# Some Paranormed Double Difference Sequence Spaces for Orlicz Functions and Bounded-Regular Matrices

Authors: S. A. Mohiuddine; Kuldip Raj; Abdullah Alotaibi
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

The aim of this paper is to introduce some new double difference sequence spaces with the help of the Musielak-Orlicz functionℱ
=
(
F
j
k
) and four-dimensional bounded-regular (shortly, RH-regular) matrices A
=
(
a
n
m
j
k
). We also make an effort to study some topological properties and inclusion relations between these double difference sequence spaces.


---

## Body

## 1. Introduction, Notations, and Preliminaries

In [1], Hardy introduced the concept of regular convergence for double sequences. Some important work on double sequences is also found by Bromwich [2]. Later on, it was studied by various authors, for example, Móricz [3], Móricz and Rhoades [4], Başarır and Sonalcan [5], Mursaleen and Mohiuddine [6–8], and many others. Mursaleen [9] has defined and characterized the notion of almost strong regularity of four-dimensional matrices and applied these matrices to establish a core theorem (also see [10, 11]). Altay and Başar [12] have recently introduced the double sequence spaces ℬ
𝒮, ℬ
𝒮
(
t
), 𝒞
𝒮
p, 𝒞
𝒮
b
p, 𝒞
𝒮
r, and ℬ
𝒱 consisting of all double series whose sequence of partial sums are in the spaces ℳ
u, ℳ
u
(
t
), 𝒞
p, 𝒞
b
p, 𝒞
r, and ℒ
u, respectively. Başar and Sever [13] extended the well-known space ℓ
q from single sequence to double sequences, denoted by ℒ
q, and established its interesting properties. The authors of [14] defined some convex and paranormed sequences spaces and presented some interesting characterization. Most recently, Mohiuddine and Alotaibi [15] introduced some new double sequences spaces for σ-convergence of double sequences and invariant mean and also determined some inclusion results for these spaces. For more details on these concepts, one can be referred to [16–18].The notion of difference sequence spaces was introduced by Kızmaz [19], who studied the difference sequence spaces l
∞
(
Δ
), c
(
Δ
), and c
0
(
Δ
). The notion was further generalized by Et and Çolak [20] by introducing the spaces l
∞
(
Δ
r
), c
(
Δ
r
), and c
0
(
Δ
r
).Letw be the space of all complex or real sequences x
=
(
x
k
) and let r and s be two nonnegative integers. Then for Z
=
l
∞
,
c
,
c
0, we have the following sequence spaces:
(1)
Z
(
Δ
s
r
)
=
{
x
=
(
x
k
)
∈
w
:
(
Δ
s
r
x
k
)
∈
Z
}
,
where Δ
s
r
x
=
(
Δ
s
r
x
k
)
=
(
Δ
s
r
-
1
x
k
-
Δ
s
r
-
1
x
k
+
1
) and Δ
0
x
k
=
x
k for all k
∈
ℕ, which is equivalent to the following binomial representation:
(2)
Δ
s
r
x
k
=
∑
v
=
0
r
‍
(
-
1
)
v
(
r
v
)
x
k
+
s
v
.We remark that fors
=
1 and r
=
s
=
1, we obtain the sequence spaces which were introduced and studied by Et and Çolak [20] and Kızmaz [19], respectively. For more details about sequence spaces see [21–27] and references therein.AnOrlicz function
F
:
[
0
,
∞
)
→
[
0
,
∞
) is continuous, nondecreasing, and convex such that F
(
0
)
=
0, F
(
x
)
>
0 for x
>
0 and F
(
x
)
→
∞ as x
→
∞. If convexity of Orlicz function is replaced by F
(
x
+
y
)
≤
F
(
x
)
+
F
(
y
), then this function is calledmodulus function. Lindenstrauss and Tzafriri [28] used the idea of Orlicz function to define the following sequence space:
(3)
ℓ
F
=
{
x
=
(
x
k
)
∈
w
:
∑
k
=
1
∞
‍
F
(
|
x
k
|
ρ
)
<
∞
,
ρ
>
0
}
,
which is known as an Orlicz sequence space. The space ℓ
F is a Banach space with the norm
(4)
∥
x
∥
=
inf
⁡
{
ρ
>
0
:
∑
k
=
1
∞
‍
F
(
|
x
k
|
ρ
)
≤
1
}
.Also it was shown in [28] that every Orlicz sequence space ℓ
F contains a subspace isomorphic to ℓ
p
(
p
≥
1
). An Orlicz function F can always be represented in the following integral form:
(5)
F
(
x
)
=
∫
0
x
‍
η
(
t
)
d
t
,
where η is known as the kernel of F, is a right differentiable for t
≥
0
,
η
(
0
)
=
0
,
η
(
t
)
>
0
,
η is nondecreasing, and η
(
t
)
→
∞ as t
→
∞.A sequenceℱ
=
(
F
k
) of Orlicz functions is said to be aMusielak-Orlicz function (see [29, 30]). A sequence 𝒩
=
(
N
k
) is defined by
(6)
N
k
(
v
)
=
sup
⁡
{
|
v
|
u
-
F
k
(
u
)
:
u
≥
0
}
,
k
=
1,2
,
…
,
which is called the complementary function of a Musielak-Orlicz function ℱ. For a given Musielak-Orlicz function ℱ, the Musielak-Orlicz sequence space t
ℱ and its subspace h
ℱ are defined as follows:
(7)
t
ℱ
=
{
x
∈
w
:
I
ℱ
(
c
x
)
<
∞
for
some
c
>
0
}
,
h
ℱ
=
{
x
∈
w
:
I
ℱ
(
c
x
)
<
∞
∀
c
>
0
}
,
where I
ℱ is a convex modular defined by
(8)
I
ℱ
(
x
)
=
∑
k
=
1
∞
‍
F
k
(
x
k
)
,
x
=
(
x
k
)
∈
t
ℱ
.We considert
ℱ equipped with the Luxemburg norm
(9)
∥
x
∥
=
inf
⁡
{
k
>
0
:
I
ℱ
(
x
k
)
≤
1
}
or equipped with the Orlicz norm
(10)
∥
x
∥
0
=
inf
⁡
{
1
k
(
1
+
I
ℱ
(
k
x
)
)
:
k
>
0
}
.A Musielak-Orlicz functionℱ
=
(
F
k
) is said to satisfy Δ
2
-
c
o
n
d
i
t
i
o
n if there exist constants a
,
K
>
0 and a sequence c
=
(
c
k
)
k
=
1
∞
∈
l
+
1 (the positive cone of l
1) such that the inequality
(11)
F
k
(
2
u
)
≤
K
F
k
(
u
)
+
c
k
holds for all k
∈
ℕ and u
∈
ℝ
+, whenever F
k
(
u
)
≤
a.A double sequencex
=
(
x
j
k
) is said to bebounded if ∥
x
∥
(
∞
,
2
)
=
sup
⁡
j
,
k
|
x
j
k
|
<
∞. We denote by l
∞
2 the space of all bounded double sequences.By the convergence of double sequencex
=
(
x
j
k
) we mean the convergence in the Pringsheim sense; that is, a double sequence x
=
(
x
j
k
) is said toconverge to the limit L in Pringsheim sense (denoted by, P
-lim
x
=
L) provided that given ϵ
>
0 there exists n
∈
ℕ such that |
x
j
k
-
L
|
<
ϵ whenever j
,
k
>
n (see [31]). We will write more briefly as P-convergent. If, in addition, x
∈
l
∞
2, then x is said to beboundedly P-convergent to L. We will denote the space of all bounded convergent double sequences (or boundedly P-convergent) by c
∞
2.LetS
⊆
ℕ
×
ℕ and let ϵ
>
0 be given. By χ
S
(
x
;
ϵ
), we denote the characteristic function of the set S
(
x
;
ϵ
)
=
{
(
j
,
k
)
∈
ℕ
×
ℕ
:
|
x
j
k
|
≥
ϵ
}.LetA
=
(
a
n
m
j
k
) be a four-dimensional infinite matrix of scalers. For all m
,
n
∈
ℕ
0, where ℕ
0
:
=
ℕ
∪
{
0
}, the sum
(12)
y
n
m
=
∑
j
,
k
=
0,0
∞
,
∞
‍
a
n
m
j
k
x
j
k
is called the A-m
e
a
n
s of the double sequence (
x
j
k
). A double sequence (
x
j
k
) is said to be A-s
u
m
m
a
b
l
e to the limit L if the A-means exist for all m
,
n in the sense of Pringsheim’s convergence:
(13)
P
-
lim
⁡
p
,
q
→
∞
∑
j
,
k
=
0,0
p
,
q
‍
a
n
m
j
k
x
j
k
=
y
n
m
,
P
-
lim
⁡
n
,
m
→
∞
y
n
m
=
L
.A four-dimensional matrixA is said to bebounded-regular (orRH-regular) if every bounded P-convergent sequence is A-summable to the same limit and the A-means are also bounded.The following is a four-dimensional analogue of the well-known Silverman-Toeplitz theorem [32].Theorem 1 (Robison [33] and Hamilton [34]).
The four-dimensional matrixA isRH-regular if and only if (RH1)
P
-
lim
⁡
n
,
m
a
n
m
j
k
=
0 for each j and k,
(RH2)
P
-
lim
⁡
n
,
m
∑
j
,
k
=
0,0
∞
,
∞
‍
|
a
n
m
j
k
|
=
1,
(RH3)
P
-
lim
⁡
n
,
m
∑
j
=
0
∞
‍
|
a
n
m
j
k
|
=
0 for each k,
(RH4)
P
-
lim
⁡
n
,
m
∑
k
=
0
∞
‍
|
a
n
m
j
k
|
=
0 for each j,
(RH5)
∑
j
,
k
=
0,0
∞
,
∞
‍
|
a
n
m
j
k
|
<
∞ for all n
,
m
∈
ℕ
0.


## 2. The Double Difference Sequence Spaces

In this section, we define some new paranormed double difference sequence spaces with the help of Musielak-Orlicz functions and four-dimensional bounded-regular matrices. Before proceeding further, first we recall the notion of paranormed space as follows.A linear topological spaceX over the real field ℝ (the set of real numbers) is said to be aparanormed space if there is a subadditive function g
:
X
→
ℝ such that g
(
θ
)
=
0, g
(
x
)
=
g
(
-
x
), and scalar multiplication is continuous; that is, |
α
n
-
α
|
→
0 and g
(
x
n
-
x
)
→
0 imply g
(
α
n
x
n
-
α
x
|
→
0 for all α’s in ℝ and all x’s in X, where θ is the zero vector in the linear space X.The linear spacesl
∞
(
p
), c
(
p
), and c
0
(
p
) were defined by Maddox [35] (also, see Simons [36]).Letℱ
=
(
F
j
k
) be a Musielak-Orlicz function; that is, ℱ is a sequence of Orlicz functions and let A
=
(
a
n
m
j
k
) be a nonnegative four-dimensional bounded-regular matrix. Then, we define the following:
(14)
W
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
)
=
{
∑
j
,
k
=
0,0
∞
,
∞
‍
x
=
(
x
j
k
)
:
1111111
P
-
lim
⁡
n
,
m
∑
j
,
k
=
0,0
∞
,
∞
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
Δ
s
r
x
j
k
|
)
p
j
k
]
=
0
}
,
W
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
)
=
{
∑
j
,
k
=
0,0
∞
,
∞
‍
x
=
(
x
j
k
)
:
1111111
P
-
lim
⁡
n
,
m
∑
j
,
k
=
0,0
∞
,
∞
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
Δ
s
r
x
j
k
-
L
|
)
p
j
k
]
1111111
=
0
for
some
L
∈
ℂ
∑
j
,
k
=
0,0
∞
,
∞
‍
}
,
where p
=
(
p
j
k
) is a double sequence of real numbers such that p
j
k
>
0 for j
,
k, sup
⁡
j
,
k
p
j
k
=
H
<
∞, and u
=
(
u
j
k
) is a double sequence of strictly positive real numbers.Remark 2.
If we takeℱ
(
x
)
=
x in W
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
) and W
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
), then we have the following spaces:
(15)
W
0
2
(
A
,
u
,
Δ
s
r
,
p
)
=
{
∑
j
,
k
=
0,0
∞
,
∞
‍
x
=
(
x
j
k
)
:
1111111
P
-
lim
⁡
n
,
m
∑
j
,
k
=
0,0
∞
,
∞
‍
a
n
m
j
k
[
(
u
j
k
|
Δ
s
r
x
j
k
|
)
p
j
k
]
=
0
}
,
W
2
(
A
,
u
,
Δ
s
r
,
p
)
=
{
∑
j
,
k
=
0,0
∞
,
∞
‍
x
=
(
x
j
k
)
:
1111111
P
-
lim
⁡
n
,
m
∑
j
,
k
=
0,0
∞
,
∞
‍
a
n
m
j
k
[
(
u
j
k
|
Δ
s
r
x
j
k
-
L
|
)
p
j
k
]
1111111
=
0
for
some
L
∈
ℂ
∑
j
,
k
=
0,0
∞
,
∞
‍
}
.Remark 3.
Letp
=
(
p
j
k
)
=
1 for all j
,
k. Then W
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
) and W
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
) are reduced to
(16)
W
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
)
=
{
∑
j
,
k
=
0,0
∞
,
∞
‍
x
=
(
x
j
k
)
:
1111111
P
-
lim
⁡
n
,
m
∑
j
,
k
=
0,0
∞
,
∞
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
Δ
s
r
x
j
k
|
)
]
=
0
}
,
W
2
(
A
,
ℱ
,
u
,
Δ
s
r
)
=
{
∑
j
,
k
=
0,0
∞
,
∞
‍
x
=
(
x
j
k
)
:
1111111
P
-
lim
⁡
n
,
m
∑
j
,
k
=
0,0
∞
,
∞
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
Δ
s
r
x
j
k
-
L
|
)
]
1111111
=
0
for
some
L
∈
ℂ
∑
j
,
k
=
0,0
∞
,
∞
‍
}
,
respectively.Remark 4.
Letu
=
(
u
j
k
)
=
1 for all j
,
k. Then, the spaces W
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
) and W
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
) are reduced to
(17)
W
0
2
(
A
,
ℱ
,
Δ
s
r
,
p
)
=
{
∑
j
,
k
=
0,0
∞
,
∞
‍
x
=
(
x
j
k
)
:
1111111
P
-
lim
⁡
n
,
m
∑
j
,
k
=
0,0
∞
,
∞
‍
a
n
m
j
k
[
F
j
k
(
|
Δ
s
r
x
j
k
|
)
p
j
k
]
=
0
}
,
W
2
(
A
,
ℱ
,
Δ
s
r
,
p
)
=
{
∑
j
,
k
=
0,0
∞
,
∞
‍
x
=
(
x
j
k
)
:
1111111
P
-
lim
⁡
n
,
m
∑
j
,
k
=
0,0
∞
,
∞
‍
a
n
m
j
k
[
F
j
k
(
|
Δ
s
r
x
j
k
-
L
|
)
p
j
k
]
1111111
=
0
for
some
L
∈
ℂ
∑
j
,
k
=
0,0
∞
,
∞
‍
}
,
respectively.Remark 5.
If we takeA
=
(
C
,
1,1
) in W
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
) and W
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
), then we have the following spaces:
(18)
W
0
2
(
ℱ
,
u
,
Δ
s
r
,
p
)
=
{
∑
j
,
k
=
0,0
∞
,
∞
‍
x
=
(
x
j
k
)
:
1111111
P
-
lim
⁡
n
,
m
∑
j
,
k
=
0,0
m
-
1
,
n
-
1
‍
[
F
j
k
(
u
j
k
|
Δ
s
r
x
j
k
|
)
p
j
k
]
=
0
}
,
W
2
(
ℱ
,
u
,
Δ
s
r
,
p
)
=
{
∑
j
,
k
=
0,0
m
-
1
,
n
-
1
‍
x
=
(
x
j
k
)
:
1111111
P
-
lim
⁡
n
,
m
∑
j
,
k
=
0,0
m
-
1
,
n
-
1
‍
[
F
j
k
(
u
j
k
|
Δ
s
r
x
j
k
-
L
|
)
p
j
k
]
1111111
=
0
for
some
L
∈
ℂ
∑
j
,
k
=
0,0
m
-
1
,
n
-
1
‍
}
.Remark 6.
If we takeA
=
(
C
,
1,1
) and ℱ
(
x
)
=
x in W
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
) and W
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
), then we have the following spaces:
(19)
W
0
2
(
u
,
Δ
s
r
,
p
)
=
{
∑
j
,
k
=
0,0
m
-
1
,
n
-
1
‍
x
=
(
x
j
k
)
:
1111111
P
-
lim
⁡
n
,
m
∑
j
,
k
=
0,0
m
-
1
,
n
-
1
‍
[
(
u
j
k
|
Δ
s
r
x
j
k
|
)
p
j
k
]
=
0
}
,
W
2
(
u
,
Δ
s
r
,
p
)
=
{
∑
j
,
k
=
0,0
m
-
1
,
n
-
1
‍
x
=
(
x
j
k
)
:
1111111
P
-
lim
⁡
n
,
m
∑
j
,
k
=
0,0
m
-
1
,
n
-
1
‍
[
(
u
j
k
|
Δ
s
r
x
j
k
-
L
|
)
p
j
k
]
1111111
=
0
for
some
L
∈
ℂ
∑
j
,
k
=
0,0
m
-
1
,
n
-
1
‍
}
.Remark 7.
Letp
j
k
=
u
j
k
=
1 for all j
,
k. If, in addition, ℱ
(
x
)
=
F
(
x
) and r
=
0, then the spaces W
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
) and W
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
) are reduced to W
0
2
(
A
,
F
) and W
2
(
A
,
F
) which were introduced and studied by Yurdakadim and Tas [37] as below:
(20)
W
0
2
(
A
,
F
)
=
{
x
=
(
x
j
k
)
:
P
-
lim
⁡
n
,
m
∑
j
,
k
‍
a
n
m
j
k
F
(
|
x
j
k
|
)
=
0
}
,
W
2
(
A
,
F
)
=
{
x
=
(
x
j
k
)
:
P
-
lim
⁡
n
,
m
∑
j
,
k
‍
a
n
m
j
k
F
(
|
x
j
k
-
L
|
)
1111111111111
,
=
0
for
some
L
∈
ℂ
∑
j
,
k
‍
}
.
Throughout the paper, we will use the following inequality: let(
a
j
k
) and (
b
j
k
) be two double sequences. Then
(21)
|
a
j
k
+
b
j
k
|
p
j
k
≤
K
(
|
a
j
k
|
p
j
k
+
|
b
j
k
|
p
j
k
)
,
where K
=
max
⁡
(
1
,
2
H
-
1
) and sup
⁡
j
,
k
p
j
k
=
H (see [15]). We will also assume throughout this paper that the symbol ℱ will denote the sublinear Musielak-Orlicz function.


## 3. Main Results

Theorem 8.
Letℱ
=
(
F
j
k
) be a sublinear Musielak-Orlicz function, A
=
(
a
n
m
j
k
) a nonnegative four-dimensional R
H-regular matrix, p
=
(
p
j
k
) a bounded sequence of positive real numbers, and u
=
(
u
j
k
) a sequence of strictly positive real numbers. Then W
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
) and W
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
) are linear spaces over the complex field ℂ.Proof.
Letx
=
(
x
j
k
)
,
y
=
(
y
j
k
)
∈
W
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
) and α
,
β
∈
ℂ. Then there exist integers M
α and N
β such that |
α
|
≤
M
α and |
β
|
≤
N
β.
Sinceℱ
=
(
F
j
k
) is a nondecreasing function, so by inequality (21), we have
(22)
∑
j
,
k
=
0,0
∞
,
∞
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
Δ
s
r
(
α
x
j
k
+
β
y
j
k
)
|
)
p
j
k
]
≤
∑
j
,
k
=
0,0
∞
,
∞
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
α
Δ
s
r
x
j
k
+
β
Δ
s
r
y
j
k
|
)
p
j
k
]
≤
K
∑
j
,
k
=
0,0
∞
,
∞
‍
a
n
m
j
k
[
F
j
k
M
α
(
u
j
k
|
Δ
s
r
x
j
k
|
)
p
j
k
]
+
K
∑
j
,
k
=
0,0
∞
,
∞
‍
a
n
m
j
k
[
F
j
k
N
β
(
u
j
k
|
Δ
s
r
y
j
k
|
)
p
j
k
]
≤
K
M
α
H
∑
j
,
k
=
0,0
∞
,
∞
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
Δ
s
r
x
j
k
|
)
p
j
k
]
+
K
N
β
H
∑
j
,
k
=
0,0
∞
,
∞
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
Δ
s
r
y
j
k
|
)
p
j
k
]
⟶
0
.
Thusα
x
+
β
y
∈
W
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
). This proves that W
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
) is a linear space. Similarly we can prove that W
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
) is also a linear space.Theorem 9.
Letℱ
=
(
F
j
k
) be a sublinear Musielak-Orlicz function, A
=
(
a
n
m
j
k
) a nonnegative four-dimensional R
H-regular matrix, p
=
(
p
j
k
) a bounded sequence of positive real numbers, and u
=
(
u
j
k
) a sequence of strictly positive real numbers. Then W
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
) and W
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
) are paranormed spaces with the paranorm
(23)
g
(
x
)
=
sup
⁡
n
,
m
∑
j
,
k
=
0,0
∞
,
∞
‍
{
a
n
m
j
k
[
F
j
k
(
u
j
k
|
Δ
s
r
x
j
k
|
)
p
j
k
]
}
1
/
M
,
where 0
<
p
j
k
≤
sup
⁡
p
j
k
=
H
<
∞ and M
=
max
⁡
(
1
,
H
).Proof.
We will prove the result forW
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
). Let x
=
(
x
j
k
)
∈
W
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
). Then for each x
=
(
x
j
k
)
∈
W
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
), g
(
x
) exists. Also it is clear that g
(
0
)
=
0
,
g
(
-
x
)
=
g
(
x
), and g
(
x
+
y
)
≤
g
(
x
)
+
g
(
y
).
We now show that the scalar multiplication is continuous. First observe the following:(24)
g
(
λ
x
)
=
sup
⁡
n
m
∑
j
,
k
=
0,0
∞
,
∞
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
λ
Δ
s
r
x
j
k
|
)
p
j
k
]
g
(
λ
x
)
≤
(
1
+
[
|
λ
|
]
)
g
(
x
)
,
where [
|
λ
|
] denotes the integer part of |
λ
|. It is also clear that if x
→
0 and λ
→
0 implies g
(
λ
x
)
→
0. For fixed λ, if x
→
0, then g
(
λ
x
)
→
0. We need to show that for fixed x
,
λ
→
0 implies g
(
λ
x
)
→
0. Let x
∈
W
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
). Thus
(25)
P
-
lim
⁡
n
,
m
∑
j
,
k
=
0,0
∞
,
∞
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
Δ
s
r
x
j
k
-
L
|
)
p
j
k
]
=
0
.
Then, forϵ
>
0 there exists N
∈
ℕ such that
(26)
∑
j
,
k
=
0,0
∞
,
∞
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
Δ
s
r
x
j
k
-
L
|
)
p
j
k
]
<
ϵ
4
for m
,
n
>
N. Also, for each m
,
n with 1
≤
m
,
n
≤
N, since
(27)
∑
j
,
k
=
0,0
∞
,
∞
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
Δ
s
r
x
j
k
-
L
|
)
p
j
k
]
<
∞
,
there exists an integer M
m
,
n such that
(28)
∑
j
,
k
>
M
m
,
n
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
Δ
s
r
x
j
k
-
L
|
)
p
j
k
]
<
ϵ
4
.
LetM
=
max
⁡
1
≤
(
m
,
n
)
≤
N
{
M
m
,
n
}. We have for each m
,
n with 1
≤
m
,
n
≤
N
(29)
∑
j
,
k
>
M
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
Δ
s
r
x
j
k
-
L
|
)
p
j
k
]
<
ϵ
4
.
Also from (26), for m
,
n
>
N, we have
(30)
∑
j
,
k
>
M
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
Δ
s
r
x
j
k
-
L
|
)
p
j
k
]
<
ϵ
4
.
ThusM is an integer independent of m
,
n such that
(31)
∑
j
,
k
>
M
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
Δ
s
r
x
j
k
-
L
|
)
p
j
k
]
<
ϵ
4
.
Since|
λ
|
p
j
k
≤
max
⁡
(
1
,
|
λ
|
H
), therefore
(32)
∑
j
,
k
=
0,0
∞
,
∞
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
λ
Δ
s
r
x
j
k
|
)
p
j
k
]
=
∑
j
,
k
=
0,0
∞
,
∞
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
λ
Δ
s
r
x
j
k
-
λ
L
+
λ
L
|
)
p
j
k
]
≤
∑
j
,
k
=
0,0
∞
,
∞
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
λ
Δ
s
r
x
j
k
-
λ
L
|
)
p
j
k
]
+
∑
j
,
k
=
0,0
∞
,
∞
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
λ
L
|
)
p
j
k
]
≤
∑
j
,
k
>
M
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
λ
Δ
s
r
x
j
k
-
λ
L
|
)
p
j
k
]
+
∑
j
,
k
≤
M
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
λ
Δ
s
r
x
j
k
-
λ
L
|
)
p
j
k
]
+
∑
j
≥
M
,
k
<
M
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
λ
Δ
s
r
x
j
k
-
λ
L
|
)
p
j
k
]
+
∑
j
<
M
,
k
≥
M
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
λ
Δ
s
r
x
j
k
-
λ
L
|
)
p
j
k
]
+
∑
j
,
k
=
0,0
∞
,
∞
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
λ
L
|
)
p
j
k
]
.
For eachm
,
n and by the continuity of F as λ
→
0, we have the following:
(33)
∑
j
,
k
≤
M
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
λ
Δ
s
r
x
j
k
-
λ
L
|
)
p
j
k
]
+
∑
j
,
k
=
0,0
∞
,
∞
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
λ
L
|
)
p
j
k
]
⟶
0
in Pringsheim’s sense. Now choose δ
<
1 such that |
λ
|
<
δ implies
(34)
∑
j
,
k
≤
M
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
λ
Δ
s
r
x
j
k
-
λ
L
|
)
p
j
k
]
+
∑
j
,
k
=
0,0
∞
,
∞
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
λ
L
|
)
p
j
k
]
<
ϵ
4
.
In the same manner, we have(35)
∑
j
≥
M
,
k
<
M
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
λ
Δ
s
r
x
j
k
-
λ
L
|
)
p
j
k
]
<
ϵ
4
,
(36)
∑
j
<
M
,
k
≥
M
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
λ
Δ
s
r
x
j
k
-
λ
L
|
)
p
j
k
]
<
ϵ
4
.
It follows from (31), (34), (35), and (36) that
(37)
∑
j
,
k
=
0,0
∞
,
∞
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
λ
Δ
s
r
x
j
k
|
)
p
j
k
]
<
ϵ
∀
m
,
n
.
Thusg
(
λ
x
)
→
0 as λ
→
0. Therefore W
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
) is a paranormed space. Similarly, we can prove that W
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
) is a paranormed space. This completes the proof.Theorem 10.
Letℱ
=
(
F
j
k
) be a sublinear Musielak-Orlicz function, A
=
(
a
n
m
j
k
) a nonnegative four-dimensional R
H-regular matrix, p
=
(
p
j
k
) a bounded sequence of positive real numbers, and u
=
(
u
j
k
) a sequence of strictly positive real numbers. Then W
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
) and W
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
) are complete topological linear spaces.Proof.
Let(
x
j
k
q
) be a Cauchy sequence in W
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
); that is, g
(
x
q
-
x
t
)
→
0 as q
,
t
→
∞. Then, we have
(38)
∑
j
,
k
=
0,0
∞
,
∞
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
Δ
s
r
x
j
k
q
-
Δ
s
r
x
j
k
t
|
)
p
j
k
]
⟶
0
.
Thus for each fixedj and k as q
,
t
→
∞, since A
=
(
a
n
m
j
k
) is nonnegative, we are granted that
(39)
F
j
k
(
u
j
k
|
Δ
s
r
x
j
k
q
-
Δ
s
r
x
j
k
t
|
)
⟶
0
,
and by continuity of ℱ
=
(
F
j
k
), (
x
j
k
q
) is a Cauchy sequence in ℂ for each fixed j and k.
Sinceℂ is complete as t
→
∞, we have x
j
k
q
→
x
j
k for each (
j
,
k
). Now from (36), we have that, for ϵ
>
0, there exists a natural number N such that
(40)
∑
j
,
k
=
0,0
q
,
t
>
N
∞
,
∞
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
Δ
s
r
x
j
k
q
-
Δ
s
r
x
j
k
t
|
)
p
j
k
]
<
ϵ
∀
m
,
n
.
Since for any fixed natural numberM, from (38) we have
(41)
∑
j
,
k
≤
M
q
,
t
>
N
∞
,
∞
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
Δ
s
r
x
j
k
q
-
Δ
s
r
x
j
k
t
|
)
p
j
k
]
<
ϵ
∀
m
,
n
.
By letting t
→
∞ in the above expression we obtain
(42)
∑
j
,
k
≤
M
q
>
N
∞
,
∞
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
Δ
s
r
x
j
k
q
-
Δ
s
r
x
j
k
|
)
p
j
k
]
<
ϵ
.
SinceM is arbitrary, by letting M
→
∞ we obtain
(43)
∑
j
,
k
=
0,0
∞
,
∞
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
Δ
s
r
x
j
k
q
-
Δ
s
r
x
j
k
|
)
p
j
k
]
<
ϵ
∀
m
,
n
.
Thusg
(
x
q
-
x
)
→
0 as q
→
∞. This proves that W
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
) is a complete topological linear space.
Now we will show thatW
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
) is a complete topological linear space. For this, since (
x
q
) is also a sequence in W
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
) by definition of W
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
), for each q, there exists L
q with
(44)
∑
j
,
k
=
0,0
∞
,
∞
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
Δ
s
r
x
j
k
q
-
Δ
s
r
L
q
|
)
p
j
k
]
⟶
0
1111111111111111111111111
as
m
,
n
⟶
∞
;
whence from the fact that sup
⁡
n
m
∑
j
,
k
=
0,0
∞
,
∞
‍
a
n
m
j
k
<
∞ and from the definition of Musielak-Orlicz function, we have F
j
k
|
Δ
s
r
L
q
-
Δ
s
r
L
|
→
0 as q
→
∞ and so L
q converges to L. Thus
(45)
∑
j
,
k
=
0,0
∞
,
∞
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
Δ
s
r
x
j
k
-
L
|
)
p
j
k
]
⟶
0
1111111111111111111111
as
m
,
n
⟶
∞
.
Hencex
∈
W
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
) and this completes the proof.Theorem 11.
Letℱ
=
(
F
j
k
) be a sublinear Musielak-Orlicz function which satisfies the Δ
2-condition. Then W
2
(
A
,
u
,
Δ
s
r
,
p
)
⊆
W
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
).Proof.
Letx
=
(
x
k
)
∈
W
2
(
A
,
u
,
Δ
s
r
,
p
); that is,
(46)
lim
⁡
n
,
m
∑
j
,
k
‍
a
n
m
j
k
[
(
u
j
k
|
Δ
s
r
x
j
k
-
L
|
)
p
j
k
]
=
0
.
Letϵ
>
0 and choose δ with 0
<
δ
<
1 such that F
j
k
(
t
)
<
ϵ for 0
≤
t
≤
δ. Write y
j
k
=
(
u
j
k
|
Δ
s
r
x
j
k
-
L
|
) and consider
(47)
∑
j
,
k
‍
a
n
m
j
k
[
F
j
k
(
y
j
k
)
p
j
k
]
=
∑
j
,
k
:
|
y
j
k
|
≤
δ
‍
a
n
m
j
k
[
F
j
k
(
y
j
k
)
p
j
k
]
∑
j
,
k
‍
a
n
m
j
k
[
F
j
k
(
y
j
k
)
p
j
k
]
=
+
∑
j
,
k
:
|
y
j
k
|
>
δ
‍
a
n
m
j
k
[
F
j
k
(
y
j
k
)
p
j
k
]
∑
j
,
k
‍
a
n
m
j
k
[
F
j
k
(
y
j
k
)
p
j
k
]
=
ϵ
∑
j
,
k
:
|
y
j
k
|
≤
δ
‍
a
n
m
j
k
∑
j
,
k
‍
a
n
m
j
k
[
F
j
k
(
y
j
k
)
p
j
k
]
=
+
∑
j
,
k
:
|
y
j
k
|
>
δ
‍
a
n
m
j
k
[
F
j
k
(
y
j
k
)
p
j
k
]
.
Fory
j
k
>
δ, we use the fact that y
j
k
<
y
j
k
/
δ
<
1
+
y
j
k
/
δ. Hence
(48)
F
j
k
(
y
j
k
)
<
F
j
k
(
1
+
y
j
k
δ
)
<
F
j
k
(
2
)
2
+
1
2
F
j
k
(
2
y
j
k
δ
)
.
Sinceℱ satisfies the Δ
2-condition, we have
(49)
F
j
k
(
y
j
k
)
<
K
y
j
k
2
δ
F
j
k
(
2
)
+
K
y
j
k
2
δ
F
j
k
(
2
)
=
K
y
j
k
δ
F
j
k
(
2
)
,
and hence
(50)
∑
j
,
k
:
|
y
j
k
|
>
δ
‍
a
n
m
j
k
[
F
j
k
(
y
j
k
)
p
j
k
]
≤
K
F
j
k
δ
(
2
)
∑
j
,
k
‍
a
n
m
j
k
[
(
u
j
k
|
Δ
s
r
x
j
k
-
L
|
)
p
j
k
]
.
SinceA is R
H-regular and x
∈
W
2
(
A
,
u
,
Δ
s
r
,
p
), we get x
∈
W
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
).Theorem 12.
Letℱ
=
(
F
j
k
) be a sublinear Musielak-Orlicz function and let A
=
(
a
n
m
j
k
) be a nonnegative four-dimensionalRH-regular matrix. Suppose that β
=
lim
⁡
t
→
∞
(
F
j
k
(
t
)
/
t
)
<
∞. Then
(51)
W
2
(
A
,
u
,
Δ
s
r
,
p
)
=
W
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
)
.Proof.
In order to prove thatW
2
(
A
,
u
,
Δ
s
r
,
p
)
=
W
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
), it is sufficient to show that W
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
)
⊂
W
2
(
A
,
u
,
Δ
s
r
,
p
). Now, let β
>
0. By definition of β, we have F
j
k
(
t
)
≥
β
t for all t
≥
0. Since β
>
0, we have t
≤
(
1
/
β
)
F
j
k
(
t
) for all t
≥
0. Let x
=
(
x
j
k
)
∈
W
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
). Thus, we have
(52)
∑
j
,
k
=
0,0
∞
,
∞
‍
a
n
m
j
k
[
(
u
j
k
|
Δ
s
r
x
j
k
-
L
|
)
p
j
k
]
≤
1
β
∑
j
,
k
=
0,0
∞
,
∞
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
Δ
s
r
x
j
k
-
L
|
)
p
j
k
]
,
which implies that x
=
(
x
j
k
)
∈
W
2
(
A
,
u
,
Δ
s
r
,
p
). This completes the proof.Theorem 13.
(i) Let0
<
inf
⁡
p
j
k
<
p
j
k
≤
1. Then
(53)
W
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
)
⊆
W
2
(
A
,
ℱ
,
u
,
Δ
s
r
)
.
(ii) Let1
≤
p
j
k
≤
sup
⁡
p
j
k
<
∞. Then
(54)
W
2
(
A
,
ℱ
,
u
,
Δ
s
r
)
⊆
W
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
)
.Proof.
(i) Letx
=
(
x
j
k
)
∈
W
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
). Then since 0
<
inf
⁡
p
j
k
<
p
j
k
≤
1, we obtain the following:
(55)
∑
j
,
k
=
0,0
∞
,
∞
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
Δ
s
r
x
j
k
-
L
|
)
]
≤
∑
j
,
k
=
0,0
∞
,
∞
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
Δ
s
r
x
j
k
-
L
|
)
p
j
k
]
.
Thusx
=
(
x
j
k
)
∈
W
2
(
A
,
ℱ
,
u
,
Δ
s
r
).
(ii) Letp
j
k
≥
1 for each j and k and sup
⁡
p
j
k
<
∞. Let x
=
(
x
j
k
)
∈
W
2
(
A
,
ℱ
,
u
,
Δ
s
r
). Then for each 0
<
ϵ
<
1 there exists a positive integer N such that
(56)
∑
j
,
k
=
0,0
∞
,
∞
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
Δ
s
r
x
j
k
-
L
|
)
]
≤
ϵ
<
1
∀
m
,
n
≥
N
.
This implies that(57)
∑
j
,
k
=
0,0
∞
,
∞
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
Δ
s
r
x
j
k
-
L
|
)
p
j
k
]
≤
∑
j
,
k
=
0,0
∞
,
∞
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
Δ
s
r
x
j
k
-
L
|
)
]
.
Thereforex
=
(
x
j
k
)
∈
W
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
). This completes the proof.Lemma 14.
Letℱ
=
(
F
j
k
) be a sublinear Musielak-Orlicz function which satisfies the Δ
2-condition and let A
=
(
a
n
m
j
k
) be a nonnegative four-dimensional R
H-regular matrix. Then W
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
)
∩
l
∞
2 is an ideal in l
∞
2.Proof.
Letx
∈
W
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
)
∩
l
∞
2 and y
∈
l
∞
2. We need to show that x
y
∈
W
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
)
∩
l
∞
2. Since y
∈
l
∞
2, there exists T
1
>
1 such that ∥
y
∥
<
T
1. In this case |
x
j
k
y
j
k
|
<
T
1
|
x
j
k
| for all j
,
k. Since ℱ is nondecreasing and satisfies Δ
2-condition, we have
(58)
[
F
j
k
(
u
j
k
|
Δ
s
r
(
x
j
k
y
j
k
)
|
)
p
j
k
]
<
[
F
j
k
(
u
j
k
T
1
|
Δ
s
r
x
j
k
|
)
p
j
k
]
[
F
j
k
(
u
j
k
|
Δ
s
r
(
x
j
k
y
j
k
)
|
)
p
j
k
]
≤
T
(
T
1
)
[
F
j
k
(
u
j
k
|
Δ
s
r
x
j
k
|
)
p
j
k
]
,
for all j
,
k and T
>
0. Therefore lim
⁡
n
,
m
∑
j
,
k
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
Δ
s
r
(
x
j
k
y
j
k
)
|
)
p
j
k
]
=
0. Thus x
y
∈
W
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
)
∩
l
∞
2. This completes the proof.Lemma 15.
LetG be an ideal in l
∞
2 and let x
=
(
x
j
k
)
∈
l
∞
2. Then x is in the closure of G in l
∞
2 if and only if χ
S
(
x
;
ϵ
)
∈
G for all ϵ
>
0.Proof.
Letx be in the closure of G and let ϵ
>
0 be given. Suppose that z
=
(
z
j
k
)
∈
G such that ∥
x
-
z
∥
<
ϵ
/
2 and observe that S
(
x
;
ϵ
)
⊆
S
(
z
;
ϵ
/
2
). Define a double sequence y
=
(
y
j
k
)
∈
l
∞
2 by
(59)
y
j
k
=
{
1
z
j
k
,
if
|
z
j
k
|
≧
ϵ
2
,
0
,
otherwise
.
Clearlyy
z
=
χ
S
(
z
;
ϵ
/
2
)
∈
G. Since S
(
x
;
ϵ
)
⊆
S
(
z
;
ϵ
/
2
) and χ
S
(
x
;
ϵ
)
∈
l
∞
2, hence χ
S
(
x
;
ϵ
)
χ
S
(
z
;
ϵ
/
2
)
=
χ
S
(
x
;
ϵ
)
∈
G.
Conversely, ifx
∈
l
∞
2 then ∥
x
-
x
χ
S
(
x
;
ϵ
)
∥
<
ϵ. It follows that χ
S
(
x
;
ϵ
)
∈
G for all ϵ
>
0; then x is in the closure of G.Lemma 16.
IfA is a nonnegative four-dimensional R
H-regular matrix, then W
0
2
(
A
,
u
,
Δ
s
r
,
p
)
∩
l
∞
2 is a closed ideal in l
∞
2.Proof.
We haveW
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
)
∩
l
∞
2
⊂
l
∞
2 and it is clear that W
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
)
∩
l
∞
2
≠
0. For x
,
y
∈
W
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
)
∩
l
∞
2, we get |
x
j
k
+
y
j
k
|
<
|
x
j
k
|
+
|
y
j
k
|. Now, we have
(60)
[
F
j
k
(
u
j
k
|
Δ
s
r
(
x
j
k
+
y
j
k
)
|
)
p
j
k
]
≤
[
F
j
k
(
u
j
k
|
Δ
s
r
x
j
k
|
+
|
Δ
s
r
y
j
k
|
)
p
j
k
]
<
1
2
[
F
j
k
(
u
j
k
2
|
Δ
s
r
x
j
k
|
)
p
j
k
]
+
1
2
[
F
j
k
(
u
j
k
2
|
Δ
s
r
y
j
k
|
)
p
j
k
]
<
1
2
K
1
[
F
j
k
(
u
j
k
|
Δ
s
r
x
j
k
|
)
p
j
k
]
+
1
2
K
2
[
F
j
k
(
u
j
k
|
Δ
s
r
y
j
k
|
)
p
j
k
]
by the Δ
2-condition and the convexity of F. Since
(61)
∑
j
,
k
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
Δ
s
r
(
x
j
k
+
y
j
k
)
|
)
p
j
k
]
≤
1
2
K
∑
j
,
k
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
Δ
s
r
x
j
k
|
)
p
j
k
]
+
1
2
K
∑
j
,
k
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
Δ
s
r
y
j
k
|
)
p
j
k
]
,
where K
=
max
⁡
{
K
1
,
K
2
}, so x
+
y
,
x
-
y
∈
W
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
)
∩
l
∞
2.
Letx
∈
W
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
)
∩
l
∞
2 and y
∈
l
∞
2. Thus, there exists a positive integer K, so that, for every j
,
k, we have |
x
j
k
y
j
k
|
≤
K
|
x
j
k
|. Therefore
(62)
[
F
j
k
(
u
j
k
|
Δ
s
r
(
x
j
k
y
j
k
)
|
)
p
j
k
]
≤
[
F
j
k
(
u
j
k
K
|
Δ
s
r
x
j
k
|
)
p
j
k
]
[
F
j
k
(
u
j
k
|
Δ
s
r
(
x
j
k
y
j
k
)
|
)
p
j
k
]
≤
T
[
F
j
k
(
u
j
k
|
Δ
s
r
x
j
k
|
)
p
j
k
]
,
and so
(63)
∑
j
,
k
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
Δ
s
r
(
x
j
k
y
j
k
)
|
)
p
j
k
]
≤
T
∑
j
,
k
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
Δ
s
r
x
j
k
|
)
p
j
k
]
.
Hencex
y
∈
W
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
)
∩
l
∞
2. So W
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
)
∩
l
∞
2 is an ideal in l
∞
2 for a Musielak-Orlicz function which satisfies the Δ
2-condition.
Now, we have to show thatW
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
)
∩
l
∞
2 is closed. Let x
∈
W
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
)
∩
l
∞
2
¯; there exists x
c
d
=
x
j
k
c
d
∈
W
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
)
∩
l
∞
2 such that x
c
d
→
x
∈
l
∞
2. For every ϵ
>
0 there exists N
1
(
ϵ
)
∈
ℕ such that, for all c
,
d
>
N
1
(
ϵ
), |
Δ
s
r
x
c
d
-
Δ
s
r
x
|
<
ϵ. Now, for ϵ
>
0, we have
(64)
∑
j
,
k
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
Δ
s
r
x
j
k
|
)
p
j
k
]
=
∑
j
,
k
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
Δ
s
r
x
j
k
-
Δ
s
r
x
j
k
c
d
+
Δ
s
r
x
j
k
c
d
|
)
p
j
k
]
≤
∑
j
,
k
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
Δ
s
r
x
j
k
-
Δ
s
r
x
j
k
c
d
|
+
|
Δ
s
r
x
j
k
c
d
|
)
p
j
k
]
≤
1
2
∑
j
,
k
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
2
|
Δ
s
r
x
j
k
-
Δ
s
r
x
j
k
c
d
|
)
p
j
k
]
+
1
2
∑
j
,
k
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
2
|
Δ
s
r
x
j
k
c
d
|
)
p
j
k
]
≤
1
2
K
F
j
k
(
ϵ
)
∑
j
,
k
‍
a
n
m
j
k
+
1
2
K
∑
j
,
k
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
Δ
s
r
x
j
k
c
d
|
)
p
j
k
]
.
Sincex
c
d
∈
W
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
)
∩
l
∞
2 and A isRH-regular, we get
(65)
lim
⁡
n
,
m
∑
j
,
k
‍
a
n
m
j
k
[
F
j
k
(
u
j
k
|
Δ
s
r
x
j
k
|
)
p
j
k
]
=
0
;
so x
∈
W
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
)
∩
l
∞
2. This completes the proof.Theorem 17.
Letx
=
(
x
j
k
) be a bounded sequence, ℱ
=
(
F
j
k
) a sublinear Musielak-Orlicz function which satisfies the Δ
2-condition, and A a nonnegative four-dimensional R
H-regular matrix. Then W
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
)
∩
l
∞
2
=
W
2
(
A
,
u
,
Δ
s
r
,
p
)
∩
l
∞
2.Proof.
Without loss of generality we may takeL
=
0 and establish
(66)
W
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
)
∩
l
∞
2
=
W
0
2
(
A
,
u
,
Δ
s
r
,
p
)
∩
l
∞
2
.
SinceW
0
2
(
A
,
u
,
Δ
s
r
,
p
)
⊆
W
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
), therefore W
0
2
(
A
,
u
,
Δ
s
r
,
p
)
∩
l
∞
2
⊆
W
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
)
∩
l
∞
2. We need to show that W
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
)
∩
l
∞
2
⊆
W
0
2
(
A
,
u
,
Δ
s
r
,
p
)
∩
l
∞
2. Notice that if S
⊂
ℕ
×
ℕ, then
(67)
∑
j
,
k
‍
a
n
m
j
k
[
F
j
k
(
χ
S
(
j
,
k
)
)
p
j
k
]
=
F
j
k
(
1
)
∑
j
,
k
‍
a
n
m
j
k
(
χ
S
(
j
,
k
)
)
p
j
k
,
for all n
,
m. Observe that χ
S
(
j
,
k
)
∈
W
0
2
(
A
,
u
,
Δ
s
r
,
p
)
∩
l
∞
2 whenever x
∈
W
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
)
∩
l
∞
2 by Lemmas 14 and 15, so
(68)
W
0
2
(
A
,
ℱ
,
u
,
Δ
s
r
,
p
)
∩
l
∞
2
⊆
W
0
2
(
A
,
u
,
Δ
s
r
,
p
)
∩
l
∞
2
.
The proof is complete.



---

*Source: 419064-2014-03-10.xml*