real number, and (X, d) be a metric space. We define an induced b(α2,β2)-hypermetric
Then 
is a b(α2,β2)-hypermetric spac
Let 
be a b(αn,βn)-hypermetric space and 
be a partition of X. For each point p ∈ X, we denote 
a point in 
containing p, and we denote the equivalent relation induced by the relation by 
.
DEFINITION 4. Let 
e a b(αn,βn) hypermetric space. Let p1, . . . , pn ∈ X, and consider 
. A quotient b(αn,βn)-hypermetric of points of 
induced by 
is the function
given by
PROPOSITION 6. The quotient b(αn,βn)-hypermetric induced by 
is well defined and is a b(αn,βn)-hypermetric on X
Proof. 
is satisfied in all properties (U1), till (U4),
Let 
be a b(αn,βn)-hypermetric space of a dimension n > 2. For any arbitrary a in X, define the function 
by 
. Then we have the following result.
PROPOSITION 7. The function 
efine a 
-hypermetric on X.
Proof. We will verify that 
satisfies the five properties of a 
-hypermetric.
PROPOSITION 8. Let Π : X → Y be an injection from a set X to a set Y. If 
is a b(αn,βn)-hypermetric on the set Y. Then 
, given by the formula 
for all x1, . . . , xn ∈ X, is a b(αn,βn)-hypermetric on the set X.
PROPOSITION 9. Let 
be any b(αn,βn)-hypermetric space and 
. Then 
is also a b(αn,βn)-hypermetric space where 
So, on the same X many intances of the b(αn,βn)-hypermetric 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, so we need the following explanation.
For any non-empty subset A of 
, and λ ∈ 
we define a set λ · A to be λ · A := {λ · a | a ∈ A}.
PROPOSITION 10. Let 
e any b(αn,βn)-hypermetric space. Let Λ be any positive real number. We define 
. Then 
is also a b(αn,βn)-hypermetric space.
A sequence {xm} in a b(αn,βn)-hypermetric space 
is said to converge to a point s in X, if for any ϵ > 0 there exists a natural number N such that for every m1, . . . , mn−1 ≥ N.
then we shall write
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 
and 
be two b(αn,βn)-hypermetric spaces. 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 5. Let 
be a b(αn,βn) -hypermetric space, 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 
and
be two b(αn,βn)-hypermetric spaces 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 6. Let 
be a b(αn,βn) -hypermetric space, then for x0 ∈ X, r > 0, the b(αn,βn) -hyperball with the centre x0 and the radius r is
PROPOSITION 13. Let 
be a b(αn,βn) -hypermetric space, then for x0 ∈ X, r > 0,
PROPOSITION 14. The set of all 
-balls, 
, forms a basis for a topology 
on X.
DEFINITION 7. Let 
be a b(αn,βn)-hypermetric space. 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, 
.
PROPOSITION 15. Let 
be a b(αn,βn)-hypermetric space. Then for a sequence {xm} ⊆ X, and a point x ∈ X the following are equivalent:
DEFINITION 8. Let 
, 
be universal hypermetric spaces of the dimensions n and m respectively; a function T : X −→ Y is b(αn,βn),(αm,βm)-continuous at the point x0 ∈ X, if 
, 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 
-topology to Y with the 
-topology.
In the sequel, for simplicity we have assumed that n = m. Since b(αn,βn)-hypermetric topologies are metric topologies, we have:
DEFINITION 9. Let 
and 
be two b(αn,βn)-hypermetric spaces and 
be a function. The function f is called b(αn,βn)-continuous at a point a ∈ X if and only if, for given ϵ > 0, there exists δ > 0 such that x1, . . . , xn−1 ∈ X and the subset relation 
implies that 
.
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 
,
be b(αn,βn)-hypermetric spaces, 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 the {xn} is b(αn,βn)-convergent to x we have (T(xn)) is b(αn,βn)-convergent to T(x).
PROPOSITION 17. Let 
be a b(αn,βn)-hypermetric space. Then the function
is jointly b(αn,βn) -continuous in all n of its variables.
DEFINITION 10. A map T : X −→ Y between b(αn,βn)-hypermetric spaces
and 
is an iso-hypermetric when 
for all x1, . . . , xn ∈ X. If the iso-b(αn,βn)-hypermetric is injective, we call it iso-b(αn,βn)-hypermetric embedding. A bijective iso-b(αn,βn)-hypermetric is called a b(αn,βn)-hypermetric isomorphism.
In a b(αn,βn-hypermetric space, the concepts of basic topological notions, such as: b(αn,βn)-Cauchy sequence, b(αn,βn)-convergent sequence and b(αn,βn)-complete b(αn,βn)-hypermetric space can be easily adopted as shown below. We discuss about the concept of b(αn,βn)-completeness of b(αn,βn)-hypermetric spaces.
DEFINITION 11. Let 
be a b(αn,βn)-hypermetric space, then a sequence {xm} ⊆ X is called b(αn,βn)-Cauchy if for every ε > 0, there exists
The next proposition follows directly from the definitions.
PROPOSITION 18. In a b(αn,βn)-hypermetric space, 
, the following are equivalent
COROLLARY 1. (i) Every b(αn,βn)-convergent sequence in a b(αn,βn)- hypermetric space is b(αn,βn)-Cauchy. (ii) If a b(αn,βn)-Cauchy sequence in a b(αn,βn)-hypermetric space 
contains a b(αn,βn)-convergent subsequence, then the sequence itself is b(αn,βn)-convergent.
DEFINITION 12. A b(αn,βn)-hypermetric space 
is called b(αn,βn)-complete if every b(αn,βn)-Cauchy sequence in 
is b(αn,βn)-convergent in 
.
PROPOSITION 19. A b(αn,βn)-hypermetric space 
is b(αn,βn)-complete if and only if 
is a complete metric space.
DEFINITION 13. Let 
and 
be two b(αn,βn)-hypermetric spaces. A function f : X −→ Y is called a b(αn,βn)-contraction if there exists a constant k ∈ [0, 1) such that 
for all x1, . . . , xn ∈ X.
It follows that f is b(αn,βn)-continuous because; 
⊆ [0, δ) with 
.
THEOREM 1. Let 
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 in X. By repeated use of the (αn, βn)-rectangle inequality and the application of the contraction property, we obtain
for all m, s1 ∈ 
which m < s1 and k ∈ [0, 1). It follows from the above that
where Γ1 = αn(xm, xs1 , . . . , xs1 ),
and 
Then we have
since
For m ≤ s1 ≤ s2 ∈ 
and (U5) it implies that
now taking a limit as m, s1, s2 → +∞, we get
Now for m ≤ s1 ≤ s2 ≤ . . . ≤ sn−1 ∈ , we will have
then {xm} is a Cauchy sequence. By completeness of 
, 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 following the b(αn,βn)-continuity of T, and
Finally, if a and b are two fixed points, then
We conclude from k < 1 that 
. Consequently, a = b and the fixed point is unique.
The objective of this paper is to bring about the study of b(αn,βn)-hypermetric spaces and to introduce certain fixed point results of mappings in the setting of b(αn,βn)-hypermetric spaces. This study presents the initial results in this topic and more refined results can be derived in the near future. Also in the future, we will consider engineering applications of the considered topic.
nmirkov@vin.bg.ac.rs
