International Journal of Analysis and Applications

Volume 16, Number 6 (2018), 882-893

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-16-2018-882

IDEAL CONVERGENT SEQUENCE SPACES WITH RESPECT TO INVARIANT

MEAN AND A MUSIELAK-ORLICZ FUNCTION OVER n-NORMED SPACES

SUNIL K. SHARMA∗

Department of Mathematics, Model Institute of Engineering & Technology,

Kot Bhalwal 181122, J & K, INDIA

∗Corresponding author: sunilksharma42@gmail.com

Abstract. In the present paper we defined I-convergent sequence spaces with respect to invariant mean

and a Musielak-Orlicz function M = (Mk) over n-normed spaces. We also make an effort to study some

topological properties and prove some inclusion relation between these spaces.

1. Introduction and preliminaries

Let σ be an injective mapping from the set of the positive integers to itself such that σp(n) 6= n for all

positive integers n and p, where σp(n) = σ(σp−1(n)). An invariant mean or a σ-mean is a continuous linear

functional defined on the space `∞ such that for all x = (xn) ∈ `∞:

(1) If xn ≥ 0 for all n, then φ(x) ≥ 0,

(2) φ(e) = 1,

(3) φ(Sx) = φ(x), where Sx = (xσ(n)).

Vσ denotes the set of bounded sequences all of whose invariant means are equal which is also called as the

space of σ-convergent sequences. In [26], it is defined by

Vσ =
{
x ∈ `∞ : lim

k
tkn(x) = `, uniformly in n,` = σ − lim x

}
,

Received 2017-09-21; accepted 2017-12-07; published 2018-11-02.

2010 Mathematics Subject Classification. 40A05,40A35, 46A45.

Key words and phrases. I-convergent, invariant mean, Orlicz function, Musielak-Orlicz function, n-normed space, A-

transform.

c©2018 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

882

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-16-2018-882


Int. J. Anal. Appl. 16 (6) (2018) 883

where tkn(x) =
xn+xσ1(n)+···+xσk(n)

k+1
.

σ-mean is called a Banach limit if σ is the translation mapping n → n + 1. In this case, Vσ becomes the set

of almost convergent sequences which is denoted by ĉ and defined in [11] as

ĉ =
{
x ∈ `∞ : lim

k
dkn(x) exists uniformly in n

}
,

where dkn(x) =
xn+xn+1+···+xn+k

k+1
.

The space of strongly almost converegnt sequences was introduced by Maddox [12] as follow:

ĉ =
{
x ∈ `∞ : lim

k
dkn(|x− `e|) exists uniformly in n for some `

}
.

The notion of ideal convergence was first introduced by P. Kostyrko [8] as a generalization of statistical

convergence which was further studied in topological spaces by Das, Kostyrko, Wilczynski and Malik see [1].

More applications of ideals can be seen in ([1], [2]). Mursaleen and Sharma [19] continue in this direction

and introduced I-convergence of generalized sequences with respect to Musielak-Orlicz function.

A family I ⊂ 2X of subsets of a non empty set X is said to be an ideal in X if

(1) φ ∈I

(2) A,B ∈I imply A∪B ∈I

(3) A ∈I, B ⊂ A imply B ∈I,

while an admissible ideal I of X further satisfies {x}∈I for each x ∈ X see [8].

A sequence (xn)n∈N in X is said to be I-convergent to x ∈ X, if for each � > 0 the set A(�) =
{
n ∈ N :

||xn −x|| ≥ �
}

belongs to I.

A sequence (xn)n∈N in X is said to be I-bounded to x ∈ X if there exists an K > 0 such that {n ∈ N :

|xn| > K} ∈ I. For more details about ideal convergence sequence spaces (see [7], [9], [15], [16], [17], [18],

[21], [25], [26], [27]) and references therein.

Let A = Aij be an infinite matrix of complex numbers aij, where i,j,∈ N. We write Ax = (Ai(x)) if

Ai(x) =

∞∑
j=1

aijxj converges for each i ∈ N. Throughout the paper, by tkn(Ax), we mean

tkn(Ax) =
An(x) + Aσ1(n)(x) + · · · ,Aσk(n)(x)

k + 1
, for all k,n ∈ N.

A sequence space X is called as solid (or normal) if (αkxk) ∈ X whenever (xk) ∈ X and (αk) is a sequence

of scalars such that |αk| ≤ 1 for all k ∈ N.

Let X be a sequence space and K = {k1 < k2 < · · ·}⊆ N. The sequence space ZXK = {(xkn) ∈ w : (xn) ∈ X}

is called K-step space of X.



Int. J. Anal. Appl. 16 (6) (2018) 884

A canonical preimage of a sequence (xkn) ∈ ZXK is a sequence (yn) ∈ w defined by

yn =


 xn, if n ∈ N;0, otherwise.

A sequence space X is monotone if it contains the canonical preimages of all its step spaces.

An Orlicz function M is a function, which is continuous, non-decreasing and convex with M(0) = 0, M(x) > 0

for x > 0 and M(x) −→∞ as x −→∞.

Lindenstrauss and Tzafriri [10] used the idea of Orlicz function to define the following sequence space. Let

w be the space of all real or complex sequences x = (xk), then

`M =
{
x ∈ w :

∞∑
k=1

M
(|xk|
ρ

)
< ∞

}
which is called as an Orlicz sequence space. The space `M is a Banach space with the norm

||x|| = inf
{
ρ > 0 :

∞∑
k=1

M
(|xk|
ρ

)
≤ 1
}
.

It is shown in [10] that every Orlicz sequence space `M contains a subspace isomorphic to `p(p ≥ 1). The

∆2−condition is equivalent to M(Lx) ≤ kLM(x) for all values of x ≥ 0, and for L > 1.

A sequence M = (Mk) of Orlicz function is called a Musielak-Orlicz function see ([13],[20]). A sequence

N = (Nk) defined by

Nk(v) = sup{|v|u− (Mk) : u ≥ 0}, k = 1, 2, · · ·

is called the complementary function of a Musielak-Orlicz function M. For a given Musielak-Orlicz function

M, the Musielak-Orlicz sequence space tM and its subspace hM are defined as follows

tM =
{
x ∈ w : IM(cx) < ∞ for some c > 0

}
,

hM =
{
x ∈ w : IM(cx) < ∞ for all c > 0

}
,

where IM is a convex modular defined by

IM(x) =

∞∑
k=1

Mk(xk),x = (xk) ∈ tM.

We consider tM equipped with the Luxemburg norm

||x|| = inf
{
k > 0 : IM

(x
k

)
≤ 1
}

or equipped with the Orlicz norm

||x||0 = inf
{1
k

(
1 + IM(kx)

)
: k > 0

}
.

For more details about sequence spaces defined by Orlicz function see ([22], [23], [24]) and reference therein.

The concept of 2-normed spaces was initially developed by Gähler[3] in the mid of 1960’s, while that of



Int. J. Anal. Appl. 16 (6) (2018) 885

n-normed spaces one can see in Misiak [14]. Since then, many others have studied this concept and obtained

various results, see Gunawan ([4],[5]) and Gunawan and Mashadi [6]. Let n ∈ N and X be a linear space

over the field K, where K is field of real or complex numbers of dimension d, where d ≥ n ≥ 2. A real valued

function ||·, · · · , ·|| on Xn satisfying the following four conditions:

(1) ||x1,x2, · · · ,xn|| = 0 if and only if x1,x2, · · · ,xn are linearly dependent in X;

(2) ||x1,x2, · · · ,xn|| is invariant under permutation;

(3) ||αx1,x2, · · · ,xn|| = |α| ||x1,x2, · · · ,xn|| for any α ∈ K, and

(4) ||x + x′,x2, · · · ,xn|| ≤ ||x,x2, · · · ,xn|| + ||x′,x2, · · · ,xn||

is called a n-norm on X, and the pair (X, ||·, · · · , ·||) is called a n-normed space over the field K.

For example, we may take X = Rn being equipped with the Euclidean n-norm ||x1,x2, · · · ,xn||E = the vol-

ume of the n-dimensional parallelopiped spanned by the vectors x1,x2, · · · ,xn which may be given explicitly

by the formula

||x1,x2, · · · ,xn||E = |det(xij)|,

where xi = (xi1,xi2, · · · ,xin) ∈ Rn for each i = 1, 2, · · · ,n. Let (X, ||·, · · · , ·||) be a n-normed space of

dimension d ≥ n ≥ 2 and {a1,a2, · · · ,an} be linearly independent set in X. Then the following function

||·, · · · , ·||∞ on Xn−1 defined by

||x1,x2, · · · ,xn−1||∞ = max{||x1,x2, · · · ,xn−1,ai|| : i = 1, 2, · · · ,n}

defines an (n− 1)-norm on X with respect to {a1,a2, · · · ,an}.

A sequence (xk) in a n-normed space (X, ||·, · · · , ·||) is said to converge to some L ∈ X if

lim
k→∞

||xk −L,z1, · · · ,zn−1|| = 0 for every z1, · · · ,zn−1 ∈ X.

A sequence (xk) in a n-normed space (X, ||·, · · · , ·||) is said to be Cauchy if

lim
k,p→∞

||xk −xp,z1, · · · ,zn−1|| = 0 for every z1, · · · ,zn−1 ∈ X.

If every cauchy sequence in X converges to some L ∈ X, then X is said to be complete with respect to the

n-norm. Any complete n-normed space is said to be n-Banach space.

In the present paper, we define some new sequence spaces by using the concept of ideal convergence, invariant

mean, Musielak-Orlicz function, n-normed and A transform as follows:

I− cσ0 (A,M,p, ||·, · · · , ·||) ={
x ∈ w :

{
k ∈ N :

[
Mk

(
||
tkn(A(x))

ρ
,z1, · · · ,zn−1||

)]pk
≥ �
}
∈I, for all n ∈ N

}
,



Int. J. Anal. Appl. 16 (6) (2018) 886

I− cσ(A,M,p, ||·, · · · , ·||) ={
x ∈ w :

{
k ∈ N :

[
Mk

(
||
tkn(A(x) −L)

ρ
,z1, · · · ,zn−1||

)]pk
≥ �
}
∈I, for all n ∈ N & for some L ∈ C

}
,

I− `σ∞(A,M,p, ||·, · · · , ·||) ={
x ∈ w : ∃ K > 0 such that

{
k ∈ N :

[
Mk

(
||
tkn(A(x))

ρ
,z1, · · · ,zn−1||

)]pk
≥ K

}
∈I, for all n ∈ N

}
.

If we take p = (pk) = 1, we get the spaces

I− cσ0 (A,M, ||·, · · · , ·||) ={
x ∈ w :

{
k ∈ N :

[
Mk

(
||
tkn(A(x))

ρ
,z1, · · · ,zn−1||

)]
≥ �
}
∈I, for all n ∈ N

}
,

I− cσ(A,M, ||·, · · · , ·||) ={
x ∈ w :

{
k ∈ N :

[
Mk

(
||
tkn(A(x) −L)

ρ
,z1, · · · ,zn−1||

)]
≥ �
}
∈I, for all n ∈ N & for some L ∈ C

}
,

I− `σ∞(A,M, ||·, · · · , ·||) ={
x ∈ w : ∃ K > 0 such that

{
k ∈ N :

[
Mk

(
||
tknA(x)

ρ
,z1, · · · ,zn−1||

)]
≥ K

}
∈I, for all n ∈ N

}
.

The following inequality will be used throughout the paper. If 0 ≤ pk ≤ sup pk = H, D = max(1, 2H−1)

then

|ak + bk|pk ≤ D{|ak|pk + |bk|pk} (1.1)

for all k and ak,bk ∈ C. Also |a|pk ≤ max(1, |a|H) for all a ∈ C.

The main goal of this paper is to introduce the sequence spaces I−cσ0 (A,M,p, ||·, · · · , ·||), I−cσ(A,M,p, ||·, · · · , ·||)

and I−`σ∞(A,M,p, ||·, · · · , ·||) defined by a Musielak-Orlicz function M = (Mk) over n-normed spaces. We

also make an effort to study some topological properties and prove some inclusion relation between these

spaces.

2. Main Results

Theorem 2.1 Let M = (Mk) be a Musielak-Orlicz function, p = (pk) be a bounded sequence of

positive real numbers. Then the spaces I − cσ0 (A,M,p, ||·, · · · , ·||), I − cσ(A,M,p, ||·, · · · , ·||) and I −

`σ∞(A,M,p, ||·, · · · , ·||) are linear.

Proof. Let x,y ∈ I − cσ0 (A,M,p, ||·, · · · , ·||) and let α,β be scalars. Then there exist positive numbers ρ1

and ρ2 such that for every � > 0

D1 =
{
k ∈ N :

[
Mk

(
||
tkn(A(x))

ρ1
,z1, · · · ,zn−1||

)]pk
≥

�

2D

}
∈I, (2.1)

D1 =
{
k ∈ N :

[
Mk

(
||
tkn(A(y))

ρ2
,z1, · · · ,zn−1||

)]pk
≥

�

2D

}
∈I, (2.2)



Int. J. Anal. Appl. 16 (6) (2018) 887

Let ρ3 = max
{

2|α|ρ1, 2|β|ρ2
}

. Since M = (Mk) is non-decreasing, convex function and so by using inequal-

ity (1.1), we have[
Mk

(
||tkn(A(αx+βy))

ρ3
,z1, · · · ,zn−1||

)]pk
≤

[
Mk

(
||
tkn(αA(x))

ρ3
,z1, · · · ,zn−1||

)]pk
+
[
Mk

(
||
tkn(βA(y))

ρ3
,z1, · · · ,zn−1||

)]pk
≤

[
Mk

(
||
tkn(A(x))

ρ1
,z1, · · · ,zn−1||

)]pk
+
[
Mk

(
||
tkn(A(y))

ρ2
,z1, · · · ,zn−1||

)]pk

Now by (2.1) and (2.2), we have{
k ∈ N :

[
Mk

(
||
tkn(A(αx + βy))

ρ3
,z1, · · · ,zn−1||

)]pk
> �
}
⊂ D1 ∪D2.

Therefore αx+βy ∈I−cσ0 (A,M,p, ||·, · · · , ·||). Hence I−cσ0 (A,M,p, ||·, · · · , ·||) is a linear space. Similarly

we can prove that I− cσ(A,M,p, ||·, · · · , ·||) and I− `σ∞(A,M,p, ||·, · · · , ·||) are linear spaces. �

Theorem 2.2 Let M = (Mk) be a Musielak-Orlicz function. Then

I− cσ0 (A,M,p, ||·, · · · , ·||) ⊂I− c
σ(A,M,p, ||·, · · · , ·||) ⊂I− `σ∞(A,M,p, ||·, · · · , ·||).

Proof. The first inclusion is obvious. For second inclusion, let x ∈ I − cσ(A,M,p, ||·, · · · , ·||). Then there

exists ρ1 > 0 such that for every � > 0

A1 =
{
k ∈ N :

[
Mk

(
||
tkn(A(x) −L)

ρ1
,z1, · · · ,zn−1||

)]pk
≥ �
}
∈I.

Let us define ρ = 2ρ1. Since M = (Mk) is non-decreasing and convex, we have

Mk

(
||
tkn(A(x))

ρ
,z1, · · · ,zn−1||

)
≤ Mk

(
||
tkn(A(x) −L)

ρ1
,z1, · · · ,zn−1||

)
+ Mk

(
||
tkn(L)

ρ1
,z1, · · · ,zn−1||

)
.

Suppose that k /∈ A1. Hence by above inequality and (1.1), we have[
Mk

(
||tkn(A(x))

ρ
,z1, · · · ,zn−1||

)]pk
≤ D

{[
Mk

(
||
tkn(A(x) −L)

ρ1
,z1, · · · ,zn−1||

)]pk
+
[
Mk

(
||
tkn(L)

ρ1
,z1, · · · ,zn−1||

)]pk}
< D

{
� +

[
Mk

(
||
tkn(L)

ρ
,z1, · · · ,zn−1||

)]pk}
.

Because of the fact that
[
Mk

(
||tkn(L)

ρ1
,z1, · · · ,zn−1||

)]pk
≤ max

{
1,
[
Mk

(
||tkn(L)

ρ1
,z1, · · · ,zn−1||

)]H}
, we

have [
Mk

(
||
tkn(L)

ρ
,z1, · · · ,zn−1||

)]pk
< ∞.



Int. J. Anal. Appl. 16 (6) (2018) 888

Put K = D
{
� +

[
Mk

(
||tkn(L)

ρ
,z1, · · · ,zn−1||

)]pk}
. It follows that

{
k ∈ N :

[
Mk

(
||
tkn(A(x))

ρ
,z1, · · · ,zn−1||

)]pk
> K

}
∈I

which means x ∈I− `σ∞(A,M,p, ||·, · · · , ·||). This completes the proof of the theorem. �

Theorem 2.3 Let M = (Mk) be a Musielak-Orlicz function, p = (pk) be a bounded sequence of positive

real numbers. Then I− `σ∞(A,M,p, ||·, · · · , ·||) is a paranormed space with paranorm defined by

g(x) = inf
{
ρ > 0 :

[
Mk

(
||
tkn(A(x))

ρ
,z1, · · · ,zn−1||

)]pk
≤ 1
}
.

Proof. It is clear that g(x) = g(−x). Since Mk(0) = 0, we get g(0) = 0. Let us take x,y ∈ I −

cσ∞(A,M,p, ||·, · · · , ·||). Let

B(x) =
{
ρ > 0 :

[
Mk

(
||
tkn(A(x))

ρ
,z1, · · · ,zn−1||

)]pk
≤ 1
}
,

B(y) =
{
ρ > 0 :

[
Mk

(
||
tkn(A(y))

ρ
,z1, · · · ,zn−1||

)]pk
≤ 1
}
.

Let ρ1 ∈ B(x) and ρ2 ∈ B(y). If ρ = ρ1 + ρ2, then we have[
Mk

(
||tkn(A(x+y))

ρ
,z1, · · · ,zn−1||

)]
≤

( ρ1
ρ1 + ρ2

)[
Mk

(
||
tkn(A(x))

ρ1
,z1, · · · ,zn−1||

)]
+
[
Mk

(
||
tkn(A(y))

ρ2
,z1, · · · ,zn−1||

)]
.

Thus
[
Mk

(
||tkn(A(x+y))

ρ1+ρ2
,z1, · · · ,zn−1||

)]pk
≤ 1 and

g(x + y) ≤ inf
{

(ρ1 + ρ2) > 0 : ρ1 ∈ B(x), ρ2 ∈ B(y)
}

≤ inf
{
ρ1 > 0 : ρ1 ∈ B(x)

}
+ inf

{
ρ2 > 0 : ρ2 ∈ B(y)

}
= g(x) + g(y).

Let ηs → η where η,ηs ∈ C and let g(xs −x) → 0 as s → ∞. We have to show that g(ηsxs −ηx) → 0 as

s →∞. Let

B(xs) =
{
ρs > 0 :

[
M
(
||
tkn(A(x

s))

ρs
,z1, · · · ,zn−1||

)]pk
≤ 1
}
,

B(xs −x) =
{
ρ′s > 0 :

[
M
(
||
tkn(A(x

s −x))
ρ′s

,z1, · · · ,zn−1||
)]pk

≤ 1
}
.



Int. J. Anal. Appl. 16 (6) (2018) 889

If ρs ∈ B(xs) and ρ′s ∈ B(xs −x) then we observe that[
Mk

(
||tkn(A(η

sxs−ηx))
ρs|ηs−η|+ρ

′
s|η|

,z1, · · · ,zn−1||
)

≤
[
Mk

(
||
tkn(A(η

sxs −ηxs))
ρs|ηs −η| + ρ

′
s|η|

+
|(ηxs −ηx)|

ρs|ηs −η| + ρ
′
s|η|

,z1, · · · ,zn−1||
)]

≤
|ηs −η|ρs

ρs|ηs −η| + ρ
′
s|η|

[
Mk

(
||
tkn(A(x

s))

ρs
,z1, · · · ,zn−1||

)]
+

|η|ρ
′

s

ρs|ηs −η| + ρ
′
s|η|

[
Mk

(
||
tkn(A(x

s −x))
ρ

′
s

,z1, · · · ,zn−1||
)
.

From the above inequality, it follows that

[
Mk

(
||
tkn(A(η

sxs −ηx))
ρs|ηs −η| + ρ

′
s|η|

,z1, · · · ,zn−1||
)]pk

≤ 1

and consequently,

g(ηsxs −ηx) ≤ inf
{(
ρs|ηs −η| + ρ

′

s|η|
)
> 0 : ρs ∈ B(xs),ρ

′

s ∈ B(x
s −x)

}
≤ (|ηs −η|) > 0 inf

{
ρ > 0 : ρs ∈ B(xs)

}
+ (|η|) > 0 inf

{
(ρ

′

s)
pn
H : ρ

′

s ∈ B(x
s −x)

}
−→ 0 as s −→∞.

This completes the proof of the theorem. �

Theorem 2.4 Let M′ = (M′k) and M
′′ = (M′′k ) are Musielak-Orlicz functions that satisfies the ∆2-

condition. Then

(i) I− cσ0 (A,M′,p, ||·, · · · , ·||) ⊆I− cσ0 (A,M′ ◦M′′,p, ||·, · · · , ·||)

(ii) I− cσ(A,M′,p, ||·, · · · , ·||) ⊆I− cσ(A,M′ ◦M′′,p, ||·, · · · , ·||)

(iii) I− lσ∞(A,M′,p, ||·, · · · , ·||) ⊆I− lσ∞θ (A,M
′ ◦M′′,p, ||·, · · · , ·||).

Proof. (i) We prove the theorem in two parts. Firstly, let M′k

(
||tkn(A(x))

ρ
,z1, · · · ,zn−1||

)
> δ. Since M′ is

nondecreasing, convex and satisfies ∆2-condition, we have[
M′′k

(
M′k

(
||tkn(A(x))

ρ
,z1, · · · ,zn−1||

))]pk
≤ (Kδ−1M′′2 (2)

pk)
[
M′k

(
||
tkn(Ax)

ρ
,z1, · · · ,zn−1||

)]pk
≤ max{1, (Kδ−1M′′k (2)

H))H
[
M′k

(
||
tkn(Ax)

ρ
,z1, · · · ,zn−1

)]pk
,



Int. J. Anal. Appl. 16 (6) (2018) 890

where K ≥ 1 and δ < 1. From the last inequality, the inclusion{
k ∈ N :

[
M′′k

(
M′k

(
||tkn(Ax)

ρ
,z1, · · · ,zn−1||

))]pk
≥ �
}

⊆
{
k ∈ N :

[
M′k

(
||
tkn(Ax)

ρ
,z1, · · · ,zn−1

)]pk
≥

�

max{1, (Kδ−1M′′k (2)H)}

}
is obtained. If x ∈I−cσ0 (M′,A,p, ||·, · · · , ·), then the set in the right side of the above inclusion belongs to

the ideal and so {
k ∈ N :

[
M′′k

(
M′k

(
||
tkn(Ax)

ρ
,z1, · · · ,zn−1||

))]pk
≥ �
}
∈I.

Secondly, suppose that M′k

(
||tkn(Ax)

ρ
,z1, · · · ,zn−1||

)
≤ δ. Since M′′k is continuous, we have

M′′k

(
M′k

(
||
tkn(Ax)

ρ
,z1, · · · ,zn−1||

))
< � for all � > 0

which implies

I− lim
k

[
M′′k

(
M′k

(
||
tkn(Ax)

ρ
,z1, · · · ,zn−1||

))]pk
= 0 as� → 0.

This completes the proof of (i) part. Similarly, we can prove other parts. �

Theorem 2.5 Let M′ = (M′k) and M
′′ = (M′′k ) are Musielak-Orlicz functions that satisfies the ∆2-

condition. Then

(i) I− cσ0 (A,M,p, ||·, · · · , ·||) ∩I− cσ0 (A,M′,p, ||·, · · · , ·||) ⊆I− cσ0 (A,M′ + M,p, ||·, · · · , ·||)

(ii) I− cσ(A,M,p, ||·, · · · , ·||) ∩I− cσ(A,M′,p, ||·, · · · , ·||) ⊆I− cσ(A,M′ + M,p, ||·, · · · , ·||)

(iii) I− lσ∞(A,M,p, ||·, · · · , ·||) ∩I− lσ∞(A,M′,p, ||·, · · · , ·||) ⊆I− lσ∞(A,M′ + M,p, ||·, · · · , ·||).

Proof. (i) Let x ∈ I − cσ0 (A,M,p, ||·, · · · , ·||) ∩I − cσ0 (A,M′,p, ||·, · · · , ·||). Then there exists K1 > 0 and

K2 > 0 such that

A1 =
{
k ∈ N :

[
Mk

(
||
tkn(A(x))

ρ1
,z1, · · · ,zn−1||

)]pk
≥ K1

}
∈I

and

A2 =
{
k ∈ N :

[
M′k

(
||
tkn(A(x))

ρ1
,z1, · · · ,zn−1||

)]pk
≥ K2

}
∈I

for some ρ > 0. Let k /∈ A1 ∪A2. Then we have[
(Mk + M

′
k)
(
||tkn(A(x))

ρ
,z1, · · · ,zn−1||

)]pk
≤ D

{(
Mk

(
||
tkn(A(x))

ρ
,z1, · · · ,zn−1||

))pk
+

(
M′k

(
||
tkn(A(x))

ρ
,z1, · · · ,zn−1||

))pk
< {K1 + K2}.



Int. J. Anal. Appl. 16 (6) (2018) 891

k /∈ B =
{
k ∈ N :

[
(M′k + Mk)

(
||
tkn(A(x))

ρ
,z1, · · · ,zn−1||

))pk
> K}. We have A1 ∪ A2 ∈ I and so

B ⊂ A1 ∪A2 which implies B ∈ I. This means that x ∈ I − cσ0 (A,M′ + M,p, ||·, · · · , ·||). This completes

the proof of (i) part of the theorem. Similarly, we can prove (ii) and (iii) part. �

Theorem 2.6 If sup
k

[
Mk(t)

]pk
< ∞ for all t > 0, then we have

I− cσ(A,M,p, ||·, · · · , ·||) ⊆I− `σ∞(A,M,p, ||·, · · · , ·||).

Proof. Let x ∈I− cσ(A,M,p, ||·, · · · , ·||). By using inequality (1.1), we have

[
Mk

(
||
tkn(A(x))

ρ

)]pk
≤ D

{[
Mk

(
||
tkn(A(x) −L)

ρ
,z1, · · · ,zn−1||

)]pk
+

[
Mk

(
||
tkn(L)

ρ
,z1, · · · ,zn−1||

)]pk}
,

where ρ = 2ρ1. Hence, we have

{
k ∈ N :

[
Mk

(
||
tkn(A(x))

ρ

)]pk
≥ K

}
⊆
{
k ∈ N :

[
Mk

(
||
tkn(A(x) −L)

ρ1
,z1, · · · ,zn−1

)]pk
≥ �
}

for all n and some K > 0. Since the set in the right side of the above inclusion belongs to the ideal, all of

its subsets are in the ideal. Hence

{
k ∈ N :

[
Mk

(
||
tkn(A(x))

ρ

)]pk
≥ K

}
∈I

which completes the proof. �

Theorem 2.7 Let 0 < pk ≤ qk < ∞ for each k ∈ N and
(
qk
pk

)
be bounded. Then following inclusions hold

(i) I− cσ0 (A,M,q, ||·, · · · , ·||) ⊆I− cσ0 (A,M,p, ||·, · · · , ·||)

(ii) I− cσ(A,M,q, ||·, · · · , ·||) ⊆I− cσ(A,M,p, ||·, · · · , ·||).

Proof. (i) Let x ∈ I − cσ0 (A,M,q, ||·, · · · , ·||). Write αk =
pk
qk

. By hypothesis, we have 0 < α ≤ αk ≤ 1. If[
Mk

(
||tkn(A(x))

ρ
,z1, · · · ,zn−1||

)]qk
≥ 1, the inequality

[
Mk

(
||
tkn(A(x))

ρ
,z1, · · · ,zn−1||

)]pk
≤
[
Mk

(
||
tkn(A(x))

ρ
,z1, · · · ,zn−1||

)]qk
holds. This implies the inclusion{
k ∈ N :

[
Mk

(
||tkn(A(x))

ρ
,z1, · · · ,zn−1||

)]pk
≥ �
}

⊆
{
k ∈ N :

[
Mk

(
||
tkn(A(x))

ρ
,z1, · · · ,zn−1||

)]qk
≥ �
}



Int. J. Anal. Appl. 16 (6) (2018) 892

and so the result is obvious. Conversely, if
[
Mk

(
||tkn(A(x))

ρ
,z1, · · · ,zn−1||

)]qk
< 1, we obtain the following

inclusion{
k ∈ N :

[
Mk

(
||tkn(A(x))

ρ
,z1, · · · ,zn−1||

)]pk
≥ �
}

⊆
{
k ∈ N :

[
Mk

(
||
tkn(A(x))

ρ
,z1, · · · ,zn−1||

)]qk
≥ �

1
α

}
since then the inequality

[
Mk

(
||
tkn(A(x))

ρ
,z1, · · · ,zn−1||

)]pk
≤
([
Mk

(
||
tkn(A(x))

ρ
,z1, · · · ,zn−1||

)]qk)α
holds. Hence we conclude that x ∈I−cσ0 (A,M,p, ||·, · · · , ·||). This completes the proof of (i) part. Similarly,

we can prove (ii) part. �

Theorem 2.8 If 0 < inf pk ≤ pk ≤ 1 for each k ∈ N. Then the following inclusions hold:

(i) I− cσ0 (A,M,p, ||·, · · · , ·||) ⊆I− cσ0 (A,M, ||·, · · · , ·||)

(ii) I− cσ(A,M,p, ||·, · · · , ·||) ⊆I− cσ(A,M, ||·, · · · , ·||).

Proof. Let x ∈ I − cσ0 (A,M,p, ||·, · · · , ·||). Suppose that k /∈
{[
Mk

(
||tkn(A(x))

ρ
,z1, · · · ,zn−1

)]pk
≥ �
}

for

0 < � < 1. By hypothesis, the inequality

Mk

(
||
tkn(A(x))

ρ
,z1, · · · ,zn−1||

)
≤
[
Mk

(
||
tkn(A(x))

ρ
,z1, · · · ,zn−1||

)]pk
holds. Then we have k /∈

{
k ∈ N : Mk

(
||tkn(A(x))

ρ
,z1, · · · ,zn−1||

)
≥ �
}

which implies{
k ∈ N : Mk

(
||tkn(A(x))

ρ
,z1, · · · ,zn−1||

)
≥ �
}

⊆
{
k ∈ N :

[
Mk

(
||
tkn(A(x))

ρ
,z1, · · · ,zn−1||

)]pk
≥ �
}
.

Hence x ∈I− cσ0 (A,M, ||·, · · · , ·||) since the set

{
k ∈ N : Mk

(
||
tkn(A(x))

ρ
,z1, · · · ,zn−1||

)
≥ �
}
∈I.

This completes the proof of (i) part. Similarly, we can prove (ii) part. �

Corollary 2.9 If 0 < inf pk ≤ pk ≤ 1 for each k ∈ N. Then the following inclusions hold:

(i) I− cσ0 (A,M, ||·, · · · , ·||) ⊆I− cσ0 (A,M,p, ||·, · · · , ·||)

(ii) I− cσ(A,M, ||·, · · · , ·||) ⊆I− cσ(A,M,p, ||·, · · · , ·||).

Proof. The proof is obvious by Theorem 2.8. �



Int. J. Anal. Appl. 16 (6) (2018) 893

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	1. Introduction and preliminaries
	2. Main Results 
	References