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Some aspects of the n-ary orthogonal and b(αn,βn)-best approximations of b(αn,βn)-hypermetric spaces over Banach algebras

  • Akbar Dehghan Nezhad , Marija Paunovic , Ana Savic EMAIL logo and Stojan Radenović
Published/Copyright: July 27, 2023
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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 56 Issue 1

Abstract

In this article, a definition of a b ( α n , β n ) -best approximations of b ( α n , β n ) -hypermetric spaces over Banach algebras is given. Our objective is to prove the concept of extension of fixed point theorems in n -ary orthogonal b ( α n , β n ) -hypermetric spaces over Banach algebras.

Keywords: b (α n ,β n )-best approximations; b (α n ,β n )-hypermetric spaces; b n -metric space; fixed point
MSC 2010: Primary 54H25; Secondary 47H10; 46B20

1 Introduction and preliminaries

In recent years, clustering problem has become an essential factor in computer science. Commonly, one of Minkowski distances (e.g., Euclidean distances) is used for the similarity measure of clusters. In general, the metric (or distance in k-means) is a mathematical term; it provides information for scientists to solve problems. Considering the ineffectiveness of existing meters for data classification, the main motivation is this new look at meters. This opinion is derived from the general understanding and application of a meter in physical reality. In this view, the distance between two points is not considered a value, but a distance with a reasonable error approximation is calculated. Article [8] is an example of using this type of idea for an application.

Generalization of the metric space structure by weakening the triangle inequality is given by Bourbaki [4] and Bakhtin [2] (see more in [5,16]). The aim of our research is the generalization of a cone n -metric spaces to the n -ary orthogonal b ( α n , β n ) -hypermetric spaces.

In this section, we recall some definitions, notations, and terminologies, which will be used to prove the main results [1,2,3,6,7,9,11,17,18,21,23] and basic definitions can be found in [12,15] and [22].

Theorem 1.1

(Brouwer’s fixed point theorem) [24]. The continuous operator T : X → X has a fixed point, provided X is a compact, convex, non-empty set in a finite-dimensional normed space over K .

Theorem 1.2

(Schauder’s fixed point theorem) [24]. The compact operator T : X → X has a fixed point, provided X is a bounded, convex, non-empty subset of a Banach space A over K .

The Schauder fixed-point theorem is an extension of the Brouwer fixed-point theorem to topological vector spaces, which may be of infinite dimension.

Proposition 1.3

Let A be a Banach algebra. If dim ( A ) < + ∞ , then the Schauder theorem is equivalent to the Brouwer theorem.

The point is that in a finite dimension, all continuous operators are compact, while in an infinite dimension, you can have continuous operators that are not compact, as the following example shows:

In l 2 ( N ) , consider the operator T ( x ) = ( 1 − ‖ x ‖ 2 , x 1 , x 2 , … ) defined for ‖ x ‖ ≤ 1 , where x = ( x 1 , x 2 , … ) and ‖ x ‖ 2 = ∑ i = 1 + ∞ ∣ x i ∣ 2 . The operator T is continuous. In fact, ‖ T ( x ) − T ( y ) ‖ 2 = ∣ 1 − ‖ x ‖ 2 − 1 − ‖ y ‖ 2 ∣ 2 + ‖ x − y ‖ 2 ≤ ∣ ‖ x ‖ 2 − ‖ y ‖ 2 ∣ + ‖ x − y ‖ 2 ≤ ( ‖ x ‖ + ‖ y ‖ ) ‖ x − y ‖ + ‖ x − y ‖ 2 ≤ 2 ‖ x − y ‖ + ‖ x − y ‖ 2 .

Moreover, T maps the closed unit ball to its boundary because ‖ T ( x ) ‖ 2 = 1 − ‖ x ‖ 2 + ‖ x ‖ 2 = 1 . The operator T does not have fixed points; by contradiction, if T ( x ) = x , we would not only have ‖ x ‖ = 1 but also ( 0 , x 1 , x 2 , … ) = ( x 1 , x 2 , … ) , i.e., x i = 0 for every i . In that case, we shall have ‖ x ‖ 2 = 0 ≠ 1 , which is a contradiction. In the finite-dimension state being compact for a set is nothing more than being bounded and closed, so to better understand the main difference.

For further details, the readers are referred to [12,13,14,16,19,20] and the references therein.

2 Main findings

In this part, we will describe our results of the b ( α n , β n ) -hypermetric spaces of dimension n .

2.1 The n -ary b ( α n , β n ) -hypermetric spaces

Some fixed point theorems in set-valued metric spaces over Banach algebra are presented below, as well as corresponding examples and applications of our findings. Our first result is Definition 2.1. Here and subsequently, for n ≥ 2 , let X n denotes the n -times Cartesian product X × … × X ︸ n -times . To simplify, we let ( x i ) i = 1 n and ( x ) 1 n stands for ( x 1 , … , x n ) and ( x ) i = 1 n . Let T be a mapping, and we will use T x instead of T ( x ) .

Let A be an ordered Banach algebra. Let P ∗ ( A ) represent the family of all non-empty subsets of A . The subset relation on power set is partial ordering. Therefore, we have that ( P ∗ ( A ) , ⊆ ) is an ordered set. Furthermore, A + will represent a set of non-negative elements of A i.e., A + = { a ∈ A : 0 A ≼ a } .

Definition 2.1

Let X be a non-empty set and α n , β n : X n ⟶ A + with 1 ≼ α n ( x i ) i = 1 n and 1 ≼ β n ( x i ) i = 1 n . Let Γ ( α n , β n ) : X n ⟶ P ∗ ( A + ) be a mapping satisfying, for all n -tuples, ( x i ) i = 1 n ∈ X n the following conditions:

  1. { 0 A } ⊆ Γ ( α n , β n ) ( x i ) i = 1 n ,

  2. Γ ( α n , β n ) ( x i ) i = 1 n = { 0 A } if x 1 = … = x n ,

  3. Γ ( α n , β n ) ( x i ) i = 1 n ⊋ { 0 A } for all ( x i ) i = 1 n ∈ X n with x i ≠ x j and for some i , j ∈ { 1 , … , n } ,

  4. Γ ( α n , β n ) ( x i ) i = 1 n = Γ ( α n , β n ) ( x π i ) i = 1 n for every permutation ( π ( 1 ) , … , π ( n ) ) of ( 1 , 2 , … , n ) ,

  5. Γ ( α n , β n ) ( ( x i ) i = 1 n − 1 , x n − 1 ) ⊆ Γ ( α n , β n ) ( x i ) i = 1 n for all x 1 , … , x n ∈ X ,

  6. Γ ( α n , β n ) ( x i ) i = 1 n ⊆ α n ( x i ) i = 1 n ⋅ Γ ( α n , β n ) ( x 1 , ( a ) 2 n ) + β n ( x i ) i = 1 n ⋅ Γ ( α n , β n ) ( a , ( x i ) i = 2 n ) for all x 1 , … , x n , a ∈ X .

Furthermore, A i are subsets of X , ( i = 1 , … , n ) , for any B , B ′ ∈ P ∗ ( A + ) and α ∈ A + . Now, we can define

Γ ( α n , β n ) ( A i ) i = 1 m = ⋃ { Γ n ( x i ) i = 1 n ∣ x i ∈ A i , i = 1 , … , n } ,

B + B ′ = { b + b ′ ∣ b ∈ B , b ′ ∈ B ′ } , and α ⋅ B = { α ⋅ b ∣ b ∈ B , α ∈ A + } .

To summarize, here we say: the function Γ n indicates an ordered b ( α n , β n ) -hypermetric over Banach algebra A of dimension. The pair ( X , Γ n ) is a b ( α n , β n ) -hypermetric space over A .

Here, for simplicity, it is assumed that A + = R + 0 ≔ [ 0 , + ∞ ) . The next useful properties of a b n -hypermetric are easily obtained from the axioms.

Remark 1

If α 2 ( x i ) i = 1 2 = β 2 ( x i ) i = 1 2 = c for c ≥ 1 and Γ 2 ( x , y ) = { d b ( x , y ) } ∈ P ∗ ( ℛ ) , then we obtain the definition of b-metric space (Czerwik [5]). Obviously for c = 1 , this b-metric becomes an ordinary metric.

Example 1

We assume that A + = R + 0 . Let X = [ 0 , 1 ] and α n , β n : X n → [ 1 , + ∞ ) , with α n ( x i ) i = 1 n = 1 + 1 1 + x 1 + ⋯ + x n and β n ( x i ) i = 1 n = 1 + 2 1 + x 1 + ⋯ + x n . Define

Ω α n , β n : X n → P ∗ ( R + 0 )

with

Ω ( α n , β n ) ( x i ) i = 1 n = 1 , 1 x 1 ⋯ x n ∪ { 0 } ; x 1 , … , x n ∈ ( 0 , 1 ] , x i ≠ x j { 0 } ; x 1 , … , x n ∈ [ 0 , 1 ] , x 1 = ⋯ = x n Ω ( α n , β n ) ( x i ) i = 1 n = 1 , 1 x j ∪ { 0 } ; x i = 0 , x j ∈ ( 0 , 1 ] Ω ( α n , β n ) ( x π i ) i = 1 n , ; for every permutation ( π ( 1 ) , … , π ( n ) ) of ( 1 , 2 , … , n ) ,

and also assume A + B = A ∪ B for all A , B ∈ P ∗ ( R + 0 ) . Then, ( X , Ω ( α n , β n ) ) is a b ( α n , β n ) -hypermetric space.

Proposition 2.2

Let ( X , Γ ( α n , β n ) ) be a b ( α n , β n ) -hypermetric space over Banach algebra A . Then, for any x 1 , … , x n , a ∈ X , it follows that

  1. if Γ ( α n , β n ) ( x i ) i = 1 n = { 0 A } , then x 1 = ⋯ = x n ,

  2. Γ ( α n , β n ) ( x i ) i = 1 n ⊆ ∑ j = 2 n Γ ( α n , β n ) ( ( x 1 ) 1 n − 1 , x j ) ,

  3. Γ ( α n , β n ) ( x i ) i = 1 n ⊆ ∑ j = 1 n Γ ( α n , β n ) ( x j , ( a ) 2 n ) ,

  4. Γ ( α n , β n ) ( x 1 , ( x 2 ) 2 n ) ⊆ ( n − 1 ) Γ ( α n , β n ) ( ( x 1 ) 1 n − 2 , x 2 ) .

Proposition 2.3

Let ( X , Γ ( α n , β n ) ) be a b ( α n , β n ) -hypermetric space over Banach algebra A . Then, { 0 A } ⊆ Γ ( α n , β n ) ( x i ) i = 1 n for every x 1 , … , x n ∈ X .

Proof

Using the condition ( G 4 ) of the definition of b ( α n , β n ) -hypermetric space, we obtain

□ { 0 A } = Γ ( α n , β n ) ( x 1 ) 1 n ⊆ Γ ( α n , β n ) ( x i ) i = 1 n .

Proposition 2.4

Every b ( α n , β n ) -hypermetric space ( X , Γ ( α n , β n ) ) over Banach algebra A defines a b ( α 2 , β 2 ) -hypermetric space ( X , Γ ( α 2 , β 2 ) ) over Banach algebra A as follows:

Γ ( α 2 , β 2 ) ( x , y ) = Γ ( α n , β n ) ( x , ( y ) 2 n ) + Γ ( α n , β n ) ( y , ( x ) 2 n ) for a l l x , y ∈ X ,

where α 2 ( x , y ) = max { α n ( x , ( y ) 2 n ) , α n ( y , ( x ) 2 n ) } and β 2 ( x , y ) = max { β n ( x , ( y ) 2 n ) , β n ( y , ( x ) 2 n ) } .

Proof

For [ ( G 0 ) ] , … , [ ( G 4 ) ] is trivial. Let us show that Γ ( α 2 , β 2 ) is satisfied in

Γ ( α 2 , β 2 ) ( x , y ) ⊆ α 2 ( x , y ) ⋅ Γ ( α 2 , β 2 ) ( x , z ) + β 2 ( x , y ) ⋅ Γ ( α 2 , β 2 ) ( z , y ) for all x , y , z ∈ X .

The proof is straightforward, by setting α 2 ( x , y ) = max { α n ( x , ( y ) 2 n ) , α n ( y , ( x ) 2 n ) } and β 2 ( x , y ) = max { β n ( x , ( y ) 2 n ) , β n ( y , ( x ) 2 n ) } and the condition ( G 5 ) of definition (2.1).□

Proposition 2.5

Let δ be an arbitrary positive real value number, and ( X , d n ) be a n-metric space. Let α n , β n : X n ⟶ A + with 1 ≼ α n ( x i ) i = 1 n and 1 ≼ β n ( x i ) i = 1 n . We define an induced b ( α n , β n ) -hypermetric over Banach algebra R as follows:

Γ ( α n , β n ) δ : X n → P ∗ ( R + 0 )

Γ ( α n , β n ) δ ( x i ) i = 1 n = ( 0 , d n ( x i ) i = 1 n + δ ) ∪ { 0 } ; 1 ≤ i , j ≤ n , x i ≠ x j , d n ( x i ) i = 1 n ≠ δ { 0 } ; 1 ≤ i , j ≤ n , x i = x j or d n ( x i ) i = 1 n = δ .

Then, ( X , Γ ( α n , β n ) δ ) is a b ( α n , β n ) -hypermetric space over Banach algebra R .

Proof

It is sufficient to show that Γ ( α n , β n ) δ is satisfied in all properties [ ( G 0 ) ] , … , [ ( G 5 ) ] . The proofs of [ ( G 0 ) ] , … , [ ( G 4 ) ] can be obtained immediately from the definition of Γ ( α n , β n ) δ . We only need to show that Γ ( α n , β n ) δ is satisfied in [ ( G 5 ) ] .

Γ ( α n , β n ) δ ( x i ) i = 1 n ⊆ α n ( x i ) i = 1 n ⋅ Γ ( α n , β n ) δ ( x 1 , ( a ) 2 n ) + β n ( x i ) i = 1 n ⋅ Γ ( α n , β n ) δ ( a , ( x i ) i = 2 n ) for all x 1 , … , x n , a ∈ X .

In the case 1 ≤ i , j ≤ n , x i = x j or d n ( x i ) i = 1 n = δ . We have Γ ( α n , β n ) δ ( x i ) i = 1 n = { 0 } . Therefore,

{ 0 } ⊆ α n ( x i ) i = 1 n ⋅ Γ ( α n , β n ) δ ( x 1 , ( a ) 2 n ) + β n ( x i ) i = 1 n ⋅ Γ ( α n , β n ) δ ( a , ( x i ) i = 2 n ) for all x 1 , … , x n , a ∈ X .

In the case 1 ≤ i , j ≤ n , x i ≠ x j , and d n ( x i ) i = 1 n ≠ δ . Clearly, according to the last definition, we have

Γ ( α n , β n ) δ ( x i ) i = 1 n = ( 0 , d n ( x i ) i = 1 n + δ ) ∪ { 0 } ⊆ { ( 0 , d n ( x 1 , ( a ) 2 n ) + δ ) + ( 0 , d n ( a , ( x i ) i = 2 n ) + δ ) } ∪ { 0 } ⊆ Γ ( α n , β n ) δ ( x 1 , ( a ) 2 n ) + Γ ( α n , β n ) δ ( a , ( x i ) i = 2 n ) ⊆ α n ( x i ) i = 1 n ⋅ Γ ( α n , β n ) δ ( x 1 , ( a ) 2 n ) + β n ( x i ) i = 1 n ⋅ Γ ( α n , β n ) δ ( a , ( x i ) i = 2 n ) , for all x 1 , … , x n , a ∈ X .

This completes the proof.□

2.2 Generalized to b ( α n , β n ) -best approximations

The subject of approximation theory is intertwined with almost all branches of mathematics; therefore, it is of great importance in the applied sciences. On the other hand, fixed point theorems are used in many cases in the approximation theory. In the approximation theory, it is interesting to know whether the approximating function inherits some beneficial properties of the function being approximated [10].

The aim of this subsection is to introduce and discuss the conception of b ( α n , β n ) -best approximation and x 0 -orthogonality in the theory of b ( α n , β n ) -hypermetric space ( X , Γ ( α n , β n ) ) over A .

Definition 2.6

Let Y be a subspace of a b ( α n , β n ) -hypermetric space ( X , Γ ( α n , β n ) ) over Banach algebra A and x ∈ X . A point x 0 is a b ( α n , β n ) -best approximation to x if the following conditions are satisfied:

  1. x 0 ∈ Y ,

  2. Γ ( α n , β n ) ( x , ( x 0 ) 1 n − 1 ) ⊆ Γ ( α n , β n ) ( x , ( y ) 1 n − 1 ) for every y ∈ Y .

In general, if X is a b ( α n , β n ) -hypermetric space over A and x 0 , x , y ∈ X . Hence x is called x 0 -orthogonal to y , which is denoted by x ⊥ Γ x 0 y , if and only if Γ ( α n , β n ) ( x , ( x 0 ) 1 n − 1 ) ⊆ Γ ( α n , β n ) ( x , ( y ) 1 n − 1 .

Let Y 1 and Y 2 be subsets of X . The Y 1 ⊥ Γ x 0 Y 2 if and only if, for all y 1 ∈ Y 1 and y 2 ∈ Y 2 one has y 1 ⊥ Γ x 0 y 2 . If x ⊥ Γ x 0 y , then it is not necessary that y ⊥ Γ x 0 x .

Let X be a b ( α n , β n ) -hypermetric space over A and let Y be a subset of X . A point y 0 ∈ Y is called a Γ -best approximation for x ∈ X if x ⊥ Γ y 0 Y or Γ ( α n , β n ) ( x , ( y 0 ) 1 n − 1 ) ⊆ Γ ( α n , β n ) ( x , ( y ) 1 n − 1 for every y ∈ Y . The set of all Γ -best approximations of x in Y is denoted by P Γ Y ( x ) .

If Y is a subset of X , then it is obvious that P Γ Y ( x ) = { y 0 ∈ Y ∣ x ⊥ Γ y 0 y , for all y ∈ Y } .

Proposition 2.7

Let ( X , Γ ( α n , β n ) ) be an n -ary b ( α n , β n ) -hypermetric space over Banach algebra A . For x , y , z , x 0 ∈ X , the following statements are true:

  1. x ⊥ Γ x 0 x if and only if x = x 0 ,

  2. x ⊥ Γ x 0 y and y ⊥ Γ x z , then x ⊥ Γ x 0 z ,

  3. x ⊥ Γ x 0 x 0 and x 0 ⊥ Γ x 0 x .

Proof

  1. If x ⊥ Γ x 0 x , then Γ ( α n , β n ) ( x , ( x 0 ) 1 n − 1 ) ⊆ Γ ( α n , β n ) ( x , ( x ) 1 n − 1 ) = 0 A . Hence, Γ ( α n , β n ) ( x , ( x 0 ) 1 n − 1 ) = 0 A , i.e., x = x 0 .

  2. Since x ⊥ Γ x 0 y and y ⊥ Γ x z , we conclude that

    Γ ( α n , β n ) ( x , ( x 0 ) 1 n − 1 ) ⊆ Γ ( α n , β n ) ( x , ( y ) 1 n − 1 ) and Γ ( α n , β n ) ( y , ( x ) 1 n − 1 ) ≼ Γ ( α n , β n ) ( y , ( z ) 1 n − 1 ) .

    Therefore, Γ ( α n , β n ) ( x , ( x 0 ) 1 n − 1 ) ⊆ Γ ( α n , β n ) ( y , ( z ) 1 n − 1 ) , i.e., x ⊥ Γ x 0 z .

  3. This statement is obvious.□

Proposition 2.8

Let ( X , Γ ( α n , β n ) ) be an n-ary b ( α n , β n ) -hypermetric space over Banach algebra A . Let Y be a subset of X . Then, the following statements are true:

  1. if x ∈ X , then x ∈ P Γ Y ( x ) ,

  2. if x ∈ Y , then x ∈ P Γ Y ( x ) = { x } .

Proof

  1. Since Γ ( α n , β n ) ( x ) 1 n = 0 A , we have x ∈ P Γ Y ( x ) .

  2. If x ∈ Y and y 0 ∈ P Γ Y ( x ) , then Γ ( α n , β n ) ( x , ( y 0 ) 2 n ) ⊆ 0 A = Γ ( α n , β n ) ( y , ( x ) 2 n ) . Therefore, x = y 0 .□

Proposition 2.9

Let ( X , Γ ( α n , β n ) ) be an n-ary b ( α n , β n ) -hypermetric space over Banach algebra A . Let Y be a subset of X and x 0 ∈ X . Then, the following statements are equivalent:

  1. x ⊥ Γ x 0 Y ,

  2. there is a function f : X ⟶ P ∗ ( [ 0 , + ∞ ) ) such that Γ ( α n , β n ) ( x , ( x 0 ) 2 n ) ⊆ f ( y ) ≼ Γ ( α n , β n ) ( x , ( y ) 2 n ) for all y ∈ Y .

Proof

The case: (m1) implies (m2).

Define a function f : X ⟶ P ∗ ( [ 0 , + ∞ ) ) by the equality f ( z ) = Γ ( α n , β n ) ( x , ( z ) 2 n ) . Suppose that y ∈ Y . Since x ⊥ Γ x 0 y , we have Γ ( α n , β n ) ( x , ( x 0 ) 2 n ) ⊆ Γ ( α n , β n ) ( x , ( y ) 2 n ) = f ( y ) .

The case: (m2) implies (m1).

By definition, we have x ⊥ Γ x 0 y for every y ∈ Y .□

Proposition 2.10

Let ( X , Γ ( α n , β n ) ) be an n -ary b ( α n , β n ) -hypermetric space over Banach algebra A . Let Y be a subset of X and y 0 ∈ Y . Then, the following statements are equivalent:

  1. y 0 P Γ Y ( x ) ,

  2. there is a function f : X ⟶ P ∗ ( [ 0 , + ∞ ) ) such that Γ ( α n , β n ) ( x , ( y 0 ) 2 n ) ⊆ f ( y ) ⊆ Γ ( α n , β n ) ( x , ( y ) 2 n ) for all y ∈ Y .

Proposition 2.11

Let ( X , Γ ( α n , β n ) ) be an n-ary b ( α n , β n ) -hypermetric space over Banach algebra A . Let Y be a subset of X , Z be a subset Y , and x , x 0 ∈ X . If there is a function f : X ⟶ P ∗ ( [ 0 , ∞ ) ) such that Γ ( α n , β n ) ( x , ( y 0 ) 2 n ) ⊆ f ( y ) ⊆ Γ ( α n , β n ) ( x , ( y ) 2 n ) for all y ∈ Z , then Z ⊆ P Γ Y ( x ) .

2.3 The n -ary orthogonal b ( α n , β n ) -hypermetric space over Banach algebra A

In this subsection, we will describe some properties and findings of the n -ary orthogonal b ( α n , β n ) -hypermetric spaces. In addition, we will give some fixed point theorems in set-valued n -ary orthogonal b ( α n , β n ) -hypermetric spaces over A . In addition, examples with the application of our main findings will be presented.

An ordered n -tuple is a set of n objects with an order associated with them. If n objects are represented by x 1 , x 2 , … , x n , hence we write the ordered n -tuple as ( x i ) i = 1 n . Let X 1 , … , X n be n sets. Then, the set of all ordered n -tuples ( x i ) i = 1 n , where x i ∈ X i for all i = 1 , … , n , is called the Cartesian product of X 1 , X 2 , … , X n and is denoted by ( X i ) i = 1 n .

Definition 2.12

An n -ary relation ϒ over sets X 1 , … , X n is an ( n + 1 ) -tuple ( X 1 , … , X n , G ) , where G is a subset of the Cartesian product ( X i ) i = 1 n called the graph of Γ .

An n -ary relation on sets X 1 , … , X n is a set of ordered n -tuples ( x i ) i = 1 n , where x i is an element of X i for all i = 1 , … , n . Thus, an n -ary relation on sets X 1 , … , X n is a subset of the Cartesian product ( X i ) i = 1 n . The set X i is called the i -th domain of ϒ . The sets X 1 , X 2 , … , X n are called the domains of the relation, and n is its degree.

We begin with the following definition. For n ≥ 2 , the set X n denotes the n -times Cartesian product X × … × X ︸ n -times and A be a Banach algebra. We denote A + as a set of non-negative elements of A . The set A + = { a ∈ A : 0 A ≼ a } and P ∗ ( A + ) denotes the family of all non-empty subsets of A + . We denote

  1. Γ min ϒ ( x i ) i = 1 n = min { Γ ( α n , β n ) ϒ ( x 1 ) 1 n , … , Γ ( α n , β n ) ϒ ( x n ) 1 n } and

  2. Γ max ϒ ( x i ) i = 1 n = max { Γ ( α n , β n ) ϒ ( x 1 ) 1 n , … , Γ ( α n , β n ) ϒ ( x n ) 1 n } .

Definition 2.13

Let X be a non-empty set and α n , β n : X n ⟶ A + . Let ( X , ϒ ) be an n -ary orthogonal set. Let Γ ( α n , β n ) ϒ : X n ⟶ P ∗ ( A + ) be a mapping (the n -ary orthogonal b ( α n , β n ) -hypermetric over A on n -ary orthogonal set ( X , ϒ ) ) satisfying the following conditions for all n -tuples ( x i ) i = 1 n ∈ X n :

  1. x = y if and only if Γ ( α n , β n ) ϒ ( x ) 1 n = Γ ( α n , β n ) ϒ ( x , ( z j ) j = 2 n − 1 , y ) = Γ ( α n , β n ) ϒ ( y ) 1 n , where z j = x or y ,

  2. Γ min ϒ ( x i ) i = 1 n ⊆ Γ ( α n , β n ) ϒ ( x i ) i = 1 n for all x 1 , … , x n ∈ X ,

  3. Γ ( α n , β n ) ϒ ( x i ) i = 1 n = Γ ( α n , β n ) ϒ ( x π i ) i = 1 n for every permutation ( π ( 1 ) , … , π ( n ) ) of ( 1 , 2 , … , n ) ,

  4. Γ ( α n , β n ) ϒ ( x i ) i = 1 n − Γ min ϒ ( x i ) i = 1 n ⊆ α n ( x i ) i = 1 n ⋅ Ω 1 + β n ( x i ) i = 1 n ⋅ Ω 2 − Γ min ϒ ( a ) i = 1 n for all x 1 , … , x n , a ∈ X

where Ω 1 = Γ ( α n , β n ) ϒ ( x 1 , ( a ) 2 n ) − Γ min ϒ ( x 1 , ( a ) 2 n ) and Ω 2 = Γ ( α n , β n ) ( a , ( x i ) i = 2 n ) − Γ min ϒ ( a , ( x i ) i = 2 n ) .

Let A i subsets of X ( i = 1 , … , n ) . For any B , B ′ ∈ P ∗ ( A + ) and α ∈ A + , we define

Γ ( α n , β n ) ϒ ( A i ) i = 1 n = ⋃ { Γ ( α n , β n ) ϒ ( x i ) i = 1 n ∣ x i ∈ A i , i = 1 , … , n } ,

B + B ′ = { b + b ′ ∣ ∈ B , b ′ ∈ B ′ } and α ⋅ B = { α ⋅ b ∣ b ∈ B , α ∈ A + } .

The pair ( X , Γ ( α n , β n ) ϒ ) is called an n -ary orthogonal b ( α n , β n ) -hypermetric space over Banach algebra A on n -ary orthogonal set ( X , ϒ ) .

For example, we can place A + = Z + 0 or R + 0 , where Z + 0 ≔ N ∪ { 0 } = { 0 , 1 , 2 , … } and R + 0 ≔ [ 0 , + ∞ ) . For simplicity, we assume that A + = R + 0 . The next helpful properties of a b n -hypermetric are simple and derived from the axioms.

Remark 2

If α n ( x i ) i = 1 n = β n ( x i ) i = 1 n = c for c ≥ 1 and n = 1 , then we have the definition of b-metric space (Czerwik [5]). It is clear that for c = 1 , this b-metric becomes a regular metric.

Definition 2.14

Let ( X , Γ ( α n , β n ) ϒ ) be an n -ary orthogonal b ( α n , β n ) -hypermetric space over Banach algebra A on n -ary orthogonal set ( X , ϒ ) . Then,

  1. a sequence { x i } i = 1 + ∞ in an n -ary orthogonal b ( α n , β n ) -hypermetric space ( X , Γ ( α n , β n ) ϒ ) converges to a point x 0 ∈ X if and only if lim k ⟶ + ∞ ( Γ ( α n , β n ) ϒ ( ( x i + k ) i = 1 n − 1 , x 0 ) − Γ min ϒ ( ( x i + k ) i = 1 n − 1 , x 0 ) ) = 0 A ;

  2. a sequence { x i } i = 1 + ∞ in an n -ary orthogonal b ( α n , β n ) -hypermetric space ( X , Γ ( α n , β n ) ϒ ) is called b ( α n , β n ) -Cauchy sequence if lim k ⟶ + ∞ ( Γ ( α n , β n ) ϒ ( x i + k ) i = 1 n − Γ min ϒ ( x i + k ) i = 1 n ) and lim k ⟶ + ∞ ( Γ max ϒ ( x i + k ) i = 1 n − Γ min ϒ ( x i + k ) i = 1 n ) exist (and are finite);

  3. an n -ary orthogonal b ( α n , β n ) -hypermetric space ( X , Γ ( α n , β n ) ϒ ) is said to be orthogonally b ( α n , β n ) -complete if every n -ary orthogonal b ( α n , β n ) -Cauchy sequence { x i } i = 1 + ∞ converges to a point x 0 ∈ X such that lim k ⟶ + ∞ ( Γ ( α n , β n ) ϒ ( ( x k ) 1 n − 1 , x 0 ) − Γ min ϒ ( ( x k ) 1 n − 1 , x 0 ) = 0 A and lim k ⟶ + ∞ ( Γ max ϒ ( ( x k ) 1 n − 1 , x 0 ) − Γ min ϒ ( ( x k ) 1 n − 1 , x 0 ) = 0 A .

Definition 2.15

Let X be a non-empty set and ϒ be an n -ary relation on sets X 1 , … , X n . Then, ( X , ϒ ) is said to be an n -ary orthogonal set (in short, O n -set) if there exists x 0 ∈ X such that for all x i ∈ X , ( 1 ≤ i ≤ n ) , ( x 0 , x 2 , … , x n ) ∈ ϒ or ⋯ or ( x 1 , … , x i − 1 , x 0 , x i + 1 , … , x n ) ∈ ϒ or ⋯ or ( x 1 , x 2 , … , x n − 1 , x 0 ) ∈ ϒ . The element x 0 is called an n -ary orthogonal element.

Definition 2.16

Let ( X , ϒ ) be an n -ary orthogonal set. A sequence { x j } is called an O n -set sequence if for all j ∈ N , ( x j , x j + 1 , … , x j + n ) ∈ ϒ or ⋯ or ( x 1 + j , … , x j − 1 , x j , x j + 1 , … , x n + j ) ∈ ϒ or ⋯ or ( x j + 1 , x 2 + j , … , x n − 1 + j , x j ) ∈ ϒ .

Definition 2.17

Let ( X , Γ ( α n , β n ) ϒ ) be an n -ary orthogonal b ( α n , β n ) -hypermetric space over Banach algebra A on n -ary orthogonal set ( X , ϒ ) . A mapping T : X ⟶ X is an n -ary orthogonal continuous (or ϒ -continuous) in x 0 ∈ X if, for each O n -set sequence, { x j } j ∈ N ∈ X such that x j ⟶ x 0 , then T ( x j ) ⟶ T ( x 0 ) . Furthermore, T is said to be ϒ -continuous on X if T is ϒ -continuous at each x 0 ∈ X .

Definition 2.18

Let ( X , Γ ( α n , β n ) ϒ ) be an n -ary orthogonal b ( α n , β n ) -hypermetric space over Banach algebra A on n -ary orthogonal set ( X , ϒ ) . Then, X is said to be n -ary orthogonally b ( α n , β n ) -complete if every b ( α n , β n ) -Cauchy sequence is convergent.

Definition 2.19

Let ( X , Γ ( α n , β n ) ϒ ) be an n -ary orthogonal b ( α n , β n ) -hypermetric space over Banach algebra A on n -ary orthogonal set ( X , ϒ ) . A mapping T : X ⟶ X is said to be ϒ -preserving if T ( ( x i ) i = 1 n ) ∈ ϒ whenever ( x i ) i = 1 n ∈ ϒ .

Theorem 2.20

Let ( X , Γ ( α n , β n ) ϒ ) be an n-ary orthogonal b ( α n , β n ) -hypermetric space and T : X ⟶ X be ϒ -preserving, ϒ -continuous, and satisfy the following condition:

Γ ( α n , β n ) ϒ ( T ( ( x i ) ) i = 1 n ) ⊆ λ Γ ( α n , β n ) ϒ ( x i ) i = 1 n for all ( x i ) i = 1 n ∈ X n with ( x i ) i = 1 n ∈ ϒ , where 0 A ≼ λ ≼ e A .

Then, T has a unique fixed point x 0 ∈ X and Γ ( α n , β n ) ϒ ( x 0 ) 1 n = { 0 A } .

Proof

From the definition of n -ary orthogonality, there exists x 0 ∈ X such that for all x i ∈ X , ( 1 ≤ i ≤ n ) , ( x 0 , x 2 , … , x n ) ∈ ϒ or ⋯ or ( x 1 , … , x i − 1 , x 0 , x i + 1 , … , x n ) ∈ ϒ or ⋯ or ( x 1 , x 2 , … , x n − 1 , x 0 ) ∈ ϒ . Hence, ( x 0 , ( T ( x 0 ) ) 2 n ) ∈ ϒ or ⋯ or ( T ( x 0 ) ) 1 j − 1 , x 0 , ( T ( x 0 ) j + 1 n ) ∈ ϒ or ⋯ or ( ( T ( x 0 ) ) 1 n − 1 , x 0 ) ∈ ϒ . Let x j = T ( x j + 1 ) for all j ∈ N since the operator T is ϒ -preserving. Therefore, ( x i ( t ) ) i = 1 ∞ is O n -set sequence. By assumption, we can see that

Γ ( α n , β n ) ϒ ( x j ) j = i i + n = Γ ( α n , β n ) ϒ ( T ( x j ) ) j = i − 1 i + n − 1 ⊆ λ n Γ ( α n , β n ) ϒ ( x j ) j = 1 n

for all i ∈ N . For any two positive integers j > i , we have

Γ ( α n , β n ) ϒ ( x i , ( x j ) 2 n ) ⊆ Γ ( α n , β n ) ϒ ( T ( x i ) , T ( ( x j − 1 ) ) 2 n ) ⊆ λ Γ ( α n , β n ) ϒ ( x i − 1 , ( x j − 1 ) 2 n ) ⊆ λ Γ ( α n , β n ) ϒ ( λ Γ ( α n , β n ) ϒ ( x i − 2 , ( x j − 2 ) 2 n ) ) ⊆ λ 2 Γ ( α n , β n ) ϒ ( ( x i − 2 , ( x j − 2 ) 2 n ) ) ⋮ ⊆ λ i Γ ( α n , β n ) ϒ ( x 0 , ( x j − i ) 2 n ) .

Consequently,

Γ ( α n , β n ) ϒ ( x i , ( x j ) 2 n ) − Γ min ϒ ( x i , ( x j ) 2 n ) ⊆ λ i ( α n ( x 0 , ( x j − i ) 2 n ) ⋅ Γ ( α n , β n ) ϒ ( x 0 , ( x 1 ) 2 n ) + ( β n ( x 0 , ( x j − i ) 2 n ) ⋅ Γ ( α n , β n ) ( x 0 , ( x i ) i = 2 n ) − Γ min ϒ ( x 0 , ( x i ) i = 2 n ) ) − Γ min ϒ ( x 0 ) 1 n ) ⊆ λ i ( α n ( x 0 , ( x j − i ) 2 n ) ⋅ Γ ( α n , β n ) ϒ ( x 0 , ( x 1 ) 2 n ) + ( β n ( x 0 , ( x j − i ) 2 n ) ⋅ Γ ( α n , β n ) ϒ ( x 1 , ( x j ) 2 n ) ) ⊆ λ i [ α n ( x 0 , ( x j − i ) 2 n ) ⋅ Γ ( α n , β n ) ϒ ( x 0 , ( x 1 ) 2 n ) + β n ( x 0 , ( x j − i ) 2 n ) ⋅ ( α n ( x 1 , ( x i ) 2 n ) ⋅ Γ ( α n , β n ) ϒ ( x 2 , ( x j ) 2 n ) + β n ( x 1 , ( x i ) 2 n ) ⋅ Γ ( α n , β n ) ϒ ( x 2 , ( x j ) 2 n ) ) ] ⋮ ⊆ Δ λ i [ 1 + Δ λ + ( Δ λ ) 2 + ⋯ + ( Δ λ ) j − i − 1 ] Γ ( α n , β n ) ϒ ( x 0 , ( x 1 ) 2 n ) = Δ λ i ( Δ λ ) j − i − 1 1 − Δ λ Γ ( α n , β n ) ϒ ( x 0 , ( x 1 ) 2 n ) ,

where Δ = sup j { α n ( x 0 , ( x j − i ) 2 n ) , β n ( x 0 , ( x j − i ) 2 n ) ⋅ α n ( x 1 , ( x i ) 2 n ) , β n ( x 0 , ( x j − i ) 2 n ) ⋅ β n ( x 1 ) , ( x i ) 2 n , … } . Since 0 A ≼ λ ≼ e A and Δ > 0 , it follows from the above inequality that Γ ( α n , β n ) ϒ ( x i , ( x j ) 2 n ) − Γ min ϒ ( x i , ( x j ) 2 n ) ⟶ 0 A as i , j ⟶ + ∞ .

A passage similar to the above implies that (similarly) Γ max ϒ ( x i , ( x j ) 2 n ) − Γ min ϒ ( x i , ( x j ) 2 n ) ⟶ 0 A as i , j ⟶ + ∞ and so Γ min ϒ ( x i , ( x j ) 2 n ) ⟶ 0 A as i ⟶ + ∞ . Therefore, we have Γ min ϒ ( x i , ( x j ) 2 n ) ⟶ { 0 A } as i ⟶ + ∞ . Thus, Γ ( α n , β n ) ϒ ( x i , ( x j ) 2 n ) = 0 A = Γ min ϒ ( x i , ( x j ) 2 n ) .

Consequently, { x i } i = 1 ∞ is a b ( α n , β n ) -Cauchy sequence in X . Since X is n -ary orthogonally b ( α n , β n ) -complete, there exists x 0 ∈ X such that Γ ( α n , β n ) ϒ ( x i , ( x j ) 2 n ) − Γ min ϒ ( x i , ( x j ) 2 n ) ⟶ { 0 A } as i ⟶ + ∞ .

Now, we present that X is a fixed point of T in X .

Γ ( α n , β n ) ϒ ( x 0 , ( T ( x 0 ) ) 2 n ) ⊆ lim j ⟶ + ∞ sup ( Γ ( α n , β n ) ϒ ( x 0 , ( x j ) 2 n ) ) + lim j ⟶ + ∞ sup ( Γ ( α n , β n ) ϒ ( x j , ( T ( x 0 ) ) 2 n ) ) ⊆ lim j ⟶ + ∞ sup ( Γ ( α n , β n ) ϒ ( x j , ( T ( x 0 ) ) 2 n ) ) ⊆ lim j ⟶ + ∞ sup ( Γ ( α n , β n ) ϒ ( T ( x j − 1 ) , ( T ( x 0 ) ) 2 n ) ) ⊆ lim j ⟶ + ∞ sup ( Γ ( α n , β n ) ϒ λ ( ( x j − 1 ) , ( x 0 ) 2 n ) ) ,

and by (O4), we have Γ ( α n , β n ) ϒ ( x 0 , ( T ( x 0 ) ) 2 n ) ⊆ lim j ⟶ + ∞ sup λ [ α n ( x j − 1 ) , ( T ( x 0 ) ) 2 n ⋅ Ω 1 + β n ( x i ) i = 1 n ⋅ Ω 2 − Γ min ϒ ( T ( x 0 ) , x 0 ) 2 n ]   ⊆   λ Γ ( α n , β n ) ϒ ( ( T ( x 0 ) ) 2 n , x 0 ) . Therefore, Γ ( α n , β n ) ϒ ( x 0 , ( T ( x 0 ) ) 2 n ) = { 0 A } , and

0 A ⊆ Γ ( α n , β n ) ϒ ( ( T ( x 0 ) ) 1 n ) ⊆ λ Γ ( α n , β n ) ϒ ( x 0 ) 1 n = { 0 A } .

Hence,

Γ ( α n , β n ) ϒ ( ( T ( x 0 ) ) 1 n ) = Γ ( α n , β n ) ϒ ( x 0 , ( T ( x 0 ) ) 2 n ) = Γ ( α n , β n ) ϒ ( x 0 ) 1 n .

From (O1), we conclude that T ( x 0 ) = x 0 . It follows that x 0 is a fixed point for the function T .

To prove the uniqueness property of a fixed point, let y ∈ X be another fixed point of T . Then, we have T i x 0 = x 0 and T i y = y for all i ∈ N .

Choosing x 0 in the first part of the proof, we obtain

( x 0 , ( y ) 2 n ) ∈ ϒ or ⋯ or ( ( y ) 1 i − 1 , x 0 , x i + 1 , ( y ) i + 1 n ) ∈ ϒ or ⋯ or ( ( y ) 1 n − 1 , x 0 ) ∈ ϒ .

Since the function T is ϒ -preserving. Then, ( T j ( x 0 ) , ( T j ( y ) ) 2 n ) ∈ ϒ or ⋯ or ( ( T j ( y ) ) 1 i − 1 , T j ( x 0 ) , ( T j ( y ) ) i + 1 n ) ∈ ϒ or ⋯ or ( ( T j ( y ) ) 1 n − 1 , T j ( x 0 ) ) ∈ ϒ for all j ∈ N .

Thus, Γ ( α n , β n ) ϒ ( x 0 , ( y ) 2 n ) = Γ ( α n , β n ) ϒ ( T j ( x 0 ) , ( T j ( y ) ) 2 n ) ⊆ λ Γ ( α n , β n ) ϒ ( x 0 , ( y ) 2 n ) ⊆ Γ ( α n , β n ) ϒ ( x 0 , ( y ) 2 n ) , which implies that Γ ( α n , β n ) ϒ ( x 0 , ( y ) 2 n ) = 0 A and so x 0 = y .

Finally, it is shown that if x 0 is a fixed point, then Γ ( α n , β n ) ϒ ( x 0 , ( y ) 2 n ) = 0 . Assume that x 0 is a fixed point of T . Then,

Γ ( α n , β n ) ϒ ( x 0 ) 1 n = Γ ( α n , β n ) ϒ ( T n ( x 0 ) , ( T n ( y ) 2 n ) ⊆ λ Γ ( α n , β n ) ϒ ( x 0 ) 1 n ⊂ Γ ( α n , β n ) ϒ ( x 0 ) 1 n ,

That is, Γ ( α n , β n ) ϒ ( x 0 ) 1 n = 0 . This completes the proof.□

Example 2

Suppose the space X = C 0 ( I ) of continuous functions defined on I = [ 0 , 1 ] with an n -ary orthogonal b ( α n , β n ) -hypermetric space over Banach algebra R given as follows:

Γ ( α n , β n ) ϒ ( x i ( t ) ) i = 1 n = 0 A , sup t ∈ [ a , b ] x 1 ( t ) + ⋯ + x n ( t ) n for all ( x i ( t ) ) i = 1 n ∈ ( C 0 ( I ) ) n .

This space can also be equipped with an n -ary orthogonal given by ( x i ( t ) ) i = 1 n ∈ ( C 0 ( I ) ) n ,

( x i ( t ) ) i = 1 n ∈ ϒ if and only if x 1 ( t ) ≼ x 2 ( t ) ≼ ⋯ ≼ x n ( t ) ,

where α n = β n = 1 . Now, we consider the following Fredholm-type integral equation:

x ( t ) = ∫ 0 1 f ( t , s , x ( s ) ) d s for all t , s ∈ [ 0 , 1 ] , where f ∈ C 0 ( [ 0 , 1 ] 2 × C 0 ( I ) , R ) .

Then, ( C 0 ( I ) , Γ ( α n , β n ) ϒ ) is an n -ary orthogonal b ( α n , β n ) -complete hypermetric space.

Proposition 2.21

Assume that, for all ( x i ) i = 1 n ∈ C 0 ( [ 0 , 1 ] , R ) ,

∑ i = 1 n f ( t , s , x i ( t ) ) ≼ λ ∣ ∑ i = 1 n x i ( t ) ∣ for a l l t , s ∈ [ 0 , 1 ] ,

where z ∈ [ 0 , 1 ) . Then the above integral equation has a unique solution.

Proof

Define T : C 0 ( I ) ⟶ C 0 ( I ) by

T ( x ( t ) ) = ∫ 0 1 f ( t , s , x ( s ) ) d s for all t , s ∈ [ 0 , 1 ] , where f ∈ C 0 ( [ 0 , 1 ] 2 × C 0 ( I ) , R ) .

Note that the existence of a fixed point of the operator T is equivalent to the existence of a solution of the above integral equation. Furthermore, for all ( x i ) i = 1 n ∈ ( C 0 ( [ 0 , 1 ] ) ) n , we have

Γ ( α n , β n ) ϒ ( T ( ( x i ) ) i = 1 n = 0 A , sup t ∈ [ a , b ] ∑ i = 1 n T ( x i ( t ) ) n = 0 A , sup t ∈ [ a , b ] ∫ 0 1 ∑ i = 1 n f ( t , s , x i ( s ) ) d s n ≼ λ 0 A , sup t ∈ [ a , b ] ∫ 0 1 ∑ i = 1 n f ( t , s , x i ( s ) ) n d s ≼ λ 0 A , sup t ∈ [ a , b ] ∫ 0 1 ∑ i = 1 n x i ( t ) n d s ≼ λ 0 A , sup t ∈ [ a , b ] x 1 ( t ) + ⋯ + x n ( t ) n ∫ 0 1 d s ≼ λ Γ ( α n , β n ) ϒ ( x i ( t ) ) i = 1 n .

Hence, the condition is satisfied. Therefore, all the conditions of the theorem are satisfied. Hence, the operator has a unique fixed point, which means that the above Fredholm integral equation has a unique solution. The proof is completed.□

3 Conclusion

The objective of this article is to study about b ( α n , β n ) -best approximations of b ( α n , β n ) -hypermetric spaces over Banach algebras and introduced certain fixed-point results of mappings in the setting of n -ary orthogonal b ( α n , β n ) -hypermetric spaces over Banach algebras. In the future, we can seek to generalize the Schauder and Brouwer fixed-point theorem on these spaces. This article is a candidate for a pioneering result, and many findings can be obtained in the near future. It can find application in engineering science [8].

Acknowledgment

The authors would like to thank the editor of the article and the referees.

  1. Conflict of interest: The authors declare that there are no conflicts of interest.

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Received: 2022-02-05
Revised: 2022-10-29
Accepted: 2023-01-20
Published Online: 2023-07-27

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  64. Special Issue on Computational and Numerical Methods for Special Functions - Part II
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  70. Normal ordering associated with λ-Stirling numbers in λ-shift algebra
  71. Analytical and numerical analysis of damped harmonic oscillator model with nonlocal operators
  72. Compositions of positive integers with 2s and 3s
  73. Kinematic-geometry of a line trajectory and the invariants of the axodes
  74. Hahn Laplace transform and its applications
  75. Discrete complementary exponential and sine integral functions
  76. Special Issue on Recent Methods in Approximation Theory - Part II
  77. On the order of approximation by modified summation-integral-type operators based on two parameters
  78. Bernstein-type operators on elliptic domain and their interpolation properties
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  80. Special Issue on Recent Advances in Fractional Calculus and Nonlinear Fractional Evaluation Equations - Part II
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  82. On some conformable boundary value problems in the setting of a new generalized conformable fractional derivative
  83. A certain class of fractional difference equations with damping: Oscillatory properties
  84. Weighted Hermite-Hadamard inequalities for r-times differentiable preinvex functions for k-fractional integrals
  85. Special Issue on Recent Advances for Computational and Mathematical Methods in Scientific Problems - Part II
  86. The behavior of hidden bifurcation in 2D scroll via saturated function series controlled by a coefficient harmonic linearization method
  87. Phase portraits of two classes of quadratic differential systems exhibiting as solutions two cubic algebraic curves
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  90. On the stability of a strongly stabilizing control for degenerate systems in Hilbert spaces
  91. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part I
  92. New conticrete inequalities of the Hermite-Hadamard-Jensen-Mercer type in terms of generalized conformable fractional operators via majorization
  93. Pell-Lucas polynomials for numerical treatment of the nonlinear fractional-order Duffing equation
  94. Impacts of Brownian motion and fractional derivative on the solutions of the stochastic fractional Davey-Stewartson equations
  95. Some results on fractional Hahn difference boundary value problems
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  97. Special Issue on Development of Fuzzy Sets and Their Extensions - Part I
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  99. Comparison of fuzzy and crisp decision matrices: An evaluation on PROBID and sPROBID multi-criteria decision-making methods
  100. Rejection and symmetric difference of bipolar picture fuzzy graph
https://doi.org/10.1515/dema-2022-0200
Keywords for this article
b (α n ,β n )-best approximations; b (α n ,β n )-hypermetric spaces; b n -metric space; fixed point
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  64. Special Issue on Computational and Numerical Methods for Special Functions - Part II
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  70. Normal ordering associated with λ-Stirling numbers in λ-shift algebra
  71. Analytical and numerical analysis of damped harmonic oscillator model with nonlocal operators
  72. Compositions of positive integers with 2s and 3s
  73. Kinematic-geometry of a line trajectory and the invariants of the axodes
  74. Hahn Laplace transform and its applications
  75. Discrete complementary exponential and sine integral functions
  76. Special Issue on Recent Methods in Approximation Theory - Part II
  77. On the order of approximation by modified summation-integral-type operators based on two parameters
  78. Bernstein-type operators on elliptic domain and their interpolation properties
  79. A class of strongly convergent subgradient extragradient methods for solving quasimonotone variational inequalities
  80. Special Issue on Recent Advances in Fractional Calculus and Nonlinear Fractional Evaluation Equations - Part II
  81. Application of fractional quantum calculus on coupled hybrid differential systems within the sequential Caputo fractional q-derivatives
  82. On some conformable boundary value problems in the setting of a new generalized conformable fractional derivative
  83. A certain class of fractional difference equations with damping: Oscillatory properties
  84. Weighted Hermite-Hadamard inequalities for r-times differentiable preinvex functions for k-fractional integrals
  85. Special Issue on Recent Advances for Computational and Mathematical Methods in Scientific Problems - Part II
  86. The behavior of hidden bifurcation in 2D scroll via saturated function series controlled by a coefficient harmonic linearization method
  87. Phase portraits of two classes of quadratic differential systems exhibiting as solutions two cubic algebraic curves
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  91. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part I
  92. New conticrete inequalities of the Hermite-Hadamard-Jensen-Mercer type in terms of generalized conformable fractional operators via majorization
  93. Pell-Lucas polynomials for numerical treatment of the nonlinear fractional-order Duffing equation
  94. Impacts of Brownian motion and fractional derivative on the solutions of the stochastic fractional Davey-Stewartson equations
  95. Some results on fractional Hahn difference boundary value problems
  96. Properties of a subclass of analytic functions defined by Riemann-Liouville fractional integral applied to convolution product of multiplier transformation and Ruscheweyh derivative
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