Acta Mathematica Academiae Paedagogicae Nyı́regyháziensis
28 (2012), 103–120
www.emis.de/journals
ISSN 1786-0091
SOME NEW LACUNARY STRONG CONVERGENT
VECTOR-VALUED MULTIPLIER DIFFERENCE SEQUENCE
SPACES DEFINED BY A MUSIELAK-ORLICZ FUNCTION
KULDIP RAJ AND SUNIL K. SHARMA
Abstract. In the present paper we introduce some vector-valued multiplier difference sequence spaces defined by a Musielak-Orlicz function, concepts of lacunary convergence and strong (A, u)-convergence, where A =
(aik ) is an infinite matrix of complex numbers and u = (ui ) be any sequence
of strictly positive real numbers. We also make an effort to study some
topological properties and some inclusion relations between these spaces.
1. Introduction and Preliminaries
The notion of difference sequence spaces was introduced by Kızmaz [9], who
studied the difference sequence spaces l∞ (∆), c(∆) and c0 (∆). The notion was
further generalized by Et and Çolak [4] by introducing the spaces l∞ (∆m ),
c(∆m ) and c0 (∆m ). Let w be the space of all real or complex sequences x =
(xk ). Let m, n be non-negative integers, then for Z = l∞ , c and c0 , we have
sequence spaces,
m
Z(∆m
n ) = {x = (xk ) ∈ w : (∆n xk ) ∈ Z},
m
m−1
where ∆m
xk − ∆m−1
xk+n ) and ∆0n xk = xk for all k ∈ N,
n x = (∆n xk ) = (∆n
n
which is equivalent to the following binomial representation
m
X
m
v
m
(−1)
xk+nv .
∆ n xk =
v
v=0
Taking m = n = 1, we get the spaces l∞ (∆), c(∆) and c0 (∆) introduced and
studied by Kızmaz [9].
An Orlicz function M : [0, ∞) → [0, ∞) is convex and continuous such that
M (0) = 0, M (x) > 0 for x > 0. Lindenstrauss and Tzafriri [11] used the idea
2010 Mathematics Subject Classification. 40A05, 40C05 40D25.
Key words and phrases. Lacunary Sequence, paranorm, Orlicz function, Musielak-Orlicz
function.
103
104
KULDIP RAJ AND SUNIL K. SHARMA
of Orlicz function to define the following sequence space,
(
)
∞
X
|xk |
`M = x ∈ w :
M
<∞
ρ
k=1
which is called as an Orlicz sequence space. The space `M is a Banach space
with the norm
(
)
∞
X
|xk |
||x|| = inf ρ > 0 :
M
≤1 .
ρ
k=1
It is shown in [11] that every Orlicz sequence space `M contains a subspace
isomorphic to `p (p ≥ 1). An Orlicz function M satisfies ∆2 -condition if and
only if for any constant L > 1 there exists a constant K(L) such that M (Lu) ≤
K(L)M (u) for all values of u ≥ 0. An Orlicz function M can always be
represented in the following integral form
Z x
M (x) =
η(t)dt,
0
where η is known as the kernel of M , is right differentiable for t ≥ 0, η(0) =
0, η(t) > 0, η is non-decreasing and η(t) → ∞ as t → ∞.
A sequence M = (Mk ) of Orlicz function is called a Musielak-Orlicz function
(see [13, 16]). A sequence N = (Nk ) defined by
Nk (v) = sup{|v|u − Mk (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 tM
and its subspace hM are defined as follows
tM = {x ∈ w : IM (cx) < ∞ for some c > 0} ,
where IM
hM = {x ∈ w : IM (cx) < ∞ for all c > 0} ,
is a convex modular defined by
∞
X
Mk (xk ), x = (xk ) ∈ tM .
IM (x) =
k=1
We consider tM equipped with the Luxemburg norm
n
x
o
||x|| = inf k > 0 : IM
≤1
k
or equipped with the Orlicz norm
1
0
||x|| = inf
(1 + IM (kx)) : k > 0 .
k
A Musielak-Orlicz function (Mk ) is said to satisfy ∆2 -condition if there exist
1
1
constants a, K > 0 and a sequence c = (ck )∞
k=1 ∈ `+ (the positive cone of ` )
such that the inequality
Mk (2u) ≤ KMk (u) + ck
SOME NEW LACUNARY STRONG CONVERGENT MULTIPLIER . . .
105
holds for all k ∈ N and u ∈ R+ , whenever Mk (u) ≤ a.
Let X 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 (σn ) is a sequence of scalars with σn → σ as n → ∞ and (xn ) is a
sequence of vectors with p(xn −x) → 0 as n → ∞, then p(σn xn −σx) →
0 as n → ∞.
A paranorm p 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 [25, Theorem
10.4.2]). For more details about sequence spaces (see [14, 17, 18, 20, 19]) and
references therein.
The space of lacunary strong convergence have been introduced by Freedman
et al. [6]. A sequence of positive integers θ = (kr ) is called lacunary if k0 =
0, 0 < kr < kr+1 and hr = kr −kr−1 → ∞, as r → ∞. The intervals determined
kr
by θ are denoted by Ir = (kr−1 , kr ] and the ratio kr−1
will be denoted by qr .
the space of lacunary strongly convergent sequences Nθ is defined by Freedman
et al. [6] as follows:
)
(
1 X
|xi − s| = 0, for some s .
Nθ = x = (xi ) : lim
r→∞ hr
i∈I
r
The space |σ1 | of strongly Cesaro summable sequences is
)
(
n
1X
|xi − L| → 0, as n → ∞ .
|σ1 | = x = (xk ) : there exists L such that
n i=1
In case, when θ = (2r ), Nθ = |σ1 |. Recently, Bilgin [1] in his paper generalized
the concept of lacunary convergence and introduced the space Nθ (A, f ), as
(
)
1 X
Nθ (A, f ) = x = (xi ) : lim
f (|Ai (x) − s|) = 0, for some s ,
r→∞ hr
i∈Ir
P∞
where f is a modulus function and A = (Ai (x)), Ai x =
k=1 aik xk converges for each i. Later Bilgin [2] generalized lacunary strongly A-convergent
sequences with respect to a sequence of modulus function F = (fi ) as follows:
(
)
X
1
fi (|Ai (x) − s|) = 0, for some s .
Nθ (A, F ) = x = (xi ) : lim
r→∞ hr
i∈I
r
By Ai (∆m
n xk ) we mean
Ai (∆m
n xk ) =
∞
X
k=1
xk+1 ).
xk − ∆m−1
aik (∆m−1
n
n
106
KULDIP RAJ AND SUNIL K. SHARMA
The main aim of this paper is to introduce the concept of lacunary strongly
(A, u)-convergence for difference sequences with the elements chosen from a
Banach space (E, k.k) over the complex field C, with respect to a MusielakOrlicz function M = (Mi ). We have studied some topological properties and
also make an effort to study some inclusion relations between below defined
sequence spaces.
Let A = (aik ) be an infinite matrix of complex numbers, M = (Mi ) be a
Musielak-Orlicz function and u = (ui ) be any sequence of strictly positive real
numbers. In the present paper we define the following sequence spaces:
∆m
n Nθ (E, A, u, M)
(
=
1 X
kAi (∆m
n xk ) − si ei k
x = (xk ) ∈ E : lim
ui M i
=0
(i)
r→∞ hr
ρ
i∈I
r
)
for some s = (s1 , s2 , . . .) ∈ E, ei ∈ C and ρ(i) > 0
and
0
∆m
n Nθ (E, A, u, M)
(
=
1 X
kAi (∆m
n xk )k
x = (xk ) ∈ E : lim
ui Mi
=0
(i)
r→∞ hr
ρ
i∈I
)
r
for some ρ
(i)
>0
If we take M(x) = x, we have
(
∆m
n Nθ (E, A, u)
=
1 X
kAi (∆m
n xk ) − si ei k
x = (xk ) ∈ E : lim
ui
=0
r→∞ hr
ρ(i)
i∈Ir
)
for some s = (s1 , s2 , . . .) ∈ E, ei ∈ C and ρ(i) > 0 .
and
0
∆m
n Nθ (E, A, u)
(
=
1 X
kAi (∆m
n xk )k
x = (xk ) ∈ E : lim
ui
=0
(i)
r→∞ hr
ρ
i∈I
r
)
for some ρ(i) > 0 ,
where
(
1,
ei =
0,
at the ith place,
otherwise.
SOME NEW LACUNARY STRONG CONVERGENT MULTIPLIER . . .
107
2. Some topological properties of the spaces ∆m
n Nθ (E, A, u, M)
m 0
and ∆n Nθ (E, A, u, M)
Theorem 2.1. Let A = (aik ) be an infinite matrix of complex numbers, M =
(Mi ) be a Musielak-Orlicz function and u = (ui ) be any sequence of strictly
m 0
positive real numbers. Then ∆m
n Nθ (E, A, u, M) and ∆n Nθ (E, A, u, M) are
linear spaces over the field of complex number C.
∆m
n
Proof. Suppose that x = (xk ), y = (yk ) ∈ ∆m
n Nθ (E, A, u, M) and (xk ) −→
∆m
n
s, (yk ) −→
t, then for some s = (s1 , s2 , . . .), t = (t1 , t2 , . . .) ∈ E, ei ∈ C, we
have
"
1 X
lim
ui Mi
r→∞ hr
i∈I
kAi (∆m
n xk ) − si ei k
"
1 X
lim
ui Mi
r→∞ hr
i∈I
kAi (∆m
n yk ) − ti ei k
r
!#
(i)
ρ1
(i)
= 0, for some ρ1 > 0
and
r
(i)
ρ2
!#
(i)
= 0, for some ρ2 > 0.
Let α, β ∈ C. Without loss of generality we may assume that there exists
(i)
(i)
P1 > 1, P2 > 1 such that |α| ≤ P1 and |β| ≤ P2 . Let ρ(i) = max(2ρ1 , 2ρ2 ).
Then
m
1 X
kAi (α∆m
n xk + β∆n yk ) − (αsi ei + βti ei k)
lim
ui Mi
r→∞ hr
ρ(i)
i∈Ir
m
1 X
kαAi (∆m
n xk ) − αsi ei k + kβAi (∆n yk ) − βti ei k
≤ lim
ui M i
r→∞ hr
ρ(i)
i∈Ir
"
!#
|α|kAi (∆m
x
)
−
s
e
k
1 X1
k
i
i
n
ui Mi
= lim
(i)
r→∞ hr
2
ρ1
i∈Ir
"
!#
y
−
t
e
)k
|β|kAi (∆m
1 X1
i i
n k
ui Mi
+ lim
(i)
r→∞ hr
2
ρ2
i∈Ir
"
!#
1 X1
P1 kAi (∆m
n xk ) − si ei k
≤ lim
ui Mi
(i)
r→∞ hr
2
ρ1
i∈Ir
"
!#
y
)
−
t
e
k
P2 kAi (∆m
1 X1
i i
n k
+ lim
ui Mi
(i)
r→∞ hr
2
ρ2
i∈I
r
108
KULDIP RAJ AND SUNIL K. SHARMA
!#
"
1 X1
kAi (∆m
x
)
−
s
e
k
i i
n k
≤K1 lim
ui Mi
(i)
r→∞ hr
2
ρ1
i∈Ir
!#
"
m
X
1
1
kAi (∆n yk ) − ti ei k
+ K2 lim
ui Mi
(i)
r→∞ hr
2
ρ2
i∈I
r
→ 0 as r → ∞.
Therefore, (αxk + βyk ) ∈ ∆m
n Nθ (E, A, u, M).
m
This proves that ∆n Nθ (E, A, u, M) is a linear space. Similarly we can prove
0
that ∆m
n Nθ (E, A, u, M) is a linear space.
Theorem 2.2. Let A = (aik ) be an infinite matrix of complex numbers, M =
(Mi ) be a Musielak-Orlicz function and u = (ui ) be any sequence of strictly
m 0
positive real numbers. Then ∆m
n Nθ (E, A, u, M) and ∆n Nθ (E, A, u, M) are
normal spaces, when E is normal.
∆m
n
Proof. Let x = (xk ) ∈ ∆m
n Nθ (E, A, u, M) and (xk ) −→ s, where s = (s1 , s2 , . . .)
∈ E, ei ∈ C. Let kyk k ≤ kxk k. Then
m
kAi (∆m
n yk ) − si ei k ≤ kAi (∆n xk ) − si ei k.
Since, M = (Mi ) is an increasing,
1 X
1 X
kAi (∆m
kAi (∆m
n yk ) − si ei k
n xk ) − si ei k
ui Mi
≤
ui M i
hr i∈I
ρ(i)
hr i∈I
ρ(i)
r
r
Consequently, y = (yk ) ∈ ∆m
n Nθ (E, A, u, M). This completes the proof of the
0
theorem. Similarly, we can prove that ∆m
n Nθ (E, A, u, M) is normal space. m 0
Theorem 2.3. The spaces ∆m
n Nθ (E, A, u, M) and ∆n Nθ (E, A, u, M) are
paranormed spaces, with respect to the paranorm
kai0 x1 k
(i)
+
kxk∆m
= inf ρ > 0 : ui Mi
n
ρ(i)
)
1 X
kAi (∆m
n xk )k
(i)
+ sup
ui Mi
≤ 1, ρ ≥ 0 .
ρ(i)
r≥1 hr
i∈I
r
It is easy to prove, so we omit the details.
3. Relation between the spaces ∆m
n Nθ (E, A, u) and
N
(E,
A,
u,
M)
∆m
θ
n
The main purpose of this section is to study some interesting relations bem
tween the spaces ∆m
n Nθ (E, A, u) and ∆n Nθ (E, A, u, M).
Theorem 3.1. Let A = (aik ) be an infinite matrix of complex numbers, M =
(Mi ) be a Musielak-Orlicz function satisfying ∆2 -condition and u = (ui ) be any
sequence of strictly positive real numbers. If x = (xk ) is ∆m
n -lacunary strong
SOME NEW LACUNARY STRONG CONVERGENT MULTIPLIER . . .
109
(A, u)-convergent to s, with respect to M and (E, k.k) is a normal Banach
m
space, then ∆m
n Nθ (E, A, u) ⊂ ∆n Nθ (E, A, u, M).
∆m
n
Proof. Let x = (xk ) ∈ ∆m
n Nθ (E, A, u) and (xk ) −→ s, where s = (s1 , s2 , . . .) ∈
E, ei ∈ C. Then
1 X
kAi (∆m
n xk ) − si ei k
lim
= 0 for some ρ > 0.
ui
r→∞ hr
ρ
i∈I
r
We define two sequences y = (yk ) and z = (zk ) such that
ui (kAi (∆m
n yk ) − si ei k)
(
m
ui (kAi (∆m
n xk ) − si ei k), if ui (kAi (∆n xk ) − si ei k) > 1,
=
0,
if ui (kAi (∆m
n xk ) − si ei k) ≤ 1,
and
ui (kAi (∆m
n zk ) − si ei k)
(
0,
if ui (kAi (∆m
n xk ) − si ei k) > 1,
=
m
ui (kAi (∆n xk ) − si ei k), if ui (kAi (∆m
n xk ) − si ei k) ≤ 1.
Hence
m
m
ui (kAi (∆m
n xk ) − si ei k) = ui (kAi (∆n yk ) − si ei k) + ui (kAi (∆n zk ) − si ei k).
Now,
m
ui (kAi (∆m
n yk ) − si ei k) ≤ ui (kAi (∆n xk ) − si ei k)
and
m
ui (kAi (∆m
n zk ) − si ei k) ≤ ui (kAi (∆n xk ) − si ei k).
110
KULDIP RAJ AND SUNIL K. SHARMA
m
Since ∆m
n Nθ (E, A, u) is normal, y = (yk ), z = (zk ) ∈ ∆n Nθ (E, A, u). Let
supi Mi (2) = T . Then
kAi (∆m
1 X
n xk ) − si ei k
ui Mi
hr i∈I
ρ(i)
r
m
1 X
kAi (∆m
n yk ) − si ei k + kAi (∆n zk ) − si ei k
=
ui M i
hr i∈I
ρ(i)
r
kAi (∆m
1
kAi (∆m
1 X
1
n yk ) − si ei k
n zk ) − s i e i k
+ Mi 2
≤
ui Mi 2
hr i∈I
2
ρ(i)
2
ρ(i)
r
1 1 X
kAi (∆m
n yk ) − si ei k
<
ui K 1
Mi (2)
2 hr i∈I
ρ(i)
r
1 1 X
kAi (∆m
n zk ) − si ei k
+
ui K 2
Mi (2)
2 hr i∈I
ρ(i)
r
1 1 X
kAi (∆m
n yk ) − si ei k
≤
ui K 1
sup Mi (2)
2 hr i∈I
ρ(i)
r
1 1 X
kAi (∆m
n zk ) − si ei k
+
ui K 2
sup Mi (2)
2 hr i∈I
ρ(i)
r
→0 as r → ∞.
Hence x = (xk ) ∈ ∆m
n Nθ (E, A, u, M). This completes the proof of the theorem.
Theorem 3.2. Let A = (aik ) be an infinite matrix of complex numbers, M =
(Mi ) be a Musielak-Orlicz function satisfying ∆2 -condition and u = (ui ) be
any sequence of strictly positive real numbers. If
lim inf
v
ui Mi ( ρ(i)
)
v
v→∞ i
> 0 for some ρ(i) > 0,
ρ(i)
m
then ∆m
n Nθ (E, A, u) = ∆n Nθ (E, A, u, M).
Proof. If limv→∞ inf i
ui Mi (
v
)
ρ(i)
v
ρ(i)
> 0 for some ρ(i) > 0, then there exists a number
γ > 0 such that
ui Mi (
v
v
) ≥ ui γ( (i) ); for all v > 0 and some ρ(i) > 0.
(i)
ρ
ρ
SOME NEW LACUNARY STRONG CONVERGENT MULTIPLIER . . .
111
∆m
n
Let x = (xk ) ∈ ∆m
n Nθ (E, A, u, M) and (xk ) −→ s, where s = (s1 , s2 , . . .) ∈
E, ei ∈ C. Then clearly
1 X
1 X
kAi (∆m
kAi (∆m
n xk ) − si ei k
n xk ) − si ei k
ui Mi
≥
ui γ
hr i∈I
ρ(i)
hr i∈I
ρ(i)
r
r
1 X
kAi (∆m
n xk ) − s i ei k
=γ
ui
hr i∈I
ρ(i)
r
Hence x = (xk ) ∈ ∆m
n Nθ (E, A, u). This completes the proof.
4. Some attractive inclusions between the spaces |∆m
n σ1 (A, u)|
m
and ∆n Nθ (E, A, u)
A sequence x = (xk ) is said to be ∆m
n -lacunary strong (A, u)-convergent
with respect to a Musielak-Orlicz function M = (Mi ) if there is a number
s = (s1 , s2 , . . .) ∈ E such that x = (xk ) ∈ ∆m
n Nθ (E, A, u, M).
We have generalized the strongly Cesaro-summable sequence spaces into
∆m
n -strongly Cesaro-summable vector-valued sequence spaces as
(
|∆m
n σ1 (A, u)| =
x = (xk ) : there exists L = (L1 , L2 , . . .) ∈ E, ei ∈ C
)
n
X
1
ui kAi (∆m
such that
n xk ) − Li ei k → 0 ,
n i=1
where A = (ank ) is a Cesaro matrix, i.e.,
(
1
, if 1 ≤ k ≤ n,
n
ank =
0, if k ≥ n.
Then it can be shown that |∆m
n σ1 (A, u)| is a paranormed space with respect
to the paranorm
!
n
1X
kxk = kx1 k + sup
ui kAi (∆m
n xk )k .
n i=1
n
In this section of the paper we study relation between the spaces |∆m
n σ(A, u)|
N
(E,
A,
u).
and ∆m
θ
n
m
Lemma 4.1. |∆m
n σ1 (A, u)| ⊂ ∆n Nθ (E, A, u) if and only if lim inf r qr > 1.
112
KULDIP RAJ AND SUNIL K. SHARMA
Proof. First, suppose that lim inf r qr > 1. Then there exist δ > 0 such that
0
1 + δ ≤ qr for all r ≥ 1. Let x ∈ |∆m
n σ1 (A, u)| . Then
kr−1
kr
1 X
1 X
1 X
m
m
ui kAi (∆n xk )k =
ui kAi (∆n xk )k −
ui kAi (∆m
n xk )k
hr i∈I
hr i=1
hr i=1
r
!
kr
kr 1 X
=
ui kAi (∆m
n xk )k
hr kr i=1
!
kr−1
X
kr−1
1
−
ui kAi (∆m
n xk )k .
hr
kr−1 i=1
Now, hr = kr − kr−1 . So we have
kr
kr
qr
1
1
δ+1
=
=
=1+
≤1+ =
.
hr
kr − kr−1
qr − 1
qr − 1
δ
δ
Also
kr−1
kr−1
1
1
=
=
≤ .
hr
kr − kr−1
qr − 1
δ
0
Since x ∈ |∆m
n σ1 (A, u)| , then
kr−1
kr
1 X
1 X
m
ui kAi (∆n xk )k → 0 and
ui kAi (∆m
n xk )k → 0,
hr i=1
hr i=1
and hence
1 X
ui kAi (∆m
n xk )k → 0
hr i∈I
r
i.e., x = (xk ) ∈
By linearity, it follows that |∆m
n σ1 (A, u)| ⊂
m
∆n Nθ (E, A, u).
Next, suppose that lim inf r qr = 1. Since θ is lacunary we can select a
subsequence krj of θ such that
0
∆m
n Nθ (E, A, u).
kr −1
k rj
1
< 1 + and j > j
krj −1
j
krj−1
where rj ≥ rj−1 + 2. Define x = (xi ) by
(
ei , if i ∈ Irj , for some j = 1, 2, . . . ,
ui ∆ m
n xi =
0,
otherwise,
where kei k = 1 and let A = I, then for any L = (L1 , L2 , · · · ) ∈ E, ei ∈ C,
1 X
kAi (∆m
kei − Li ei k
k1 − Li k
n xk ) − Lk
ui
=
=
for j = 1, 2, . . .
(i)
(i)
hrj i∈I
ρ
ρ
ρ(i)
r
SOME NEW LACUNARY STRONG CONVERGENT MULTIPLIER . . .
113
and
1 X
kAi (∆m
kei k
1
n xk )k
ui
= (i) = (i) .
(i)
hr i∈I
ρ
ρ
ρ
r
So, x = (xk ) ∈
/ ∆m
n Nθ (E, A, u). But, x = (xk ) is strongly Cesaro-summable,
since if t is sufficiently large integer we can find the unique j for which krj −1 <
t ≤ krj+1 −1 and hence
t
t
kr −1 + hrj
1X
1 X
1
1 1
2
m
ui (kAi (∆n xk )k) <
1≤
k rj ≤ j
< + =
t i=1
krj −1 i=1
krj −1
krj −1
j j
j
as t → ∞, it follows that also j → ∞. Hence x = (xk ) ∈ |∆m
n σ1 (A, u)|.
m
Lemma 4.2. ∆m
n Nθ (E, A, u) ⊂ |∆n σ1 (A, u)| if and only if lim supr qr < ∞.
Proof. First, suppose that if lim supr qr < ∞, there exists M > 0 such that
qr < M for all r ≥ 1. Let x = (xk ) ∈ ∆m
n Nθ (E, A, u) and > 0. Then
kAi (∆m
1 X
n xk )k
ui
= 0 for some ρ > 0.
lim
r→∞ hr
ρ(i)
i∈I
r
Then we can find R > 0 and K > 0 such that
1 X
kAi (∆m
n xk )k
sup
ui
<
ρ(i)
j≥R hj
I
j
and
1 X
kAi (∆m
n xk )k
ui
< K for all i = 1, 2, . . . .
hj I
ρ(i)
j
Then if t is any integer with
kr−1 ≤ t ≤ kr , where r > R,
114
KULDIP RAJ AND SUNIL K. SHARMA
then
kAi (∆m
n xk )k
ρ(i)
kr
1 X
kAi (∆m
n xk )k
≤
ui
kr−1 i=1
ρ(i)
X kAi (∆m xk )k X kAi (∆m xk )k 1
n
n
ui
=
ui
+
+ ···
(i)
kr−1 I
ρ(i)
ρ
I2
1
!
m
X
kAi (∆n xk )k
+
ui
ρ(i)
Ir
k1 1 X
kAi (∆m
k2 − k1 1 X
kAi (∆m
n xk )k
n xk )k
=
ui
+
ui
+ ···
kr−1 h1 I
ρ(i)
kr−1 h2 I
ρ(i)
1
2
kAi (∆m
kR − kR−1 1 X
x
)k
n k
ui
+
(i)
kr−1 hR I
ρ
R
kAi (∆m
kR+1 − kR 1 X
n xk )k
ui
+
kr−1 hR+1 I
ρ(i)
R+1
kr − kr−1 1 X
kAi (∆m
n xk )k
+ ··· +
ui
kr−1 hr I
ρ(i)
1X
ui
t j=1
t
r
1 X
kR
kAi (∆m
kr − kR 1 X
kAi (∆m
n xk )k
n xk )k
≤
sup
ui
+
ui
kr−1 i≥1 hi I
ρ(i)
kr−1 hr I
ρ(i)
r
i
kR
kR
kR
kR
<K
+ qr −
<K
+ qr < K
+ M.
kr−1
kr−1
kr−1
kr−1
Since kr−1 → ∞ as r → ∞, it follows that
1 X kAi (∆m
n xk )k
ui
→0
t j=1
ρ(i)
t
and hence x = (xk ) ∈ |∆m
n σ1 (A, u)|.
Next, suppose that lim supr qr = ∞. Now, we construct a sequence in
m
∆m
n Nθ (E, A, u) that is not Cesaro ∆n -summable. By the idea of Freedman
[6] we can construct a subsequence krj of the lacunary sequence θ = (kr ) such
that qrj > j, and then define a bounded difference sequence x = (xi ) by
(
ei , if krj −1 < i < 2krj −1 ,
ui ∆ m
n xi =
0,
otherwise,
SOME NEW LACUNARY STRONG CONVERGENT MULTIPLIER . . .
115
where kei k = 1. Let A = I and ρ = 1. Then
2krj −1 − krj −1
krj −1
1 X
1
ui (kAi (∆m
=
<
n xk )k) =
hrj I
krj − krj −1
krj − krj −1
j−1
rj
and if r 6= rj ,
1 X
ui (kAi (∆m
n xk )k) = 0.
hrj I
rj
Thus x = (xk ) ∈
1, 2, . . . , krj
1 X
k rj
∆m
n Nθ (E, A, u).
For the above sequence and for i =
ui (kAi (∆m
n xk ) − ei k) >
i
1
(2krj −1 − krj −1 )
k rj
=1−
2
2
>1− ,
qr j
j
this converges to 1, but for i = 1, 2, . . . , 2krj −1
krj −1
2 X
1
ui (kAi (∆m
= .
n xk )k) ≥
krj−1 i
2krj −1
2
m
It proves that x = (xk ) ∈
/ |∆m
n σ1 (A, u)|, since any sequence in |∆n σ1 (A, u)|
consisting of 0’s and ei ’s has an limit only 0 or 1.
Theorem 4.3. Let θ be a lacunary sequence. Then
m
|∆m
n σ1 (A, u)| = ∆n Nθ (E, A, u)
if and only if
1 ≤ lim inf qr ≤ lim sup qr < ∞.
r
r
The proof of this theorem follows from Lemma 4.1 and Lemma 4.2.
5. Statistical Convergence
The notion of statistical convergence was introduced by Fast [5] and Schoenberg [23] 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 [7], Connor
[3], Salat [21], Mursaleen [15], Isık [8], Savaş [22], Malkosky and Savaş [14],
Kolk [10], Maddox [12], Tripathy and Sen [24] 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-Cech compactification of
natural numbers. Moreover, statistical convergence is closely related to the
116
KULDIP RAJ AND SUNIL K. SHARMA
concept of convergence in probability. The notion depends on the density of
subsets of the set N of natural numbers.
A subset E of N is said to have the natural density δ(E) if the following
limit exists:
n
1X
δ(E) = lim
χE (k),
n→∞ n
k=1
where χE 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 sequence x = (xk ) is said to be statistically convergent to the number L
if for every > 0
1 lim k ≤ n : |xk − L| ≥ = 0.
n→∞ n
Bilgin [2] also introduced the concept of statistical convergence in N0 (A, F )
and proved some inclusion relation.
Let θ be a lacunary sequence and A = (aik ) be an infinite matrix of complex
m
numbers. Then a sequence x = (xk ) ∈ ∆m
n Nθ (E, A, u, M) is said to be ∆n lacunary (A, u)-statistically convergent to a number s = (s1 , s2 , . . .) ∈ E, ei ∈
C if for any > 0,
1
ui |∆m
lim
n A0 ()| = 0,
r→∞ hr
where
kAi (∆m
n xk ) − si ei k
m
∆n A0 () = i ∈ Ir : ui Mi
≥ .
ρ(i)
∆m − stat
We denote it as (xk ) n −→
s. The vertical bar denotes the cardinality of
the set. The set of all ∆m
-lacunary
(A, u)-statistical convergent sequences is
n
m
denoted by ∆n Sθ (A, u).
In this section we study some relations between the spaces |∆m
n Sθ (A, u)| and
m
∆n Nθ (E, A, u, M).
Theorem 5.1. Let M = (Mi ) be a Musielak-Orlicz function and (Mi ) be
pointwise convergent. Then
m
∆m
n Nθ (E, A, u, M) ⊂ ∆n Sθ (A, u)
if and only if
lim ui Mi (
i
v
) > 0 for some v > 0, ρ(i) > 0.
(i)
ρ
Proof. Let > 0 and x = (xk ) ∈ ∆m
n Nθ (E, A, u, M).
∆m
n
v
s, where s = (s1 , s2 , . . .) ∈ E, ei ∈ C. Since limi ui Mi ( ρ(i)
Let (xk ) −→
) > 0,
there exists a number c > 0 such that
v
ui Mi ( (i) ) ≥ c for v > .
ρ
SOME NEW LACUNARY STRONG CONVERGENT MULTIPLIER . . .
Let
Ir1
=
117
kAi (∆m
n xk ) − si ei k
i ∈ I r : ui M i
≥ .
ρ(i)
Then
1 X
kAi (∆m
1 X
kAi (∆m
n xk ) − si ei k
n xk ) − si ei k
ui Mi
ui Mi
≥
hr i∈I
ρ(i)
hr 1
ρ(i)
i∈Ir
r
1 m
|∆ A0 ()|.
hr n
Hence, it follows that x = (xk ) ∈ ∆m
n Sθ (A, u).
Conversely, let us assume that the condition does not hold. Then there is a
v
number v > 0 such that limi ui Mi ( ρ(i)
) = 0 for some ρ > 0. Now, we select a
v
lacunary sequence θ = (kr ) such that ui Mi ( ρ(i)
) < 2−r for any i > kr .
Let A = I, define the sequence x = (xk ) by putting
(
v, if kr−1 < i ≤ kr +k2 r−1 ,
m
∆n xi =
0, if kr +k2 r−1 < i ≤ kr .
≥c
Therefore,
m 1 X
k∆n xi k
1
Mi
=
(i)
hr i∈I
ρ
hr
r
X
Mi
kr +kr−1
kr−1 <i≤
2
v
ρ(i)
1 1
kr + kr−1
<
− kr−1
hr 2r−1
2
1
= r → 0 as r → ∞.
2
m 0
Thus, we have x = (xk ) ∈ ∆n Nθ (E, A, u, M). But,
1 k∆xi k
=
lim
i
∈
I
:
M
≥
r
i
r→∞ hr ρ(i)
1 kr + kr−1
v
)
:
M
≥
= lim
i
∈
(k
,
i
r−1
r→∞ hr 2
ρ(i)
1 kr − kr−1
1
= lim
= .
r→∞ hr
2
2
m
So x = (xk ) ∈
/ ∆n Sθ (A, u).
Theorem 5.2. Let M = (Mi ) be a Musielak-Orlicz function and u = (ui ) be
any sequence of strictly positive real numbers. Then
m
∆m
n Sθ (A, u) ⊂ ∆n Nθ (E, A, u, M)
if and only if
sup sup ui Mi (
v
i
v
) < ∞.
ρ(i)
118
KULDIP RAJ AND SUNIL K. SHARMA
∆m
n − stat
−→
s. Suppose h(v) =
Proof. Let x = (xk ) ∈ ∆m
S
(A,
u)
and
(x
)
θ
k
n
v
supi ui Mi ( ρ(i)
) and h = supv h(v). Let
kAi (∆m
n xk ) − si ei k
2
Ir = i ∈ Ir : Mi
< .
ρ(i)
Now, Mi (v) ≤ h for all i, v > 0. So
1 X
kAi (∆m
n xk ) − si ei k
ui Mi
=
hr i∈I
ρ(i)
r
kAi (∆m
1 X
n xk ) − si ei k
=
ui Mi
hr 1
ρ(i)
i∈Ir
1 X
kAi (∆m
n xk ) − si ei k
+
ui Mi
hr 2
ρ(i)
i∈Ir
≤h
1 m
|∆ A0 ()| + h().
hr n
Hence, as → 0, it follows that x = (xk ) ∈ ∆m
n Nθ (E, A, u, M).
Conversely, suppose that
v
sup sup ui Mi ( (i) ) = ∞.
ρ
v
i
Then we have
0 < v1 < v2 < . . . < vr−1 < vr < . . . ,
r
so that Mkr ( ρv(i)
) ≥ hr for r ≥ 1. Let A = I. We set a sequence x = (xi ) by
(
vr , if i = kr for some r = 1, 2, . . . ,
ui ∆ m
n xi =
0,
otherwise.
Then
m 1 k∆n xi k
= lim 1 = 0.
lim
i
∈
I
:
u
M
≥
r
i
i
r→∞ hr
r→∞ hr ρ(i)
∆m − stat
Hence (xk ) n −→ 0 and hence x = (xk ) ∈ ∆m
n Sθ (A, u).
But,
1 X
kAi (∆m
1
vr − si ei k
n xi ) − si ei k
lim
ui Mi
= lim
M kr
(i)
r→∞ hr
r→∞ hr
ρ
ρ(i)
i∈I
r
1
hr = 1.
r→∞ hr
≥ lim
So, x = (xk ) ∈
/ ∆m
n Nθ (E, A, u, M).
SOME NEW LACUNARY STRONG CONVERGENT MULTIPLIER . . .
119
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120
KULDIP RAJ AND SUNIL K. SHARMA
Received December 27, 2011.
School of Mathematics,
Shri Mata Vaishno Devi University,
Katra-182320, India
E-mail address: kuldeepraj68@rediffmail.com
E-mail address: sunilksharma42@yahoo.co.in