INTERMATHS, VOL. 4, NO. 2 (2023), 9–27

https://doi.org/10.22481/intermaths.v4i2.13991

Article

cb licença creative commons

On the reachability tube of non-Newtonian ﬁrst-

order linear diﬀerential equations

R. Temoltzi-Ávila

Área Académica de Matemáticas y Física, Universidad Autónoma del Estado de Hidalgo, Pachuca-Tulancingo km 4.5, 42184, Mineral de la Reforma,

Hidalgo, México

* Correspondence: temoltzi@uaeh.edu.mx

Abstract: A problem of practical interest is the determination of the reachability sets of

ordinary diﬀerential equations with an external perturbation, or with a control. This problem

can be extended to non-Newtonian spaces generated by continuous and injective functions

This paper presents a method to determine the reachability tube of a family of non-Newtonian

ﬁrst-order linear diﬀerential equations with an external perturbation, or with a control, that

belongs to a set of functions that are

-continuous and

-bounded. The reachability tube is

determined explicitly in three non-Newtonian spaces that are associated with three

-generators.

The results obtained are illustrated numerically.

Keywords:

-generators; non-Newtonian calculus; non-Newtonian diﬀerential equations;

external perturbations; reachability tube.

Classiﬁcation MSC: 26A24; 34A26; 93B03

1 Introduction

Newtonian calculus emerged in the second half of the 17th century under the contri-

butions of mathematicians such as Isaac Newton, Gottfried Wilhelm Leibnitz, Jakob

Bernoulli, Johann Bernoulli, and others, who established the bases of diﬀerential and

integral calculus, see [

]. The basic operations of Newtonian calculus, integration and

diﬀerentiation, are the inﬁnitesimal versions of the arithmetic operations on the set

of real numbers, addition and subtraction. In this context, the Newtonian calculus is

sometimes called an additive calculus, indicating that the basic operation is addition.

The use of additive calculus is quite intuitive when we want to interpret some

properties that characterize the set of real numbers. For example, given two real

numbers

and

, the distance between these numbers is related to the inverse

operation of addition: subtraction. In such a case, the distance is represented as the

absolute value of the real number

− a

, that is,

− a

. The generalization to the

case of spaces of greater dimension brings with it a non-essential complication. For

example, the distance between two vectors in the plane

= (

, a

) and

= (

, b

)

cannot be interpreted with the argument given in the set of real numbers. The reasoning

Submitted 28 November 2023; Accepted 20 December 2023; Available online: 30 December 2023.

that allows for an interpretation of the distance between these two vectors involves

the square root function

(

) =

√

and its inverse

−1

(

) =

. In this case, we

observe that

(

−1

(

− a

) +

−1

(

− a

)) =

− a

)

+ (b

− a

)

. According

to the deﬁnition of Euclidean distance between two vectors, we can consider using

(

−1

(

− a

) +

−1

(

− a

)) to deﬁne the distance between the vectors

= (

, a

)

and b = (b

, b

), that is,

∥b − a∥ = α(α

−1

− a

) + α

−1

− a

)).

Naturally, the extension to more dimensions can be obtained under the same argument;

see [

]. On the other hand, this method constitutes a generalization of the one-

dimensional case, since if we choose

= (

0) and

= (

0), then obviously we obtain

the distance on the real line: α(α

−1

− a

) + α

−1

− a

)) =

− a

)

= |b

− a

The reasoning that has been presented can be used to extend other properties that

characterize the set of real numbers. In particular, it is possible to generalize the

arithmetic operations and the order that is deﬁned in the set of real numbers. This

generates the so-called non-Newtonian calculus.

Non-Newtonian calculus was developed in the works of Grossman and Katz in a series

of papers which are summarized in [

]. Recently, non-Newtonian calculus has provided

a wide variety of mathematical tools for use in science, engineering, and mathematics,

and appears to have considerable potential for use as an alternative to the calculus of

Newton and Leibniz; see [20, 23, 24]. The basic principles are summarized as follows.

Let

be a non-empty subset of

and let

α: X → R

be an injective function such that

the range of this function is a subset

⊂ R

. The function

is called an

-generator of

-arithmetic over

if the following

-operations are well deﬁned for each

a, b ∈ R

⊕ b = α(α

−1

(a) + α

−1

(b)), α-addition, (1)

⊖ b = α(α

−1

(a) − α

−1

(b)), α-substraction, (2)

⊙ b = α(α

−1

(a) · α

−1

(b)), α-multiplication, (3)

⊘ b = α

−1

(a)

−1

(b)

, α-division; (4)

see [1]. The set R

is called set of non-Newtonian real numbers.

The function

allows us to establish an

-order on the set

as follows:

< b

if,

and only if, α

−1

(a) < α

−1

(b); equivalently: a

≤ b if, and only if, α

−1

(a) ≤ α

−1

(b).

In the particular case where

and

(

) =

(

), where

id: R → R

deﬁnes the

identity function:

(

) =

for each

x ∈ R

, the

-operations are reduced to the usual

operations on real numbers:

⊕ b = a + b, a

⊖ b = a − b, a

⊙ b = a · b, a

⊘ b =

We note that, naturally, the

-order coincides with the usual order of real numbers:

< b if, and only if, a < b.

For other choices of the function

and the set

, an inﬁnite number of

-arithmetic

can be obtained, on which Grossman and Katz developed the study of non-Newtonian

calculus. We describe two particular cases that are of practical interest.

10 | https://doi.org/10.22481/intermaths.v4i2.13991 R. Temoltzi-Ávila

If we choose the functions

(

) =

exp

(

) and

−1

(

) =

(

) for each

x ∈ R

, then

the exp-operations (1)–(4) are deﬁned by:

exp

⊕ b = ab, a

exp

⊖ b =

, a

exp

⊙ b = a

ln(b)

= b

ln(a)

, a

exp

⊘ b = a

ln(b)

, b ̸= 1.

The non-Newtonian calculus obtained is called multiplicative calculus, or geometric

calculus. A review of the main properties of the multiplicative calculus can be consulted

in [

]. We should mention that this is probably one of the most popular non-

Newtonian calculus due to the amount of research and applications reported in the

literature. Some applications can be consulted in [12, 19, 20, 23].

On the other hand, if we choose

(

) =

−1

sinh

(

) and

−1

(

) =

arcsinh

(

κx

) for

each

x ∈ R

, where

κ ∈

(

−

1), then the corresponding non-Newtonian calculus is called

Kaniadakis κ-calculus. The κ-operations (1)–(4) are deﬁned by

⊕ b = κ

−1

sinh(arcsinh(κa) + arcsinh(κb)) = x

1 + κ

+ y

√

1 + κ

⊖ b = κ

−1

sinh(arcsinh(κa) − arcsinh(κb)) = x

1 + κ

− y

√

1 + κ

⊙ b = κ

−1

sinh



−1

arcsinh(κa) arcsinh(κb)



⊘ b = κ

−1

sinh

arcsinh(κa)

arcsinh(κb)

The resulting non-Newtonian calculus describes a generalized form of arithmetic that is

generally used in statistical physics; see [

]. Some applications of the Kaniadakis

κ-calculus can be consulted in [11, 22, 24].

Other aspects about particular cases of non-Newtonian calculus and their applications

can be consulted in [2–4, 21].

The rest of the paper is organized as follows. In Sect. 2 the basic preliminaries of

non-Newtonian calculus are presented: its algebraic and topological properties. In Sect. 3

some known properties of non-Newtonian diﬀerential calculus and their relationship with

Newtonian diﬀerential calculus are presented. The problem of ﬁnding the reachability

tube of the family of non-Newtonian ﬁrst-order linear diﬀerential equations is presented

and solved in Sect. 4. The conclusions are formulated in Sect. 5.

2 Preliminaries of non-Newtonian calculus

In this section, we present a brief discussion on the algebraic and topological structure

of non-Newtonian real numbers, for which we assume that

and that

α: X → R

is a continuous and injective function. The results presented are an adaptation of those

discussed in [14].

2.1 Algebraic properties of non-Newtonian numbers

One of the main characteristics of non-Newtonian real numbers is that this set is an

ordered ﬁeld.

We remember that an ordered ﬁeld is a system consisting of a set

, four binary

operations (

◦

⊕,

◦

⊖,

◦

⊙,

◦

⊘

) deﬁned in

and an order

◦

of the set

. The binary operations

R. Temoltzi-Ávila INTERMATHS, 4(2), 9–27, December 2023 | 11

deﬁned in

behave in the same way as the binary operations (+

, −, ·, /

) deﬁned in

and the order of the set

, behaves in the same way as the order

of the set

; see [

An ordered ﬁeld is called an arithmetic ﬁeld when it is a subset of R.

The following results show that

is an ordered ﬁeld under the operations (1)–(4)

and, therefore, is also an arithmetic ﬁeld.

Theorem 2.1. (R

⊕) is an abelian group.

Proof.

We note that from the deﬁnition of

-addition it follows that

⊕ b ∈ R

for

all

a, b ∈ R

, that is, the set

is closed under

-addition. On the other hand, if

a, b, c ∈ R

, then from the deﬁnition of the α-addition we observe that

⊕ b)

⊕ c = α(α

−1

⊕ b) + α

−1

(c)) = α(α

−1

(α(α

−1

(a) + α

−1

(b))) + α

−1

(c))

= α(α

−1

(a) + α

−1

(b) + α

−1

(c))

= α(α

−1

(a) + α

−1

(α(α

−1

(b) + α

−1

(c))))

= α(α

−1

(a) + α

−1

⊕ c)) = a

⊕ (b

⊕ c).

Therefore, the

-addition is associative. On the other hand, we note that for all

a ∈ R

⊕ α

(0) =

(

−1

(

)+0) =

(0+

−1

(

)) =

(0)

⊕ a

, it follows that

(0) is the zero

element of

with respect to the

-addition. Furthermore, for

a ∈ R

we observe that

⊕ α

(

−α

−1

(

)) =

(

−1

(

)

−α

−1

(

)) =

(0) =

(

−α

−1

(

) +

−1

(

)) =

(

−α

−1

(

))

⊕

, that is,

(

−α

−1

(

))

∈ R

is the additive inverse of

a ∈ R

. It follows that (

⊕

)

is a group, and since

⊕ b

(

−1

(

) +

−1

(

)) =

(

−1

(

) +

−1

(

)) =

⊕ a

for all

a, b ∈ R

, we conclude that this is an abelian group.

Theorem 2.2. (R

\ {α(0)},

⊙) is an abelian group.

Proof.

From the deﬁnition of

-multiplication we observe that

⊙ b

is an element of

\{α

(0)

}

for all

a, b ∈ R

, that is, the set

\{α

(0)

}

is closed under

-multiplication.

Let a, b, c ∈ R

\ {α(0)}. It follows from deﬁnition of α-multiplication that

⊙ b)

⊙ c = α(α

−1

⊙ b) · α

−1

(c)) = α(α

−1

(α(α

−1

(a) · α

−1

(b))) · α

−1

(c))

= α(α

−1

(a) · α

−1

(b) · α

−1

(c))

= α(α

−1

(a) · α

−1

(α(α

−1

(b) · α

−1

(c))))

= α(α

−1

(a) · α

−1

⊙ c)) = a

⊙ (b

⊙ c).

Therefore,

-multiplication is associative. Furthermore, given that for

a ∈ R

\ {α

(0)

}

it holds

⊙ α

(1) =

(

−1

(

)

· α

−1

(

(1))) =

(

−1

(

(1))

· α

−1

(

)) =

(1)

⊙ a

, we

conclude that

(1) is the identity of

\{α

(0)

}

. On the other hand, if

a ∈ R

\{α

(0)

}

then

⊙ α

(

−1

(a)

) =

(

−1

(

)

−1

(a)

) =

(1) =

(

−1

(a)

· α

−1

(

)) =

(

−1

(a)

)

⊙ a

therefore

a ∈ R

\ {α

(0)

}

has multiplicative inverse

(

−1

(a)

)

∈ R

\ {α

(0)

}

. It follows

that (

\{α

(0)

⊙

) is a group. We conclude that

\{α

(0)

}

is an abelian group, since

⊙ b = α(α

−1

(a) · α

−1

(b)) = α(α

−1

(b) · α

−1

(a)) = b

⊙ a for all a, b ∈ R

\ {α(0)}.

Theorem 2.3. α-multiplication is distributive over α-addition.

12 | https://doi.org/10.22481/intermaths.v4i2.13991 R. Temoltzi-Ávila

Proof. Let a, b, c ∈ R

. we observe that

⊙ (b

⊕ c) = α(α

−1

(a) · α

−1

⊕ c)) = α(α

−1

(a) · (α

−1

(b) + α

−1

(c)))

= α(α

−1

(a) · α

−1

(b) + α

−1

(a) · α

−1

(c))

= (α(α

−1

(a) · α

−1

(b)))

⊕ (α(α

−1

(a) · α

−1

(c)))

= (a

⊙ b)

⊕ (a

⊙ c).

Therefore, the identity

⊙

(

⊕ c

) = (

⊙ b

)

⊕

(

⊙ c

) holds in the set of non-

Newtonian real numbers

. We can verify that (

⊕ c

)

⊙ a

= (

⊙ a

)

⊕

(

⊙ a

) by the

same method.

As a consequence of Theorems 2.1, 2.2 and 2.3, we obtain the following result.

Theorem 2.4. (

⊕,

⊙,

) is an ordered ﬁeld and therefore is also an arithmetic ﬁeld.

The following result shows the relationship that exists between the arithmetic ﬁelds

(R, +, ·, <) and (R

⊕,

⊙,

<).

Theorem 2.5. The ﬁelds (R, +, ·, <) and (R

⊕,

⊙,

<) are isomorphic.

Proof. It is enough to see that if φ: R → R

is deﬁned by φ(x) = α(x), then it holds:

φ(a + b) = α(α

−1

(α(a)) + α

−1

(α(b))) = α(a)

⊕ α(b) = φ(a)

⊕ φ(b),

φ(a · b) = α(α

−1

(α(a)) · α

−1

(α(b))) = α(a)

⊙ α(b) = φ(a)

⊕ φ(b).

Equivalently, if ψ : R

→ R is deﬁned by ψ(y) = α

−1

(y), then it holds:

ψ(a

⊕ b) = α

−1

(α(α

−1

(a) + α

−1

(b))) = α

−1

(a) + α

−1

(b) = ψ(a) + ψ(b),

ψ(a

⊙ b) = α

−1

(α(α

−1

(a) · α

−1

(b))) = α

−1

(a) · α

−1

(b) = ψ(a) · ψ(b).

This shows the result.

From an algebraic point of view, we observe that some properties of the set of

non-Newtonian real numbers can be obtained isomorphically from the corresponding

properties of the set of Newtonian real numbers. In the next section we show that this

situation can be extended to a topology of the set of non-Newtonian real numbers.

2.2 Topological properties of non-Newtonian numbers

In this section we show that the topology of the set of non-Newtonian real numbers

is obtained from the topology of the set of Newtonian real numbers. A more detailed

study on these properties can be found in [14, 18].

The set of non-Newtonian natural numbers is deﬁned as

= {α(n) ∈ R

| n ∈ N}.

We observe that this set is closed under the

-addition, since if

(

) and

(

) are

chosen in

, then

(

)

⊕ α

(

) =

(

−1

(

)) +

−1

(

))) =

(

)

∈ N

. In

particular, it is clear that

(

)

⊕ α

(1) =

(

+ 1). Therefore

(

) =

α(1)

⊕ ···

⊕ α(1)

| {z }

n-times

R. Temoltzi-Ávila INTERMATHS, 4(2), 9–27, December 2023 | 13

(

)

> α

(1), then

(

)

⊖ α

(1) =

(

n −

1). This observation allows us to introduce

the set of non-Newtonian integers, denoted by Z

, as the set

= {α(n) ∈ R

| n ∈ Z}.

The set of rational numbers Q

is naturally deﬁned as:

= {α(n)

⊘ α(m) ∈ R

| α(n), α(m) ∈ Z

and α(m) ̸= α(0)}.

, a

, . . . , a

are arbitrary non-Newtonian real numbers, then we use the following

notation to denote the sum of these numbers:

k=1

⊕ a

⊕ ···

⊕ a

. Using

(1) and induction it can be veriﬁed that:

⊕ a

= α(α

−1

) + α

−1

)),

⊕ a

= α(α

−1

(α(α

−1

) + α

−1

))) + α

−1

)),

= α(α

−1

) + α

−1

) + α

−1

)),

···

⊕ a

⊕ ···

⊕ a

= α(α

−1

) + α

−1

) + ··· + α

−1

)),

that is,

k=1

= α

k=1

−1

)

For each

a ∈ R

and each

(

)

∈ N

, we denote by

α(m)

the

(

)-th power of

which is obtained by α-multiplying m-times the number a with itself:

α(2)

= a

⊙ a = α(α

−1

(a) · α

−1

(a)) = α((α

−1

(a))

α(3)

= a

α(2)

⊙ a = α(α

−1

(α((α

−1

(a))

)) · α

−1

(a)) = α((α

−1

(a))

···

α(m)

= a

α(m−1)

⊙ a = α((α

−1

(a))

that is,

α(m)

= α((α

−1

(a))

), m ∈ N. (5)

We observe from (5) that

α(1)

(

−1

(

)) =

and that

α(0)

(1). We also

observe that

α(−1)

(

−1

(a)

), that is,

α(−1)

is the multiplicative inverse of

, see

Theorem 2.2. Therefore, we conclude that:

⊙ a

α(−1)

⊙ a

(1) for all

a ∈ R \ {α(0)}.

An application that is obtained as a consequence of (5) is that for each

a ∈ R

and

each m ∈ N, the equation b

α(m)

= a has the solution

b = α



−1

(a)



which is veriﬁed directly by substituting in the corresponding equation. This allows us

to introduce the

(

)-th root of a non-Newtonian real number

a ∈ R

, which is denoted

√

and is deﬁned as the number

√

= α



−1

(a)



. (6)

14 | https://doi.org/10.22481/intermaths.v4i2.13991 R. Temoltzi-Ávila

As a particular case, we observe that the α-square root of a ∈ R

is deﬁned as

√

= α



−1

(a)



In the same way, we introduce the

-absolute value of a non-Newtonian real number

a ∈ R

, which is denoted by |a|

and is deﬁned as

|a|

√

α(2)

= α



−1

(α((α

−1

(a))

))



= α(|α

−1

(a)|),

that is,

|a|

= α(|α

−1

(a)|). (7)

If we consider the possible cases for

a ∈ R

, we obtain as a consequence of the

representation (7), the deﬁnition of the function

-absolute value

|·|

: R

→ R

, which

is deﬁned as

|a|











a, if a

> α(0),

α(0), if a = α(0),

α(0)

⊖ a, if a

< α(0).

(8)

The particularity of the representations (7)–(8) is due to the fact that the

-absolute

value allows us to characterize the non-Newtonian real numbers

. Let

a ∈ R

. We

say that

is a positive non-Newtonian real number if it satisﬁes

≥ α

(0),

is a negative

non-Newtonian real number if it satisﬁes

≤ α

(0) and, ﬁnally, we say that

is an

unsigned non-Newtonian real number if it satisﬁes

(0). Therefore, the set of

non-Newtonian real numbers

admits the decomposition

∪ R

−

∪ {α

(0)

}

where

= {a ∈ R

| a

> α(0)}, R

−

= {a ∈ R

| a

< α(0)}.

We call

the set of positive non-Newtonian real numbers and

−

the set of negative

non-Newtonian real numbers. It is clear from the above deﬁnitions that

⊂ R

Furthermore, we call the set

∪ {α

(0)

}

the set of non-negative non-Newtonian

real numbers.

A property of the absolute value in the set of real numbers is that

· b

| · |b

for each

, b

∈ R

. This property also holds for the set of non-Newtonian real numbers

with respect to

-absolute value. To see that this is so, let

, b

∈ R

. Then it follows

that

⊙ b

= α(|α

−1

⊙ b

)|) = α(



α

−1

(α(α

−1

) · α

−1

)))



)

= α(|α

−1

) · α

−1

)|)

= α(|α

−1

)| · |α

−1

)|)

= α(α

−1

(α(|α

−1

)|)) · α

−1

(α(|α

−1

)|)))

= α(|α

−1

)|)

⊙ α(|α

−1

)|) = |b

⊙ |b

that is,

⊙ b

= |b

⊙ |b

. (9)

Another property of the absolute value in the set of real numbers is the triangular

inequality:

| ≤ |b

for each

, b

∈ R

. This property also holds for the set

R. Temoltzi-Ávila INTERMATHS, 4(2), 9–27, December 2023 | 15

of non-Newtonian real numbers. To see that this is so, let

, b

∈ R

. Then we observe

from (7) that

⊕ b

= α(|α

−1

⊕ b

)|) = α(|α

−1

(α(α

−1

)+α

−1

)))|) = α(|α

−1

)+α

−1

)|).

Therefore, if we apply α

−1

on both sides of this equality, we obtain

−1

(|b

⊕ b

) = |α

−1

) + α

−1

)| ≤ |α

−1

)| + |α

−1

)|,

and, since α is injective, we obtain

⊕ b

≤ α(|α

−1

)| + |α

−1

)|)

≤ α(α

−1

(α(|α

−1

)|)) + α

−1

(α(|α

−1

)|)))

= α(|α

−1

)|)

⊕ α(|α

−1

)|) = |b

⊕ |b

Therefore, for all b

, b

∈ R

we conclude that:

⊕ b

≤ |b

⊕ |b

. (10)

The distance between two numbers

, a

∈ R

is deﬁned as

⊖ a

. We observe

that this distance is an element of

, which follows from the representation (7), and

from observing that:

⊖ a

= α(|α

−1

⊖ a

)|) = α(|α

−1

(α(α

−1

) − α

−1

)))|)

= α(|α

−1

) − α

−1

)|),

that is,

⊖ a

= α(|α

−1

) − α

−1

)|). (11)

We observe that the inequality

⊖ a

≥ α

(0) is satisﬁed, since this is equivalent to

|α

−1

(

)

− α

−1

(

)

| ≥

0. Furthermore, it is clear that

⊖ a

(0) if, and only if,

−1

(

) =

−1

(

). On the other hand, it follows that the distance is symmetric, in the

sense that

⊖ a

(

|α

−1

(

)

− α

−1

(

)

) =

(

|α

−1

(

)

− α

−1

(

)

) =

⊖ a

Finally, for each

, a

∈ R

it holds

⊖ a

≤ |a

⊖ a

⊕ |a

⊖ a

, which is

obtained from (10), and from observing that if we choose

⊖ a

and

⊖ a

then

⊕ b

= (a

⊖ a

)

⊕ (a

⊖ a

) = α(α

−1

⊖ a

) + α

−1

⊖ a

))

= α(α

−1

) − α

−1

) + α

−1

) − α

−1

))

= α(α

−1

) − α

−1

)) = a

⊖ a

As a consequence, the function ϱ

: R

× R

→ R

deﬁned by

, a

) = |a

⊖ a

induces a metric on R

, which satisﬁes for all a

, a

∈ R

the following conditions:

1. ϱ

, a

)

≥ α(0).

2. ϱ

, a

) = α(0) if, and only if, a

= a

16 | https://doi.org/10.22481/intermaths.v4i2.13991 R. Temoltzi-Ávila

3. ϱ

, a

) = ϱ

, a

4. ϱ

, a

)

≤ ϱ

, a

)

⊕ ϱ

, a

The following result is obtained:

Theorem 2.6. The non-Newtonian space

is a metric space with metric

deﬁned

, a

) = |a

⊖ a

= α



|α

−1

) − α

−1



. (12)

The metric space (

, ϱ

) is called non-Newtonian metric space and

is called

non-Newtonian metric.

We observe that if in

we consider the standard metric

ϱ: R × R → R

deﬁned by

ϱ(b

, b

) = |b

− b

|, then

, a

) = α



ϱ(α

−1

), α

−1

))



, a

∈ R

, (13)

which is obtained from (11). This property allows us to observe that the topology

for

, which is induced by the metric

, is obtained from the topology

for

induced

by the metric ϱ.

Let

and

be two arbitrary non-Newtonian real numbers. We denote by (

a, b

)

the set

of non-Newtonian real numbers

x ∈ R

that satisfy the following inequalities

< x

< b

that is, the open interval (

a, b

)

{x ∈ R

| a

< x

< b}

. Equivalently, we identify this

set with the set of real numbers

−1

(

)

∈ R

that satisfy

−1

(

)

< α

−1

(

)

< α

−1

(

that is, with the open interval of real numbers (

−1

(

)

, α

−1

(

)). The above allows us to

express an open interval of non-Newtonian real numbers as

(a, b)

= α



(α

−1

(a), α

−1

(b))



. (14)

Similarly, we denote by [

a, b

]

{x ∈ R

≤ x

≤ b}

a closed interval of non-Newtonian

real numbers. Under an argument similar to the one used, we express a closed interval

of non-Newtonian real numbers as

[a, b]

= α



[α

−1

(a), α

−1

(b)]



. (15)

Let

a ∈ R

be arbitrary and

ϵ ∈ R

such that

> α

(0). An open non-Newtonian ball of

center

and radius

, which is denoted by

(

a, ϵ

), is deﬁned as the set of non-Newtonian

real numbers

x ∈ R

such that

(

x, a

)

< ϵ

, that is,

(

a, ϵ

) =

{x ∈ R

| ϱ

(

x, a

)

< ϵ}

According to (7) and (11), we observe that if

x ∈ B

(

a, ϵ

), then

(

|α

−1

(

⊖ a

)

< ϵ

, and

−1

(

⊖ a

) =

−1

(

)

−α

−1

(

), then we can express the last inequality in the following

equivalent form

−1

(

)

−α

−1

(

)

< α

−1

(

)

< α

−1

(

) +

−1

(

). As a consequence, we can

express a non-Newtonian open ball with center a ∈ R

and radius ϵ

> α(0) as

(a, ϵ) = α



B(α

−1

(a), α

−1

(ϵ))



, (16)

where

(

−1

(

)

, α

−1

(

)) is a open ball of center

−1

(

)

∈ R

and radius

−1

(

)

0, that

is, if

(

−1

(

)

, α

−1

(

)) represents the open interval (

−1

(

)

− α

−1

(

)

, α

−1

(

) +

−1

(

)).

R. Temoltzi-Ávila INTERMATHS, 4(2), 9–27, December 2023 | 17

Following the same reasoning, we observe that if the set

(

−1

(

)

, α

−1

(

)) describes

a closed ball of center

−1

(

)

∈ R

and radius

−1

(

)

0, that is, if

(

−1

(

)

, α

−1

(

))

represents the closed interval [

−1

(

)

− α

−1

(

)

, α

−1

(

) +

−1

(

)], then we can express a

closed non-Newtonian ball of center a and radius ϵ as

(a, ϵ) = α



B(α

−1

(a), α

−1

(ϵ))



. (17)

The expressions (16) and (17) allow us to observe that there is a dependence between

the topologies of

and

. We recall that a basis for a topology on

is a family

open sets such that each open subset of

is expressed as a union of some members of

; see [

]. In particular, it is well known that if

(

a, ϵ

)

| a ∈ R and ϵ > 0}

, then

is a basis for the usual

topology of

, which is induced by the usual metric

. Furthermore, it is well known that if

is a continuous function, then

−1

is also a

continuous function, since α is injective. We observe that from this it follows that

= {α(B(a, ϵ)) | B(a, ϵ) ∈ B}, (18)

is a basis for a topology T

for R

. This allows us to obtain the following result.

Theorem 2.7. Let

be deﬁned as in (18). Then

is a basis for a topology

for

that is induced by the function

from the usual topology

of the set of real numbers

3 Preliminaries of non-Newtonian differential calculus

The importance of the Theorem 2.7 allows us to reformulate the basic concepts for func-

tions of a variable in non-Newtonian calculus:

-limit,

-continuity,

-diﬀerentiability,

etc. These concepts allow us to introduce the basic principles of non-Newtonian diﬀer-

ential calculus.

Let

f : R

→ R

be a function and

x ∈ R

We say that

ℓ ∈ R

is the

-limit of the

function f as x tends to a ∈ R

, which is denoted by

lim

x→a

f(x) = ℓ, (19)

if for all

> α

(0) there exists a non-Newtonian real number

> α

(0) such that if

(0)

< ϱ

(

x, a

)

< δ

then

(

)

, ℓ

)

< ϵ

. Equivalently,

ℓ ∈ R

is the non-Newtonian

limit of the function

tends to

a ∈ R

, if for all

> α

(0) exists a non-Newtonian

real number δ

> α(0) such that if x ∈ B

(a, δ) \ {a}, then f(x) ∈ B

(ℓ, ϵ).

From the deﬁnition of

-limit of a function, we can verify that if

lim

x→a

(

) =

ℓ

and

lim

x→a

(x) = ℓ

, then

lim

x→a

(x)

⊕ f

(x)) = ℓ

⊕ ℓ

lim

x→a

(x)

⊖ f

(x)) = ℓ

⊖ ℓ

lim

x→a

(x)

⊙ f

(x)) = ℓ

⊙ ℓ

and if ℓ

̸= α(0), then

lim

x→a

(x)

⊘ f

(x)) = ℓ

⊘ ℓ

We can also consider functions

f : R

→ R

, where

is a set of non-Newtonian real numbers generated by a function

and

is a set of non-Newtonian real numbers

generated by a function σ. Some general cases can be consulted in [16].

18 | https://doi.org/10.22481/intermaths.v4i2.13991 R. Temoltzi-Ávila

According to the previous deﬁnitions, we can establish the meaning of the expression

lim

y→x

(

) =

(

) and introduce the concept of

-continuity of the function

at a

point

x ∈ R

. We say that

-continuous in

x ∈ R

, if for all

> α

(0) there exists a

non-Newtonian real number

> α

(0) such that if

(

x, z

)

< δ

then

(

)

, f

(

))

< ϵ

Equivalently, we say that

-continuous in

x ∈ R

, if for all

> α

(0) there exists a

non-Newtonian real number

> α

(0) such that

(

x, δ

))

⊂ B

(

)

, ϵ

). Additionally,

we can verify that f is α-continuous in x ∈ R

if, and only if,

lim

h→α(0)

f(x

⊕ h) = f(x).

We say that the function

-continuous over [

a, b

]

-continuous for

all

x ∈

[

a, b

]

. We denote by

[

a, b

] the set of functions

f :

[

a, b

]

→ R

that are

α-continuous over [a, b]

Analogously, we say that the function

-diﬀerentiable in

x ∈ R

if the following

limit exists:

lim

h→α(0)

((f(x

⊕ h)

⊖ f (x))

⊘ h).

In such a case, the α-derivative of f at x ∈ R

is denoted by

f(x)

lim

h→α(0)

((f(x

⊕ h)

⊖ f (x))

⊘ h). (20)

We say that the function

-diﬀerentiable over [

a, b

]

-diﬀerentiable for

all

x ∈

[

a, b

]

. We denote by

[

a, b

] the set of functions

f :

[

a, b

]

→ R

that are

α-diﬀerentiable over [a, b]

and such that its α-derivative is α-continuous over [a, b].

We note that if

-diﬀerentiable in

x ∈ R

, then

is also

-continuous in

, which

follows from observe that

lim

h→α(0)

(f(x

⊕ h)

⊖ f (x)) =

lim

h→α(0)

(((f(x

⊕ h)

⊖ f (x))

⊘ h)

⊙ h)



lim

h→α(0)

((f(x

⊕ h)

⊖ f (x))

⊘ h)



⊙



lim

h→α(0)



f(x)

⊙ α(0) = α(0),

which shows the result.

We consider an arbitrary function

f : R

→ R

and let

f : R → R

be the function

such that the following diagram is commutative:

R R

−1

(21)

that is, such that

(

−1

(

)) = (

−1

◦ f ◦ α

)(

−1

(

)) is satisﬁed for each

x ∈ R

. We

note that if

α: R → R

is continuous, then from the diagram (21) it follows that

R. Temoltzi-Ávila INTERMATHS, 4(2), 9–27, December 2023 | 19

lim

x→a

(

) =

ℓ

if, and only if,

lim

−1

(x)→α

−1

(a)

(

−1

(

)) =

−1

(

ℓ

). The above means

that

lim

x→a

f(x) = α

lim

−1

(x)→α

−1

(a)

f(α

−1

(x))

. (22)

Furthermore, if

α: R → R

is continuous and

f : R

→ R

-diﬀerentiable in

x ∈ R

then from diagram (21) it follows once again that:

f(x)

lim

h→α(0)

((f(x

⊕ h)

⊖ f (x))

⊘ h)

= lim

−1

(h)→0

f(α

−1

(x) + α

−1

(h)) −

f(α

−1

(x))

−1

(h)

= α

f(α

−1

(x))

dα

−1

(x)

(23)

whenever

is diﬀerentiable in

−1

(

). Other properties analogous to (22) and (23) can

be consulted in [21].

The properties that have been described show that there is a dependence between

Newtonian diﬀerential calculus and non-Newtonian diﬀerential calculus, which follows

from the dependence that exists between the topology of the space of Newtonian real

numbers and the topology of the space of non-Newtonian real numbers.

There is a similar dependence between Newtonian integral calculus and non-Newtonian

integral calculus. We brieﬂy mention this dependence. Let

f : R

→ R

. The non-

Newtonian

-integral, denoted by

(

) d

, is deﬁned by the requirement that, under

typical assumptions parallel to those of the fundamental theorem of Newtonian calculus,

it must be satisﬁed

f(y) dy = f(x),

f(x)

dx = f(b)

⊖ f (a),

which uniquely implies that

f(x) dx = α

−1

(b)

−1

(a)

f(α

−1

(x)) dα

−1

(x)

where

f is the function obtained from the commutative diagram (21); see [21].

Now we propose to verify that there is also a dependence in the set of Newtonian

diﬀerential equations and non-Newtonian diﬀerential equations, as a particular case, we

verify that this property is conserved in the set of ﬁrst-order linear diﬀerential equations.

Reachability tube of the non-Newtonian ﬁrst-order linear differential

equations

Using the

-derivative, we can consider ﬁnding the solutions of the non-Newtonian

ﬁrst-order linear diﬀerential equation with an external perturbation

y(x)

⊕ µ

⊙ y(x) = u(x), y(α(0)) = α(0), (24)

20 | https://doi.org/10.22481/intermaths.v4i2.13991 R. Temoltzi-Ávila

where

is a non-Newtonian real number and

(

) is an external perturbation that

belongs to the set of functions

= {u(x) ∈ C

[α(0), b] | δ

−

≤ u(x)

≤ δ

}, (25)

where

> α

(0) is ﬁxed,

−

and

are positive non-Newtonian real numbers that satisfy

the inequality δ

−

< α(0)

< δ

We note that (24)–(25) describes a family of non-Newtonian ﬁrst-order linear diﬀeren-

tial equations whose elements are indexed by an inclusion

(

)

∈ U

. In this family we

consider the following set:

= {(x, y(x))

⊤

∈ R

| y(x) satisﬁes (24) for some u(x) ∈ U

with x ∈ [α(0), b]

which is called the reachability tube of the family of non-Newtonian ﬁrst-order linear

diﬀerential equations (24)–(25). The importance of the reachability tube

in control

theory is due to the following geometric interpretation: this set contains all trajectories

of the solutions of the family of non-Newtonian ﬁrst-order linear diﬀerential equations

(27)–(28).

In what follows we propose to determine the set Q

If we consider the functions

ˆy

and

ˆu

that are associated with

and

, and such that

the following two diagrams are commutative

R R

−1

ˆy

R R

−1

ˆu

(26)

that is, such that

ˆy

(

−1

(

)) = (

−1

◦y◦α

)(

−1

(

)) and

ˆu

(

−1

(

)) = (

−1

◦u◦α

)(

−1

(

))

for each

x ∈ R

, and if we use the relation (23), then we can rewrite the non-Newtonian

ﬁrst-order diﬀerential equation (23) as

dˆy(ξ)

dξ

+ α

−1

(µ)ˆy(ξ) = ˆu(ξ), ˆy(0) = 0, (27)

where

−1

(

) for each

x ∈ R

. On the other hand, we observe that if

(

)

∈ U

then ˆu(ξ) ∈

U, where

U = {ˆu(ξ) ∈ C[0, α

−1

(b)] | α

−1

(δ

−

) ≤ ˆu(ξ) ≤ α

−1

(δ

)}. (28)

We observe that the resulting family of Newtonian ﬁrst-order linear diﬀerential

equations (27)–(28) is equivalent to the family of non-Newtonian ﬁrst-order linear

diﬀerential equations (24)–(25) under the action of the

-generator. Therefore, we can

formulate the problem of determining the reachability tube for the resulting family of

Newtonian ﬁrst-order linear diﬀerential equations (27)–(28), which is described by

Q = {(ξ, ˆy(ξ))

⊤

∈ R

| ˆy(ξ) satisﬁes (27) for some ˆu(ξ) ∈

with ξ ∈ [0, α

−1

(b)]},

This reachability tube is well known and has been used to establish a robust stability

criterion for a family of Newtonian ﬁrst-order linear diﬀerential equations similar to

R. Temoltzi-Ávila INTERMATHS, 4(2), 9–27, December 2023 | 21

(27)–(28), and which have been used to establish a robust stability criterion in some

cases of the heat equation; see [25].

We list some of its main properties.

We ﬁrst observe that if we choose an arbitrary external perturbation ˆu(ξ) ∈

, and

if we substitute this into (27), then we obtain the solution

ˆy(ξ) =

−α

−1

(µ)η

ˆu(ξ − η) dη, ξ ∈ [0, α

−1

(b)].

On the other hand, if we choose constant external perturbations

ˆu

(

) =

−1

(

), and

if we substitute these into (27), then the corresponding solutions are

ˆy

(ξ) = α

−1

(δ

)

−α

−1

(µ)η

dη =

−1

(δ

)

−1

(µ)



1 − e

−α

−1

(µ)ξ



, ξ ∈ [0, α

−1

(b)].

We observe that the functions

ˆy

(

) describe the boundary of the set

, since if

ˆu(ξ) ∈

is any arbitrary perturbation, then the following inequality is satisﬁed

ˆu

−

(η) ≤ ˆu(η) ≤ ˆu

−

(η), η ∈ [0, α

−1

(b)],

Therefore, if we multiply the members of this inequality by

−α

−1

(µ)η

, and if we then

integrate them from 0 to ξ, then we obtain that

ˆy

−

(ξ) ≤ ˆy(ξ) ≤ ˆy

(ξ), ξ ∈ [0, α

−1

(b)].

As a consequence of this property, the external perturbations

ˆu

(

) are called Newtonian

worst external perturbations. We also observe that the set of trajectories obtained from

the family of ﬁrst-order linear diﬀerential equations (27)-(28) ﬁll the set

, since if

(

, ν

) is a point such that

∈

, α

−1

(

)] and

ˆy

−

(

)

≤ ν

≤ ˆy

(

), then there exists

λ ∈

1] such that

λˆy

−

(

) + (1

−λ

)

ˆy

(

). In this case, if we choose the external

perturbation

ˆu

∗

(

) =

λˆu

−

(

) + (1

−λ

)

ˆu

(

)

∈

, and we substitute it in (27), then the

corresponding solution is described by

ˆy

∗

(

) =

λˆy

−

(

) + (1

− λ

)

ˆy

(

), which satisﬁes

ˆy

∗

(ξ

) = ν

. This shows that (ξ

, η

) is a point that belongs to

The above description shows that

Q = {(ξ, λˆy

−

(ξ) + (1 − λ)ˆy

(ξ))

⊤

∈ R

| with λ ∈ [0, 1] and ξ ∈ [0, α

−1

(b)]}.

Furthermore, we observe that

Q ⊂

−1

(δ

−

)

−1

(µ)

−1

(δ

)

−1

(µ)

since lim

ξ→+∞

ˆy

(ξ) =

−1

(δ

)

−1

(µ)

On the other hand, from the ﬁrst diagram of (26) we observe that if

(

) is a solution

associated with an external perturbation

(

)

∈ U

, then

(

) =

(

ˆy

(

−1

(

))) for each

x ∈ [α(0), b], that is

y(x) = α

−1

(x)

−α

−1

(µ)η

ˆu(α

−1

(x) − η) dη

, x ∈ [α(0), b].

22 | https://doi.org/10.22481/intermaths.v4i2.13991 R. Temoltzi-Ávila

Furthermore, if

is a continuous function, then we obtain the equality

(

) and,

as a particular case, we note that the boundary of the reachability tube

is described

by the functions y

(x) = α(ˆy

(α

−1

(x))), where

(x) = α

−1

(δ

)

−1

(µ)



1 − e

−α

−1

(µ)α

−1

(x)



Equivalently,

(x) = (δ

⊘ µ)

⊙ α(1 − e

−α

−1

(µ)α

−1

(x)

). (29)

The solutions in (29) are obtained as an eﬀect of the external perturbations

(

) =

for each

x ∈

[

(0)

, b

]

. We also note that

(

) =

(

ˆu

(

−1

(

))) for each

x ∈

[

(0)

, b

]

Furthermore, due to the order of the non-Newtonian real numbers, it follows that if

(

)

∈ U

is an arbitrary external perturbation, then the following inequalities are

satisﬁed

−

(x)

≤ y(x)

≤ y

(x), x ∈ [α(0), b].

We note that as a consequence of this property, the external perturbations

(

) will be

called worst non-Newtonian external perturbations. On the other hand, as a consequence

of the previous inequalities, we can observe that

Q = {(x, λ

⊙ y

−

(x)

⊕ (α(1)

⊖ λ)

⊙ y

(ξ))

⊤

∈ R

with λ ∈ [α(0), α(1)]

and x ∈ [α(0), b]}.

Finally, we observe that

⊂

−

⊘ µ, δ

⊘ µ

, since

lim

x→α(+∞)

(

) =

⊘ µ, where α(+∞) = lim

ξ→+∞

α(ξ).

We summarize the results obtained as follows.

Theorem 4.1. Consider the non-Newtonian ﬁrst-order linear diﬀerential equation

y(x)

⊕ µ

⊙ y(x) = u(x), y(α(0)) = α(0),

where

is a non-Newtonian real number and

(

)

∈ U

is an external perturbation.

Then the solution to this non-Newtonian ﬁrst-order linear diﬀerential equation is

y(x) = α

−1

(x)

−α

−1

(µ)η

ˆu(α

−1

(x) − η) dη

, x ∈ [α(0), b].

On the other hand, the boundary of the reachability tube

is represented by the graph

of the functions

(x) = α

−1

(δ

)

−1

(µ)



1 − e

−α

−1

(µ)α

−1

(x)



, x ∈ [α(0), b].

An analogous result can be obtained if we consider an initial condition

(

(0))

(0).

The following example shows the reachability tube obtained in a non-Newtonian

ﬁrst-order linear diﬀerential equation using some particular cases of non-Newtonian

calculus.

R. Temoltzi-Ávila INTERMATHS, 4(2), 9–27, December 2023 | 23

Example 4.2. We choose

(2) and

(

1), and consider as a particular

case the following non-Newtonian ﬁrst-order linear diﬀerential equation deﬁned on

[α(0), α(2)]

y(x)

⊕ α(2)

⊙ y(x) = u(x), y(α(0)) = α(0),

with

(

)

∈ U

, where

(

)

∈ C

[

(0)

, α

(2)]

| δ

−

≤ u

(

)

≤ δ

}

. We obtain the

following three particular cases of the functions

(

) that describe the boundary of the

reachability tube

when

x ∈

[

(0)

, α

(2)]. We note that all three graphs are plotted

in R

to compare the structure of the reachability tube Q

in each case.

If we choose

(

) =

(

), that is, if we consider a Newtonian calculus, then the

boundary of the reachability tube Q

is described by the functions

(x) = ±(1 − e

−2x

), x ∈ [0, 2]. (30)

An illustration of the corresponding reachability tube is shown in Figure 1.

0 1 2 3 4 5 6 7 8

−2

−1

(x)

Fig. 1. Reachability tube in a Newtonian calculus with boundary described by the functions (30).

On the other hand, if we assume that

(

) =

exp

(

), that is, if we consider a

multiplicative calculus, or geometric calculus, then the boundary of the reachability

tube Q

is described by functions

(x) = e



1−



, x ∈ [1, e

]

exp

. (31)

An illustration of the corresponding reachability tube is shown in Figure 2.

0 1 2 3 4 5 6 7 8

−2

−1

(x)

Fig. 2. Reachability tube in a multiplicative calculus, or geometric calculus, with boundary described

by the functions (31).

Finally, if we consider that

(

) =

−1

sinh

(

κx

), that is, if we consider a Kaniadakis

-calculus, then the boundary of the reachability tube

is described by the functions

(x) = ±κ

−1

sinh



1 − e

−2κ sinh(κx)



, x ∈

0, κ

−1

sinh(2κ)

. (32)

24 | https://doi.org/10.22481/intermaths.v4i2.13991 R. Temoltzi-Ávila

An illustration of the corresponding reachability tube is shown in Figure 3 with

0 1 2 3 4 5 6 7 8

−2

−1

(x)

Fig. 3. Reachability tube in a Kaniadakis

-calculus with boundary described by the functions (32).

Other values for the parameters and other

-generators can be chosen to obtain

similar illustrations of the corresponding reachability tube Q

5 Conclusion

This paper presented the algebraic and topological properties of non-Newtonian real

numbers, as well as some elements of non-Newtonian diﬀerential calculus. Using the

deﬁnition of a non-Newtonian derivative, a non-Newtonian ﬁrst-order linear diﬀerential

equation that admits external perturbations has been considered, and for this, the

problem of determining the reachability tube has been posed.

ORCID

R.Temoltzi-Ávila https://orcid.org/0000-0003-4462-2197

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