Vol 8, No 5 (2017)

Available online throug

ISSN 2229 – 5046

GENERALIZED gIC

λ

-RATE SEQUENCE SPACES OF DIFFERENCE SEQUENCE

OF MODAL INTERVAL NUMBERS DEFINED BY A MODULUS FUNCTION

1

_{S. ZION CHELLA RUTH*,}

2

_{G. RAJANIRANJANA}

1

_{Assistant Professor,}

2

_{M. Phil Scholar,}

1&2

_{Department of Mathematics,}

Aditanar College of Arts and science, Tiruchendur, Tamilnadu. India.

(Received On: 24-04-17; Revised & Accepted On: 19-05-17)

ABSTRACT

I

n this paper we introduce and study the concepts of generalized gICλ-Rate sequence spaces of difference sequence of

modal interval numbers and prove some inclusion relations.

Keywords: FK-spaces, Modulus Function, Rate sequence space, Modal Interval Numbers, Difference Sequence Space,

𝐶𝜆-Summability Method.

Mathematics Subject Classification 2000: 40C05, 40D25, 40G05, 42A05. 42A10.

1. INTRODUCTION

Many mathematical structures have been constructed with real or complex numbers. In recent years, these mathematical structures were replaced by fuzzy numbers or interval numbers and these mathematical structures have been very popular since 1965. Interval arithmetic is a tool in numerical computing where the rules for the arithmetic of intervals are explicity stated. Interval arithmetic was first suggested by P.S.Dwyer[3] in 1951. Development of interval arithmetic as a formal system and evidence of its value as a computational device was provided by R.E.Moore[13] in 1962. Probably the most important paper for the development of interval arithmetic has been published by the Japanese scientist Sunaga [19]. Recently, the sequence spaces of modal interval numbers 𝑔𝐼𝐶_𝑐𝜆_0𝜋(∆𝑚,𝑓,𝑝,𝑞),𝑔𝐼𝐶_𝑐𝜆_𝜋(∆𝑚,𝑓,𝑝,𝑞) and 𝑔𝐼𝐶_(ℓ

∞)𝜋

𝜆 _(∆𝑚_{,𝑓,𝑝,𝑞) using a modulus function 𝑓 and more general 𝐶}

𝜆-method in view of Armitage and Maddox

[12]. Several properties of these spaces, and some inclusion relation have been examined.

2. PRELIMINARIES

Definition 2.1: A set consisting of a closed interval of real numbers

x

such that

a

≤

x

≤

b

is called an interval number. A real interval can also be considered as a set. We denote the set of all real valued closed intervals by

I

ℜ

. Any elements of

I

ℜ

is called closed interval and denoted by

x

. That is

x

=

[

x

_l

,

x

_r

]

=

{

x

∈

ℜ

:

x

_l

≤

x

≤

x

_r

}.

An

interval number

x

is a closed subset of real numbers. Let

x

_l

and

x

_r be respectively referred to as the infimum (lower bound) and supremum (upper bound) of theinterval number

x

. If

x

=

[

0

,

0

]

_{, then}

x

is said to be a zero

interval. It is denoted by

0

_{.Chiao [11] introduced sequence of interval numbers and defined usual convergence of}

sequences of interval number. When

x

>

x

,

x

ˆ

is not an interval number. But in modal analysis,

[

x

,

x

]

is a valid interval.

Definition 2.2: A modal interval number

~

x

=

{[

x

,

x

]

:

x

,

x

∈

ℜ

}

is defined by a pair of real numbers

x

,

x

and it is

denoted by

gI

and |𝑥�| = max {�𝑥�, |𝑥|}. If 𝑥�= [0,0], then 𝑥� is said to be a zero modal interval.

Corresponding Author:

1

_{S. Zion Chella Ruth*}

1

_{Assistant Professor,}

1

_{Department of Mathematics,}

Definition 2.3: For

x

~

₁

,

x

~

₂

∈

gI

,

]

,

[

~

1

a

a

x

=

(Degenerate modal interval number)

2 1

~

~

_x

x

=

if and only if

x

₁

=

x

₂

and

x

₁

=

x

₂

.

)}

:

{

~

~

2 1 2 1 2

1

x

x

x

x

x

x

x

x

+

=

∈

ℜ

+

≤

≤

+

)}

,

,

,

max(

)

,

,

,

min(

:

{

~

~

2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

1

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

×

=

∈

ℜ

≤

≤













×

=

2 2 1 2 1

1

,

1

~

~

/

~

x

x

x

x

x

Definition 2.4: The distance between the two modal interval numbers

~

x

₁

,

~

x

₂is defined by

{

1 2 1 2

}

2

1

)

max

,

~

,

~

(

x

x

x

x

x

x

d

=

−

−

. Clearly

d

is a metric on

gI

.

Definition 2.5: Let us define transformation

f

:

N

→

gI

,

k

→

f

(

k

)

=

~

x

_k, then

x



=

(

x



_k

)

is called sequence of modal interval number.

x

~

_kis called the

k

thterm of sequence

~

x

=

(

~

x

k

)

,

ω

(

gI

)

denote the set of all sequence of

modal interval number with real terms.

Definition 2.6: Let

x



=

(

x



k

)

=

([

x

k

,

x

k

])

∈

ω

(

gI

)

.If

x

k

=

x

k, for all

k

∈

N

,

then the sequence

~

x

=

(

~

x

k

)

is

called degenerate sequence of modal interval number.

Definition 2.7: A sequence

x



=

(

x



k

)

of modal interval number is said to be convergent to a modal interval number

~

x

0 if for each

ε

>

0

there exists a positive integer

k

₀ such that

d

(

x

~

_k

,

~

x

₀

)

<

ε

for all

k

≥

k

₀ and we denote it by

0

~

~

lim

x

_k

x

k

=

. Equivalently

lim

~

x

_k

~

x

₀

k

=

if and only if

lim

x

x

₀

and

lim

x

_k

x

₀

k k

k

=

=

.

Definition 2.8: A sequence of modal interval numbers

~

x

=

(

~

x

k

)

is said to be bounded if there exist a positive

number A such that

d

(

x

~

_k

,

~

0

)

≤

A

for all

k

∈

N

_.

Definition 2.9: A modulus f is a function from [0, ∞) to [0,∞) such that i) f(x) = 0 if and only if x = 0,

ii) f(x + y) ≤ f(x) + f(y) for x,y≥ 0,

iii) f is increasing,

iv) f is continuous from the right at 0.

It follows that f must be continuous everywhere on [0, ∞). Maddox [12] used a modulus function to construct some

sequence spaces. Later on using a modulus different sequence spaces have been studied by Altın and et al. [1],

et al. [5], Nuray and Savas [15], Tripathy and Chandra [20] and many others.

Let 𝜋= (𝜋_𝑛) be a sequence of positive number i.e., 𝜋_𝑛> 0,∀𝑛 ∈ ℕ and X is FK- space. We shall consider the sets of sequences of modal interval numbers 𝑥�= (𝑥�_𝑛)

𝑋𝜋(𝑔𝐼) =�𝑥� ∈ 𝜔: (𝑔𝐼)�𝑥�_𝜋𝑛

𝑛� ∈ 𝑋(𝑔𝐼)�.

The set 𝑋_𝜋(𝑔𝐼) may be considered as FK-space. We shall call them as rate spaces of modal interval numbers (see, [17]).

Let F be an infinite subset of ℕ and F as the range of a strictly increasing sequence of positive integers, say

𝐹= {𝜆(𝑛)}𝑛=1∞ . The Cesaro submethod 𝐶𝜆 is defined as

(𝐶𝜆𝑥�)𝑛=_𝜆(𝑛)1 � 𝑥�𝑘 𝜆(𝑛)

𝑘=1

, (𝑛= 1,2, , … )

where {𝑥�_𝑘} is a sequence of a modal interval numbers. Therefore, the 𝐶_𝜆-method yields a subsequence of the Cesaro method 𝐶₁ and hence it is regular for any 𝜆. 𝐶_𝜆is obtained by deleting a set of rows from Cesaro matrix. If 𝜆(𝑛) =𝑛 is taken, then 𝐶_𝜆=𝐶₁ is obtained. On a range of sequences

We will write 𝐶_𝜆~𝐶₁. The basic properties of 𝐶_𝜆-method can be found in [2] and [18].

Let 𝑝= (𝑝_𝑘) be a sequence of positive real numbers with 𝐺=𝑠𝑢𝑝_𝑘𝑝𝑘 and 𝐷 =𝑚𝑎𝑥(1, 2𝐺−1). Then it is well known that for all 𝑎_𝑘,𝑏_𝑘 ∈ ℝ, the field of real numbers, for all 𝑘 ∈ ℕ,

|𝑎𝑘+𝑏𝑘|𝑝𝑘≤ 𝐷(|𝑎𝑘|𝑝𝑘+ |𝑏𝑘|𝑝𝑘) (1)

Also for any real 𝜇,

𝜇𝑝𝑘≤ 𝑚𝑎𝑥(1,𝜇𝐺) (2)

Let X(gI) be a sequence space of modal interval numbers. Then X(gI) is called;

i) Solid (or normal) if (𝛼_𝑘𝑥�_𝑘)∈ 𝑋(𝑔𝐼) whenever (𝑥�_𝑘)∈ 𝑋(𝑔𝐼) for all sequences (𝛼_𝑘) of scalars with |𝛼_𝑘|≤1. ii) Symmetric if (𝑥�_𝑘)∈ 𝑋(𝑔𝐼) implies �𝑥�_𝜋(𝑘)� ∈ 𝑋(𝑔𝐼) where 𝜋 is a permutation of ℕ,

iii) Sequence algebra if 𝑋(𝑔𝐼) is closed under multiplication.

3. MAIN RESULTS

Let f be a modulus function, X(gI) be a locally convex Hausdorff topological linear space whose topology is determined by a set Q of continuous seminorms 𝑞 and 𝑝= (𝑝_𝑘) be a sequence of positive real numbers. Then defined the following sequence spaces of modal interval numbers

𝑔𝐼𝐶𝑐𝜆0𝜋(∆𝑚,𝑓,𝑝,𝑞) =�𝑥� ∈ 𝜔(𝑔𝐼):

1

𝜆(𝑛)� �𝑓 �𝑞 ��∆𝑚 𝑥�𝑘 𝜋𝑘����

𝑝𝑘

→0 𝑎𝑠 𝑛 → ∞ 𝜆(𝑛)

𝑘=1

�

𝑔𝐼𝐶𝑐𝜆𝜋(∆𝑚,𝑓,𝑝,𝑞) =�𝑥� ∈ 𝜔(𝑔𝐼):

1

𝜆(𝑛)� �𝑓 �𝑞 ��∆𝑚 𝑥�𝑘

𝜋𝑘− 𝐿���� 𝑝𝑘

→0 𝜆(𝑛)

𝑘=1

𝑎𝑠 𝑛 → ∞ 𝑓𝑜𝑟 𝑠𝑜𝑚𝑒 𝐿

�

𝑔𝐼𝐶(𝜆ℓ∞)𝜋(∆𝑚,𝑓,𝑝,𝑞) =�𝑥� ∈ 𝜔(𝑔𝐼):

𝑠𝑢𝑝

𝑛 𝜆(𝑛)1 � �𝑓 ���∆𝑚 𝑥�𝑘 𝜋𝑘����

𝑝𝑘

<∞ 𝜆(𝑛)

𝑘=1 𝑎𝑠 𝑛 → ∞

�

where ∆0�𝑥�𝑘

𝜋𝑘�=

𝑥𝑘

𝜋𝑘,∆

𝑚_�𝑥�𝑘

𝜋𝑘�=�∆

𝑚−1 𝑥�𝑘

𝜋𝑘− ∆

𝑚−1 𝑥�𝑘+1

𝜋𝑘+1� and ∆

𝑚_�𝑥�𝑘

𝜋𝑘�=∑ (−1)

𝑣_�𝑚 𝑣 � 𝑚

𝑣=0 _𝜋𝑥�𝑘+𝑣_𝑘+𝑣.

For 𝑓(𝑥) =𝑥 we shall write 𝑔𝐼𝐶_𝑐𝜆_0𝜋(∆𝑚,𝑝,𝑞),𝑔𝐼𝐶_𝑐𝜆_𝜋(∆𝑚,𝑝,𝑞)and 𝑔𝐼𝐶_(ℓ ∞)𝜋

𝜆 _(∆𝑚_{,𝑝,𝑞) instead of 𝑔𝐼𝐶}

𝑐𝜆0𝜋(∆𝑚,𝑓,𝑝,𝑞),

𝑔𝐼𝐶𝑐𝜆𝜋(∆𝑚,𝑓,𝑝,𝑞) and 𝑔𝐼𝐶(𝜆ℓ∞)𝜋(∆𝑚,𝑓,𝑝,𝑞) respectively.

Theorem 3.1: Let 𝑝= (𝑝_𝑘) be bounded, then 𝑔𝐼𝐶_𝑐𝜆_0𝜋(∆𝑚,𝑓,𝑝,𝑞),𝑔𝐼𝐶_𝑐𝜆_𝜋(∆𝑚,𝑓,𝑝,𝑞) and 𝑔𝐼𝐶₍𝜆_ℓ∞)𝜋(∆𝑚,𝑓,𝑝,𝑞) are linear spaces of modal interval numbers.

Theorem 3.2: 𝑔𝐼𝐶_𝑐𝜆_0𝜋(∆𝑚,𝑓,𝑝,𝑞)is a paranormed space of modal interval number (not totally paranormed), paranormed by

𝑔∆(𝑥�) =𝑠𝑢𝑝_{𝑛 �𝜆(𝑛)}1 � 𝑓 �𝑞 ��∆𝑚_𝜋𝑥�𝑘 𝑘���

𝑝𝑘

𝜆(𝑛)

𝑘=1

�

1 𝑀

where 𝑀= max (1,𝐺=𝑠𝑢𝑝 𝑝_𝑘).

Theorem 3.3: Let 𝑓,𝑓₁,𝑓₂ are modulus function and 0 <ℎ= inf𝑝_𝑘 ≤sup_𝑘𝑝_𝑘=𝐺 then (i) 𝑔𝐼𝐶_𝑐𝜆_0𝜋(∆𝑚,𝑓₁,𝑝,𝑞)⊆ 𝑔𝐼𝐶_𝑐𝜆_0𝜋(∆𝑚,𝑓 ∘ 𝑓₁,𝑝,𝑞)

(ii) 𝑔𝐼𝐶_𝑐𝜆_0𝜋(∆𝑚,𝑓₁,𝑝,𝑞)∩ 𝑔𝐼𝐶_𝑐𝜆_0𝜋(∆𝑚,𝑓₂,𝑝,𝑞)⊆ 𝑔𝐼𝐶_𝑐𝜆_0𝜋(∆𝑚,𝑓₁+𝑓₂,𝑝,𝑞)

Proof: (i) Let 𝑥�=�𝑥�𝑘

𝜋𝑘� ∈ 𝑔𝐼𝐶𝑐0𝜋

𝜆 _(∆𝑚_,𝑓

1,𝑝,𝑞). Let 𝜀> 0 and choose 𝛿 with 0 <𝛿< 1 such that 𝑓(𝑡) <𝜀 for

0≤ 𝑡 ≤ 𝛿.Write 𝑦_𝑘 =𝑓₁�𝑞 ��∆𝑚 𝑥�𝑘

𝜋𝑘��� and consider

�[𝑓(𝑦𝑘)]𝑝𝑘 𝜆(𝑛)

𝑘=1

=�[𝑓(𝑦𝑘)]𝑝𝑘 1

+�[𝑓(𝑦𝑘)]𝑝𝑘 2

where the first summation is over 𝑦_𝑘 ≤ 𝛿 and the second over 𝑦_𝑘 >𝛿. Since f is continuous, we get

and for 𝑦_𝑘>𝛿 we use the fact that

𝑦𝑘 <𝑦_𝛿𝑘 < 1 +𝑦_𝛿𝑘.

By the definition of f, we have 𝑦_𝑘>𝛿,

𝑓(𝑦𝑘)≤ 𝑓(1)�1 +�𝑦_{𝛿 �� ≤}𝑘 2𝑓(1)𝑦_𝛿𝑘.

Hence

1

𝜆(𝑛)∑2[𝑓(𝑦𝑘)]𝑝𝑘< max�1,�

2𝑓(1)

𝛿 � 𝐺

�_𝜆(1_𝑛)∑𝑘=1𝜆(𝑛)[𝑦𝑘]𝑝𝑘→0 (4)

By (3) and (4) we have 𝑔𝐼𝐶_𝑐𝜆_0𝜋(∆𝑚,𝑓₁,𝑝,𝑞)⊆ 𝑔𝐼𝐶_𝑐𝜆_0𝜋(∆𝑚,𝑓 ∘ 𝑓₁,𝑝,𝑞).

(ii) Let 𝑥�=�𝑥�𝑘

𝜋𝑘� ∈ 𝑔𝐼𝐶𝑐0𝜋

𝜆 _(∆𝑚_,𝑓

1,𝑝,𝑞)∩ 𝑔𝐼𝐶𝑐𝜆0𝜋(∆𝑚,𝑓2,𝑝,𝑞). Then there exist f1 and f2 such that 1

𝜆(𝑛)∑ �𝑓1�𝑞 ��∆𝑚 𝑥�𝜋𝑘𝑘����

𝑝𝑘

𝜆(𝑛)

𝑘=1 →0 𝑎𝑠 𝑛 → ∞ (5)

1

𝜆(𝑛)∑ �𝑓2�𝑞 ��∆𝑚 𝑥�𝜋𝑘𝑘����

𝑝𝑘

𝜆(𝑛)

𝑘=1 →0 𝑎𝑠 𝑛 → ∞ (6)

Then using (i) it can be shown that

1

𝜆(𝑛)� �(𝑓1+𝑓2)�𝑞 ��∆𝑚 𝑥�𝑘 𝜋𝑘����

𝑝𝑘

𝜆(𝑛)

𝑘=1

=_𝜆(𝑛)1 � �𝑓1�𝑞 ��∆𝑚_𝜋𝑥�𝑘 𝑘����

𝑝𝑘

𝜆(𝑛)

𝑘=1

+_𝜆(𝑛)1 � �𝑓2�𝑞 ��∆𝑚𝑥�_𝜋𝑘 𝑘����

𝑝𝑘

𝜆(𝑛)

𝑘=1

≤ 𝐷_𝜆(𝑛)1 � �𝑓1�𝑞 ��∆𝑚_𝜋𝑥�𝑘 𝑘����

𝑝𝑘

𝜆(𝑛)

𝑘=1

+𝐷_𝜆(𝑛)1 � �𝑓2�𝑞 ��∆𝑚_𝜋𝑥�𝑘 𝑘����

𝑝𝑘

𝜆(𝑛)

𝑘=1 →0 𝑎𝑠 𝑛 → ∞ .

Therefore 𝑥�=�𝑥�𝑘

𝜋𝑘� ∈ 𝑔𝐼𝐶𝑐0𝜋

𝜆 _(∆𝑚_,𝑓

1+𝑓2,𝑝,𝑞).

Hence, 𝑔𝐼𝐶_𝑐𝜆_0𝜋(∆𝑚,𝑓₁,𝑝,𝑞)∩ 𝑔𝐼𝐶_𝑐𝜆_0𝜋(∆𝑚,𝑓₂,𝑝,𝑞)⊆ 𝑔𝐼𝐶_𝑐𝜆_0𝜋(∆𝑚,𝑓₁+𝑓₂,𝑝,𝑞).

The proof of the following result is a routine work of the above theorem.

Corollary 3.4: Let f, f1, f2 are modulus function then (i) 𝑔𝐼𝐶_𝑐𝜆_𝜋(∆𝑚,𝑓₁,𝑝,𝑞)⊆ 𝑔𝐼𝐶_𝑐𝜆_𝜋(∆𝑚,𝑓 ∘ 𝑓₁,𝑝,𝑞),

(ii) 𝑔𝐼𝐶_𝑐𝜆_𝜋(∆𝑚,𝑓₁,𝑝,𝑞)∩ 𝑔𝐼𝐶_𝑐𝜆_𝜋(∆𝑚,𝑓₂,𝑝,𝑞)⊆ 𝑔𝐼𝐶_𝑐𝜆_𝜋(∆𝑚,𝑓₁+𝑓₂,𝑝,𝑞), (iii)𝑔𝐼𝐶_(ℓ

∞)𝜋

𝜆 _(∆𝑚_,𝑓

1,𝑝,𝑞)⊆ 𝑔𝐼𝐶(𝜆ℓ∞)𝜋(∆𝑚,𝑓 ∘ 𝑓1,𝑝,𝑞), (iv)𝑔𝐼𝐶_(ℓ

∞)𝜋

𝜆 _(∆𝑚_,𝑓

1,𝑝,𝑞)∩ 𝑔𝐼𝐶(𝜆ℓ∞)𝜋(∆𝑚,𝑓2,𝑝,𝑞)⊆ 𝑔𝐼𝐶(𝜆ℓ∞)𝜋(∆𝑚,𝑓1+𝑓2,𝑝,𝑞).

Proposition 3.5: 𝑔𝐼𝐶_𝑐𝜆_𝜋(∆𝑚−1,𝑓,𝑝,𝑞)⊆ 𝑔𝐼𝐶_𝑐𝜆_𝜋(∆𝑚,𝑓,𝑝,𝑞).

Theorem 3.6: Let 𝑚 ≥1, then the following inclusion are strict (i) 𝑔𝐼𝐶_𝑐𝜆_0𝜋(∆𝑚−1,𝑓,𝑞)⊆ 𝑔𝐼𝐶_𝑐𝜆_0𝜋(∆𝑚,𝑓,𝑞),

(ii) 𝑔𝐼𝐶_𝑐𝜆_𝜋(∆𝑚−1,𝑓,𝑞)⊆ 𝑔𝐼𝐶_𝑐𝜆_𝜋(∆𝑚,𝑓,𝑞), (iii)𝑔𝐼𝐶_(ℓ

∞)𝜋

𝜆 _(∆𝑚−1_{,𝑓,𝑞)⊆ 𝑔𝐼𝐶}

(𝜆ℓ∞)𝜋(∆𝑚,𝑓,𝑞).

Example 3.7: Let 𝑓(𝑥) =𝑥 and 𝑞(𝑥) = |𝑥|. Consider the sequences (𝑥�_𝑘) = ([0,𝑘𝑚+𝛼]) and (𝜋𝑘) = (𝑘𝛼+1), where 𝑥�=�𝑥�𝑘

𝜋𝑘� and 𝑚 ∈ ℕ,𝛼 ∈ ℝ. Then 𝑥� ∈ 𝑔𝐼𝐶𝑐0𝜋

𝜆 _(∆𝑚_{,𝑓,𝑞) but 𝑥� ∉ 𝑔𝐼𝐶}

𝑐𝜆0𝜋(∆𝑚−1,𝑓,𝑞), since ∆𝑚 𝑥�_𝜋𝑘_𝑘= 0,

∆𝑚−1 𝑥�𝑘

𝜋𝑘= [0, (−1)

𝑚−1_{(𝑚 −1)!] ∀𝑘 ∈ ℕ.}

Theorem 3.8: For any two sequence 𝑝= (𝑝_𝑘) and 𝑡= (𝑡_𝑘) of positive real numbers and any two seminorms q1, q2 we have

Proof: Since the zero element belongs to each of the above classes of sequences, thus the intersection is nonempty.

The following result is a consequence of Theorem 3.3 (i) and Corollary 3.4 (i) and (iii).

Theorem 3.9: Let f be a modulus function. Then (i) 𝑔𝐼𝐶_𝑐𝜆_0𝜋(∆𝑚,𝑝,𝑞)⊆ 𝑔𝐼𝐶_𝑐𝜆_0𝜋(∆𝑚,𝑓,𝑝,𝑞), (ii) 𝑔𝐼𝐶_𝑐𝜆_𝜋(∆𝑚,𝑝,𝑞)⊆ 𝑔𝐼𝐶_𝑐𝜆_𝜋(∆𝑚,𝑓,𝑝,𝑞), (iii)𝑔𝐼𝐶_(ℓ

∞)𝜋

𝜆 _(∆𝑚_{,𝑝,𝑞)⊆ 𝑔𝐼𝐶}

(𝜆ℓ∞)𝜋(∆𝑚,𝑓,𝑝,𝑞).

Theorem 3.10: The sequence spaces 𝑔𝐼𝐶_𝑐𝜆_0𝜋(∆𝑚,𝑓,𝑝,𝑞),𝑔𝐼𝐶_𝑐𝜆_𝜋(∆𝑚,𝑓,𝑝,𝑞) and 𝑔𝐼𝐶_(ℓ ∞)𝜋

𝜆 _(∆𝑚_{,𝑓,𝑝,𝑞) are neither solid}

nor symmetric, nor sequence algebras for m≥1.

Proof: Let m = 1, pk= 1 for all k∈ ℕ,𝑓(𝑥) =𝑥 𝑎𝑛𝑑 𝑞(𝑥) = |𝑥|. If the sequences (𝑥�𝑘) = ([0,𝑘𝑛+1]) and (𝜋𝑘) = (𝑘𝑛) are taken, then the sequence �𝑥�𝑘

𝜋𝑘� belongs to 𝑔𝐼𝐶(ℓ∞)𝜋

𝜆 _{(△) and 𝑔𝐼𝐶}

𝑐𝜆𝜋(△) where n∈ ℝ. Let 𝛼𝑘 = (−1)𝑘, then (𝛼𝑘𝑥�) does not belong to 𝑔𝐼𝐶_(ℓ

∞)𝜋

𝜆 _{(△) and 𝑔𝐼𝐶}

𝑐𝜆𝜋(△). Hence 𝑔𝐼𝐶𝑐𝜆𝜋(∆𝑚,𝑓,𝑝,𝑞) and 𝑔𝐼𝐶(𝜆ℓ∞)𝜋(∆𝑚,𝑓,𝑝,𝑞) are not solid. The other cases can be proved on considering similar examples.

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Source of support: Nil, Conflict of interest: None Declared.