An. Şt. Univ. Ovidius Constanţa
Vol. 20(1), 2012, 417–430
On some lacunary difference sequence spaces
defined by a sequence of Orlicz functions and
q-lacunary ∆nm -statistical convergence
Binod Chandra Tripathy and Hemen Dutta
Abstract
In this article, we introduce the lacunary difference sequence spaces
n
n
w0 (M , θ, ∆n
m , p, q), w1 (M , θ, ∆m , p, q) and w∞ (M , θ, ∆m , p, q) using a
sequence M = (Mk ) of Orlicz functions and investigate some relevant
properties of these spaces. Then, we define and study the notion of qlacunary ∆n
m -statistical convergent sequences. Further, we study the relationship between q-lacunary ∆n
m -statistical convergent sequences and
n
the spaces w0 (M , θ, ∆n
m , p, q) and w1 (M , θ, ∆m , p, q).
1
Introduction
The notion of difference sequence space was introduced by Kizmaz [10], who
studied the difference sequence spaces ℓ∞ (∆), c(∆) and c0 (∆). The notion
was further generalized by Et and Colak [4] by introducing the spaces ℓ∞ (∆n ),
c(∆n ) and c0 (∆n ). Another type of generalization of the difference sequence
spaces is due to Tripathy and Esi [23], who studied the spaces ℓ∞ (∆m ), c(∆m )
and c0 (∆m ).
Tripathy, Esi and Tripathy [24] generalized the above notions and unified
these as follows:
Key Words: Orlicz function, lacunary sequence, seminorm, statistical convergence
2010 Mathematics Subject Classification: Primary 40A05, 46A45; Secondary 40C05,
40A35.
Received: February, 2011.
Revised: April, 2011.
Accepted: February, 2012.
417
418
Binod Chandra Tripathy and Hemen Dutta
Let m, n be non-negative integers, then for Z a given sequence space we
have
Z(∆nm ) = {x = (xk ) ∈ w : (∆nm xk ) ∈ Z},
n−1
n−1
xk+m ) and ∆0m xk = xk , for all
xk − ∆m
where ∆nm x = (∆nm xk ) = (∆m
k ∈ N , which is equivalent to the following binomial representation:
n
X
v n
n
xk+mv .
(−1)
∆m xk =
v
v=0
The notion of difference sequences was investigated from different aspects
by Tripathy [16], Tripathy, Altin and Et [17], Tripathy and Baruah [18], Tripathy and Borgogain [20], Tripathy, Choudhary and Sarma [21], Tripathy and
Dutta [22], Tripathy and Mahanta [27] are a few to be named.
The notion of statistical convergence was studied at the initial stage by Fast
[5] and Schoenberg [13] independently. Later on, it was further investigated
by Fridy [6], Rath and Tripathy [12], Šalàt [14], Tripathy ([15], [16]), Tripathy
and Baruah [19], Tripathy and Sarma [28], Tripathy and Sen [32] and many
others.
n
P
A subset E of N is said to have density δ(E) if δ(E) = lim n1
χE (k)
n→∞
k=1
exists, where χE is the characteristic function of E.
A sequence (xk ) is said to be statistically convergent to L if for every ε > 0,
δ({k ∈ N : |xk − L| ≥ ε}) = 0. For L = 0, we say (xk ) is statistically null.
By a lacunary sequence θ = (kr ); r = 1, 2, 3, . . . , where k0 = 0, we mean
an increasing sequence of non-negative integers with hr = (kr − kr−1 ) → ∞
kr
, for r = 1, 2, 3, . . . . The
as r → ∞. We denote Ir = (kr−1 , kr ] and ηr = kr−1
space of lacunary strongly convergent sequence Nθ was defined by Freedman,
Sember and Raphael [7] as follows:
1 X
Nθ = {x = (xk ) : lim
|xk − L| = 0, for some L}.
r→∞ hr
k∈Ir
The space Nθ is a BK-space with the norm
1 X
kxkθ = sup
|xk |.
r hr
k∈Ir
Nθ0 denotes the subset of those sequences in Nθ for which L = 0. (Nθ0 , k.kθ )
is also a BK-space. Freedman, Sember and Raphael [7] also defined the space
|σ1 | of strongly Cesàro summable sequences as follows:
n
1X
|xk − L| = 0, for some L}.
n→∞ n
|σ1 | = {x = (xk ) : lim
k=1
ON SOME LACUNARY DIFFERENCE SEQUENCE SPACES DEFINED BY A
SEQUENCE OF ORLICZ FUNCTIONS AND q-LACUNARY ∆n
m -STATISTICAL
CONVERGENCE
419
In the special case when θ = (2r ), Nθ = |σ1 |.
The notion of lacunary convergence has been investigated by Colak, Tripathy and Et [2], Tripathy and Baruah [19], Tripathy and Mahanta [27] and
many others.
An Orlicz function is a function M : [0, ∞) −→ [0, ∞), which is continuous,
non-decreasing and convex with M (0) = 0, M (x) > 0, for x > 0 and M (x) →
∞, as x → ∞.
Lindenstrauss and Tzafriri [11] used the Orlicz function and introduced
the sequence space ℓM as follows:
)
(
∞
X
|xk |
< ∞, for some ρ > 0 .
M
ℓM = (xk ) ∈ w :
ρ
k=1
They proved that ℓM is a Banach space normed by
(
)
∞
X
|xk |
M
k(xk )k = inf ρ > 0 :
≤1 .
ρ
k=1
In the recent past the notion of Orlicz function was investigated from different aspects and sequence spaces have been studied by Altin, Et and Tripathy
[1], Et, Altin, Choudhary and Tripathy [3], Hudzik, Kamińska and Mastylo
[8], Isik, Et and Tripathy [9], Tripathy, Altin and Et [17], Tripathy and Borgogain [20], Tripathy and Dutta [22], Tripathy and Hazarika [26], Tripathy
and Mahanta [27], Tripathy and Sarma ([29], [30], [31]) and many others.
Remark 1.1. An Orlicz function M satisfies the inequality M (λx) ≤ λM (x),
for all λ with 0 < λ < 1.
The following inequality will be used throughout the article. Let p = (pk )
be a positive sequence of real numbers with 0 < pk ≤ sup pk = G, D =
max(1, 2G−1 ). Then for all ak , bk ∈ C for all k ∈ N , we have
|ak + bk |pk ≤ D {|ak |pk + |bk |pk } .
The notion of paranormed sequences has been investigated from sequence
space point of view and linked with summability theory by Rath and Tripathy [12], Tripathy [16], Tripathy and Dutta [22], Tripathy and Hazarika [25],
Tripathy and Sen ([32], [33]) and many others.
Definition 1.1. Two non-negative functions f, g are called equivalent, whenever C1 f ≤ g ≤ C2 f , for some Cj > 0, j = 1, 2 and in this case we write
f ≈ g.
420
2
Binod Chandra Tripathy and Hemen Dutta
Definition and Preliminaries
Lemma 2.1. (Işik, Et and T ripathy [9], Lemma1.1) Let p and q be seminorms on a linear space X. Then p is stronger than q if and only if there
exists a constant M such that q(x) ≤ M p(x) for all x ∈ X.
Let M =(Mk ) be a sequence of Orlicz functions, p = (pk ) be a bounded
sequence of positive real numbers and X be a seminormed space over the field
C of complex numbers with the seminorm q. w(X) denotes the space of all
sequences x = (xk ), where xk ∈ X, for all k ∈ N . We define the following
sequence spaces:
P h
n ipk
∆m x k
Mk q
= 0,
ρ
P h
n
ipk
∆m xk −L
Mk q
ρ
1
r→∞ hr k∈I
r
w0 (M , θ, ∆nm , p, q) = {x ∈ w(X) : lim
for some ρ > 0},
1
r→∞ hr k∈I
r
w1 (M , θ, ∆nm , p, q) = {x ∈ w(X) : lim
= 0, for some ρ > 0 and L ∈ X},
w∞ (M , θ, ∆nm , p, q) = {x ∈ w(X) : sup h1r
r
for some ρ > 0}.
P h
k∈Ir
n ipk
∆m x k
Mk q
< ∞,
ρ
If Mk (x) = x, for all x ∈ [0, ∞), for all k ∈ N , pk = 1, for all k ∈ N ,
X = C, q(x) = |x|, for all x ∈ X and n = 0 so that ∆0m xk = xk , for all k ∈ N ,
then w1 (M , θ, ∆nm , p, q) = Nθ and w0 (M , θ, ∆nm , p, q) = Nθ0 . If in addition,
we take θ = (2r ), then w1 (M , θ, ∆nm , p, q) = |σ1 |.
3
Main Results
In this section, we investigate the results of this paper involving the spaces
w0 (M , θ, ∆nm , p, q), w1 (M , θ, ∆nm , p, q) and w∞ (M , θ, ∆nm , p, q).
Theorem 3.1. Let M = (Mk ) be a sequence of Orlicz functions. Then
w0 (M, θ, ∆nm , p, q) ⊂ w1 (M, θ, ∆nm , p, q) ⊂ w∞ (M, θ, ∆nm , p, q).
Proof. It is obvious that w0 (M , θ, ∆nm , p, q) ⊆ w1 (M , θ, ∆nm , p, q). We shall
prove that w1 (M , θ, ∆nm , p, q) ⊆ w∞ (M , θ, ∆nm , p, q).
ON SOME LACUNARY DIFFERENCE SEQUENCE SPACES DEFINED BY A
SEQUENCE OF ORLICZ FUNCTIONS AND q-LACUNARY ∆n
m -STATISTICAL
CONVERGENCE
421
Let (xk ) ∈ w1 (M , θ, ∆nm , p, q). Then there exist some ρ > 0 and L ∈ X
such that
n
pk
1 X
∆m xk − L
lim
Mk q
= 0.
r→∞ hr
ρ
k∈Ir
On taking ρ1 = 2ρ, we have
1
hr
P h
k∈Ir
n ipk
∆m x k
Mk q
ρ1
n
h
ipk
P
∆m xk −L
1
≤ hDr
+
2 Mk q
ρ
k∈Ir
≤
D
hr
D
hr
k∈Ir
2 Mk
ipk
q Lρ
n
h
iH ipk
∆m xk −L
1
,
+D max 1, sup 2 Mk q Lρ
2 Mk q
ρ
P h1
k∈Ir
P h1
where sup pk = G, H = max(1, G) and D = max(1, 2G−1 ).
k
Thus we get (xk ) ∈ w∞ (M , θ, ∆nm , p, q).
The inclusions are strict follows from the following examples.
Example 3.1. Let m = n = 2, θ = (3r ), pk = 1, for all k ∈ N , X = C 2 ,
q(x) = max(|x1 |, |x2 |), for x = (x1 , x2 ) ∈ C 2 and Mk (x) = x2 , for all x ∈
[0, ∞) and k ∈ N . Consider the sequence (xk ) defined by xk = (k 2 , k 2 ) for each
fixed k ∈ N . Then (xk ) ∈ w1 (M, θ, ∆nm , p, q), but (xk ) ∈
/ w0 (M, θ, ∆nm , p, q).
Example 3.2. Let m = n = 2, θ = (2r ), pk = 2, for all k odd and pk = 3, for
all k even, X = C 3 , q(x) = max(|x1 |, |x2 |, |x3 |), for x = (x1 , x2 , x3 ) ∈ C 3 and
Mk (x) = x4 , for all x ∈ [0, ∞) and k ∈ N . Consider the sequence (xk ) defined
by xk = (k, k, k) for each fixed k ∈ N . Then (xk ) ∈ w∞ (M, θ, ∆nm , p, q), but
(xk ) ∈
/ w1 (M, θ, ∆nm , p, q).
Corollary 3.2. w0 (M, θ, ∆nm , p, q) and w1 (M, θ, ∆nm , p, q) are nowhere dense
subsets of w∞ (M, θ, ∆nm , p, q).
Proof. Proof is a consequence of Theorem 3.1.
Proof of the following theorem is easy, so omitted.
Theorem 3.3. The spaces w0 (M, θ, ∆nm , p, q), w1 (M, θ, ∆nm , p, q) and
w∞ (M, θ, ∆nm , p, q) are linear.
Theorem 3.4. The spaces w0 (M, θ, ∆nm , p, q), w1 (M, θ, ∆nm , p, q) and
422
Binod Chandra Tripathy and Hemen Dutta
w∞ (M, θ, ∆nm , p, q) are paranormed spaces paranormed by
n mn
X
pr
∆m xk
q(xi ) + inf ρ H : sup Mk q
g(x) =
≤ 1, ρ > 0, r ∈ N ,
ρ
k
i=1
where H = max(1, sup pr ).
r
Proof.
g(x) = g(−x). Since Mk (0) = 0, for all k ∈ N , we get
pr Clearly
inf ρ H = 0 for x = θ. Now let x, y ∈ w0 (M , θ, ∆nm , p, q) and choose ρ1 , ρ2 >
0 such that
n n ∆m yk
∆m xk
≤ 1 and sup Mk q
≤1
sup k Mk q
ρ
ρ2
1
k
k
Let ρ = ρ1 + ρ2 . Then we have
i
h
n
∆m (xk +yk )
sup Mk q
ρ
k
n i n i
h
h
∆m x k
∆m y k
ρ2
1
q
q
≤ ρ1ρ+ρ
sup
+
sup
M
M
k
k
ρ1
ρ1 +ρ2
ρ2
2
k
k
ρ1
ρ2
≤ ρ1 +ρ2 + ρ1 +ρ2 = 1.
Hence g(x + y) ≤ g(x) + g(y).
Finally let λ be a given non-zero scalar, then the continuity of the scalar
multiplication follows from the following equality
i
n
h
mn
P
pr
∆m (λxk )
≤1
q(λxi ) + inf ρ H : sup Mk q
g(λx) =
ρ
k
i=1
i
h
mn
P
pr
∆n
m (xk )
q(xi ) + inf (|λ|s) H : sup Mk q
= |λ|
≤
1
, where s =
s
ρ
|λ| .
i=1
k
This completes the proof.
Proof of the following result is easy, so omitted.
Theorem 3.5. Let M = (Mk ) and T = (Tk ) be sequences of Orlicz functions
and Z = w0 , w1 and w∞ . Then for any two sequences p = (pk ) and t = (tk )
of bounded positive real numbers and for any two seminorms q1 and q2 , we
have
(i) If q1 is stronger than
θ, ∆nm , p, q1 ) ⊂ Z(M, θ, ∆nm , p, q2 ),
T q2 , then Z(M,
n
n
Z(M, θ, ∆m , p, q2 ) ⊂ Z(M, θ, ∆nm , p, q1 + q2 ),
(ii) Z(M, θ, ∆m , p, q1 )
ON SOME LACUNARY DIFFERENCE SEQUENCE SPACES DEFINED BY A
SEQUENCE OF ORLICZ FUNCTIONS AND q-LACUNARY ∆n
m -STATISTICAL
CONVERGENCE
423
T
(iii) Z(M, θ, ∆nm , p, q1 ) T Z(T, θ, ∆nm , p, q1 ) ⊂ Z(M + T, θ, ∆nm , p, q1 ),
Z(M, θ, ∆nm , t, q2 ) 6= φ,
(iv) Z(M, θ, ∆nm , p, q1 )
n−1
(v) The inclusions Z(M, θ, ∆m
, p, q1 ) ⊂ Z(M, θ, ∆nm , p, q1 ) are strict. In
i
general Z(M, θ, ∆m , p, q1 ) ⊂ Z(M, θ, ∆nm , p, q1 ) for i = 1, 2, . . . , n-1 and the
inclusion is strict.
Theorem 3.6. Let Z = w0 , w1 and w∞ . Then we have the followings.
(i) Let 0 < inf pk ≤ pk ≤ 1. Then Z(M, θ, ∆nm , p, q) ⊂ Z(M, θ, ∆nm , q),
(ii) Let 1 ≤ pk sup pk < ∞. Then
θ, ∆nm , q) ⊂ Z(M, θ, ∆nm , p, q),
Z(M,
(iii) Let 0 < pk ≤ tk and ptkk be bounded. Then Z(M, θ, ∆nm , t, q) ⊆
Z(M, θ, ∆nm , p, q).
Proof. Proof of the parts (i) and (ii) is easy and so omitted. We prove the
part (iii) for Z = w1 and for Z = w0 , w∞ , it will follow on applying similar
technique.
n
h
it
∆m xk −L
We write Sk = Mk q
and µk =
ρ
k
′
pk
tk
so that 0 < µ ≤ µk ≤ 1.
′′
Define Sk = Sk if Sk ≥ 1
= 0 if Sk < 1,
Sk = 0 if Sk ≥ 1
= Sk if Sk < 1
′
′′
Then Sk = Sk + Sk , Skµk = Skµk + Sk µk .
′
′′
′
′′
′′
Now it follows that Skµk ≤ Sk ≤ Sk , Sk µk ≤ Sk µ .
′
We have the following inequality
1
hr
P
k∈Ir
Skµk ≤
1
hr
P
k∈Ir
Sk +
1
hr
P
k∈Ir
′′
Sk µ .
Therefore if (xk ) ∈ w1 (M, θ, ∆nm , t, q), then (xk ) ∈ w1 (M, θ, ∆nm , p, q).
The following Theorem is a direct consequence of Definition 1.1.
Theorem 3.7. Let M = (Mk ) and T = (Tk ) be two sequences of Orlicz functions such that Mk ≈ Tk , for each k ∈ N . Then for Z = w0 , w1 and w∞ , we
have Z(M, θ, ∆nm , p, q)=Z(T, θ, ∆nm , p, q).
Theorem 3.8. Let M = (Mk ) be a sequence of Orlicz functions and Z =
w0 , w1 and w∞ . Then Z(M, θ, ∆nm , p, q)=Z(θ, ∆nm , p, q), if the following conditions hold
424
Binod Chandra Tripathy and Hemen Dutta
lim
t→0
Mk (t)
t
> 0 and lim
t→0
Mk (t)
t
< ∞, for each k ∈ N .
Proof. If the given conditions are satisfied, we have Mk (t) = t, for each k ∈ N .
Then the proof from using Theorem 3.7.
q-Lacunary ∆nm -Statistical Convergence
4
In this section, we define the notion of q-lacunary ∆nm -statistical convergence
and investigate some of its properties. Further, we establish some relations between q-lacunary ∆nm -statistical convergence and the spaces w0 (M , θ, ∆nm , p, q)
and w1 (M , θ, ∆nm , p, q).
Definition 4.1. Let θ be a lacunary sequence, then the sequence x = (xk )
is said to be q-lacunary ∆nm -statistical convergent to the number L provided
that for every ε > 0,
lim h1r .card {k ∈ Ir : q (∆nm xk − L) ≥ ε} = 0.
r→∞
In this case, we write xk → L (Sθq (∆nm )) or Sθq (∆nm )-lim xk = L and we
define
Sθq (∆nm ) = {x ∈ w(X) : Sθq (∆nm ) − lim xk = L, for some L}.
In the case θ = (2r ), we write S q (∆nm ) instead of Sθq (∆nm ).
If X = C, q(x) = |x|, we write Sθ (∆nm ) instead of Sθq (∆nm ) and if θ = (2r )
we write S(∆nm ) instead of Sθ (∆nm ).
q
In the special case L = 0, we denote it by S0θ
(∆nm ).
Theorem 4.1. Let θ be a lacunary sequence and 0 < p < ∞.
(i) If xk → L(wθq (∆nm )), then xk → L(Sθq (∆nm )),
(ii) If x ∈ ℓ∞ (q, ∆nm ) and xk → L(Sθq (∆nm )), then xk → L(wθq (∆nm )),
where ℓ∞ (q, ∆nm )=x ∈ w(X) : sup q(∆nm xk ) < ∞ and
k
)
(
P
p
q
1
n
n
(q (∆m xk − L)) = 0, for some L .
wθ (∆m )= x ∈ w(X) : lim hr
r
(iii)
ℓ∞ (q, ∆nm )
∩
k∈Ir
Sθq (∆nm )=ℓ∞ (q, ∆nm )
∩ wθq (∆nm ).
Proof.
(i) Let xk → L(wθq (∆nm )) and ε > 0. Then we have
P
p
(q (∆nm xk − L)) ≥ εp card{k ∈ Ir : q (∆nm xk − L) ≥ ε}.
k∈Ir
ON SOME LACUNARY DIFFERENCE SEQUENCE SPACES DEFINED BY A
SEQUENCE OF ORLICZ FUNCTIONS AND q-LACUNARY ∆n
m -STATISTICAL
CONVERGENCE
425
Hence xk → L(Sθq (∆nm )).
(ii) Suppose x ∈ ℓ∞ (q, ∆nm ) and xk → L(Sθq (∆nm )). Let ε > 0 be given and
n0 (ε) ∈nN such that
1 o
ε p
1
n
< 2Kε p for all r > n0 (ε), where K =
hr card k ∈ Ir : q (∆m xk − L) ≥ 2
n
1 o
sup (q (∆nm xk − L)) and we set Lr = k ∈ Ir : q (∆nm xk − L) ≥ 2ε p .
k
Now for all r > n0 , we have
1
hr
P
k∈Ir
p
(q (∆nm xk − L)) =
1
hr
+
1
hr
≤
Hence xk → L(wθq (∆nm )).
P
k∈Ir ,k∈Lr
P
(q (∆nm xk − L))
p
p
(q (∆nm xk − L))
+ h1r hr 2ε = ε.
k∈Ir ,k∈L
/ r p
hr ε
1
hr 2K p K
(iii) The proof follows from (i) and (ii).
Theorem 4.2. Let θ be a lacunary sequence.
(i) If lim inf r ηr > 1, then S q (∆nm ) ⊆ Sθq (∆nm ),
(ii) If lim supr ηr < ∞, then Sθq (∆nm ) ⊆ S q (∆nm ),
(iii) If 1 < lim inf r ηr ≤ lim supr ηr < ∞, then Sθq (∆nm ) = S q (∆nm ).
Proof. (i) If lim inf r ηr > 1, then there exists a δ > 0 such that 1 + δ ≤ ηr
for sufficiently large r. Since hr = kr − kr−1 , we have hkrr ≤ 1+δ
δ . Let
(xk ) ∈ L (S q (∆nm )). Then for every ε > 0, we have
1
kr card{k
≤ kr : q (∆nm xk − L) ≥ ε} ≥ k1r card{k ∈ Ir : q (∆nm xk − L) ≥ ε}
δ
1
n
≥ δ+1
hr card{k ∈ Ir : q (∆m xk − L) ≥ ε}.
Thus xk → L (Sθq (∆nm )). Hence S q (∆nm ) ⊆ Sθq (∆nm ).
(ii) Suppose lim supr ηr < ∞. Then there exists M > 0 such that ηr < M
for all r ≥ 1.
Let xk → L (Sθq (∆nm )) and ε > 0. Suppose Er =card{k ∈ Ir : q (∆nm xk − L) ≥ ε},
then there exists n0 ∈ N such that h1r Er < ε for all r > n0 . Let K = max Er
1≤r≤n0
and choose n such that kr−1 < n ≤ Kr , then we have
426
Binod Chandra Tripathy and Hemen Dutta
1
n card{k
1
card{k ≤ kr : q (∆nm xk − L) ≥ ε}
≤ n : q (∆nm xk − L) ≥ ε} ≤ kr−1
1
≤ kr−1
{E1 + · · · + En0 + · · · + Er }
o
n
En0 +1
Er
1
K
≤ kr−1 n0 + kr−1
hn0 +1 hn0 +1 + · · · + hr hr
Er
1
K
≤ kr−1 n0 + kr−1 sup hr {hn0 +1 + · · · + hr }
r>n0
K
k −k
≤
≤
r
n0
kr−1 n0 + ε kr−1
K
kr−1 n0 + εηr
≤
K
kr−1 n0
+ εM .
Since kr−1 → ∞ as n → ∞, it follows that xk → L (S q (∆nm )). Hence
⊆ S q (∆nm ).
Sθq (∆nm )
(iii) The proof follows from (i) and (ii).
Theorem 4.3. (i) w1 (M, θ, ∆nm , p, q) ⊆ Sθq (∆nm ),
q
(∆nm ).
(ii) w0 (M, θ, ∆nm , p, q) ⊆ S0θ
Proof. (i) Let (xk ) ∈ w1 (M , θ, ∆nm , p, q). Then there exist some ρ > 0 and
L ∈ X such that
n
pk
∆m xk − L
1 X
Mk q
= 0.
lim
r→∞ hr
ρ
k∈Ir
denote the sum over k ∈ Ir such that q(∆nm −L) ≥ ε
Let ε > 0 be given and
1
P
denote the sum over k ∈ Ir such the q(∆nm − L) < ε. Then
and
2
n
n
ipk
ipk
P h
Ph
∆m xk −L
∆m xk −L
1
1
q
q
M
M
=
k
k
hr
ρ
hr
ρ
k∈Ir
1 n
h
ipk
P
∆m xk −L
Mk q
+ h1r
ρ
2
P
ε
1
pk
≥ hr [Mk (ε1 )] , where ρ = ε1
1
P
min [Mk (ε1 )]inf pk , [Mk (ε1 )]G
≥ h1r
1
≥ h1r card{k ∈ Ir : q(∆nm xk − L) ≥ ε} min [Mk (ε1 )]inf pk , [Mk (ε1 )]G .
P
Hence (xk ) ∈ Sθq (∆nm ).
(ii) Proof is similar to that of part (i).
ON SOME LACUNARY DIFFERENCE SEQUENCE SPACES DEFINED BY A
SEQUENCE OF ORLICZ FUNCTIONS AND q-LACUNARY ∆n
m -STATISTICAL
CONVERGENCE
427
Theorem 4.4. (i) ℓ∞ (q, ∆nm ) ∩ Sθq (∆nm ) = ℓ∞ (q, ∆nm ) ∩ w1 (M, θ, ∆nm , p, q),
q
(∆nm ) = ℓ∞ (q, ∆nm ) ∩ w0 (M, θ, ∆nm , p, q).
(ii) ℓ∞ (q, ∆nm ) ∩ S0θ
Proof. (i) Using Theorem 4.3, it is enough to show that ℓ∞ (q, ∆nm ) ∩ Sθq (∆nm )
⊆ ℓ∞ (q, ∆nm ) ∩ w1 (M , θ, ∆nm , p, q). P
Let (xkP
) ∈ ℓ∞ (q, ∆nm ) ∩ Sθq (∆nm ) and tk
q
n
n
be the same as in the proof of
and
= (∆m xk − L) → 0 (Sθ (∆m )). Let
2
1
n
the previous
Theorem. Since (xk ) ∈ ℓ∞ (q, ∆m ), there exists K > 0 such that
Mk q
tk
ρ
≤ K for all k ∈ N . Then given ε > 0, we have
1 X
1 X
1 X
tk
tk
tk
=
+
Mk q
Mk q
Mk q
hr
ρ
hr 1
ρ
hr 2
ρ
k∈Ir
≤
K
hr card{k
∈ Ir : q(tk ) ≥ ερ} +
1
hr
P
Mk
k∈Ir
Hence (xk ) ∈ ℓ∞ (q, ∆nm ) ∩ w1 (M , θ, ∆nm , p, q).
ε
ρ
.
(ii) Proof is similar to that of part (i).
References
[1] Y. Altin, M. Et and B.C. Tripathy, The sequence space |N̄p |(M, r, q, s) on
seminormed spaces, Applied Mathematics and Computation 154 (2004),
423-430.
[2] R. Colak, B.C. Tripathy and M. Et, Lacunary strongly summable sequences and q-lacunary almost statistical convergence, Vietnam J. Math.
34(2) (2006), 129-138.
[3] M. Et, Y. Altin, B. Choudhary and B.C. Tripathy, On some classes of
sequences defined by sequences of Orlicz functions, Mathematical Inequalities and Applications 9(2) (2006), 335-342.
[4] M. Et and R. Colak, On generalized difference sequence spaces, Soochow
J. Math. 21(4) (1995), 377-386.
[5] H. Fast, Surla convergence statistique, Colloq. Math. 2 (1951), 241-244.
[6] J. A. Fridy, On statistical convergence, Analysis 5 (1985), 301-313.
[7] A.R. Freedman, J.J. Sember and M. Raphael, Some Cesàro-type summability spaces, Proc. Lond. Math. Soc. 37(3) (1978), 508-520.
428
Binod Chandra Tripathy and Hemen Dutta
[8] H. Hudzik, A. Kamińska and M. Mastylo, On the dual of Orlicz-Lorentz
space, Proc. Amer. Math. Soc. 130(6) (2002), 1645-1654.
[9] M. Işik, M. Et and B.C. Tripathy, On some new seminormed sequence
spaces defined by Orlicz functions, Thai J. Math. 2(1) (2004), 141-149.
[10] H. Kizmaz, On certain sequence spaces, Canad. Math. Bull. 24(2) (1981),
169-176.
[11] J. Lindenstrauss and L. Tzafriri, On Orlicz sequence spaces, Israel J.
Math. 10 (1971), 379-390.
[12] D. Rath and B.C. Tripathy, Matrix maps on sequence spaces associated
with sets of integers, Indian J. Pure Appl. Math. 27(2) (1996) 197-206.
[13] I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66 (1959), 361-375.
[14] T. S̆alàt, On statistically convergent sequences of real numbers, Math.
Slovaca, 30 (1980), 139-150.
[15] B.C. Tripathy, Matrix transformations between some classes of sequences,
J. Math. Analysis Appl. 206 (1997), 448-450.
[16] B.C. Tripathy, On generalized difference paranormed statistically convergent sequences, Indian J. Pure Appl. Math. 35(5) (2004),655-663.
[17] B.C. Tripathy, Y. Altin and M. Et, Generalized difference sequences
spaces on seminormed spaces defined by Orlicz functions, Math. Slovaca
58(3) (2008), 315-324
[18] B.C. Tripathy and A. Baruah, New type of difference sequence spaces of
fuzzy real numbers, Mathematical Modelling and Analysis 14(3) (2009),
391-397.
[19] B.C. Tripathy and A. Baruah, Lacunary statistically convergent and lacunary strongly convergent generalized difference sequences of fuzzy real
numbers, Kyungpook Math. J. 50(4)(2010), 565-574..
[20] B.C. Tripathy and S. Borgogain, The sequence space m(M, φ, ∆nm , p)F ,
Mathematical Modelling and Analysis 13(4) (2008), 577-586.
[21] B.C. Tripathy; B. Choudhary and B. Sarma, On some new type generalized difference sequence spaces, Kyungpook Math. J. 48(4) (2008),
613-622.
ON SOME LACUNARY DIFFERENCE SEQUENCE SPACES DEFINED BY A
SEQUENCE OF ORLICZ FUNCTIONS AND q-LACUNARY ∆n
m -STATISTICAL
CONVERGENCE
429
[22] B.C. Tripathy and H. Dutta, On some new paranormed difference sequence spaces defined by Orlicz functions, Kyungpook Math. J. 50 (2010),
59-69.
[23] B. C. Tripathy and A. Esi, A new type of difference sequence spaces, Int.
J. Sci. Technol. 1(1) (2006), 11-14.
[24] B .C. Tripathy, A. Esi and B. K. Tripathy, On a new type of generalized
difference Cesàro Sequence spaces, Soochow J. Math. 31:3 (2005), 333340.
[25] B.C. Tripathy and B. Hazarika: Paranormed I-convergent sequences
spaces; Math. Slovaca; 59(4)(2009), 485-494.
[26] B.C. Tripathy and B. Hazarika, I-convergent sequences spaces defined by
Orlicz function, Acta Mathematica Applicatae Sinica; 27(1)(2011) 149154.
[27] B. C. Tripathy and S. Mahanta, On a class of generalized lacunary difference sequence spaces defined by Orlicz function, Acta Mathematica
Applicata Sinica 20(2) (2004), 231-238.
[28] B.C. Tripathy and B. Sarma, Statistically convergent difference double
sequence spaces, Acta Mathematica Sinica 24(5) (2008), 737-742.
[29] B.C. Tripathy and B. Sarma, Sequence spaces of fuzzy real numbers defined by Orlicz functions, Math. Slovaca 58(5) (2008), 621-628.
[30] B.C. Tripathy and B. Sarma, Vector valued double sequence spaces defined
by Orlicz function, Math. Slovaca 59(6) (2009), 767-776.
[31] B.C. Tripathy and B. Sarma, Double sequence spaces of fuzzy numbers
defined by Orlicz function, Acta Mathematica Scientia 31B(1) (2011),
134-140.
[32] B. C. Tripathy and M. Sen, On generalized statistically convergent sequence spaces, Indian J. Pure Appl. Math. 32(11) (2001), 1689-1694.
[33] B.C. Tripathy and M. Sen, Characterization of some matrix classes involving paranormed sequence spaces, Tamkang J. Mathematics 37(2)
(2006), 155-162.
Binod Chandra TRIPATHY,
Mathematical Sciences Division,
Institute of Advanced Study in Science and Technology,
Paschim Boragaon, Garchuk, Guwahati-781013, Assam, India.
Email: tripathybc@yahoo.com; tripathybc@rediffmail.com
430
Hemen DUTTA,
Department of Mathematics,
Gauhati University Campus,
Kokrajhar-783370, Assam, India.
Email: hemen− dutta08@rediffmail.com
Binod Chandra Tripathy and Hemen Dutta