SOME PROPERTIES OF A TRIPLE SEQUENCE SPACES OF FUZZY REAL NUMBERS BY DOUBLE ORLICZ FUNCTIONS

THE USA JOURNALS

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PUBLISHED DATE: - 08-10-2024
DOI: -

https://doi.org/10.37547/tajssei/Volume06Issue10-05

PAGE NO.: - 47-55

SOME PROPERTIES OF A TRIPLE SEQUENCE
SPACES OF FUZZY REAL NUMBERS BY
DOUBLE ORLICZ FUNCTIONS

Alaa Fawzi Dabbas

Ministry of Education, Najaf Education Directorate, Iraq

INTRODUCTION

Zadeh established the idea of fuzzy set theory [1]. Based on this, other authors have introduced and
examined sequences of fuzzy numbers, studying significant properties. In Bromwich, the work on the
double sequence of real numbers was discovered [2]. In addition, a number of people, including Moricz
[3], Basarir and Sonalncan [4], and others, exacted the double sequence.

𝐸

𝐹

3

, denoting the family of all

ℝ

or

𝕔

triple sequences, is used throughout this article.

A triple sequence in [5][6] is a function from (

ℕ

)to(

ℝ

or

𝕔

). Using a function

Υ

where

Υ =

(Υ

1

(𝔛). Υ

2

(𝔐)).

novel outcomes of triple sequence spaces would be examined by double Orlicz function.

Assume that

(𝔛. 𝔐) = (𝔛

𝔥.𝔘.𝔑

. . 𝔐

𝔥.𝔘.𝔑

).

symbolizes a triple infinite array of elements with the name

(𝔛

𝔥.𝔘.𝔑

) . (𝔐

𝔥.𝔘.𝔑

)

.

𝔛 = (𝔛

𝔥.𝔘.𝔑

)

be an infinite array of elements

𝔛

𝔥.𝔘.𝔑

, and

𝔐 = (𝔐

𝔥.𝔘.𝔑

)

be an endless

array of elements

𝔐

𝔥.𝔘.𝔑

.

respectively.

Prof. Wlayshaw Roman Orlicz, a Polish scholar, established the concept of the Orlicz function and gave
it his name; consequently, he created the Orlicz space [7].

2)Definitions and preliminaries

Definition 2.1

Assume that

the sequence 𝔛 = (𝔛

𝔥.𝔘.𝔑

). 𝔐 = ( 𝔐

𝔥.𝔘.𝔑

)

are Cauchy, then

(𝔛. 𝔐) = (𝔛

𝔥.𝔘.𝔑

. 𝔐

𝔥.𝔘.𝔑

)

is

said to be a triple Cauchy sequence if for every

𝜖 > 0

there exists

𝑁 ∈ ℕ

such that

𝒹̅ ((𝔛

𝔥.𝔘.𝔑

. 𝔛

𝑖.𝔧.𝑒

). ( 𝔐

𝔥.𝔘.𝔑

. 𝔐

𝑖.𝔧.𝑒

)) < 𝜖.

for all

𝑖 ≥ 𝔥 ≥ 𝑁. 𝔧 ≥ 𝔘 ≥ 𝑁.

𝑒 ≥ 𝔑 ≥ 𝑁

where

𝒹̅(𝔛

𝔥.𝔘.𝔑

. 𝔛

𝑖.𝔧.𝑒

) <

𝜖

and

𝒹̅( 𝔐

𝔥.𝔘.𝔑

. 𝔐

𝑖.𝔧.𝑒

) < 𝜖

.

Definition 2.2

A Triple sequence space

𝐸

𝐹

3

is said to be

convergence free if

(𝔅

𝔥.𝔘.𝔑

. 𝔔

𝔥.𝔘.𝔑

) ∈ 𝐸

𝐹

3

.

whenever

RESEARCH ARTICLE

Open Access

Abstract

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(𝔛

𝔥.𝔘.𝔑

. 𝔐

𝔥.𝔘.𝔑

) ∈ 𝐸

𝐹

3

and

(𝔛

𝔥.𝔘.𝔑

. 𝔐

𝔥.𝔘.𝔑

) = (0.0)

implies

(𝔅

𝔥.𝔘.𝔑

. 𝔔

𝔥.𝔘.𝔑

) = (0.0)

.

Definition 2.3

If

(𝔛

𝔥.𝔘.𝔑

. 𝔐

𝔥.𝔘.𝔑

). (𝔅

𝔥.𝔘.𝔑

. 𝔔

𝔥.𝔘.𝔑

) ∈ 𝐸

𝐹

3

suchthat

𝒹̅ ((𝔛

𝔥.𝔘.𝔑

. 0̅). (𝔐

𝔥.𝔘.𝔑

. 0̅)) ≤ 𝒹̅ ((𝔅

𝔥.𝔘.𝔑

. 0̅). (𝔔

𝔥.𝔘.𝔑

. 0̅))

for all

𝔥. 𝔘. 𝔑 ∈ ℕ.

then

𝐸

𝐹

3

is said to be solid.

Definition 2.4

A canonical pre-image of a triple sequence of

(𝔛

𝔥.𝔘.𝔑

. 𝔐

𝔥.𝔘.𝔑

) ∈ 𝐸

𝐹

3

is a triple sequence

(𝔅

𝔥.𝔘.𝔑

. 𝔔

𝔥.𝔘.𝔑

) ∈ 𝐸

𝐹

3

defined as follows:

(𝔅

𝔥.𝔘.𝔑

. 𝔔

𝔥.𝔘.𝔑

) = {

(𝔛

𝔥.𝔘.𝔑

. 𝔐

𝔥.𝔘.𝔑

). If (𝔥. 𝔘. 𝔑) ∈ 𝐾.

(0̅. 0̅).

otherwise.

Definition 2.5:

If all of the canonical pre-images of a triple sequence space

𝐸

𝐹

3

are included within it, then

𝐸

𝐹

3

is

considered monoton.

Definition 2.6:

For any

(𝔛. 𝔐) = (𝔛

𝔥.𝔘.𝔑

. 𝔐

𝔥.𝔘.𝔑

) ∈ 𝐸

𝐹

3

, a

𝐸

𝐹

3

is considered symmetric if

𝐺(𝔛. 𝔐) ⊆ 𝐸

𝐹

3

where

𝐺(𝔛. 𝔐)

=

{(𝔛

𝜋(𝔥)𝜋(𝔘)𝜋(𝔑)

. 𝔐

𝜋(𝔥)𝜋(𝔘)𝜋(𝔑)

): 𝜋 is a permutation of ℕ × ℕ × ℕ}

.

We now define the following classes of triple sequences

:

(𝑙

∞

)

𝐹

3

Υ

=

{

(𝔛

𝔰𝔥.𝔘.𝔑

. 𝔐

𝔥.𝔘.𝔑

) ∈ 𝐸

𝐹

3

: 𝑠𝑢𝑝

𝔥.𝔘.𝔑

{Υ

1

(

𝒹̅(𝔛

𝔥.𝔘.𝔑

. 0̅)

𝜌

) ∨ Υ

2

(

𝒹̅(𝔐

𝔥.𝔘.𝔑

. 0̅)

𝜌

)} < ∞

for some 𝜌 > 0.

}

.

where

(𝑙

∞

)

𝐹

3

Υ

1

=

{

(𝔛

𝔥.𝔘.𝔑

) ∈ 𝐸

𝐹

3

: 𝑠𝑢𝑝

𝔥.𝔘.𝔑

{Υ

1

(

𝒹̅(𝔛

𝔥.𝔘.𝔑

. 0̅)

𝜌

) } < ∞

for some 𝜌 > 0.

}

.

and

(𝑙

∞

)

𝐹

3

Υ

2

=

{

(𝔐

𝔥.𝔘.𝔑

) ∈ 𝐸

𝐹

3

: 𝑠𝑢𝑝

𝔥.𝔘.𝔑

{Υ

2

(

𝒹̅(𝔐

𝔥.𝔘.𝔑

. 0̅)

𝜌

)} < ∞

for some 𝜌 > 0.

}

.

(𝒸)

𝐹

3

Υ

=

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{

(𝔛

𝔥.𝔘.𝔑

. 𝔐

𝔥.𝔘.𝔑

) ∈ 𝐸

𝐹

3

: 𝑙𝑖𝑚

𝔥.𝔘.𝔑

{Υ

1

(

𝒹̅(𝔛

𝔥.𝔘.𝔑

. ℓ

1

)

𝜌

) ∨ Υ

2

(

𝒹̅(𝔐

𝔥.𝔘.𝔑

. ℓ

2

)

𝜌

)} = 0

for some 𝜌 > 0.

}

.

where

(𝒸)

𝐹

3

Υ

1

=

{

(𝔛

𝔥.𝔘.𝔑

) ∈ 𝐸

𝐹

3

: 𝑙𝑖𝑚

𝔥.𝔘.𝔑

{Υ

1

(

𝒹̅(𝔛

𝔥.𝔘.𝔑

. ℓ

1

)

𝜌

)} = 0

for some 𝜌 > 0.

}

and

(𝒸)

𝐹

3

Υ

2

=

{

(𝔐

𝔥.𝔘.𝔑

) ∈ 𝐸

𝐹

3

: 𝑙𝑖𝑚

𝔥.𝔘.𝔑

{Υ

2

(

𝒹̅(𝔐

𝔥.𝔘.𝔑

. ℓ

2

)

𝜌

)} = 0

for some 𝜌 > 0.

}

.

(𝒸

0

)

𝐹

3

Υ

=

{

(𝔛

𝔥.𝔘.𝔑

. 𝔐

𝔥.𝔘.𝔑

) ∈ 𝐸

𝐹

3

: 𝑙𝑖𝑚

𝔥.𝔘.𝔑

{Υ

1

(

𝒹̅(𝔛

𝔥.𝔘.𝔑

. 0̅)

𝜌

) ∨ Υ

2

(

𝒹̅(𝔐

𝔥.𝔘.𝔑

. 0̅)

𝜌

)} = 0

for some 𝜌 > 0.

}

.

where

(𝒸

0

)

𝐹

3

Υ

1

{

(𝔛

𝔥.𝔘.𝔑

) ∈ 𝐸

𝐹

3

: 𝑙𝑖𝑚

𝔥.𝔘.𝔑

{Υ

1

(

𝒹̅(𝔛

𝔥.𝔘.𝔑

. 0̅)

𝜌

) } = 0

for some 𝜌 > 0.

}

.

and

(𝒸

0

)

𝐹

3

Υ

2

{

(𝔐

𝔥.𝔘.𝔑

) ∈ 𝐸

𝐹

3

: 𝑙𝑖𝑚

𝔥.𝔘.𝔑

{Υ

2

(

𝒹̅(𝔐

𝔥.𝔘.𝔑

. 0̅)

𝜌

)} = 0

𝜌 > 0.

}

.

Moreover, The triple sequence classes are defined by us.

(𝑚)

𝐹

3

= (𝒸)

𝐹

3

(Υ) ∩ (𝑙

∞

)

𝐹

3

(Υ).

(𝑚

0

)

𝐹

3

= (𝒸

0

)

𝐹

3

(Υ) ∩ (𝑙

∞

)

𝐹

3

(Υ).

3) Main Results
Theorem 3.1:

Let

(𝓅

𝔥.𝔘.𝔑

)

be bounded. Then the classes of triple sequences

(𝑙

∞

)

𝐹

3

(Υ . 𝓅)

is complete

metric spaces with respect to the distance defined by

𝐺((𝔛. 𝔐). (𝔅. 𝔔))

=

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𝑖𝑛𝑓 {𝜌

𝓅𝔥.𝔘.𝔑

𝒥

> 0: 𝑠𝑢𝑝

𝔥.𝔘.𝔑

{(Υ

1

(

𝒹̅(𝔛

𝔥.𝔘.𝔑

.𝔐

𝔥.𝔘.𝔑

)

𝜌

)) ∨ (Υ

2

(

𝒹̅(𝔅

𝔥.𝔘.𝔑

.𝔔

𝔥.𝔘.𝔑

)

𝜌

))} ≤ 1}

𝒥 = max (1. 2

𝑇−1

)

where

𝐺(𝔛. 𝔐) = 𝑖𝑛𝑓 {𝜌

𝓅

𝔥.𝔘.𝔑

𝒥

> 0: 𝑠𝑢𝑝

𝔥.𝔘.𝔑

(Υ

1

(

𝒹̅(𝔛

𝔥.𝔘.𝔑

. 𝔐

𝔥.𝔘.𝔑

)

𝜌

)) ≤ 1}

𝐺(𝔅. 𝔔) = 𝑖𝑛𝑓 {𝜌

𝓅

𝔰.𝔯.𝓋

𝒥

> 0: 𝑠𝑢𝑝

𝔥.𝔘.𝔑

(Υ

2

(

𝒹̅(𝔅

𝔥.𝔘.𝔑

. 𝔔

𝔥.𝔘.𝔑

)

𝜌

)) ≤ 1}

Proof

: Let us consider the case

(𝑙

∞

)

𝐹

3

(Υ. 𝓅)

and the other cases can be established next similar

techniques..
Let

(𝔛

𝑖

). (𝔐

𝑖

)

be any Cauchy sequences in

(𝑙

∞

)

𝐹

3

(Υ

1

. 𝓅).

(𝑙

∞

)

𝐹

3

(Υ

2

. 𝓅)

respectively, hence

(𝔛

𝑖

. 𝔐

𝑖

) =

(𝔛

𝔥.𝔘.𝔑

𝑖

. 𝔐

𝔥.𝔘.𝔑

𝑖

)

be a triple Cauchy sequence

Let

𝜖 > 0

,

𝔛

0

. 𝑟 > 0

be fixed. Then

∀

there exists a positive integer N such that

𝐺

Υ

1

(𝔛

𝑖

. 𝔛

𝑗

) <

𝜖

𝑟𝔛

0

and

𝐺

Υ

2

(𝔐

𝑖

. 𝔐

𝑗

) <

𝜖

𝑟𝔛

0

.

for

𝑖. 𝑗 ≥ 𝑁.

and consequently,

𝐺

Υ

((𝔛

𝑖

. 𝔛

𝑗

). (𝔐

𝑖

. 𝔐

𝑗

)) = (𝐺

Υ

1

(𝔛

𝑖

. 𝔛

𝑗

). 𝐺

Υ

2

(𝔐

𝑖

. 𝔐

𝑗

)) <

𝜖

𝑟𝔛

0

,

for all

𝑖. 𝑗 ≥ 𝑁.

By definition of

𝐺.

we obtain

𝑖𝑛𝑓 {𝜌 > 0: 𝑠𝑢𝑝

𝔥.𝔘.𝔑

{Υ

1

(

𝒹̅ (𝔛

𝔥.𝔘.𝔑

𝑖

. 𝔛

𝔥.𝔘.𝔑

𝑗

)

𝜌

) ∨ Υ

2

(

𝒹̅ (𝔐

𝔥.𝔘.𝔑

𝑖

. 𝔐

𝔥.𝔘.𝔑

𝑗

)

𝜌

)} ≤ 1}.

Thus,

𝑠𝑢𝑝

𝔥.𝔘.𝔑

{Υ

1

(

𝒹̅ (𝔛

𝔥.𝔘.𝔑

𝑖

. 𝔛

𝔥.𝔘.𝔑

𝑗

)

𝜌

) ∨ Υ

2

(

𝒹̅ (𝔐

𝔥.𝔘.𝔑

𝑖

. 𝔐

𝔥.𝔘.𝔑

𝑗

)

𝜌

)} ≤ 1.

for all

𝑖. 𝑗 ≥ 𝑁.

⟹ 𝑠𝑢𝑝

𝔥.𝔘.𝔑

{Υ

1

(

𝒹̅ (𝔛

𝔥.𝔘.𝔑

𝑖

. 𝔛

𝔥.𝔘.𝔑

𝑗

)

𝐺

Υ

1

(𝔛

𝔥.𝔘.𝔑

𝑖

. 𝔛

𝔥.𝔘.𝔑

𝑗

)

) ∨ Υ

2

(

𝒹̅ (𝔐

𝔥.𝔘.𝔑

𝑖

. 𝔐

𝔥.𝔘.𝔑

𝑗

)

𝐺

Υ

2

(𝔐

𝔥.𝔘.𝔑

𝑖

. 𝔐

𝔥.𝔘.𝔑

𝑗

)

)} ≤ 1.

for each

𝑖. 𝑗 ≥ 𝑁.

Since

𝓅

𝔥.𝔘.𝔑

bounded it follows that

{Υ

1

(

𝒹̅ (𝔛

𝔥.𝔘.𝔑

𝑖

. 𝔛

𝔥.𝔘.𝔑

𝑗

)

𝐺

Υ

1

(𝔛

𝔥.𝔘.𝔑

𝑖

. 𝔛

𝔥.𝔘.𝔑

𝑗

)

) ∨ Υ

2

(

𝒹̅ (𝔐

𝔥.𝔘.𝔑

𝑖

. 𝔐

𝔥.𝔘.𝔑

𝑗

)

𝐺

Υ

2

(𝔐

𝔥.𝔘.𝔑

𝑖

. 𝔐

𝔥.𝔘.𝔑

𝑗

)

)} ≤ 1.

for each

𝔰. 𝔯. 𝓋 ≥ 1

and for all

𝑖. 𝑗 ≥ 𝑁.

Hence one can find

𝑟 > 0

with

Υ

1

(

𝑟𝔛

0

2

) ≥ 1

and

Υ

2

(

𝑟𝔛

0

2

) ≥ 1

, such that

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Υ

1

(

𝒹̅ (𝔛

𝔥.𝔘.𝔑

𝑖

. 𝔛

𝔥.𝔘.𝔑

𝑗

)

𝐺

Υ

1

(𝔛

𝔥.𝔘.𝔑

𝑖

. 𝔛

𝔥.𝔘.𝔑

𝑗

)

) ≤ Υ

1

(

𝑟𝔛

0

2

) and Υ

2

(

𝒹̅ (𝔐

𝔥.𝔘.𝔑

𝑖

. 𝔐

𝔥.𝔘.𝔑

𝑗

)

𝐺

Υ

2

(𝔐

𝔥.𝔘.𝔑

𝑖

. 𝔐

𝔥.𝔘.𝔑

𝑗

)

) ≤ Υ

2

(

𝑟𝔛

0

2

)

Hence,

Υ (

𝑟𝔛

0

2

.

𝑟𝔛

0

2

)

=

(Υ

1

(

𝑟𝔛

0

2

) . Υ

2

(

𝑟𝔛

0

2

)) ≥ (1.1)

therefore,

{Υ

1

(

𝒹̅ (𝔛

𝔥.𝔘.𝔑

𝑖

. 𝔛

𝔥.𝔘.𝔑

𝑗

)

𝐺

Υ

1

(𝔛

𝔥.𝔘.𝔑

𝑖

. 𝔛

𝔥.𝔘.𝔑

𝑗

)

) . Υ

2

(

𝒹̅ (𝔐

𝔥.𝔘.𝔑

𝑖

. 𝔐

𝔥.𝔘.𝔑

𝑗

)

𝐺

Υ

2

(𝔐

𝔥.𝔘.𝔑

𝑖

. 𝔐

𝔥.𝔘.𝔑

𝑗

)

)} ≤ (Υ

1

(

𝑟𝔛

0

2

) . Υ

2

(

𝑟𝔛

0

2

)).

This implies that.

𝒹̅ (𝔛

𝔥.𝔘.𝔑

𝑖

. 𝔛

𝔥.𝔘.𝔑

𝑗

) ≤

𝑟𝔛

0

2

∙ 𝐺

Υ

1

((𝔛

𝑖

. 𝔛

𝑗

))

for all

𝑖. 𝒿 ≥ 𝑛

0

.

𝒹̅ (𝔛

𝔥.𝔘.𝔑

𝑖

. 𝔛

𝔥.𝔘.𝔑

𝑗

) ≤

𝑟𝔛

0

2

∙

𝜖

𝑟𝔛

0

=

𝜖
2

𝑖. 𝒿 ≥ 𝑛

0

.

⟹ 𝑑̅ (𝔛

𝔥.𝔘.𝔑

𝑖

. 𝔛

𝔥.𝔘.𝔑

𝑗

) ≤

𝜖
2

𝑖. 𝒿 ≥ 𝑛

0

.

and

𝒹̅ (𝔐

𝔥.𝔘.𝔑

𝑖

. 𝔐

𝔥.𝔘.𝔑

𝑗

) ≤

𝑟𝔛

0

2

∙ 𝐺

Υ

2

((𝔐

𝑖

. 𝔐

𝑗

))

𝑖. 𝒿 ≥ 𝑛

0

.

𝒹̅ (𝔐

𝔥.𝔘.𝔑

𝑖

. 𝔐

𝔥.𝔘.𝔑

𝑗

) ≤

𝑟𝔛

0

2

∙

𝜖

𝑟𝔛

0

=

𝜖
2

𝑖. 𝒿 ≥ 𝑛

0

⟹ 𝒹̅ (𝔐

𝔥.𝔘.𝔑

𝑖

. 𝔐

𝔥.𝔘.𝔑

𝑗

) ≤

𝜖
2

𝑖. 𝒿 ≥ 𝑛

0

.

then

𝒹̅ ((𝔛

𝔥.𝔘.𝔑

𝑖

. 𝔛

𝔥.𝔘.𝔑

𝑗

) . (𝔐

𝔥.𝔘.𝔑

𝑖

. 𝔐

𝔥.𝔘.𝔑

𝑗

)) ≤

𝑟𝔛

0

2

.

𝜖

𝑟𝔛

0

=

𝜖
2

𝑖. 𝒿 ≥ 𝑛

0 .

Hence (

𝔛

𝔥.𝔘.𝔑

𝑖

. 𝔐

𝔥.𝔘.𝔑

𝑖

)

is a triple Cauchy sequence in

𝑅

3

.

Thus,
For each

(0 < 𝜖 < 1)

,there exists a positive integer

𝑁

such that

𝒹̅ ((𝔛

𝔥.𝔘.𝔑

𝑖

. 𝔛). (𝔐

𝔥.𝔘.𝔑

𝑖

. 𝔐))

< 𝜖

for

all

𝑖. 𝑗 ≥ 𝑁.

where

𝒹̅(𝔛

𝑖

. 𝔛) < 𝜖

and

𝒹

̅ (𝔐

𝑖

. 𝔐) < 𝜖

for all

𝑖. 𝑗 ≥ 𝑁.

Taking

𝒿 → ∞

and fixing

𝑖.

so by using the continuity of

Υ = (Υ

1

. Υ

2

)

we get

𝑠𝑢𝑝

𝔥.𝔘.𝔑

{Υ

1

(

𝒹̅(𝔛

𝔥.𝔘.𝔑

𝑖

. lim

𝑗→∞

𝔛

𝔥.𝔘.𝔑

𝑗

)

𝜌

) ∨ Υ

2

(

𝒹̅(𝔐

𝔥.𝔘.𝔑

𝑖

. lim

𝑗→∞

𝔐

𝔥.𝔘.𝔑

𝑗

)

𝜌

)} ≤ 1

Thus,

𝑠𝑢𝑝

𝔥.𝔘.𝔑

{Υ

1

(

𝒹̅(𝔛

𝔥.𝔘.𝔑

𝑖

. 𝔛)

𝜌

) ∨ Υ

2

(

𝒹̅(𝔐

𝔥.𝔘.𝔑

𝑖

. 𝔐)

𝜌

)} ≤ 1.

On taking the infimum of such

𝜌

' s, we get,

𝑖𝑛𝑓 {𝜌 > 0: 𝑠𝑢𝑝

𝔥.𝔘.𝔑

{Υ

1

(

𝒹̅(𝔛

𝔥.𝔘.𝔑

𝑖

. 𝔛)

𝜌

) ∨ Υ

2

(

𝒹̅(𝔐

𝔥.𝔘.𝔑

𝑖

. 𝔐)

𝜌

)} ≤ 1} ≤ 𝜖

for all

𝑖 ≥ 𝑁

and

𝑗 → ∞.

Since

(𝔛

𝑖

. 𝔐

𝑖

) ∈ (𝑙

∞

)

𝐹

3

(Υ . 𝓅)

and

Υ

is continuous, it follows that

(𝔛. 𝔐) ∈ (𝑙

∞

)

𝐹

3

(Υ . 𝓅)

.

Proposition 3.1:

The class of triple sequences

(𝑙

∞

)

𝐹

3

(Υ)

is symmetric but the class of triple sequences

(𝒸)

𝐹

3

(Υ).

(𝒸

0

)

𝐹

.

(Υ

), are not symmetric.

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Proof:

Noticeably the class of triple sequence

(𝑙

∞

)

𝐹

3

(Υ)

is symmetric. However, other the class of triple

sequences, could be indicated by the example below.

Example 3.1 :

Let's say the class of triple sequences

(𝒸)

𝐹

3

(Υ)

. Consider

Υ (𝔛. 𝔐) = (𝔛. 𝔐)

and suppose the triple sequence

(𝔛

𝔥.𝔘.𝔑

. 𝔐

𝔥.𝔘.𝔑

)

be defined by

𝔭

where

(𝔛

1𝔘

)(𝔭) = {

(𝔭 + 1). for − 1 ≤ 𝔭 ≤ 0;

(−𝔭 + 1). for 0 ≤ 𝔭 ≤ 1;

0. otherwise,

and

(𝔐

1𝔘

)(𝔭) = {

(𝔭 + 1). for − 1 ≤ 𝔭 ≤ 0;

(−𝔭 + 1). for 0 ≤ 𝔭 ≤ 1;

0. otherwise.

For

𝔥 > 1

, we have

(𝔛

𝔥.𝔘.𝔑

. 𝔐

𝔥.𝔘.𝔑

)(𝔭) = {

(𝔭 + 2. 𝔭 + 2)

for − 2 ≤ 𝔭 ≤ −1;

(−𝔭. −𝔭)

for − 1 ≤ 𝔭 ≤ 0;

(0.0)

otherwise.

where

(𝔛

𝔥.𝔘.𝔑

)(𝔭) = {

𝔭 + 2. for − 2 ≤ 𝔭 ≤ −1;
−𝔭. for − 1 ≤ 𝔭 ≤ 0;

0. otherwise.

and

(𝔐

𝔥.𝔘.𝔑

)(𝔭) = {

𝔭 + 2. for − 2 ≤ 𝔭 ≤ −1;

−𝔭. for − 1 ≤ 𝔭 ≤ 0;
0. otherwise.

Let

(𝔅

𝔥.𝔘.𝔑

)

,

(𝔔

𝔥.𝔘.𝔑

)

be a rearrangement of

(𝔛

𝔥.𝔘.𝔑

)

,

(𝔐

𝔥.𝔘.𝔑

)

respectively which is defined by

(𝔅

𝔥.𝔥

)(𝔭) = {

𝔭 + 1. for − 1 ≤ 𝔭 ≤ 0;

−𝔭 + 1. for 0 ≤ 𝔭 ≤ 1;

0. otherwise.

and

(𝔔

𝔥.𝔥

)(𝔭) = {

𝔭 + 1. for − 1 ≤ 𝔭 ≤ 0 ;

−𝔭 + 1. for 0 ≤ 𝔭 ≤ 1;

0. otherwise.

Therefore,

(𝔅

𝔥.𝔥

. 𝔔

𝔥.𝔥

)

can be defined by

(𝔅

𝔥.𝔥

. 𝔔

𝔥.𝔥

)(𝔭) = {

(𝔭 + 1. 𝔭 + 1).

for − 1 ≤ 𝔭 ≤ 0;

(−𝔭 + 1. −𝔭 + 1).

for 0 ≤ 𝔭 ≤ 1;

(0.0). otherwise.

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and for

𝔥 ≠ 𝔘.

we have

(𝔅

𝔥.𝔘.𝔑

. 𝔔

𝔥.𝔘.𝔑

)(𝔭)

=

{

(𝔭 + 2 . 𝔭 + 2). for − 2 ≤ 𝔭 ≤ −1.

(−𝔭 . −𝔭). for − 1 ≤ 𝔭 ≤ −1.

( 0 . 0). otherwise.

}

where

(𝔅

𝔥.𝔘.𝔑

)(𝔭) = {

𝔭 + 2. for − 2 ≤ 𝔭 ≤ −1;

−𝔭. for − 1 ≤ 𝔭 ≤ 0;

0. otherwise.

and

(𝔔

𝔥.𝔘.𝔑

)(𝔭) = {

𝔭 + 2. for − 2 ≤ 𝔭 ≤ −1;

−𝔭. for − 1 ≤ 𝔭 ≤ 0;

0. otherwise.

Thus,

(𝔛

𝔥.𝔘.𝔑

. 𝔐

𝔥.𝔘.𝔑

) ∈ (𝒸)

𝐹

3

(Υ)

but

(𝔅

𝔥.𝔘.𝔑

. 𝔔

𝔥.𝔘.𝔑

) ∉ (𝒸)

𝐹

3

(Υ).

Hence

(𝒸)

𝐹

3

(Υ)

is not symmetric. In same sense,

it could

be indicated that other spaces of triple sequences are

not

symmetric too.

Proposition 3.2:

The

classes

of triple sequences

(𝑙

∞

)

𝐹

3

(Υ).

(𝒸

0

)

𝐹

3

(Υ)

and

(𝑚

0

)

𝐹

3

(Υ)

are solid

.

Proof :

Consider

(𝑙

∞

)

𝐹

3

(Υ)

the class of triple sequences.

So

(𝔛

𝔥.𝔘.𝔑

. 𝔐

𝔥.𝔘.𝔑

)

∈ (𝑙

∞

)

𝐹

3

(Υ)

and

(𝔅

𝔥.𝔘.𝔑

. 𝔔

𝔥.𝔘.𝔑

)

be such that

.

𝒹̅(𝔅

𝔥.𝔘.𝔑

. 0̅) ≤ 𝒹̅(𝔛

𝔥.𝔘.𝔑

. 0̅)

and

𝒹̅(𝔔

𝔥.𝔘.𝔑

. 0̅) ≤ 𝒹̅(𝔐

𝔥.𝔘.𝔑

. 0̅)

and consequently

𝒹̅ ((𝔅

𝔥.𝔘.𝔑

. 0̅). (𝔔

𝔥.𝔘.𝔑

. 0̅)) ≤ 𝒹̅ ((𝔛

𝔥.𝔘.𝔑

. 0̅). (𝔐

𝔥.𝔘.𝔑

. 0̅))

as

Υ = (Υ

1

. Υ

2

)

is increasing , we have

𝑠𝑢𝑝

𝔥.𝔘.𝔑

{Υ

1

(

𝑑̅ ((𝔅

𝔥.𝔘.𝔑

. 0̅). (𝔔

𝔥.𝔘.𝔑

. 0̅))

𝜌

)}

≤ 𝑠𝑢𝑝

𝔥.𝔘.𝔑

{Υ

2

(

𝑑̅ ((𝔛

𝔥.𝔘.𝔑

. 0̅). (𝔐

𝔥.𝔘.𝔑

. 0̅))

𝜌

)}

Hence, the classes of triple sequences

(𝑙

∞

)

𝐹

3

(Υ)

is solid. In same way, we could recognize other spaces

are solid too by following same sense .

∎

Proposition 3.3:

The classes of triple sequences

(𝒸)

𝐹

3

(Υ). (𝑚)

𝐹

3

(Υ)

are not

monotone and hence

not

solid

.

Corollary 3.1

𝑍(Υ

1

) ∩ 𝑍(Υ

2

) ⊆ 𝑍(Υ

1

+ Υ

2

).

for

𝑍 = (𝑙

∞

)

𝐹

3

(Υ).

(𝒸)

𝐹

3

(Υ )

Corollary 3.2:

Let

Υ

and

Υ

1

be two Orlicz function then

𝑍(Υ

1

) ⊆ 𝑍(Υ ∘ Υ

1

).

for

𝑍

=

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(𝑙

∞

)

𝐹

3

. (𝒸)

𝐹

3

. (𝒸

0

)

𝐹

3

. (𝑚)

𝐹

3

.

and

(𝑚

0

)

𝐹

3

.

Proof :

The result will be proven for the case

𝑍 = (𝒸

0

)

𝐹

3

.

The other cases can be prove by using the same

technique. Take

𝜖 > 0.

there exists

𝑛 > 0.

such that

𝜖 = Υ(𝑛).

Let

(𝔛

𝔥.𝔘.𝔑

. 𝔐

𝔥.𝔘.𝔑

) ∈ 𝑍( Υ

1

).

then, there

exist

𝑘

0

. 𝑙

0

∈ ℕ.

such that

Υ

1

[

𝑑

̅ ((𝔛

𝔥.𝔘.𝔑

. 0̅). (𝔐

𝔥.𝔘.𝔑

. 0̅))

𝜌

] < 𝑛. for some 𝜌 > 0.

Let (𝔅

𝔥.𝔘.𝔑

. 𝔔

𝔥.𝔘.𝔑

) = Υ

1

[

𝒹̅ ((𝔛

𝔥.𝔘.𝔑

. 0̅). (𝔐

𝔥.𝔘.𝔑

. 0̅))

𝜌

] . for some 𝜌 > 0.

Since

𝛶

is continuous and non-decreasing, we get

Υ (𝔅

𝔥.𝔘.𝔑

. 𝔔

𝔥.𝔘.𝔑

) = Υ [Υ

1

[

𝒹̅ ((𝔛

𝔥.𝔘.𝔑

. 0̅). (𝔐

𝔥.𝔘.𝔑

. 0̅))

𝜌

]] < Υ(𝑛) = 𝜖.

for some 𝜌 > 0.

Which implies that,

(𝔛

𝔥.𝔘.𝔑

. 𝔐

𝔥.𝔘.𝔑

) ∈ 𝑍(Υ ∘ Υ

1

).

∎

Proposition 3.4

The class of triple sequences

(𝑙

∞

)

𝐹

3

(Υ). (𝒸)

𝐹

3

(Υ). (𝒸

0

)

𝐹

3

(Υ). (𝑚)

𝐹

3

(Υ)

are not

convergent

free

.

Proof:

The following Example will lead to such result.

Example 3.2:

Consider the classes of triple sequences

(𝒸)

𝐹

3

(Υ).

Suppose

Υ(𝔛. 𝔐) = (𝔛. 𝔐)

and consider the triple

sequence

(𝔛

𝔥.𝔘.𝔑

. 𝔐

𝔥.𝔘.𝔑

)

defined by

(𝔛

1𝔘

. 𝔐

1𝔘

) = (0̅. 0̅)

and for other values,

(𝔛

𝔥.𝔘.𝔑

. 𝔐

𝔥.𝔘.𝔑

)(𝔭)

=

{

(1.1).

for 0 ≤ 𝔭 ≤ 1;

(−𝔥𝔭(𝔥 + 1)

−1

+ (2𝔥 + 1)(𝔥 + 1)

−1

. −𝔥𝔭 + (2𝔥 + 1)(𝔥 + 1)

−1

). for 1 ≤ 𝔭 ≤ 2 + 𝔥

−1

;

(0.0).

otherwise.

where

(𝔛

𝔥.𝔘.𝔑

)(𝔭)

=

{−

1.

for 0 ≤ 𝔭 ≤ 1;

𝔥𝔭(𝔥 + 1)

−1

+ (2𝔥 + 1)(𝔥 + 1)

−1

. for1 ≤ 𝔭 ≤ 2 + 𝔥

−1 .

0.

otherwise.

and

(𝔛

𝔥.𝔘.𝔑

)(𝔭)

=

{−

1.

for 0 ≤ 𝔭 ≤ 1;

𝔥𝔭(𝔥 + 1)

−1

+ (2𝔥 + 1)(𝔥 + 1)

−1

. for1 ≤ 𝔭 ≤ 2 + 𝔥

−1.

0.

otherwise.

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Let the triple sequence

(𝔅

𝔥.𝔘.𝔑

. 𝔔

𝔥.𝔘.𝔑

).

be defined by

(𝔅

1𝔘.

. 𝔔

1𝔘

) = (0̅. 0̅).

and for other values,

(𝔅

𝔥.𝔘.𝔑

. 𝔔

𝔥.𝔘.𝔑

)

can be defined as

(𝔅

𝔥.𝔘.𝔑

. 𝔔

𝔥.𝔘.𝔑

)(𝔭)

=

{

(1.1).

for 0 ≤ 𝔭 ≤ 1;

((𝔥 − 𝔭)(𝔥 − 1)

−1

. (𝔥 − 𝔭)(𝔥 − 𝔭)

−1

). for 1 ≤ 𝔭 ≤ 𝔥

(0.0).

otherwise.

where

(𝔅

𝔥.𝔘.𝔑

)(𝔭) = {

1.

for 0 ≤ 𝔭 ≤ 1;

(𝔥 − 𝔭)(𝔥 − 𝔭)

−1

. for 1 ≤ 𝔭 ≤ 𝔥;

0.

otherwise.

and

(𝔔

𝔥.𝔘.𝔑

)(𝔭) = {

1.

for 0 ≤ 𝔭 ≤ 1;

(𝔥 − 𝔭)(𝔥 − 𝔭)

−1

. for 1 ≤ 𝔭 ≤ 𝔥;

0.

otherwise.

Then

(𝔛

𝔥.𝔘.𝔑

) ∈ (𝒸)

𝐹

3

(Υ).

and

(𝔐

𝔰.𝔱.𝔞

) ∈ (𝒸)

𝐹

3

(Υ).

⟹

(𝔛

𝔥.𝔘.𝔑

. 𝔐

𝔥.𝔘.𝔑

) ∈ (𝒸)

𝐹

3

(Υ)

but

(𝔅

𝔥.𝔘.𝔑

)

∉ (𝒸)

𝐹

3

(Υ).

and

(𝔔

𝔥.𝔘.𝔑

) ∉ (𝒸)

𝐹

3

(Υ).

⟹

(𝔅

𝔥.𝔘.𝔑

. 𝔔

𝔥.𝔘.𝔑

) ∉ (𝒸)

𝐹

3

(Υ).

Therefore,

(𝒸)

𝐹

3

(Υ)

isn't convergent free. Likewise, the other spaces are also not convergent free.

REFERENCES
1.

LA. Zadeh, Fuzzy sets. Lnform and Control 8 (1965) 338-353.

2.

T.J.I' A. Bromwich, An Introduction to the Theory of Infinite Series, New York, MacMillan and Co,
(1965).

3.

F. Moricz, Extension of spaces

𝑐

and

𝑐

0

from single to double sequences, Acta Math. Hung. 57(1-2)

(1991) 129-136.

4.

M. Basarir. O. Solancan, On some double sequence spaces, J. Indian Acad. Math. 21, (1999) 193-200.

5.

A.J.Datta , A. Esi, B.C. Tripathy, “Statistically convergent triple sequence spaces defined by Orlicz
function”. J. Math. Anal.,4(2) (2013) 16

-22.

6.

J. Lindenstrauss and L. Tzafriri, On Orlicz sequence spaces, Israel J. Math;101, (1971),379-390.

7.

P. Kostyrko, T. Salat, W. Wilczynski, '' -convergen'', Real Analysis Exchang, (2000), 26(2):669-686.