# Some New Lacunary Strong Convergent Vector-Valued Sequence Spaces

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

---

## Abstract

We introduce some vector-valued sequence spaces defined by a Musielak-Orlicz function and the concepts of lacunary convergence and strong (A)-convergence, where A
=
(
a
i
k
) is an infinite matrix of complex numbers. We also make an effort to study some topological properties and some inclusion relations between these spaces.


---

## Body

## 1. Introduction and Preliminaries

An Orlicz functionM
:
[
0
,
∞
)
→
[
0
,
∞
) is convex and continuous such that M
(
0
)
=
0, M
(
x
)
>
0 for x
>
0. Lindenstrauss and Tzafriri [1] used the idea of Orlicz function to define the following sequence space:
(1)
l
M
=
{
x
∈
w
:
∑
k
=
1
∞
‍
M
(
|
x
k
|
ρ
)
<
∞
}
,
which is called as an Orlicz sequence space. The space l
M is a Banach space with the norm
(2)
∥
x
∥
=
inf
⁡
{
ρ
>
0
:
∑
k
=
1
∞
‍
M
(
|
x
k
|
ρ
)
≤
1
}
.
It is shown in [1] that every Orlicz sequence space l
M contains a subspace isomorphic to l
p
(
p
≥
1
). An Orlicz function M satisfies Δ
2-condition if and only if, for any constant L
>
1, there exists a constant K
(
L
) such that M
(
L
u
)
≤
K
(
L
)
M
(
u
) for all values of u
≥
0. An Orlicz function M can always be represented in the following integral form:
(3)
M
(
x
)
=
∫
0
x
‍
η
(
t
)
d
t
,
where η is known as the kernel of M and is right differentiable for t
≥
0, η
(
0
)
=
0, η
(
t
)
>
0; η is nondecreasing and η
(
t
)
→
∞ as t
→
∞.A sequenceM
=
(
M
k
) of Orlicz functions is called a Musielak-Orlicz function; see ([2, 3]). A sequence N
=
(
N
k
) defined by
(4)
N
k
(
v
)
=
sup
⁡
{
|
v
|
u
-
M
k
(
u
)
:
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 t
M and its subspace h
M are defined as follows:
(5)
t
M
=
{
x
∈
w
:
I
M
(
c
x
)
<
∞
for
some
c
>
0
}
,
h
M
=
{
x
∈
w
:
I
M
(
c
x
)
<
∞
,
∀
c
>
0
}
,
where I
M is a convex modular defined by
(6)
I
M
(
x
)
=
∑
k
=
1
∞
‍
M
k
(
x
k
)
,
x
=
(
x
k
)
∈
t
M
.
We consider t
M equipped with the Luxemburg norm
(7)
∥
x
∥
=
inf
⁡
{
k
>
0
:
I
M
(
x
k
)
≤
1
}
or equipped with the Orlicz norm
(8)
∥
x
∥
0
=
inf
⁡
{
1
k
(
1
+
I
M
(
k
x
)
)
:
k
>
0
}
.
A Musielak-Orlicz function (
M
k
) is said to satisfy Δ
2-condition 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
(9)
M
k
(
2
u
)
≤
K
M
k
(
u
)
+
c
k
holds for all k
∈
N and u
∈
R
+, whenever M
k
(
u
)
≤
a.LetX be a linear metric space. A function p: X
→
R is called paranorm, if(1)
p
(
x
)
≥
0, for all x
∈
X;
(2)
p
(
-
x
)
=
p
(
x
), for all x
∈
X;
(3)
p
(
x
+
y
)
≤
p
(
x
)
+
p
(
y
), for all x, y
∈
X;
(4)
if(
σ
n
) is a sequence of scalars with σ
n
→
σ as n
→
∞ and (
x
n
) is a sequence of vectors with p
(
x
n
-
x
)
→
0
as
n
→
∞, then p
(
σ
n
x
n
-
σ
x
)
→
0 as n
→
∞.A paranormp for which p
(
x
)
=
0 implies x
=
0 is called total paranorm and the pair (
X
,
p
) is called a total paranormed space. It is well known that the metric of any linear metric space is given by some total paranorm (see [4], Theorem 10.4
.
2, P-183). For more details about sequence spaces, see [5–11] and references therein.The space of lacunary strong convergence has been introduced by Freedman et al. [12]. A sequence of positive integers θ
=
(
k
r
) is called “lacunary” if k
0
=
0, 0
<
k
r
<
k
r
+
1 and h
r
=
k
r
-
k
r
-
1
→
∞, as r
→
∞. The intervals determined by θ are denoted by I
r
=
(
k
r
-
1
,
k
r
] and the ratio k
r
/
k
r
-
1 will be denoted by q
r. The space of lacunary strongly convergent sequences N
θ is defined by Freedman et al. [12] as follows:
(10)
N
θ
=
{
x
=
(
x
i
)
:
lim
⁡
r
→
∞
1
h
r
∑
i
∈
I
r
‍
|
x
i
-
s
|
=
0
,
for
some
s
}
.
The space |
σ
1
| of strongly Cesàro summable sequences is
(11)
|
σ
1
|
=
{
∑
i
=
1
n
‍
x
=
(
x
k
)
:
there
exists
L
such
that
1
n
∑
i
=
1
n
‍
|
x
i
-
L
|
⟶
0
,
as
n
⟶
∞
}
.
In case, when θ
=
(
2
r
), N
θ
=
|
σ
1
|. Recently, Bilgin [13] in his paper generalized the concept of lacunary convergence and introduced the space N
0
(
A
,
f
), as
(12)
N
0
(
A
,
f
)
=
{
x
=
(
x
i
)
:
lim
⁡
r
→
∞
1
h
r
∑
i
∈
I
r
‍
f
(
|
A
i
(
x
)
-
s
|
)
=
0
,
for
some
s
}
,
where f is a modulus function and A
=
(
A
i
(
x
)
); A
i
x
=
∑
k
=
1
∞
‍
a
i
k
x
k converges for each i. Later Bilgin [14] generalized lacunary strongly A-convergent sequences with respect to a sequence of modulus function F
=
(
f
i
) as follows:
(13)
N
0
(
A
,
F
)
=
{
x
=
(
x
i
)
:
lim
⁡
r
→
∞
1
h
r
∑
i
∈
I
r
‍
f
i
(
|
A
i
(
x
)
-
s
|
)
=
0
,
for
some
s
}
.
We write θ for the zero sequences.Mursaleen and Noman [15] introduced the notion of λ-convergent and λ-bounded sequences as follows.Letλ
=
(
λ
k
)
k
=
1
∞ be a strictly increasing sequence of positive real numbers tending to infinity, that is,
(14)
0
<
λ
0
<
λ
1
<
⋯
,
λ
k
⟶
∞
as
k
⟶
∞
,
and said that a sequence x
=
(
x
k
)
∈
w is λ-convergent to the number L, called the λ-limit of x if Λ
m
(
x
)
→
L as m
→
∞, where
(15)
λ
m
(
x
)
=
1
λ
m
∑
k
=
1
m
‍
(
λ
k
-
λ
k
-
1
)
x
k
.
The sequence x
=
(
x
k
)
∈
w is λ-bounded if sup
⁡
m
|
Λ
m
(
x
)
|
<
∞. It is well known [15] that if lim
⁡
m
x
m
=
a in the ordinary sense of convergence, then
(16)
lim
⁡
m
(
1
λ
m
(
∑
k
=
1
m
‍
(
λ
k
-
λ
k
-
1
)
|
x
k
-
a
|
)
)
=
0
.
This implies that
(17)
lim
⁡
m
|
Λ
m
(
x
)
-
a
|
=
lim
⁡
m
|
1
λ
m
∑
k
=
1
m
‍
(
λ
k
-
λ
k
-
1
)
(
x
k
-
a
)
|
=
0
,
which yields that lim
⁡
m
Λ
m
(
x
)
=
a and hence x
=
(
x
k
)
∈
w is λ-convergent to a.We now introduce the concept of lacunary stronglyA-convergence for sequences with the elements chosen from a Banach space (
E
,
∥
·
∥
) over the complex field C, with respect to Musielak-Orlicz functions M
=
(
M
i
).LetA
=
(
a
i
k
) be an infinite matrix of complex numbers and M
=
(
M
i
) be a Musielak-Orlicz function. In the present paper we define the following sequence spaces:
(18)
N
θ
(
E
,
A
,
Λ
,
M
)
=
{
∑
i
∈
I
r
‍
x
=
(
x
k
)
:
x
k
∈
E
,
lim
⁡
r
→
∞
1
h
r
∑
i
∈
I
r
‍
[
M
i
(
∥
A
i
(
Λ
i
(
x
)
-
s
i
e
i
)
∥
ρ
(
i
)
)
]
=
0
for
some
s
=
(
s
1
,
s
2
,
…
)
∈
E
,
e
i
∈
C
,
ρ
(
i
)
>
0
∑
i
∈
I
r
‍
}
,
N
θ
0
(
E
,
A
,
Λ
,
M
)
=
{
∑
i
∈
I
r
‍
x
=
(
x
k
)
:
x
k
∈
E
,
lim
⁡
r
→
∞
1
h
r
∑
i
∈
I
r
‍
[
M
i
(
∥
A
i
(
Λ
i
(
x
)
)
∥
ρ
(
i
)
)
]
=
0
for
some
ρ
(
i
)
>
0
∑
i
∈
I
r
‍
}
.If we takeM
i
(
x
)
=
x, for all i
∈
N, we have
(19)
N
θ
(
E
,
A
,
Λ
)
=
{
∑
i
∈
I
r
‍
x
=
(
x
k
)
:
x
k
∈
E
,
lim
⁡
r
→
∞
1
h
r
∑
i
∈
I
r
‍
[
∥
A
i
(
Λ
i
(
x
)
-
s
i
e
i
)
∥
ρ
(
i
)
]
=
0
for
some
s
=
(
s
1
,
s
2
,
…
)
∈
E
,
e
i
∈
C
,
ρ
(
i
)
>
0
∑
i
∈
I
r
‍
}
,
N
θ
0
(
E
,
A
,
Λ
)
=
{
∑
i
∈
I
r
‍
x
=
(
x
k
)
:
x
k
∈
E
,
lim
⁡
r
→
∞
1
h
r
∑
i
∈
I
r
‍
[
∥
A
i
(
Λ
i
(
x
)
)
∥
ρ
(
i
)
]
=
0
for
some
ρ
(
i
)
>
0
∑
i
∈
I
r
‍
}
,
where
(20)
e
i
=
{
1
,
at
the
i
th
place
,
0
,
otherwise
.
A sequence x is said to be Λ-lacunary strong A-convergent with respect to M if there is a number s
=
(
s
1
,
s
2
,
…
)
∈
E, such that x
∈
N
θ
(
E
,
A
,
Λ
,
M
).We have generalized the strongly Cesàro-summable sequence space intoΛ-strongly Cesàro-summable vector-valued sequence space as
(21)
|
σ
1
(
E
,
A
,
Λ
)
|
=
{
∑
i
=
1
n
‍
x
=
x
k
:
there
exists
L
=
(
L
1
,
L
2
,
…
)
∈
E
,
e
i
∈
C
such
that
1
n
∑
i
=
1
n
‍
∥
A
i
(
Λ
i
(
x
)
-
L
i
e
i
)
∥
⟶
0
}
,
where A
=
(
a
n
k
) is a Cesàro matrix, that is,
(22)
a
n
k
=
{
1
n
,
if
1
≤
k
≤
n
,
0
,
if
k
≥
n
.
Then it can be shown that |
σ
1
(
E
,
A
,
Λ
)
| is a paranormed space with respect to the paranorm
(23)
∥
x
∥
=
∥
x
1
∥
+
sup
⁡
n
(
1
n
∑
i
=
1
n
‍
∥
A
i
(
Λ
i
(
x
)
)
∥
)
.


## 2. Topological Properties of the SpacesN
θ
(
E
,
A
,
Λ
,
M
) and N
θ
0
(
E
,
A
,
Λ
,
M
)

Theorem 1.
LetA
=
(
a
i
k
) be an infinite matrix of complex numbers and let M
=
(
M
i
) be a Musielak-Orlicz function. Then N
θ
(
E
,
A
,
Λ
,
M
) and N
θ
0
(
E
,
A
,
Λ
,
M
) are linear spaces over the field of complex number C.Proof.
It is easy to prove.Theorem 2.
LetA
=
(
a
i
k
) be an infinite matrix of complex numbers and let M
=
(
M
i
) be a Musielak-Orlicz function. Then N
θ
0
(
E
,
A
,
Λ
,
M
) is normal spaces, when E is normal.Proof.
Letx
∈
N
θ
0
(
E
,
A
,
Λ
,
M
). Let ∥
y
∥
≤
∥
x
∥. Then
(24)
∥
A
i
(
Λ
i
(
y
)
)
∥
≤
∥
A
i
(
Λ
i
(
x
)
)
∥
.
Since M
=
(
M
i
) is increasing,
(25)
1
h
r
∑
i
∈
I
r
‍
[
M
i
(
∥
A
i
(
Λ
i
(
y
)
)
∥
ρ
(
i
)
)
]
≤
1
h
r
∑
i
∈
I
r
‍
[
M
i
(
∥
A
i
(
Λ
i
(
x
)
)
∥
ρ
(
i
)
)
]
.
Consequently, y
∈
N
θ
0
(
E
,
A
,
Λ
,
M
). This completes the proof of the theorem.Theorem 3.
The spacesN
θ
(
E
,
A
,
Λ
,
M
) and N
θ
0
(
E
,
A
,
Λ
,
M
) are paranormed spaces, with respect to the paranorm
(26)
∥
x
∥
=
inf
⁡
{
ρ
(
i
)
>
0
:
M
i
(
∥
a
i
0
x
1
∥
ρ
(
i
)
)
+
sup
⁡
r
≥
1
1
h
r
∑
i
∈
I
r
‍
[
M
i
(
∥
A
i
(
Λ
i
(
x
)
)
∥
ρ
(
i
)
)
]
≤
1
,
ρ
(
i
)
≥
0
∑
i
∈
I
r
‍
}
.Proof.
It is easy to prove, so we omit the details.


## 3. Relation between the SpacesN
θ
(
E
,
A
,
Λ
) and N
θ
(
E
,
A
,
Λ
,
M
)

The main purpose of this section is to study relation betweenN
θ
(
E
,
A
,
Λ
) and N
θ
(
E
,
A
,
Λ
,
M
).Theorem 4.
LetA
=
(
a
i
k
) be an infinite matrix of complex numbers and let M
=
(
M
i
) be a Musielak-Orlicz function satisfying Δ
2-condition. If x is Λ-lacunary strong A-convergent to s, with respect to M and (
E
,
∥
·
∥
) is a normal Banach space, then N
θ
(
E
,
A
,
Λ
)
⊂
N
θ
(
E
,
A
,
Λ
,
M
).Proof .
Letx
∈
N
θ
(
E
,
A
,
Λ
) and x
→
Λ
s, where s
=
(
s
1
,
s
2
,
…
)
∈
E
,
e
i
∈
C. Then
(27)
lim
⁡
r
→
∞
1
h
r
∑
i
∈
I
r
‍
(
∥
A
i
(
Λ
i
(
x
)
-
s
i
e
i
)
∥
ρ
)
=
0
for
some
ρ
>
0
.
We define two sequences y and z such that
(28)
(
∥
A
i
(
Λ
i
(
y
)
-
s
i
e
i
)
∥
)
=
{
(
∥
A
i
(
Λ
i
(
x
)
-
s
i
e
i
)
∥
)
,
if
(
∥
A
i
(
Λ
k
(
x
)
-
s
i
e
i
)
∥
)
>
1
,
θ
,
if
(
∥
A
i
(
Λ
i
(
x
)
-
s
i
e
i
)
∥
)
≤
1
,
(
∥
A
i
(
Λ
i
(
z
)
-
s
i
e
i
)
∥
)
=
{
θ
,
if
(
∥
A
i
(
Λ
i
(
x
)
-
s
i
e
i
)
∥
)
>
1
,
(
∥
A
i
(
Λ
i
(
x
)
-
s
i
e
i
)
∥
)
,
if
(
∥
A
i
(
Λ
k
(
x
)
-
s
i
e
i
)
∥
)
≤
1
.
Hence,
(29)
(
∥
A
i
(
Λ
i
(
x
)
-
s
i
e
i
)
∥
)
=
(
∥
A
i
(
Λ
i
(
y
)
-
s
i
e
i
)
∥
)
+
(
∥
A
i
(
Λ
i
(
z
)
-
s
i
e
i
)
∥
)
.
Now,
(30)
(
∥
A
i
(
Λ
i
(
y
)
-
s
i
e
i
)
∥
)
≤
(
∥
A
i
(
Λ
i
(
x
)
-
s
i
e
i
)
∥
)
,
(
∥
A
i
(
Λ
i
(
z
)
-
s
i
e
i
)
∥
)
≤
(
∥
A
i
(
Λ
i
(
x
)
-
s
i
e
i
)
∥
)
.
Since N
θ
(
E
,
A
,
Λ
) is normal, y
,
z
∈
N
θ
(
E
,
A
,
Λ
). Let sup
⁡
i
M
i
(
2
)
=
T. Then
(31)
1
h
r
∑
i
∈
I
r
‍
[
M
i
(
∥
A
i
(
Λ
i
(
x
)
-
s
i
e
i
)
∥
ρ
(
i
)
)
]
=
1
h
r
∑
i
∈
I
r
‍
[
M
i
(
∥
A
i
(
Λ
i
(
y
)
-
s
i
e
i
)
∥
+
∥
A
i
(
Λ
i
(
z
)
-
s
i
e
i
)
∥
ρ
(
i
)
)
]
≤
1
h
r
∑
i
∈
I
r
‍
[
1
2
M
i
(
2
∥
A
i
(
Λ
i
(
y
)
-
s
i
e
i
)
∥
ρ
(
i
)
)
+
1
2
M
i
(
2
∥
A
i
(
Λ
i
(
z
)
-
s
i
e
i
)
∥
ρ
(
i
)
)
]
<
1
2
1
h
r
∑
i
∈
I
r
‍
K
1
(
∥
A
i
(
Λ
i
(
y
)
-
s
i
e
i
)
∥
ρ
(
i
)
)
M
i
(
2
)
+
1
2
1
h
r
∑
i
∈
I
r
‍
K
2
(
∥
A
i
(
Λ
i
(
z
)
-
s
i
e
i
)
∥
ρ
(
i
)
)
M
i
(
2
)
≤
1
2
1
h
r
∑
i
∈
I
r
‍
K
1
(
∥
A
i
(
Λ
i
(
y
)
-
s
i
e
i
)
∥
ρ
(
i
)
)
sup
⁡
M
i
(
2
)
+
1
2
1
h
r
∑
i
∈
I
r
‍
K
2
(
∥
A
i
(
Λ
i
(
z
)
-
s
i
e
i
)
∥
ρ
(
i
)
)
sup
⁡
M
i
(
2
)
⟶
0
as
r
⟶
∞
.
Hence x
∈
N
θ
(
E
,
A
,
Λ
,
M
). This completes the proof of the theorem.Theorem 5.
LetA
=
(
a
i
k
) be an infinite matrix of complex numbers and let M
=
(
M
i
) be a Musielak-Orlicz function satisfying Δ
2-condition. If
(32)
lim
⁡
u
→
∞
inf
⁡
i
M
i
(
v
/
ρ
(
i
)
)
v
/
ρ
(
i
)
>
0
f
o
r
s
o
m
e
ρ
(
i
)
>
0
,
then N
θ
(
E
,
A
,
Λ
)
=
N
θ
(
E
,
A
,
Λ
,
M
).Proof .
Iflim
⁡
v
→
∞
inf
⁡
i
(
M
i
(
v
/
ρ
(
i
)
)
/
v
/
ρ
(
i
)
)
>
0
for
some
ρ
(
i
)
>
0, then there exists a number γ
>
0 such that
(33)
M
i
(
v
ρ
(
i
)
)
≥
γ
(
v
ρ
(
i
)
)
∀
v
>
0
and
some
ρ
(
i
)
>
0
.
Let x
∈
N
θ
(
E
,
A
,
Λ
,
M
) and x
→
Λ
s, where s
=
(
s
1
,
s
2
,
…
)
∈
E, e
i
∈
C. Then clearly
(34)
1
h
r
∑
i
∈
I
r
‍
[
M
i
(
∥
A
i
(
Λ
i
(
x
)
-
s
i
e
i
)
∥
ρ
(
i
)
)
]
≥
1
h
r
∑
i
∈
I
r
‍
u
i
[
γ
(
∥
A
i
(
Λ
i
(
x
)
-
s
i
e
i
)
∥
ρ
(
i
)
)
]
=
γ
1
h
r
∑
i
∈
I
r
‍
(
∥
A
i
(
Λ
i
(
x
)
-
s
i
e
i
)
∥
ρ
(
i
)
)
.
Hence x
∈
N
θ
(
E
,
A
,
Λ
). This completes the proof.


## 4.  Relation between the spaces|
σ
1
(
E
,
A
,
Λ
)
| and N
θ
(
E
,
A
,
Λ
)

In this section of the paper we study relation between the spaces|
σ
1
(
E
,
A
,
Λ
)
| and N
θ
(
E
,
A
,
Λ
).Lemma 6.
|
σ
1
(
E
,
A
,
Λ
)
|
0
⊂
N
θ
(
E
,
A
,
Λ
) if and only if lim
⁡
inf
⁡
r
q
r
>
1.Proof .
First suppose thatlim
⁡
inf
⁡
r
q
r
>
1. Then there exist δ
>
0 such that 1
+
δ
≤
q
r for all r
≥
1. Let x
∈
|
σ
1
(
E
,
A
,
Λ
)
|
0. Then
(35)
1
h
r
∑
i
∈
I
r
‍
∥
A
i
(
Λ
i
(
x
)
)
∥
=
1
h
r
∑
i
=
1
k
r
‍
∥
A
i
(
Λ
i
(
x
)
)
∥
-
1
h
r
∑
i
=
1
k
r
-
1
‍
∥
A
i
(
Λ
i
(
x
)
)
∥
=
k
r
h
r
(
1
k
r
∑
i
=
1
k
r
‍
∥
A
i
(
Λ
i
(
x
)
)
∥
)
-
k
r
-
1
h
r
(
1
k
r
-
1
∑
i
=
1
k
r
-
1
‍
∥
A
i
(
Λ
i
(
x
)
)
∥
)
.
Now, h
r
=
k
r
-
k
r
-
1. So we have
(36)
k
r
h
r
=
k
r
k
r
-
k
r
-
1
=
q
r
q
r
-
1
=
1
+
1
q
r
-
1
≤
1
+
1
δ
=
δ
+
1
δ
.
Also
(37)
k
r
-
1
h
r
=
k
r
-
1
k
r
-
k
r
-
1
=
1
q
r
-
1
≤
1
δ
.
Since x
∈
|
σ
1
(
E
,
A
,
Λ
)
|
0, then
(38)
1
k
r
∑
i
=
1
k
r
‍
∥
A
i
(
Λ
i
(
x
)
)
∥
⟶
0
,
1
k
r
-
1
∑
i
=
1
k
r
-
1
‍
∥
A
i
(
Λ
i
(
x
)
)
∥
⟶
0
,
and hence
(39)
1
h
r
∑
i
∈
I
r
‍
∥
A
i
(
Λ
i
(
x
)
)
∥
⟶
0
;
that is, x
∈
N
θ
0
(
E
,
A
,
Λ
). By linearity, it follows that |
σ
1
(
E
,
A
,
Λ
)
|
0
⊂
N
θ
(
E
,
A
,
Λ
).
Next, suppose thatlim
⁡
inf
⁡
r
q
r
=
1. Since θ is lacunary we can select a subsequence k
r
j of θ such that
(40)
k
r
j
k
r
j
-
1
<
1
+
1
j
,
k
r
j
-
1
k
r
j
-
1
>
j
,
where r
j
≥
r
j
-
1
+
2. Define x
=
(
x
i
) by
(41)
Λ
i
(
x
)
=
{
e
i
,
if
i
∈
I
r
j
,
for
some
j
=
1,2
,
…
,
θ
,
otherwise
,
where ∥
e
i
∥
=
1 and let A
=
I, and then for any L
=
(
L
1
,
L
2
,
…
)
∈
E, e
i
∈
C,
(42)
1
h
r
j
∑
i
∈
I
r
‍
(
∥
A
i
(
Λ
i
(
x
)
-
L
)
∥
ρ
)
=
∥
e
i
-
L
i
e
i
∥
ρ
=
∥
1
-
L
i
∥
ρ
for
j
=
1,2
,
…
,
1
h
r
∑
i
∈
I
r
‍
(
∥
A
i
(
Λ
i
(
x
)
)
∥
ρ
)
=
∥
e
i
∥
ρ
=
1
ρ
.
So, x
∉
N
θ
(
E
,
A
,
Λ
). But x is strongly Cesàro-summable, since if t is sufficiently large integer we can find the unique j for which k
r
j
-
1
<
t
≤
k
r
j
+
1
-
1 and hence
(43)
1
t
∑
i
=
1
t
‍
(
∥
A
i
(
Λ
i
(
x
)
)
∥
)
<
1
k
r
j
-
1
∑
i
=
1
t
‍
1
≤
1
k
r
j
-
1
k
r
j
≤
k
r
j
-
1
+
h
r
j
k
r
j
-
1
<
1
j
+
1
j
=
2
j
,
as
t
⟶
∞
,
and it follows that also j
→
∞. Hence x
∈
|
σ
1
(
E
,
A
,
Λ
)
|
0.Lemma 7.
N
θ
(
E
,
A
,
Λ
)
⊂
|
σ
1
(
E
,
A
,
Λ
)
| if and only if lim
⁡
sup
⁡
r
q
r
<
∞.Proof.
First suppose that iflim
⁡
sup
⁡
r
q
r
<
∞, there exists M
>
0 such that q
r
<
M for all r
≥
1. Let x
∈
N
θ
(
E
,
A
,
Λ
) and ϵ
>
0. Then
(44)
lim
⁡
r
→
∞
1
h
r
∑
i
∈
I
r
‍
(
∥
A
i
(
Λ
i
(
x
)
)
∥
ρ
)
=
0
for
some
ρ
>
0
.
Then we can find R
>
0 and K
>
0 such that
(45)
sup
⁡
j
≥
R
1
h
j
∑
I
j
‍
(
∥
A
i
(
Λ
i
(
x
)
)
∥
ρ
)
<
ϵ
,
1
h
j
∑
I
j
‍
(
∥
A
i
(
Λ
i
(
x
)
)
∥
ρ
)
<
K
∀
i
=
1,2
,
…
Then if t is any integer with
(46)
k
r
-
1
≤
t
≤
k
r
,
where
r
>
R
,
then
(47)
1
t
∑
j
=
1
t
‍
(
∥
A
i
(
Λ
i
(
x
)
)
∥
ρ
)
≤
1
k
r
-
1
∑
i
=
1
k
r
‍
(
∥
A
i
(
Λ
i
(
x
)
)
∥
ρ
)
=
1
k
r
-
1
(
∑
I
1
‍
(
∥
A
i
(
Λ
i
(
x
)
)
∥
ρ
)
+
∑
I
2
‍
(
∥
A
i
(
Λ
i
(
x
)
)
∥
ρ
)
+
⋯
+
∑
I
r
‍
(
∥
A
i
(
Λ
i
(
x
)
)
∥
ρ
)
)
=
k
1
k
r
-
1
1
h
1
∑
I
1
‍
(
∥
A
i
(
Λ
i
(
x
)
)
∥
ρ
)
+
k
2
-
k
1
k
r
-
1
1
h
2
∑
I
2
‍
(
∥
A
i
(
Λ
i
(
x
)
)
∥
ρ
)
+
⋯
+
k
R
-
k
R
-
1
k
r
-
1
1
h
R
∑
I
R
‍
(
∥
A
i
(
Λ
i
(
x
)
)
∥
ρ
)
+
k
R
+
1
-
k
R
k
r
-
1
1
h
R
+
1
∑
I
R
+
1
‍
(
∥
A
i
(
Λ
i
(
x
)
)
∥
ρ
)
+
⋯
+
k
r
-
k
r
-
1
k
r
-
1
1
h
r
∑
I
r
‍
(
∥
A
i
(
Λ
i
(
x
)
)
∥
ρ
)
≤
k
R
k
r
-
1
sup
⁡
i
≥
1
1
h
i
∑
I
i
‍
(
∥
A
i
(
Λ
i
(
x
)
)
∥
ρ
)
+
k
r
-
k
R
k
r
-
1
1
h
r
∑
I
r
‍
(
∥
A
i
(
Λ
i
(
x
)
)
∥
ρ
)
<
K
k
R
k
r
-
1
+
ϵ
(
q
r
-
k
R
k
r
-
1
)
<
K
k
R
k
r
-
1
+
ϵ
q
r
<
K
k
R
k
r
-
1
+
ϵ
M
.
Since k
r
-
1
→
∞ as r
→
∞, it follows that
(48)
1
t
∑
j
=
1
t
‍
(
∥
A
i
(
Λ
i
(
x
)
)
∥
ρ
)
⟶
0
and hence x
∈
|
σ
1
(
E
,
A
,
Λ
)
|.
Next, suppose thatlim
⁡
sup
⁡
r
q
r
=
∞. We construct a sequence in N
θ
(
E
,
A
,
Λ
) that is not Cesaro Λ-summable. By the idea of Freedman et al. [12] we can construct a subsequence k
r
j of the lacunary sequence θ
=
(
k
r
) such that q
r
j
>
j, and then define a bounded difference sequence x by
(49)
Λ
i
(
x
)
=
{
e
i
,
if
k
r
j
-
1
<
i
<
2
k
r
j
-
1
,
θ
,
otherwise
,
where ∥
e
i
∥
=
1. Let A
=
I and ρ
=
1. Then,
(50)
1
h
r
j
∑
I
r
j
‍
(
∥
A
i
(
Λ
i
(
x
)
)
∥
)
=
2
k
r
j
-
1
-
k
r
j
-
1
k
r
j
-
k
r
j
-
1
=
k
r
j
-
1
k
r
j
-
k
r
j
-
1
<
1
j
-
1
and if r
≠
r
j,
(51)
1
h
r
j
∑
I
r
j
‍
(
∥
A
i
(
Λ
i
(
x
)
)
∥
)
=
0
.
Thus x
∈
N
θ
(
E
,
A
,
Λ
). For the above sequence and for i
=
1,2
,
…
,
k
r
j
(52)
1
k
r
j
∑
i
‍
(
∥
A
i
(
Λ
i
(
x
)
-
e
i
)
∥
)
>
1
k
r
j
(
2
k
r
j
-
1
-
k
r
j
-
1
)
=
1
-
2
q
r
j
>
1
-
2
j
,
this converges to 1, but for i
=
1,2
,
…
,
2
k
r
j
-
1,
(53)
2
k
r
j
-
1
∑
i
‍
(
∥
A
i
(
Λ
i
(
x
)
)
∥
)
≥
k
r
j
-
1
2
k
r
j
-
1
=
1
2
.
It proves that x
∉
|
σ
1
(
E
,
A
,
Λ
)
|, since any sequence in |
σ
1
(
E
,
A
,
Λ
)
| consisting of θ’s and e
i’s has a limit only 0 or 1.Theorem 8.
Letθ be a lacunary sequence. Then |
σ
1
(
E
,
A
,
Λ
)
|
=
N
θ
(
E
,
A
,
Λ
) if and only if 1
≤
lim
⁡
inf
⁡
r
q
r
≤
lim
⁡
sup
⁡
r
q
r
<
∞.Proof .
The proof of the theorem follows from Lemmas6 and 7.


## 5. Statistical Convergence

The notion of statistical convergence was introduced by Fast [16] and Schoenberg [17] independently. Over the years and under different names, statistical convergence has been discussed in the theory of Fourier analysis, ergodic theory, and number theory. Later on, it was further investigated from the sequence space point of view and linked with summability theory by Fridy [18], Connor [19], Šalát [20], Mursaleen and Edely [21], Isk [22], Mohiuddine and Alghamdi [23], Hazarika et al. [24], Kolk [25], Maddox [26], Alotaibi and Mursaleen [27], Mohiuddine et al. [28], Mohiuddine and Aiyub [29], and many others. In recent years, generalizations of statistical convergence have appeared in the study of strong integral summability and the structure of ideals of bounded continuous functions on locally compact spaces. Statistical convergence and its generalizations are also connected with subsets of the Stone-C
˘ech compactification of natural numbers. Moreover, statistical convergence is closely related to the concept of convergence in probability. The notion depends on the density of subsets of the set N of natural numbers.A subsetE of N is said to have the natural density δ
(
E
) if the following limit exists:
(54)
δ
(
E
)
=
lim
⁡
n
→
∞
1
n
∑
k
=
1
n
‍
χ
E
(
k
)
,
where χ
E is the characteristic function of E. It is clear that any finite subset of N has zero natural density and δ
(
E
c
)
=
1
-
δ
(
E
).A sequencex
=
(
x
k
) is said to be statistically convergent to the number L if for every ϵ
>
0
(55)
lim
⁡
n
→
∞
1
n
|
{
k
≤
n
:
|
x
k
-
L
|
≥
ϵ
}
|
=
0
.
Bilgin [14] also introduced the concept of statistical convergence in N
0
(
A
,
F
) and proved some inclusion relation.Letθ be a lacunary sequence and let A
=
(
a
i
k
) be an infinite matrix of complex numbers. Then a sequence x
∈
N
θ
(
E
,
A
,
Λ
,
M
) is said to be Λ-lacunary A-statistically convergent to a number s
=
(
s
1
,
s
2
,
…
)
∈
E, e
i
∈
C if for any ϵ
>
0,
(56)
lim
⁡
r
→
∞
1
h
r
|
Λ
k
A
0
(
ϵ
)
|
=
0
,
where
(57)
Λ
A
0
(
ϵ
)
=
{
i
∈
I
r
:
M
i
(
∥
A
i
(
Λ
k
(
x
)
-
s
i
e
i
)
∥
ρ
(
i
)
)
≥
ϵ
}
.
We denote it as x
→
Λ
-stat
s. The vertical bar denotes the cardinality of the set. The set of all Λ-lacunary A-statistical convergent sequences is denoted by S
θ
(
E
,
A
,
Λ
,
M
).In this section we study some relation between the spacesS
θ
(
E
,
A
,
Λ
,
M
) and N
θ
(
E
,
A
,
Λ
,
M
).Theorem 9.
LetM
=
(
M
i
) be a Musielak-Orlicz function and let (
M
i
) be pointwise convergent. Then N
θ
(
E
,
A
,
Λ
,
M
)
⊂
S
θ
(
E
,
A
,
Λ
,
M
) if and only if lim
⁡
i
M
i
(
v
/
ρ
(
i
)
)
>
0 for some v
>
0
,
ρ
(
i
)
>
0.Proof .
Letϵ
>
0 and x
∈
N
θ
(
E
,
A
,
Λ
,
M
). Let x
→
Λ
s, where s
=
(
s
1
,
s
2
,
…
)
∈
E,  e
i
∈
C. Since lim
⁡
i
M
i
(
v
/
ρ
)
>
0, there exists a number c
>
0 such that
(58)
M
i
(
v
ρ
)
≥
c
for
v
>
ϵ
.
Let
(59)
I
r
1
=
{
i
∈
I
r
:
[
M
i
(
∥
A
i
(
Λ
i
(
x
)
-
s
i
e
i
)
∥
ρ
(
i
)
)
]
≥
ϵ
}
.
Then
(60)
1
h
r
∑
i
∈
I
r
‍
[
M
i
(
∥
A
i
(
Λ
i
(
x
)
-
s
i
e
i
)
∥
ρ
(
i
)
)
]
≥
1
h
r
∑
i
∈
I
r
1
‍
[
M
i
(
∥
A
i
(
Λ
i
(
x
)
-
s
i
e
i
)
∥
ρ
(
i
)
)
]
≥
c
1
h
r
|
Λ
k
A
0
(
ϵ
)
|
.
Hence it follows that x
∈
S
θ
(
E
,
A
,
Λ
,
M
).
Conversely, let us assume that the condition does not hold good. Then there is a numberv
>
0 such that lim
⁡
i
M
i
(
v
/
ρ
)
=
0 for some ρ
>
0. Now, we select a lacunary sequence θ
=
(
k
r
) such that M
i
(
v
/
ρ
)
<
2
-
r for any i
>
k
r.
LetA
=
I, and define the sequence x by putting
(61)
Λ
i
(
x
)
=
{
v
,
if
k
r
-
1
<
i
≤
k
r
+
k
r
-
1
2
,
θ
,
if
k
r
+
k
r
-
1
2
<
i
≤
k
r
.
Therefore,
(62)
1
h
r
∑
i
∈
I
r
‍
M
i
(
∥
Λ
i
(
x
)
∥
ρ
(
i
)
)
=
1
h
r
∑
k
r
-
1
<
i
≤
(
k
r
+
k
r
-
1
)
/
2
‍
M
i
(
v
ρ
(
i
)
)
<
1
h
r
1
2
r
-
1
[
k
r
+
k
r
-
1
2
-
k
r
-
1
]
=
1
2
r
⟶
0
as
r
⟶
∞
.
Thus, we have x
∈
N
θ
0
(
E
,
A
,
Λ
,
M
). But
(63)
lim
⁡
r
→
∞
1
h
r
|
{
i
∈
I
r
:
M
i
(
∥
Λ
i
(
x
)
∥
ρ
(
i
)
)
≥
ϵ
}
|
=
lim
⁡
r
→
∞
1
h
r
|
{
i
∈
(
k
r
-
1
,
k
r
+
k
r
-
1
2
)
:
M
i
(
v
ρ
(
i
)
)
≥
ϵ
}
|
=
lim
⁡
r
→
∞
1
h
r
k
r
-
k
r
-
1
2
=
1
2
.
So x
∉
S
θ
(
E
,
A
,
Λ
,
M
).Theorem 10.
LetM
=
(
M
i
) be a Musielak-Orlicz function. Then S
θ
(
E
,
A
,
Λ
,
M
)
⊂
N
θ
(
E
,
A
,
Λ
,
M
) if and only if sup
⁡
v
sup
⁡
i
M
i
(
v
/
ρ
)
<
∞.Proof .
Letx
∈
S
θ
(
E
,
A
,
Λ
,
M
) and x
→
Λ
-stat
s. Suppose h
(
v
)
=
sup
⁡
i
M
i
(
v
/
ρ
) and h
=
sup
⁡
v
h
(
v
). Let
(64)
I
r
2
=
{
i
∈
I
r
:
M
i
(
∥
A
i
(
Λ
i
(
x
)
-
s
i
e
i
)
∥
ρ
(
i
)
)
<
ϵ
}
.
Now, M
i
(
v
)
≤
h for all i, v
>
0. So
(65)
1
h
r
∑
i
∈
I
r
‍
[
M
i
(
∥
A
i
(
Λ
i
(
x
)
-
s
i
e
i
)
∥
ρ
(
i
)
)
]
=
1
h
r
∑
i
∈
I
r
1
‍
[
M
i
(
∥
A
i
(
Λ
i
(
x
)
-
s
i
e
i
)
∥
ρ
(
i
)
)
]
+
1
h
r
∑
i
∈
I
r
2
‍
[
M
i
(
∥
A
i
(
Λ
i
(
x
)
-
s
i
e
i
)
∥
ρ
(
i
)
)
]
≤
h
1
h
r
|
Λ
k
A
0
(
ϵ
)
|
+
h
(
ϵ
)
.
Hence, as ϵ
→
0, it follows that x
∈
N
θ
(
E
,
A
,
Λ
,
M
).
Conversely, suppose that(66)
sup
⁡
v
sup
⁡
i
M
i
(
v
ρ
)
=
∞
.
Then we have
(67)
0
<
v
1
<
v
2
<
⋯
<
v
r
-
1
<
v
r
<
⋯
,
so that M
k
r
(
v
r
/
ρ
)
≥
h
r for r
≥
1. Let A
=
I. We set a sequence x
=
(
x
i
) by
(68)
Λ
i
(
x
)
=
{
v
r
,
if
i
=
k
r
for
some
r
=
1,2
,
…
,
θ
,
otherwise
.
Then
(69)
lim
⁡
r
→
∞
1
h
r
|
{
i
∈
I
r
:
[
M
i
(
∥
Λ
i
(
x
)
∥
ρ
(
i
)
)
]
≥
ϵ
}
|
=
lim
⁡
r
→
∞
1
h
r
=
0
.
Hence x
→
Λ
-stat
0 and hence x
∈
S
θ
(
E
,
A
,
Λ
,
M
).
But(70)
lim
⁡
r
→
∞
1
h
r
∑
i
∈
I
r
‍
[
M
i
(
∥
A
i
(
Λ
i
(
x
)
-
s
i
e
i
)
∥
ρ
(
i
)
)
]
=
lim
⁡
r
→
∞
1
h
r
[
M
k
r
(
∥
v
r
-
s
i
e
i
∥
ρ
(
i
)
)
]
≥
lim
⁡
r
→
∞
1
h
r
h
r
=
1
.
So, x
∉
N
θ
(
E
,
A
,
Λ
,
M
).



---

*Source: 858504-2014-07-24.xml*