# New Classes of Generalized Seminormed Difference Sequence Spaces

Authors: M. Mursaleen; A. Alotaibi; Sunil K. Sharma
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

The purpose of this paper is to introduce new classes of generalized seminormed difference sequence spaces defined by a Musielak-Orlicz function. We also study some topological properties and prove some inclusion relations between resulting sequence spaces.


---

## Body

## 1.  Introduction and Preliminaries

Letl
0 denote the space of all real sequences x
=
{
x
k
}. Let C denote the space whose elements are the sets of distinct positive integers. Given any element σ of C, we denote by c
(
σ
) the sequence {
c
n
(
σ
)
} such that c
n
(
σ
)
=
1 if n
∈
σ, and c
n
(
σ
)
=
0 otherwise. Further
(1)
C
s
=
{
σ
∈
C
:
∑
n
=
1
∞
‍
c
n
(
σ
)
≤
s
}
,
the set of those σ  whose support has cardinality at most s, and
(2)
Φ
=
{
(
ϕ
k
k
)
ϕ
=
{
ϕ
k
}
∈
l
0
:
ϕ
1
>
0
,
Δ
ϕ
k
≥
0
,
j
j
j
j
j
j
j
j
j
j
j
Δ
(
ϕ
k
k
)
≤
0
(
k
=
1,2
,
…
)
}
,
where Δ
ϕ
k
=
ϕ
k
-
ϕ
k
-
1.Forϕ
∈
Φ, Sargent [1] defined the following sequence space:
(3)
m
(
ϕ
)
=
{
x
=
{
x
k
}
∈
l
0
:
sup
⁡
s
≥
1
sup
⁡
σ
∈
C
s
(
1
ϕ
s
∑
k
∈
σ
‍
|
x
k
|
)
<
∞
}
,
which was further studied in [2–4].The spacem
(
ϕ
) was extended to m
(
ϕ
,
p
) by Tripathy and Sen [5] as follows:
(4)
m
(
ϕ
,
p
)
=
{
x
=
{
x
k
}
∈
l
0
:
sup
⁡
s
≥
1
sup
⁡
σ
∈
C
s
(
1
ϕ
s
∑
k
∈
σ
‍
|
x
k
|
p
)
<
∞
}
.The notion of the difference sequence space was introduced by Kızmaz [6] which was generalized by Mursaleen [7]. It was further generalized by Et and Çolak [8] as follows: Z
(
Δ
μ
)
=
{
x
=
(
x
k
)
∈
ω
:
(
Δ
μ
x
k
)
∈
z
} for z
=
l
∞, c, and c
0, where μ is a nonnegative integer and
(5)
Δ
μ
x
k
=
Δ
μ
-
1
x
k
-
Δ
μ
-
1
x
k
+
1
,
Δ
0
x
k
=
x
k
∀
k
∈
N
or equivalent to the following binomial representation:
(6)
Δ
μ
x
k
=
∑
v
=
0
μ
‍
(
-
1
)
v
(
μ
v
)
x
k
+
v
.
These sequence spaces were generalized by Et and Basarir [9] for z
=
l
∞
(
p
), c
(
p
), and c
0
(
p
).Dutta [10] introduced the following difference sequence spaces using a new difference operator:
(7)
Z
(
Δ
(
η
)
)

=

{
x

=

(
x
k
)

∈

ω
:
Δ
(
η
)
x

∈

z
}
for
z
=
l
∞
,
c
,
and
c
0
,
where Δ
(
η
)
x
=
(
Δ
(
η
)
x
k
)
=
(
x
k
-
x
k
-
η
) for all k
,
η
∈
N.In [11], Dutta introduced the sequence spaces c
-
(
∥
·
,
·
∥
,
Δ
(
η
)
μ
,
p
)
,
c
-
0
(
∥
·
,
·
∥
,
Δ
(
η
)
μ
,
p
), l
∞
(
∥
·
,
·
∥
,
Δ
(
η
)
μ
,
p
),   m
(
∥
·
,
·
∥
,
Δ
(
η
)
μ
,
p
), and m
0
(
∥
·
,
·
∥
,
Δ
(
η
)
μ
,
p
), where η
,
μ
∈
N and Δ
(
η
)
μ
x
k
=
(
Δ
(
η
)
μ
x
k
)
=
(
Δ
(
η
)
μ
-
1
x
k
-
Δ
(
η
)
μ
-
1
x
k
-
η
) and Δ
(
η
)
0
x
k
=
x
k for all k
,
η
∈
N, which is equivalent to the following binomial representation:
(8)
Δ
(
η
)
μ
x
k
=
∑
v
=
0
μ
‍
(
-
1
)
v
(
μ
v
)
x
k
-
η
v
.The difference sequence spaces have been studied by several authors [12–19] and references therein. Başar and Altay [20] introduced the generalized difference matrix B
=
(
b
m
k
)
k
,
m
∈
N by
(9)
b
m
k
=
{
r
,
k
=
m
s
,
k
=
m
-
1
0
,
(
k
>
m
)
or
(
0
≤
k
<
m
-
1
)
.Başarir and Kayikçi [21] defined the matrix B
μ
(
b
m
k
μ
) which reduces to the difference matrix Δ
(
1
)
μ if r
=
1
,
s
=
-
1. The generalized B
μ-difference operator is equivalent to the following binomial representation:
(10)
B
μ
x
=
B
μ
(
x
k
)
=
∑
v
=
0
μ
‍
(
μ
v
)
r
μ
-
v
s
v
x
k
-
v
.Let∧
=
(
∧
k
) be a sequence of nonzero scalars. Then, for a sequence space E, the multiplier sequence space E
∧, associated with the multiplier sequence ∧, is defined as
(11)
E
∧
=
{
x
=
(
x
k
)
∈
ω
:
(
∧
k
x
k
)
∈
E
}
.Letω
(
X
) denote the space of all sequences with elements in (
X
,
q
), where (
X
,
q
) denotes a seminormed space, seminormed by q. The zero sequence is denoted by θ
=
(
0,0
,
0
,
…
).An Orlicz functionM is a function, M
:
[
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 [22] used the idea of Orlicz function to define the following sequence space:
(12)
l
M
=
{
x
∈
ω
:
∑
k
=
1
∞
‍
M
(
|
x
k
|
ρ
)
<
∞
}
,
which is called an Orlicz sequence space. The space l
M is a Banach space with the norm
(13)
∥
x
∥
=
inf
⁡
{
ρ
>
0
:
∑
k
=
1
∞
‍
M
(
|
x
k
|
ρ
)
≤
1
}
.
It is shown in [22] that every Orlicz sequence space l
M contains a subspace isomorphic to l
p
(
p
≥
1
). The Δ
2-condition is equivalent to M
(
L
x
)
≤
K
L
M
(
x
) for all values of x
≥
0 and for L
>
1.A sequenceM
=
(
M
k
) of Orlicz functions is called a Musielak-Orlicz function. A sequence N
=
(
N
k
) defined by
(14)
N
k
(
v
)
=
sup
⁡
{
|
v
|
u
-
M
k
(
u
)
:
u
≥
0
}
,
k
=
1,2
,
…
,
is called the complimentary function of a Musielak-Orlicz function (see [23, 24]). For a given Musiclak-Orlicz function M, the Musielak-Orlicz sequence space t
M and its subspace h
M are defined as follows:
(15)
t
M
=
{
x
∈
ω
:
I
M
(
c
x
)
<
∞
for
some
c
>
0
}
,
h
M
=
{
x
∈
ω
:
I
M
(
c
x
)
<
∞
∀
c
>
0
}
,
where I
M is a convex modular defined by
(16)
I
M
(
x
)
=
∑
k
=
1
∞
‍
M
k
(
x
k
)
,
x
=
(
x
k
)
∈
t
M
.
We consider t
M equipped with the Luxemburg norm,
(17)
∥
x
∥
=
inf
⁡
{
k
>
0
:
I
M
(
x
k
)
≤
1
}
,
or equipped with the Orlicz norm,
(18)
∥
x
∥
0
=
inf
⁡
{
1
k
(
1
+
I
M
(
k
x
)
)
:
k
>
0
}
.
A sequence space E is said to be solid if (
α
k
x
k
)
∈
E, whenever (
x
k
)
∈
E for all sequences (
α
k
) of scalars such that |
α
k
|
≤
1 for all k
∈
N.A sequence spaceE is said to be monotone if E contains the canonical preimages of all its step spaces.Remark 1.
It is well known that a sequence space is solid implies that it is monotone (see Kamthan and Gupta [25]).The sequence spacem
(
ϕ
) was introduced by Sargent [1]. He studied some of its properties and obtained its relationship with the space l
p. Later on, it was investigated from sequence space point of view and related with summability theory by Bilgin [26], Esi [27], Tripathy and Mahanta [28], and many others.The main goal of the present paper is to introduce new classes of generalized seminormed difference sequence spaces defined by Musielak-Orlicz function.For a given infinite matrixA
=
(
a
i
k
)
i
,
k
≥
1. The A-transform of a sequence x
=
(
x
k
)
k
≥
1 is the sequence A
x
=
(
A
i
) (i
≥
1), where
(19)
A
i
(
x
)
=
∑
k
=
1
∞
‍
a
i
k
x
k
,
provided that the series on the right converges for each i
≥
1.Let(
X
,
q
) be a seminormed space, M
=
(
M
i
) a Musielak-Orlicz function, and p
=
(
p
i
) a bounded sequence of positive real numbers. Then we define the following classes of sequences:
(20)
l
∞
(
M
,
A
,
B
Λ
μ
,
q
,
p
)
=
{
x
=
(
x
k
)
∈
w
(
X
)
:
sup
⁡
i
≥
1
M
i
(
q
(
A
i
(
B
Λ
μ
x
k
)
ρ
)
)
p
i
<
∞
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
for
some
ρ
>
0
M
i
(
q
(
A
i
(
B
Λ
μ
x
k
)
ρ
)
)
p
i
}
,
l
1
(
M
,
A
,
B
Λ
μ
,
q
,
p
)
=
{
x
=
(
x
k
)
∈
w
(
X
)
:
∑
i
=
1
∞
‍
M
i
(
q
(
A
i
(
B
Λ
μ
x
k
)
ρ
)
)
p
i
<
∞
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
for
some
ρ
>
0
(
q
(
A
i
(
B
Λ
μ
x
k
)
ρ
)
)
p
i
}
,
m
(
M
,
A
,
B
Λ
μ
,
ϕ
,
q
,
p
)
=
{
(
q
(
A
i
(
B
Λ
μ
x
k
)
ρ
)
)
p
i
x
=
(
x
k
)
∈
w
(
X
)
:
sup
⁡
s
≥
1
,
σ
∈
C
s
1
ϕ
s
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
x
k
)
ρ
)
)
p
i
<
∞
,
hhhhhhhhhhhhhhhhhhhhhh
for
some
ρ
>
0
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
x
k
)
ρ
)
)
p
i
}
.The following inequality will be used throughout the paper. If0
<
h
=
inf
⁡
p
k
≤
p
k
≤
sup
⁡
p
k
=
H and D
=
max
⁡
(
1
,
2
H
-
1
) then
(21)
|
a
k
+
b
k
|
p
k
≤
D
{
|
a
k
|
p
k
+
|
b
k
|
p
k
}
,
for all k and a
k
,
b
k
∈
C. Also |
a
|
p
k
≤
max
⁡
(
1
,
|
a
|
H
) for all a
∈
C.We study here some topological properties and establish inclusion relations between these sequence spaces.


## 2. Main Results

Theorem 2.
LetM
=
(
M
i
) be a Musielak-Orlicz function and p
=
(
p
i
) a bounded sequence of positive real numbers. Then the spaces l
∞
(
M
,
A
,
B
Λ
μ
,
q
,
p
), l
1
(
M
,
A
,
B
Λ
μ
,
q
,
p
), and m
(
M
,
A
,
B
Λ
μ
,
ϕ
,
q
,
p
) are linear spaces over the field of complex number C.Proof.
Letx
=
(
x
k
),  y
=
(
y
k
)
∈
m
(
M
,
A
,
B
Λ
μ
,
ϕ
,
q
,
p
), and α
,
β
∈
C. Then there exist positive real numbers ρ
1
,
ρ
2
>
0 such that
(22)
sup
⁡
s
≥
1
,
σ
∈
C
s
1
ϕ
s
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
x
k
)
ρ
1
)
)
p
i
<
∞
,
sup
⁡
s
≥
1
,
σ
∈
C
s
1
ϕ
s
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
y
k
)
ρ
2
)
)
p
i
<
∞
.
Define ρ
3
=
max
⁡
(
2
|
α
|
ρ
1
,
2
|
β
|
ρ
2
). Since (
M
i
) is a nondecreasing, convex function and so by using inequality (21), we have
(23)
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
(
α
x
+
β
y
)
)
ρ
3
)
)
p
i
≤
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
α
x
)
ρ
3
)
+
q
(
A
i
(
B
Λ
μ
β
y
)
ρ
3
)
)
p
i
≤
D
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
α
x
)
ρ
1
)
)
p
i
+
D
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
β
y
)
ρ
2
)
)
p
i
.
Thus
(24)
sup
⁡
s
≥
1
,
σ
∈
C
s
1
ϕ
s
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
(
α
x
+
β
y
)
)
ρ
3
)
)
p
i
≤
sup
⁡
s
≥
1
,
σ
∈
C
s
1
ϕ
s
D
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
α
x
)
ρ
1
)
)
p
i
+
sup
⁡
s
≥
1
,
σ
∈
C
s
1
ϕ
s
D
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
β
y
)
ρ
2
)

)
p
i
<
∞
.
Thus (
α
x
+
β
y
)
∈
m
(
M
,
A
,
B
Λ
μ
,
ϕ
,
q
,
p
). Hence m
(
M
,
A
,
B
Λ
μ
,
ϕ
,
q
,
p
) is a linear space. Similarly, we can prove that the spaces l
∞
(
M
,
A
,
B
Λ
μ
,
q
,
p
) and l
1
(
M
,
A
,
B
Λ
μ
,
q
,
p
) are linear spaces. This completes the proof of the theorem.Theorem 3.
LetM
=
(
M
i
) be a Musielak-Orlicz function and p
=
(
p
i
) a bounded sequence of positive real numbers. Then l
1
(
M
,
A
,
B
Λ
μ
,
q
,
p
)
⊂
m
(
M
,
A
,
B
Λ
μ
,
ϕ
,
q
,
p
)
⊂
l
∞
(
M
,
A
,
B
Λ
μ
,
q
,
p
).Proof.
Letx
=
(
x
k
)
∈
l
1
(
M
,
A
,
B
Λ
μ
,
q
,
p
). Then, for some ρ
>
0, we have
(25)
∑
i
=
1
∞
‍
M
i
(
q
(
A
i
(
B
Λ
μ
x
k
)
ρ
)
)
p
i
<
∞
.
Since (
ϕ
n
) is a monotonic increasing, we have
(26)
1
ϕ
s
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
x
k
)
ρ
)
)
p
i
≤
1
ϕ
1
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
x
k
)
ρ
)
)
p
i
≤
1
ϕ
1
∑
i
=
1
∞
‍
M
(
q
(
A
i
(
B
Λ
μ
x
k
)
ρ
)
)
p
i
<
∞
.
Hence,
(27)
sup
⁡
s
≥
1
,
σ
∈
C
s
1
ϕ
s
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
x
k
)
ρ
)
)
p
i
<
∞
.
Thus, x
=
(
x
k
)
∈
m
(
M
,
A
,
B
Λ
μ
,
ϕ
,
q
,
p
). Therefore, l
1
(
M
,
A
,
B
Λ
μ
,
q
,
p
)
⊂
m
(
M
,
A
,
B
Λ
μ
,
ϕ
,
q
,
p
).
Next, letx
=
(
x
k
)
∈
m
(
M
,
A
,
B
Λ
μ
,
ϕ
,
q
,
p
). Then, for some ρ
>
0, we have
(28)
sup
⁡
s
≥
1
,
σ
∈
C
s
1
ϕ
s
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
x
k
)
ρ
)
)
p
i
<
∞
.
Hence,
(29)
sup
⁡
s
≥
1
,
σ
∈
C
s
1
ϕ
s
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
x
k
)
ρ
)
)
p
i
<
∞
h
h
h
h
(
on
⁡
taking
cardinality
of
σ
to
be
1
)
.
Thus, x
=
(
x
k
)
∈
l
∞
(
M
,
A
,
B
Λ
μ
,
q
,
p
). Therefore, m
(
M
,
A
,
B
Λ
μ
,
ϕ
,
q
,
p
)
⊂
l
∞
(
M
,
A
,
B
Λ
μ
,
q
,
p
). This completes the proof of the theorem.Theorem 4.
LetM
=
(
M
i
) be a Musielak-Orlicz function and p
=
(
p
i
) a bounded sequence of positive real numbers. Then the space m
(
M
,
A
,
B
Λ
μ
,
φ
,
q
,
p
) is a seminormed space, seminormed by
(30)
g
(
x
)
=
inf
⁡
{
ρ
>
0
:
sup
⁡
s
≥
1
,
σ
∈
C
s
1
ϕ
s
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
x
)
ρ
)
)
p
i
≤
1
}
.Proof.
Clearly,g
(
x
)
≥
0 for all x
=
(
x
k
)
∈
m
(
M
,
A
,
B
Λ
μ
,
ϕ
,
q
,
p
) and g
(
θ
)
=
0. Let x
=
(
x
k
), y
=
(
y
k
)
∈
m
(
M
,
A
,
B
Λ
μ
,
ϕ
,
q
,
p
). Then there exist ρ
1
>
0 and ρ
2
>
0 such that
(31)
sup
⁡
s
≥
1
,
σ
∈
C
s
1
ϕ
s
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
x
)
ρ
1
)
)
p
i
≤
1
,
sup
⁡
s
≥
1
,
σ
∈
C
s
1
ϕ
s
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
y
)
ρ
2
)
)
p
i
≤
1
.
Letρ
=
ρ
1
+
ρ
2. Thus, we have
(32)
sup
⁡
s
≥
1
,
σ
∈
C
s
⁡
1
ϕ
s
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
(
x
+
y
)
)
ρ
)
)
p
i
=
sup
⁡
s
≥
1
,
σ
∈
C
s
1
ϕ
s
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
(
x
+
y
)
)
ρ
1
+
ρ
2
)
)
p
i
≤
sup
⁡
s
≥
1
,
σ
∈
C
s
1
ϕ
s
∑
i
∈
σ
‍
{
ρ
1
ρ
1
+
ρ
2
M
i
(
q
(
A
i
(
B
Λ
μ
(
x
)
)
ρ
1
)
)
hhhhhhhhhhhhhh
+
ρ
2
ρ
1
+
ρ
2
M
i
(
q
(
A
i
(
B
Λ
μ
(
y
)
)
ρ
2
)
)
}
p
i
≤
(
ρ
1
ρ
1
+
ρ
2
)
sup
⁡
s
≥
1
,
σ
∈
C
s
1
ϕ
s
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
x
)
ρ
1
)
)
p
i
+
(
ρ
2
ρ
1
+
ρ
2
)
sup
⁡
s
≥
1
,
σ
∈
C
s
1
ϕ
s
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
y
)
ρ
2
)
)
p
i
≤
1
.
Since the ρ’s are nonnegative, so we have
(33)
g
(
x
+
y
)
=
inf
⁡


{
ρ
>
0
:


sup
⁡
s
≥
1
,
σ
∈
C
s
1
ϕ
s
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
(
x
+
y
)
)
ρ
)
)
p
i


≤

1

}
≤
inf
⁡
{
ρ
1
>
0
:
sup
⁡
s
≥
1
,
σ
∈
C
s
1
ϕ
s
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
x
)
ρ
1
)
)
p
i
≤
1
}
+
inf
⁡
{
ρ
2
>
0
:


sup
⁡
s
≥
1
,
σ
∈
C
s
1
ϕ
s
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
y
)
ρ
2
)
)
p
i

≤

1

}


.
Thus, g
(
x
+
y
)
≤
g
(
x
)
+
g
(
y
). Next, for λ
∈
C, without loss of generality, λ
≠
0, then
(34)
g
(
ν
x
)
=
inf
⁡
{
ρ
>
0
:


sup
⁡
s
≥
1
,
σ
∈
C
s
1
ϕ
s
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
(
ν
x
)
)
ρ
)
)
p
i

≤
1
}
=
inf
⁡
{
ρ
>
0
:
sup
⁡
s
≥
1
,
σ
∈
C
s
1
ϕ
s
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
x
)
r
)
)
p
i
≤
1
}
,
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
j
j
j
k
k
k
k
k
k
k
k
k
gg
g
where
r
=
ρ
|
ν
|
.
This completes the proof of the theorem.Theorem 5.
LetM
=
(
M
i
) be a Musielak-Orlicz function and p
=
(
p
i
) a bounded sequence of positive real numbers. Then (i)
the spacel
∞
(
M
,
A
,
B
Λ
μ
,
q
,
p
) is a seminormed space, seminormed by
(35)
f
(
x
)
=
inf
⁡
{
ρ
>
0
:
sup
⁡
i
≥
1
M
i
(
q
(
A
i
(
B
Λ
μ
x
)
ρ
)
)
p
i
≤
1
}
,
(ii)
the spacel
1
(
M
,
A
,
B
Λ
μ
,
q
,
p
) is a seminormed space, seminormed by
(36)
h
(
x
)
=
inf
⁡
{
ρ
>
0
:
∑
i
=
1
∞
‍
M
i
(
q
(
A
i
(
B
Λ
μ
x
)
ρ
)
)
p
i
≤
1
}
.Proof.
It is easy to prove in view of Theorem4, so we omit the details.Theorem 6.
LetM
=
(
M
i
) be a Musielak-Orlicz function and p
=
(
p
i
) a bounded sequence of positive real numbers. Then m
(
M
,
A
,
B
Λ
μ
,
ϕ
,
q
,
p
)
⊂
m
(
M
,
A
,
B
Λ
μ
,
ψ
,
q
,
p
) if and only if sup
⁡
s
≥
1
(
φ
s
/
ψ
s
)
<
∞.Proof.
Supposesup
⁡
s
≥
1
(
φ
s
/
ψ
s
)
<
∞ and x
=
(
x
k
)
∈
m
(
M
,
A
,
B
Λ
μ
,
ϕ
,
q
,
p
). Then, we have for some ρ
>
0
(37)
sup
⁡
s
≥
1
,
σ
∈
C
s
1
ϕ
s
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
x
)
ρ
)
)
p
i
<
∞
.
Thus,
(38)
sup
⁡
s
≥
1
,
σ
∈
C
s
1
ψ
s
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
x
)
ρ
)
)
p
i
≤
(
sup
⁡
s
≥
1
ϕ
s
ψ
s
)
(
sup
⁡
s
≥
1
,
σ
∈
C
s
1
ϕ
s
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
x
)
ρ
)
)
p
i
)
<
∞
.
Therefore, x
=
(
x
k
)
∈
m
(
M
,
A
,
B
Λ
μ
,
ψ
,
q
,
p
). Hence, m
(
M
,
A
,
B
Λ
μ
,
ϕ
,
q
,
p
)
⊂
m
(
M
,
A
,
B
Λ
μ
,
ψ
,
q
,
p
).
Conversely, letm
(
M
,
A
,
B
Λ
μ
,
ϕ
,
q
,
p
)
⊂
m
(
M
,
A
,
B
Λ
μ
,
ψ
,
q
,
p
). Suppose that sup
⁡
s
≥
1
(
ϕ
s
/
ψ
s
)
=
∞. Then there exists a sequence of naturals {
s
i
} such that lim
⁡
i
→
∞
(
ϕ
s
i
/
ψ
s
i
)
=
∞. Let x
=
(
x
k
)
∈
m
(
M
,
A
,
B
Λ
μ
,
ϕ
,
q
,
p
). Then there exists ρ
>
0 such that
(39)
sup
⁡
s
≥
1
,
σ
∈
C
s
1
ϕ
s
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
x
)
ρ
)
)
p
i
<
∞
.
Now, we have
(40)
sup
⁡
s
≥
1
,
σ
∈
C
s
1
ψ
s
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
x
)
ρ
)
)
p
i
≥
(
sup
⁡
s
≥
1
ϕ
s
ψ
s
)
(
sup
⁡
s
≥
1
,
σ
∈
C
s
1
ϕ
s
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
x
)
ρ
)
)
p
i
)
=
∞
.
Therefore, x
=
(
x
k
)
∉
m
(
M
,
A
,
B
Λ
μ
,
ψ
,
q
,
p
), which is a contradiction. Hence sup
⁡
s
≥
1
(
ϕ
s
/
ψ
s
)
<
∞.We get the following corollary as a consequence of Theorem6.Corollary 7.
LetM
=
(
M
i
) be a Musielak-Orlicz function and p
=
(
p
i
) a bounded sequence of positive real numbers. Then m
(
M
,
A
,
B
Λ
μ
,
ϕ
,
q
,
p
)
=
m
(
M
,
A
,
B
Λ
μ
,
ψ
,
q
,
p
) if and only if s
u
p
s
≥
1
(
ϕ
s
/
ψ
s
)
<
∞ and s
u
p
s
≥
1
(
ψ
s
/
ϕ
s
)
<
∞ for all s
=
1,2
,
3
,
….Theorem 8.
LetM
′
=
(
M
i
)
′
,
M
′′
=
(
M
i
)
′′ be Musielak-Orlicz functions which satisfy Δ
2-conditions and p
=
(
p
i
) a bounded sequence of positive real numbers. Then (i)
m
(
M
Λ
′
μ
,
ϕ
,
q
,
p
)
⊆
m
(
M
∘
M
Λ
′
μ
,
A
,
ϕ
,
q
,
p
);
(ii)
m
(
M
Λ
′
μ
,
ϕ
,
q
,
p
)
∩
m
(
M
Λ
′′
μ
,
ϕ
,
q
,
p
)
⊆
m
(
M
′
+
M
Λ
′′
μ
,
ϕ
,
q
,
p
).Proof.
(i) Letx
=
(
x
k
)
∈
m
(
M
Λ
′
μ
,
ϕ
,
q
,
p
). Then there exists ρ
>
0 such that
(41)
sup
⁡
s
≥
1
,
σ
∈
C
s
1
ϕ
s
∑
i
∈
σ
‍
M
i
′
(
q
(
A
i
(
B
Λ
μ
x
)
ρ
)
)
p
i
<
∞
.
Let 0
<
ε
<
1 and 0
<
δ
<
1 such that M
i
(
t
)
<
ε for 0
≤
t
<
δ. Let y
k
=
M
i
′
(
q
(
A
i
(
B
Λ
μ
x
)
/
ρ
)
)
p
i and, for any σ
∈
C
s, let ∑
i
∈
σ
‍
M
i
(
y
k
)
=
∑
1
‍
M
i
(
y
k
)
+
∑
2
‍
M
i
(
y
k
), where the first summation is over y
k
≤
δ and the second summation is over y
k
>
δ. Since (
M
i
) satisfies Δ
2-condition, we have
(42)
∑
1
‍
M
i
(
y
k
)
≤
M
i
(
1
)
∑
1
‍
y
k
≤
M
i
(
2
)
∑
1
‍
y
k
.
For y
k
>
δ
(43)
y
k
<
y
k
δ
≤
1
+
y
k
δ
.
Since (
M
i
) is nondecreasing and convex, so
(44)
M
(
y
k
)
<
M
(
1
+
y
k
δ
)
<
1
2
M
(
2
)
+
1
2
M
i
(
2
y
k
δ
)
.
Since (
M
i
) also satisfies Δ
2-condition, so
(45)
M
i
(
y
k
)
<
1
2
K
y
k
δ
M
i
(
2
)
+
1
2
K
y
k
δ
M
i
(
2
)
=
K
y
k
δ
M
i
(
2
)
.
Hence,
(46)
∑
2
‍
M
i
(
y
k
)
≤
max
⁡
(
1
,
K
δ
-
1
M
i
(
2
)
)
∑
2
‍
y
k
.
By (42) and (46), we have x
=
(
x
k
)
∈
m
(
M
∘
M
Λ
′
μ
,
ϕ
,
q
,
p
). Hence
(47)
m
(
M
Λ
′
μ
,
ϕ
,
q
,
p
)
⊆
m
(
M
∘
M
Λ
′
,
ϕ
,
q
,
p
)
.
(ii) Letx
=
(
x
k
)
∈
m
(
M
Λ
′
μ
,
ϕ
,
q
,
p
)
∩
m
(
M
Λ
′′
μ
,
ϕ
,
q
,
p
). Then there exists ρ
>
0 such that
(48)
sup
⁡
s
≥
1
,
σ
∈
C
s
1
ϕ
s
∑
i
∈
σ
‍
M
i
′
(
q
(
A
i
(
B
Λ
μ
x
)
ρ
)
)
p
i
<
∞
,
sup
⁡
s
≥
1
,
σ
∈
C
s
1
ϕ
s
∑
i
∈
σ
‍
M
i
′′
(
q
(
A
i
(
B
Λ
μ
x
)
ρ
)
)
p
i
<
∞
.
The rest of the proof follows from the equality
(49)
∑
i
∈
σ
‍
(
M
i
′
+
M
i
′′
)
(
q
(
A
i
(
x
)
ρ
)
)
p
i
=
∑
i
∈
σ
‍
M
i
′
(
q
(
A
i
(
B
Λ
μ
x
)
ρ
)
)
p
i
+
∑
i
∈
σ
‍
M
i
′′
(
q
(
A
i
(
B
Λ
μ
x
)
ρ
)
)
p
i
.
This completes the proof of the theorem.Corollary 9.
LetM
=
(
M
i
) be a Musielak-Orlicz function and p
=
(
p
i
) a bounded sequence of positive real numbers. Then, we have m
(
A
,
B
Λ
μ
,
φ
,
q
,
p
)
⊆
m
(
M
,
A
,
B
Λ
μ
,
φ
,
q
,
p
).Proof.
It follows from Theorem8(i) on considering M
′
(
x
)
=
x, for all x
∈
[
0
,
∞
).The following result is a consequence of Theorem8 and Corollary 9.Corollary 10.
LetM
=
(
M
i
) be a Musielak-Orlicz function and p
=
(
p
i
) a bounded sequence of positive real numbers. Then m
(
A
,
B
Λ
μ
,
φ
,
q
,
p
)
⊆
m
(
M
,
A
,
B
Λ
μ
,
ψ
,
q
,
p
) if and only if s
u
p
s
≥
1
(
φ
s
/
ψ
s
)
<
∞.Theorem 11.
LetM
=
(
M
i
) be a Musielak-Orlicz function and p
=
(
p
i
) a bounded sequence of positive real numbers. Then the space m
(
M
,
A
,
B
Λ
μ
,
ϕ
,
q
,
p
) is solid.Proof.
Letx
=
(
x
k
)
∈
m
(
M
,
A
,
B
Λ
μ
,
ϕ
,
q
,
p
). Then
(50)
sup
⁡
s
≥
1
,
σ
∈
C
s
1
ϕ
s
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
x
)
ρ
)
)
p
i
<
∞
.
Let (
α
k
) be a sequence of scalars with |
α
k
|
≤
1 for all k
∈
N. Then the result follows from (50) and the following inequality
(51)
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
α
x
)
ρ
)
)
p
i
≤
∑
i
∈
σ
‍
|
α
|
M
i
(
q
(
A
i
(
B
Λ
μ
x
)
ρ
)
)
p
i
≤
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
x
)
ρ
)
)
p
i
.
This completes the proof of the theorem.In view of the above result, we get the following corollaries.Corollary 12.
LetM
=
(
M
i
) be a Musielak-Orlicz function and p
=
(
p
i
) a bounded sequence of positive real numbers. Then the space m
(
M
,
A
,
B
Λ
μ
,
ϕ
,
q
,
p
) is monotone.We formulate the following result which can be established following the technique of Theorem11 and Corollary 12.Corollary 13.
LetM
=
(
M
i
) be a Musielak-Orlicz function and p
=
(
p
i
) a bounded sequence of positive real numbers. Then the spaces l
∞
(
M
,
A
,
B
Λ
μ
,
q
,
p
) and l
1
(
M
,
A
,
B
Λ
μ
,
q
,
p
) are solid and monotone.Theorem 14.
If(
X
,
q
) is complete, then m
(
M
,
A
,
B
Λ
μ
,
ϕ
,
q
,
p
) is also complete.Proof.
Let(
x
j
) be a Cauchy sequence in m
(
M
,
A
,
B
Λ
μ
,
ϕ
,
q
,
p
), where x
j
=
(
x
k
j
)
=
(
x
1
j
,
x
2
j
,
x
3
j
,
…
)
∈
m
(
M
,
A
,
B
Λ
μ
,
ϕ
,
q
,
p
) for each j
∈
N. Let r
>
0 and x
0
>
0 be fixed. Then for each ε
/
r
x
0
>
0, there exists a positive integer n
0 such that
(52)
g
(
x
j
-
x
l
)
<
ε
r
x
0
,
∀
j
,
l
≥
n
0
.
This implies
(53)
inf
⁡
{
ρ
:
sup
⁡
s
≥
1
,
σ
∈
C
s
1
ϕ
s
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
(
x
j
-
x
l
)
)
ρ
)
)
p
i
≤
1
}
<
ε
r
x
0
,
∀
j
,
l
≥
n
0
.
We have for all j, l
≥
n
0 and by (53)
(54)
sup
⁡
s
≥
1
,
σ
∈
C
s
1
ϕ
s
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
(
x
j
-
x
l
)
)
h
(
x
j
-
x
l
)
)
)
p
i
≤
1
⟹
1
ϕ
1
M
i
(
q
(
A
i
(
B
Λ
μ
(
x
j
-
x
l
)
)
h
(
x
j
-
x
l
)
)
)
p
i
≤
1
⟹
M
i
(
q
(
A
i
(
B
Λ
μ
(
x
j
-
x
l
)
)
h
(
x
j
-
x
l
)
)
)
p
i
≤
ϕ
1
,
∀
j
,
l
≥
n
0
.
We can find r
>
0 such that (
r
x
0
/
2
)
η
(
x
0
/
2
)
>
ϕ
1, where η is the kernel associated with Musielak-Orlicz function M, such that
(55)
M
i
(
q
(
A
i
(
B
Λ
μ
(
x
j
-
x
l
)
)
h
(
x
j
-
x
l
)
)
)
p
i
≤
r
x
0
2
η
(
x
0
2
)
⟹
q
(
A
i
(
B
Λ
μ
(
x
j
-
x
l
)
)
)
p
i
<
r
x
0
2
·
ε
r
x
0
=
ε
2
.
Hence A
i
(
B
Λ
μ
x
j
)
j
≥
1 is a Cauchy sequence in (
X
,
q
), which is complete. Therefore, for each k
∈
N, there exist x
k
∈
X and x
=
(
x
k
) such that q
(
A
i
(
B
Λ
μ
(
x
j
-
x
)
)
)
p
i
→
0 as j
→
∞. Using the continuity of M, so for some ρ
>
0, we have
(56)
sup
⁡
s
≥
1
,
σ
∈
C
s
1
ϕ
s
∑
i
∈
σ
‍
M
i
(
q
(
lim
⁡
i
→
∞
⁡
A
i
(
B
Λ
μ
(
x
j
-
x
l
)
)
ρ
)
)
p
i
≤
1
⟹
sup
⁡
s
≥
1
,
σ
∈
C
s
1
ϕ
s
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
(
x
j
-
x
l
)
)
ρ
)
)
p
i
≤
1
.
Now, taking the infimum of such ρ’s by (53), we get
(57)
inf
⁡
⁡
{
(
q
(
A
i
(
B
Λ
μ
(
x
j
-
x
l
)
)
ρ
)
)
p
i
ρ
>
0
:
sup
⁡
s
≥
1
,
σ
∈
C
s
1
ϕ
s
∑
i
∈
σ
‍
M
i
(
q
(
A
i
(
B
Λ
μ
(
x
j
-
x
l
)
)
ρ
)
)
p
i


≤
1
}

<

ε
,
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
∀
j
≥
n
0
.
Since m
(
M
,
A
,
B
Λ
μ
,
ϕ
,
q
,
p
) is a linear space and (
x
-
x
j
) are in m
(
M
,
A
,
B
Λ
μ
,
ϕ
,
q
,
p
), so it follows that x
=
x
j
+
(
x
-
x
j
)
∈
m
(
M
,
A
,
B
Λ
μ
,
ϕ
,
q
,
p
). Hence m
(
M
,
A
,
B
Λ
μ
,
ϕ
,
q
,
p
) is complete. This completes the proof of the theorem.



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*Source: 461081-2014-06-01.xml*