# Some New Classes of Generalized Difference Strongly Summablen-Normed Sequence Spaces Defined by Ideal Convergence and Orlicz Function

Authors: Adem Kılıçman; Stuti Borgohain
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We study some new generalized difference strongly summablen-normed sequence spaces using ideal convergence and an Orlicz function in connection with de la Vallèe Poussin mean. We give some relations related to these sequence spaces also.


---

## Body

## 1. Introduction

Letl
∞, c, and c
0 be the Banach space of bounded, convergent, and null sequences x
=
(
x
k
), respectively, with the usual norm ∥
x
∥
=
sup
⁡
n
|
x
n
|.A sequencex
∈
l
∞ is said to be almost convergent if all of its Banach limits coincide.Letc
^ denote the space of all almost convergent sequences.Lorentz in [1] proved that
(1)
c
^
=
{
x
∈
l
∞
:
lim
⁡
m
⁡
t
m
,
n
(
x
)
exists
uniformly
in
n
}
,
where
(2)
t
m
,
n
(
x
)
=
x
n
+
x
n
+
1
+
⋯
+
x
m
+
n
m
+
1
.
The following space of strongly almost convergent sequence was introduced by Maddox in [2]:
(3)
[
c
^
]
=
{
x
∈
l
∞
:
lim
⁡
m
⁡
t
m
,
n
(
|
x
-
L
e
|
)
exists
uniformly
in
n
for
some
L
x
∈
l
∞
:
lim
⁡
m
⁡
t
m
,
n
(
|
x
-
L
e
|
)
}
,
where e
=
(
1,1
,
…
).Letσ be a one-to-one mapping from the set of positive integers into itself such that σ
m
(
n
)
=
σ
m
-
1
(
σ
(
n
)
), m
=
1,2
,
3
,
…, where σ
m
(
n
) denotes the mth iterative of the mapping σ in n.A continuous linear functionalϕ on l
∞ is said to be an invariant mean or a σ-mean, if and only if it satisfies the following conditions:(1)
ϕ
(
x
)
≥
0, when the sequence x
=
(
x
n
) is such that x
n
≥
0 for all n;
(2)
ϕ
(
e
)
=
1, where e
=
(
1,1
,
…
);
(3)
ϕ
(
x
σ
(
n
)
)
=
ϕ
(
x
), for all x
∈
l
∞.For a certain kind of mapping σ, we get that every invariant mean ϕ extends the functional limit on the space c, such that ϕ
(
x
)
=
lim
⁡
⁡
x for all x
∈
c. Consequently, we get that c
⊂
V
σ, where V
σ is the set of bounded sequences with equal σ-means.Schaefer in [3] proved that
(4)
V
σ
=
{
x
∈
l
∞
:
lim
⁡
k
⁡
t
k
m
(
x
)
=
L
uniformly
in
m
for
some
L
=
σ
-
lim
⁡
⁡
x
}
,
where
(5)
t
k
m
(
x
)
=
x
+
x
σ
(
m
)
+
⋯
+
x
σ
k
(
m
)
k
+
1
,
t
-
1
,
m
=
0
.
Thus we say that a bounded sequence x
=
(
x
k
) is σ-convergent, if and only if x
∈
V
σ such that σ
k
(
n
)
≠
n for all n
≥
0, k
≥
1.Note that similarly as the concept of almost convergence leads naturally to the concept of strong almost convergence, theσ-convergence leads naturally to the concept of strong σ-convergence.A sequencex
=
(
x
k
) is said to be strongly σ-convergent (Mursaleen [4]), if there exists a number l such that
(6)
1
k
∑
i
=
1
k
‍
|
x
σ
i
(
m
)
-
l
|
⟶
0
,
as
k
⟶
∞
uniformly
in
m
.
We write [
V
σ
] to denote the set of all strong σ-convergent sequences and when (6) holds, we write [
V
σ
]
-
lim
⁡
x
=
l.Takingσ
(
m
)
=
m
+
1, we obtain [
V
σ
]
=
[
c
^
]. Then the strong σ-convergence generalizes the concept of strong almost convergence.We also note that(7)
[
V
σ
]
⊂
V
σ
⊂
l
∞
.
The notion of ideal convergence was first introduced by Kostyrko et al. [5] as a generalization of statistical convergence which was later studied by many other authors.An Orlicz function is a functionM
:
[
0
,
∞
)
→
[
0
,
∞
), which is continuous, nondecreasing, and convex with M
(
0
)
=
0, M
(
x
)
>
0, for x
>
0, and M
(
x
)
→
∞, as x
→
∞.Lindenstrauss and Tzafriri [6] used the idea of Orlicz function to construct the sequence space:
(8)
l
M
=
{
(
x
k
)
∈
w
:
∑
k
=
1
∞
‍
M
(
|
x
|
ρ
)
<
∞
,
for
some
ρ
>
0
}
.
The space l
M with the norm
(9)
∥
x
∥
=
inf
⁡
{
ρ
>
0
:
∑
k
=
1
∞
‍
M
(
|
x
k
|
ρ
)
≤
1
}
becomes a Banach space which is called an Orlicz sequence space.Kızmaz [7] studied the difference sequence spaces l
∞
(
Δ
), c
(
Δ
), and c
0
(
Δ
) of crisp sets. The notion is defined as follows:
(10)
Z
(
Δ
)
=
{
x
=
(
x
k
)
:
(
Δ
x
k
)
∈
Z
}
,
for Z
=
l
∞
,
c and c
0, where Δ
x
=
(
Δ
x
k
)
=
(
x
k
-
x
k
+
1
), for all k
∈
N.The above spaces are Banach spaces, normed by(11)
∥
x
∥
Δ
=
|
x
1
|
+
sup
⁡
k
|
Δ
x
k
|
.
Later the idea of Kızmaz [7] was applied to introduce different types of difference sequence spaces and study their different properties by many others later on.The generalized difference notion is defined as follows. Form
≥
1 and n
≥
1,
(12)
Z
(
Δ
m
n
)
=
{
x
=
(
x
k
)
:
(
Δ
m
n
x
k
)
∈
Z
}

,
for
Z
=
l
∞
,
c
and
c
0
.
This generalized difference has the following binomial representation:
(13)
Δ
m
n
x
k
=
∑
r
=
0
n
‍
(
-
1
)
r
(
n
r
)
x
k
+
r
m
.
The concept of 2-normed space was initially introduced by Gähler [8], in the mid of 1960s, while that of n-normed spaces can be found in Misiak [9]. Since then, many other authors have used this concept and obtained various results. Recently, several various activities have been initiated to study summability, sequence spaces, and related topics in these spaces. The notion of ideal convergence in 2-normed spaces was initially introduced by Gurdal [10]. Later on, it was extended to n-normed spaces by Gurdal and Sahiner in [11].


## 2. Definitions and Preliminaries

Letn
∈
N and X be a real vector space. A real valued function on X
n satisfies the following four properties:(1)
∥
(
z
1
,
z
2
,
…
,
z
n
)
∥
n
=
0 if and only if z
1
,
z
2
,
…
,
z
n are linearly dependent;
(2)
∥
(
z
1
,
z
2
,
…
,
z
n
)
∥
n is invariant under permutation;
(3)
∥
(
z
1
,
z
2
,
…
,
z
n
-
1
,
α
z
n
)
∥
n
=
|
α
|
∥
z
1
,
z
2
,
…
,
z
n
∥
n, for all α
∈
R;
(4)
∥
(
z
1
,
z
2
,
…
,
z
n
-
1
,
x
+
y
)
∥
n
≤
∥
(
z
1
,
z
2
,
…
,
z
n
-
1
,
x
)
∥
n
+
∥
(
z
1
,
z
2
,
…
,
z
n
-
1
,
y
)
∥
nis called an n-norm on X and the pair (
X
,
∥
·
,
·
∥
n
) is called an n-normed space.LetX be a nonempty set. Then a family of sets I
⊆
2
X (power sets of X) is said to be an i
d
e
a
l if I is additive that is A
,
B
∈
I
⇒
A
∪
B
∈
I and hereditary that is A
∈
I
,
B
⊆
A
⇒
B
∈
I.A sequence(
x
k
) in a normed space (
X
,
∥
·
,
·
,
⋯
∥
n
) is said to be I
-convergent to x
0
∈
X with respect to n-norm, if for each ɛ
>
0, the set
(14)
E
(
ɛ
)
=
{
∈
X
k
∈
N
:
∥
x
k
-
x
0
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
≥
ɛ
,
for
every
z
1
,
z
2
,
…
,
z
n
-
1
∈
X
}
belongs
to
I
.
The generalized de la Vallée Poussin mean is defined by
(15)
t
n
(
x
)
=
1
λ
n
∑
k
∈
I
n
x
k
,
where I
n
=
[
n
-
λ
n
+
1
,
n
] for n
=
1,2
….Then a sequencex
=
(
x
k
) is said to be (
V
,
λ
)-summable to a number L, if t
n
(
x
)
→
L as n
→
∞, and we write
(16)
[
V
,
λ
]
0
=
{
x
:
lim
⁡
n
1
λ
n
∑
k
∈
I
n
‍
|
x
k
|
=
0
}
,
[
V
,
λ
]
=
{
x
:
x
-
l
e
∈
[
V
,
λ
]
0
for
some
l
∈
C
}
,
[
V
,
λ
]
∞
=
{
x
:
sup
⁡
n
1
λ
n
∑
k
∈
I
n
‍
|
x
k
|
<
∞
}
for the sets of sequences that are, respectively, strongly summable to zero, strongly summable, and strongly bounded by de la Vallée Poussin method.Maddox introduced and studied the special case, whereλ
n
=
n, for n
=
1,2
,
3
,
…; the sets [
V
,
λ
]
0, [
V
,
λ
], and [
V
,
λ
]
∞ reduce to the sets w
0, w, and w
∞.In this paper, we define some new sequence spaces inn-normed spaces by using Orlicz function with notion of generalized de la Vallèe Poussin mean, generalized difference sequences, and ideals. We will also introduce and examine certain new sequence spaces using the above tools as also the n-norm.


## 3. Main Results

LetI be an admissible ideal of N, let M be an Orlicz function, and let (
X
,
∥
·
,
⋯
∥
n
) be a n-normed space. Further, let s
=
(
s
k
) be a bounded sequence of positive real numbers. By S
(
n
-
X
), we denote the space of all sequences defined over (
X
,
∥
·
,
⋯
∥
n
).In this paper, we have introduced the following sequence spaces:(17)
(
[
V
σ
,
λ
,
Δ
p
q
,
M
,
r
]
0
I
,
∥
·
,
·
,
⋯
∥
n
)
=
{
{
n
∈
N
:
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
x
σ
k
(
m
)
,
z
1
,
z
2
,
n
-
1
∥
n
ρ
)
}
r
k
≥
ɛ
}
∈
I
x
:
∀
ɛ
>
0
×
{
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
x
σ
k
(
m
)
,
z
1
,
z
2
,
…
z
n
-
1
∥
n
ρ
)
}
r
k
≥
ɛ
n
∈
N
:
i
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
x
σ
k
(
m
)
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
ρ
)
}
r
k
i
≥
ɛ
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
x
σ
k
(
m
)
,
z
1
,
z
2
,
…
z
n
-
1
∥
n
ρ
)
}
r
k
}
∈
I
}
,
for
some
ρ
>
0
and
each
z
∈
X
,
(
[
V
σ
,
λ
,
Δ
p
q
,
M
,
r
]
I
,
∥
·
,
·
,
⋯
∥
n
)
=
{
{
n
∈
N
:
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
x
σ
k
(
m
)
-
L
,
z
1
,
z
2
,
…
z
n
-
1
∥
n
ρ
)
}
r
k
≥
ɛ
}
∈
I
x
:
∀
ɛ
>
0
×
{
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
x
σ
k
(
m
)
-
L
,
z
1
,
z
2
,
…
z
n
-
1
∥
n
ρ
)
}
r
k
n
∈
N
:
i
1
λ
n
∑
k
∈
I
n
‍
{

M
(

∥
Δ
p
q
x
σ
k
(
m
)
-
L
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
ρ

)

}
r
k
i
≥
ɛ
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
x
σ
k
(
m
)
-
L
,
z
1
,
z
2
,
…
z
n
-
1
∥
n
ρ
)
}
r
k
}
∈
I
}
,
for
some
ρ
>
0
,
L
∈
X
and
each
z
∈
X
,
(
[
V
σ
,
λ
,
Δ
p
q
,
M
,
r
]
∞
I
,
∥
·
,
·
,
⋯
∥
n
)
=
{
s
.
t
.
{
s
u
p
n
,
m
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
x
σ
k
(
m
)
,
z
1
,
z
2
,
n
-
1
∥
n
ρ
)
}
r
k
≥
K
}
∈
I
x
:
∃
K
>
0
,
s
.
t
.

{

sup
⁡
n
,
m
1
λ
n
∑
k
∈
I
n
‍

{

M

(


∥
Δ
p
q
x
σ
k
(
m
)
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
ρ

)

}
r
k
i
≥
K
sup
⁡
n
,
m
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
x
σ
k
(
m
)
,
z
1
,
z
2
,
…
z
n
-
1
∥
n
ρ
)
}
r
k
}
∈
I
}
,
for
some
ρ
>
0
and
each
z
∈
X
.
In particular, if we take r
k
=
1, for all k, we have
(18)
(
[
V
σ
,
λ
,
Δ
p
q
,
M
]
0
I
,
∥
·
,
⋯
∥
n
)
=
{
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
x
σ
k
(
m
)
,
z
1
,
z
2
,
…
z
n
-
1
∥
n
ρ
)
}
x
:
∀
ɛ
>
0
i
×
{
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
x
σ
k
(
m
)
,
z
1
,
z
2
,
…
z
n
-
1
∥
n
ρ
)
}
n
∈
N
:
i
i
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
x
σ
k
(
m
)
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
ρ
)
}
i
i
≥
ɛ
,
uniformly
in
m
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
x
σ
k
(
m
)
,
z
1
,
z
2
,
…
z
n
-
1
∥
n
ρ
)
}
}
∈
I
}
,
for
some
ρ
,
∀
z
1
,
z
2
,
…
,
z
n
-
1
∈
X
,
(
[
V
σ
,
λ
,
Δ
p
q
,
M
]
I
,
∥
·
,
·
,
⋯
∥
n
)
=
{
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
x
σ
k
(
m
)
-
L
,
z
1
,
z
2
,
n
-
1
∥
n
ρ
)
}
x
:
∀
ɛ
>
0
i
×
{
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
x
σ
k
(
m
)
-
L
,
z
1
,
z
2
,
n
-
1
∥
n
ρ
)
}
n
∈
N
:
i
i
1
λ
n
∑
k
∈
I
n
‍
{

M
(
∥
Δ
p
q
x
σ
k
(
m
)
-
L
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
ρ
)

}
i
i
≥
ɛ
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
x
σ
k
(
m
)
-
L
,
z
1
,
z
2
,
n
-
1
∥
n
ρ
)
}
}
∈
I
}
,
for
some
ρ
>
0
,
L
∈
X
and
each
z
∈
X
,
(
[
V
σ
,
λ
,
Δ
p
q
,
M
]
∞
I
,
∥
·
,
·
,
⋯
∥
n
)
=
{
s
u
p
n
,
m
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
x
σ
k
(
m
)
,
z
1
,
z
2
,
n
-
1
∥
n
ρ
)
}
x
:
∃
K
>
0
×
{
sup
⁡
n
,
m
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
x
σ
k
(
m
)
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
ρ
)
}
≥
K
s
u
p
n
,
m
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
x
σ
k
(
m
)
,
z
1
,
z
2
,
n
-
1
∥
n
ρ
)
}
}
∈
I
}
.
Similarly, when σ
(
m
)
=
m
+
1, then (
[
V
^
,
λ
,
Δ
p
q
,
M
,
r
]
0
I
,
∥
⋯
∥
n
), (
[
V
σ
,
λ
,
Δ
p
q
,
M
,
r
]
I
,
∥
⋯
∥
n
), and (
[
V
σ
,
λ
,
Δ
p
q
,
M
,
r
]
∞
I
,
∥
⋯
∥
n
) are reduced to
(19)
(
[
V
^
,
λ
,
Δ
p
q
,
M
,
r
]
0
I
,
∥
⋯
∥
n
)
=
{
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
x
k
+
m
,
z
1
,
z
2
,
…
z
n
-
1
∥
n
ρ
)
}
r
k
x
:
∀
ɛ
>
0
×
{
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
x
k
+
m
,
z
1
,
z
2
,
…
z
n
-
1
∥
n
ρ
)
}
r
k
n
∈
N
:
1
λ
n
∑
k
∈
I
n
‍
{
M
(

∥
Δ
p
q
x
k
+
m
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
ρ

)
}
r
k
≥
ɛ
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
x
k
+
m
,
z
1
,
z
2
,
…
z
n
-
1
∥
n
ρ
)
}
r
k
}
∈
I
}
,
for
some
ρ
>
0
and
each
z
∈
X
,
(
[
V
^
,
λ
,
Δ
p
q
,
M
,
r
]
I
,
∥
·
,
⋯
∥
n
)
=
{
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
x
k
+
m
-
L
,
z
1
,
z
2
,
…
z
n
-
1
∥
n
ρ
)
}
r
k
x
:
∀
ɛ
>
0
×
{
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
x
k
+
m
-
L
,
z
1
,
z
2
,
…
z
n
-
1
∥
n
ρ
)
}
r
k
n
∈
N
:
1
λ
n
∑
k
∈
I
n
‍
{
M
(

∥
Δ
p
q
x
k
+
m
-
L
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
ρ

)
}
r
k
≥
ɛ
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
x
k
+
m
-
L
,
z
1
,
z
2
,
…
z
n
-
1
∥
n
ρ
)
}
r
k
}
∈
I
}
,
for
some
ρ
>
0
,
L
∈
X
and
each
z
∈
X
,
(
[
V
^
,
λ
,
Δ
p
q
,
M
,
r
]
∞
I
,
∥
·
,
⋯
∥
n
)
=
{
s
u
p
n
,
m
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
x
k
+
m
,
z
1
,
z
2
,
…
z
n
-
1
∥
n
ρ
)
}
r
k
x
:
∃
K
>
0
,
i
s
.
t
.

{
sup
⁡
n
,
m
1
λ
n
∑
k
∈
I
n
‍

{

M

(

∥
Δ
p
q
x
k
+
m
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
ρ

)

}
r
k
≥
k
s
u
p
n
,
m
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
x
k
+
m
,
z
1
,
z
2
,
…
z
n
-
1
∥
n
ρ
)
}
r
k
}
∈
I
}
,
for
some
ρ
>
0
,
and
each
z
∈
X
.
In particular, if we put r
k
=
r, for all k, then we have the spaces
(20)
(
[
V
^
,
λ
,
Δ
p
q
,
M
,
r
]
0
I
,
∥
·
,
⋯
∥
n
)
=
(
[
V
^
,
λ
,
Δ
p
q
,
M
]
0
I
,
∥
·
,
·
∥
n
)
,
(
[
V
^
,
λ
,
Δ
p
q
,
M
,
r
]
I
,
∥
·
,
⋯
∥
n
)
=
(
[
V
^
,
λ
,
Δ
p
q
,
M
]
I
,
∥
·
,
·
∥
n
)
,
(
[
V
^
,
λ
,
Δ
p
q
,
M
,
r
]
∞
I
,
∥
·
,
⋯
∥
n
)
=
(
[
V
^
,
λ
,
Δ
p
q
,
M
]
∞
I
,
∥
·
,
·
∥
n
)
.
Further when λ
n
=
n, for n
=
1,2
,
…, the sets (
[
V
^
,
λ
,
Δ
p
q
,
M
]
0
I
,
∥
·
,
·
,
∥
n
) and (
[
V
^
,
λ
,
Δ
p
q
,
M
]
I
,
∥
·
,
·
,
∥
n
) are reduced to (
[
c
^
0
(
M
,
Δ
p
q
)
]
I
,
∥
·
,
·
,
⋯
∥
n
) and (
[
c
^
(
M
,
Δ
p
q
)
]
I
,
∥
·
,
·
,
⋯
∥
n
), respectively.Now, if we considerM
(
x
)
=
x, then we can easily obtain
(21)
(
[
V
σ
,
λ
,
Δ
p
q
,
r
]
0
I
,
∥
·
,
⋯
∥
n
)
=
{
1
λ
n
∑
k
∈
I
n
‍
(
∥
Δ
p
q
x
σ
k
(
m
)
,
,
z
1
,
z
2
,
…
z
n
-
1
∥
n
)
r
k
x
:
∀
ɛ
>
0
×
{
1
λ
n
∑
k
∈
I
n
‍
(
∥
Δ
p
q
x
σ
k
(
m
)
,
,
z
1
,
z
2
,
…
z
n
-
1
∥
n
)
r
k
n
∈
N
:
1
λ
n
∑
k
∈
I
n
‍
(
∥
Δ
p
q
x
σ
k
(
m
)
,
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
)
r
k
≥
ɛ
,
uniformly
in
m
1
λ
n
∑
k
∈
I
n
‍
(
∥
Δ
p
q
x
σ
k
(
m
)
,
,
z
1
,
z
2
,
…
z
n
-
1
∥
n
)
r
k
}
∈
I
}
,
(
[
V
σ
,
λ
,
Δ
p
q
,
r
]
I
,
∥
·
,
⋯
∥
n
)
=
{
1
λ
n
∑
k
∈
I
n
‍
(
∥
Δ
p
q
x
σ
k
(
m
)
-
L
,
z
1
,
z
2
,
…
z
n
-
1
∥
n
)
r
k
x
:
∀
ɛ
>
0
×
{
1
λ
n
∑
k
∈
I
n
‍
(
∥
Δ
p
q
x
σ
k
(
m
)
-
L
,
z
1
,
z
2
,
…
z
n
-
1
∥
n
)
r
k
n
∈
N
:
1
λ
n
∑
k
∈
I
n
‍
(
∥
Δ
p
q
x
σ
k
(
m
)
-
L
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
)
r
k
≥
ɛ
1
λ
n
∑
k
∈
I
n
‍
(
∥
Δ
p
q
x
σ
k
(
m
)
-
L
,
z
1
,
z
2
,
…
z
n
-
1
∥
n
)
r
k
}
∈
I
}
,
for
L
∈
X
and
each
z
∈
X
,
(
[
V
σ
,
λ
,
Δ
p
q
,
r
]
∞
I
,
∥
·
,
⋯
∥
n
)
=
{
sup
⁡
n
,
m
1
λ
n
∑
k
∈
I
n
‍
(
∥
Δ
p
q
x
σ
k
(
m
)
,
z
1
,
z
2
,
…
z
n
-
1
∥
n
)
r
k
x
:
∃
K
>
0
,
i
s
.
t
.
{
sup
⁡
n
,
m
1
λ
n
∑
k
∈
I
n
‍
(
∥
Δ
p
q
x
σ
k
(
m
)
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
)
r
k
≥
K
sup
⁡
n
,
m
1
λ
n
∑
k
∈
I
n
‍
(
∥
Δ
p
q
x
σ
k
(
m
)
,
z
1
,
z
2
,
…
z
n
-
1
∥
n
)
r
k
}
∈
I
}
.
If x
∈
(
[
V
σ
,
λ
,
Δ
p
q
,
M
,
r
]
I
,
∥
⋯
∥
n
), with {
(
1
/
λ
n
)
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
x
σ
k
(
m
)
-
L
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
/
ρ
)
}
r
k
≥
ɛ
}
∈
I as n
→
∞ uniformly in m, then we write x
k
→
L
∈
(
[
V
σ
,
λ
,
Δ
p
q
,
M
,
r
]
I
,
∥
⋯
∥
n
).The following well-known inequality will be used later.If0
≤
r
k
≤
sup
⁡
r
k
=
H and C
=
max
⁡
(
1
,
2
H
-
1
), then
(22)
|
a
k
+
b
k
|
r
k
≤
C
{
|
a
k
|
r
k
+
|
b
k
|
r
k
}
,
for all k and a
k
,
b
k
∈
C.Lemma 1.
Letr
k
>
0 and s
k
>
0. Then c
0
(
s
)
⊂
c
0
(
r
), if and only if lim
⁡
k
→
∞
inf
⁡
⁡
(
r
k
/
s
k
)
>
0, where c
0
(
r
)
=
{
x
:
|
x
k
|
r
k
→
0
a
s
k
→
∞
}.
Note that no other relation between(
r
k
) and (
s
k
) is needed in Lemma 1.Theorem 2.
Letlim
⁡
k
→
∞
inf
⁡
r
k
>
0. Then, x
k
→
L implies x
k
→
L
∈
(
[
V
σ
,
λ
,
Δ
p
q
,
M
,
r
]
I
,
∥
⋯
∥
n
). Let lim
⁡
k
→
∞
r
k
=
r
>
0. If x
k
→
L
∈
(
[
V
σ
,
λ
,
Δ
p
q
,
M
,
r
]
I
,
∥
⋯
∥
n
), then L is unique.Proof.
Letx
k
→
L.
By the definition of Orlicz function, we have, for allɛ
>
0,
(23)
{
lim
⁡
n
1
λ
n
∑
k
∈
I
n
‍
M
(
∥
Δ
p
q
x
σ
k
(
m
)
-
L
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
ρ
)
≥
ɛ
lim
⁡
n
1
λ
n
∑
k
∈
I
n
‍
M
(
∥
Δ
p
q
x
σ
k
(
m
)
-
L
,
z
1
,
z
2
,
…
z
n
-
1
∥
n
ρ
)
}
∈
I
.
Since lim
⁡
k
→
∞
inf
⁡
r
k
>
0, it follows that
(24)
{
lim
⁡
n
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
x
σ
k
(
m
)
-
L
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
ρ
)
}
r
k
≥
ɛ
lim
⁡
n
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
x
σ
k
(
m
)
-
L
,
z
1
,
z
2
,
…
z
n
-
1
∥
n
ρ
)
}
r
k
}
∈
I
.
And consequently, x
k
→
L
∈
[
V
σ
,
λ
,
Δ
p
q
,
M
,
r
]
I.
Letlim
⁡
k
→
∞
r
k
=
r
>
0. Suppose that x
k
→
L
1
∈
[
V
σ
,
λ
,
Δ
p
q
,
M
,
r
]
I, x
k
→
L
2
∈
[
V
σ
,
λ
,
Δ
p
q
,
M
,
r
]
I, and (
∥
L
1
-
L
2
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
)
r
k
=
a
>
0.
Now, from (22) and the definition of Orlicz, we have
(25)
1
λ
n
lim
⁡
k
∈
I
n
M
(
∥
L
1
-
L
2
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
ρ
)
r
k
≤
C
λ
n
∑
k
∈
I
n
‍
M
(
∥
Δ
p
q
x
σ
k
(
m
)
-
L
1
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
ρ
)
r
k
+
C
λ
n
M
(
∥
Δ
p
q
x
σ
k
(
m
)
-
L
2
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
ρ
)
r
k
,
since
(26)
{
1
λ
n
∑
k
∈
I
n
‍
M
(
∥
Δ
p
q
x
σ
k
(
m
)
-
L
1
,
z
1
,
z
2
,
…
z
n
-
1
∥
n
ρ
)
r
k
≥
ɛ
n
∈
N
:
1
λ
n
∑
k
∈
I
n
‍
M
(
∥
Δ
p
q
x
σ
k
(
m
)
-
L
1
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
ρ
)
r
k
≥
ɛ
:
1
λ
n
∑
k
∈
I
n
‍
M
(
∥
Δ
p
q
x
σ
k
(
m
)
-
L
1
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
ρ
)
r
k
}
∈
I
,
{
1
λ
n
∑
k
∈
I
n
‍
M
(
∥
Δ
p
q
x
σ
k
(
m
)
-
L
2
,
z
1
,
z
2
,
…
z
n
-
1
∥
n
ρ
)
r
k
≥
ɛ
n
∈
N
:
1
λ
n
∑
k
∈
I
n
‍
M
(
∥
Δ
p
q
x
σ
k
(
m
)
-
L
2
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
ρ
)
r
k
≥
ɛ
:
1
λ
n
∑
k
∈
I
n
‍
M
(
∥
Δ
p
q
x
σ
k
(
m
)
-
L
2
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
ρ
)
r
k
}
∈
I
.
Hence,
(27)
{
n
∈
N
:
1
λ
n
∑
k
∈
I
n
‍
M
(
∥
Ł
1
-
L
2
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
ρ
)
r
k

≥
ɛ
}

∈
I
.
Further, M
(
∥
L
1
-
L
2
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
/
ρ
)
r
k
→
M
(
a
/
ρ
)
r as k
→
∞, and therefore
(28)
lim
⁡
n
→
∞
1
λ
n
∑
k
∈
I
n
‍
M
(
∥
L
1
-
L
2
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
ρ
)
r
k
=
M
(
a
ρ
)
r
.
From (27) and (28), it follows that M
(
a
/
ρ
)
=
0 and by the definition of an Orlicz function, we have a
=
0.
Hence,L
1
=
L
2 and this completes the proof.Theorem 3.
(i) Let0
<
inf
⁡
k
r
k
≤
r
k
≤
1. Then,
(29)
[
V
σ
,
λ
,
Δ
p
q
,
M
,
r
]
I
⊂
[
V
σ
,
λ
,
Δ
p
q
,
M
]
I
.
(ii) Let 0
<
r
k
≤
sup
⁡
k
r
k
<
∞. Then,
(30)
[
V
σ
,
λ
,
Δ
p
q
,
M
]
I
⊂
[
V
σ
,
λ
,
Δ
p
q
,
M
,
r
]
I
.Theorem 4.
LetX
(
V
σ
,
λ
,
Δ
p
q
-
1
) stand for (
[
V
σ
,
λ
,
Δ
p
q
-
1
,
M
,
r
]
0
I
,
∥
·
,
·
,
⋯
∥
n
), (
[
V
σ
,
λ
,
Δ
p
q
-
1
,
M
,
r
]
I
,
∥
·
,
·
,
⋯
∥
n
), or (
[
V
σ
,
λ
Δ
p
q
-
1
,
M
,
r
]
∞
I
,
∥
·
,
·
,
⋯
∥
n
) and m
≥
1. Then the inclusion X
(
V
σ
,
λ
,
Δ
p
q
-
1
)
⊂
X
(
V
σ
,
λ
,
Δ
p
q
) is strict. In general, X
(
V
σ
,
λ
,
Δ
p
i
)
⊂
X
(
V
σ
,
λ
,
Δ
p
q
) for all i
=
1,2
,
3
,
…
,
p
-
1 and the inclusion is strict.Proof.
Let us take(
[
V
σ
,
λ
,
Δ
p
q
-
1
,
M
,
r
]
0
I
,
∥
·
,
·
,
⋯
∥
n
).
Letx
=
(
x
k
)
∈
(
[
V
σ
,
λ
,
Δ
p
q
-
1
,
M
,
r
]
0
I
,
∥
·
,
·
,
⋯
∥
n
). Then for given ɛ
>
0, we have
(31)
{
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
-
1
x
σ
k
(
m
)
,
z
1
,
z
2
,
…
z
n
-
1
∥
n
ρ
)
}
r
k
≥
ɛ
n
∈
N
:
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
-
1
x
σ
k
(
m
)
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
ρ
)
}
r
k
≥
ɛ
:
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
-
1
x
σ
k
(
m
)
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
ρ
)
}
r
k
}
∈
I
,
for
some
ρ
>
0
.
Since M is nondecreasing and convex, it follows that
(32)
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
x
σ
k
(
m
)
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
ρ
)
}
r
k
=
1
λ
n


×


∑
k
∈
I
n
‍


{

M

(

∥
Δ
p
q
-
1
x
σ
k
+
1
(
m
)
-
Δ
p
q
-
1
x
σ
k
(
m
)
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
ρ


)

}
r
k
≤
D
1
λ
n
∑
k
∈
I
n
‍
(
[
1
2
M
(
∥
Δ
p
q
-
1
x
σ
k
+
1
(
m
)
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
ρ
)
]
r
k
i
+
[
1
2
M
(
∥
Δ
p
q
-
1
x
σ
k
(
m
)
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
ρ
)
]
r
k

)
≤
D
1
λ
n
∑
k
∈
I
n
‍
(
[
M
(
∥
Δ
p
q
-
1
x
σ
k
+
1
(
m
)
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
ρ
)
]
r
k
i
+
[
M
(
∥
Δ
p
q
-
1
x
σ
k
(
m
)
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
ρ
)
]
r
k
)
.
Hence we have
(33)
{
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
x
σ
k
(
m
)
,
z
1
,
z
2
,
…
z
n
-
1
∥
n
ρ
)
}
r
k
≥
ɛ
n
∈
N
:
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
x
σ
k
(
m
)
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
ρ
)
}
r
k
≥
ɛ
}
⊆
{
D
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
-
1
x
σ
k
+
1
(
m
)
,
z
1
,
z
2
,
…
z
n
-
1
∥
n
ρ
)
}
r
k
≥
ɛ
2
n
∈
N
:
D
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
-
1
x
σ
k
+
1
(
m
)
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
ρ
)
}
r
k
≥
ɛ
2
:
D
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
-
1
x
σ
k
+
1
(
m
)
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
ρ
)
}
r
k
}
∪
{
D
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
-
1
x
σ
k
(
m
)
,
z
1
,
z
2
,
…
z
n
-
1
∥
n
ρ
)
}
r
k
≥
ɛ
2
n
∈
N
:
D
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
-
1
x
σ
k
(
m
)
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
ρ
)
}
r
k
≥
ɛ
2
:
D
1
λ
n
∑
k
∈
I
n
‍
{
M
(
∥
Δ
p
q
-
1
x
σ
k
(
m
)
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
ρ
)
}
r
k
}
.
Since the set on the right hand side belongs to I, so does the left hand side. The inclusion is strict as the sequence x
=
(
k
r
), for example, belongs to (
[
V
σ
,
λ
,
Δ
p
q
,
M
]
0
I
,
∥
·
,
·
,
⋯
∥
n
) but does not belong to (
[
V
σ
,
λ
,
Δ
p
q
-
1
,
M
]
0
I
,
∥
·
,
·
,
⋯
∥
n
) for M
(
x
)
=
x and r
k
=
1 for all k.Theorem 5.
(
[
V
σ
,
λ
,
Δ
p
q
,
M
,
r
]
0
I
,
∥
·
,
·
,
⋯
∥
n
) and (
[
V
σ
,
λ
,
Δ
p
q
,
M
,
r
]
I
,
∥
·
,
·
,
⋯
∥
n
) are complete linear topological spaces, with paranorm g, where g is defined by
(34)
g
(
x
)
=
∑
m
=
1
p
q
‍
∥
x
σ
k
(
m
)
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
+
inf
⁡
{
sup
⁡
m
,
n
(
1
λ
n
∑
k
∈
I
n
‍
(
M
(
∥
Δ
p
q
x
σ
k
(
m
)
,
z
1
,
z
2
,
…
z
n
-
1
∥
n
ρ
)
)
)
H
ρ
r
k
/
H
:
sup
⁡
m
,
n
(


1
λ
n
∑
k
∈
I
n
‍


(


M

(

∥
Δ
p
q
x
σ
k
(
m
)
,
z
1
,
z
2
,
…
,
z
n
-
1
∥
n
ρ


)


)


)
H

}


,
where H
=
max
⁡
(
1
,
(
sup
⁡
k
r
k
)
).



---

*Source: 621383-2014-07-23.xml*