General Mathematics Vol. 17, No. 2 (2009), 53–66
Generalized Difference Sequence Spaces
Defined by Orlicz Functions1
Ayhan Esi
Abstract
The idea of difference sequences was first introduced by Kizmaz
[4]. In this first paper we define some generalized difference sequence
combining lacunary sequences and Orlicz function. We establish
some relations between these sequence space.
2000 Mathematics Subject Classification: 40A05, 40C05A, 46A45
Key words and phrases: Lacunary sequence, difference sequence, Orlicz
function, strongly almost convergence.
1
Definitions and notations
Let l∞ , c and c0 be the sequence spaces of bounded, convergent and null
sequences x = (xi ), respectively.
A sequence x = (xi ) ∈ l∞ is said to be almost convergent [2] if all Banach
limits of x = (xi ) coincide. In [2], it was shown that
(
)
n
1X
ĉ = x = (xi ) : lim
xi+s exits, unif ormly in s
n→∞ n
i=1
1
Received 5 May, 2008
Accepted for publication (in revised form) 5 June, 2008
53
54
Ayhan Esi
In [3,4], Maddox defined a sequence x = (xi ) to strongly almost convergent
to a number L if
n
1X
|xi+s − L| = 0, unif ormly in s
lim
n→∞ n
i=1
By a lacunary sequence θ = (kr ), r = 0, 1, 2, . . . , where k0 = 0, we shall
mean an increasing sequence of non-negative integers hr = (kr − kr−1 ) →
∞(r → ∞). The intervals determined by θ are denoted by Ir = (kr−1 , kr ]
kr
will be denoted by qr . The space of lacunary strongly
and the ratio kr−1
convergent sequence Nθ was defined by Freedman et al. [10] as follows:
(
)
1 X
Nθ = x = (xi ) : lim
|xi − L| = 0, f ot some L .
r→∞ hr
i∈I
r
In [1], Kizmaz defined the sequence spaces Z(∆) = {x = (xi ) : (∆xi ) ∈
Z} for Z = l∞ , c and c0 , where ∆x = (∆xi ) = (xi − xi+1 ). After Et and
Çolak [8] generalized the difference sequence spaces to the sequence spaces
Z(∆m ) = {x = (xi ) : (∆m xi ) ∈ Z} for Z = l∞ , c and c0 , where m ∈
N, ∆0 x = (xi ), ∆x = (xi − xi+1 ), ∆m x = (∆m xi ) = (∆m−1 xi − ∆m−1 xi+1 )
and so that
!
m
X
m
xi+v .
(−1)v
∆m x i =
v
v=0
An Orlicz function is a function M : [0, ∞) → [0, ∞) which is continuous,
nondecreasing an convex with M (0) = 0, M (x) > 0 for x > 0 and M (x) →
∞ as x →= ∞.
An Orlicz function M is said to satisfy ∆2 -condition for all values of u,
if there exists a constant T > 0, such that M (2u) ≤ T M (u)(u ≥ 0). The
∆2 -condition is equivalent to M (Lu) ≤ T LM (u), for all values of u and for
L > 1.
An Orlicz function M can be always be represented (see[6]) in the inRx
tegral form M (x) = 0 q(t)dt, where q known as the kernel of M , is right
differentiable for t ≥ 0, q(t) > 0 for t > 0, q is non-decreasing and q(t) → ∞
as t → ∞.
55
Generalized Difference Sequence Spaces ...
Remark 1.1. An Orlicz function satisfies the inequality M (λu) ≤ λM (u)
for all λ with 0 < λ < 1.
Lindenstrauss and Tzafriri [7] used the idea of Orlicz function to construct Orlicz sequence space,
(
)
X |xi | lM = x = (xi ) :
M
< ∞, f or someρ > 0 .
ρ
i
The sequence space lm with the norm
(
)
X |xi | ||x|| = inf ρ > 0 :
M
≤1
ρ
i
becomes a Banach space which is called an Orlicz Sequence Space. The
space lM is closely related to the space lp , which is an Orlicz sequence space
with M (x) = xp for 1 ≤ p < ∞.
Let M be an Oriclz function and p = (pi ) be any sequence of strictly
positive real numbers. Güngör and Et[3] defined the following sequence
spaces:
(
pi
m
n 1X
|∆ xi+s − L|
m
[ĉ, M, p](∆ ) = x = (xi ) : lim
M
= 0,
n→∞ n
ρ
i=1
unif ormly in s, f or some ρ > 0 and L > 0
[ĉ, M, p]0 =
,
x = (xi ) :
)
pi
m
n 1X
|∆ xi+s |
M
= 0, unif ormly in s, f or some ρ > 0 ,
lim
n→∞ n
ρ
i=1
m
[ĉ, M, p]∞ (∆ ) = x = (xi ) :
)
pi
m
n |∆ xi+s |
1X
< ∞, f or some ρ > 0 .
sup
M
ρ
n,s n
i=1
56
Ayhan Esi
The purpose of this paper is to introduce and study a concept of lacunary
almost generalized ∆m − convergence function and to examine some properties of these new sequence spaces which also generalize the well known Orlicz
sequence space lM and strongly summable sequence [C, 1, p], [C, 1, p]0 and
[C, 1, p]∞ [9].
In the present paper we introduce and examine the following spaces
defined by Orlicz function.
Definition 1.1. Let M be an Orlicz function and p = (pi ) be any bounded
sequence of strictly positive real numbers. We have
(
pi
m
|∆ xi+s − L|
1 X
∞
m
= 0,
M
[ĉ, M, p] (∆ ) = x = (xi ) : lim
r→∞ hr
ρ
i∈I
r
unif ormly in s, f or some ρ > 0 and L > 0
[ĉ, M, p]∞
0
=
,
x = (xi ) :
)
pi
m
1 X
|∆ xi+s |
lim
M
= 0, unif ormly in s, f or some ρ > 0 ,
r→∞ hr
ρ
i∈I
r
m
[ĉ, M, p]∞ (∆ ) =
x = (xi ) :
)
pi
m
n 1 X
|∆ xi+s |
< ∞, f or some ρ > 0 .
sup
M
ρ
r,s hr
i=1
If x = (xi ) ∈ [ĉ, M, p]θ (∆m ), we say that x = (xi ) is lacunary almost
∆m -convergence to L with respect to Orlicz function M .
When M (x) = x, then we write [ĉ, p]θ (∆m ), [ĉ, p]θ0 (∆m ) for the spaces
[ĉ, M, p]θ (∆m ), [ĉ, M, p]θ0 (∆m ) and [ĉ, M, p]θ∞ (∆m ), respectively. If pi = 1 for
all i ∈ N , then [ĉ, M, p]θ (∆m ), [ĉ, M, p]θ0 (∆m ) and [ĉ, M, p]θ∞ (∆m ) reduce to
[ĉ, M ]θ (∆m ), [ĉ, M ]θ0 (∆m ) and [ĉ, M ]θ∞ (∆m ), respectively.
57
Generalized Difference Sequence Spaces ...
The following inequality will be used throughout the paper,
|xi + yi |pi ≤ K(|xi |pi + |yi |pi )
(1.1)
where xi and yi are complex numbers, k = max(1, 2H−1 ) and H = supi pi <
∞.
2
Main Result
In this section we prove some results involving the sequence spaces [ĉ, M, p]θ (∆m ),
[ĉ, M, p]θ0 (∆m ) and [ĉ, M, p]θ∞ (∆m ).
Theorem 2.1. Let M be an Orlicz function and p = (pi ) be a bounded
sequence of strictly real numbers. Then [ĉ, M, p]θ (∆m ), [ĉ, M, p]θ0 (∆m ) and
[ĉ, M, p]θ∞ (∆m ) are linear spaces over the set of complex numbers C
Proof. Let x = (xi ), y = (yi ) ∈ [ĉ, M, p]θ0 (∆m ) and α, β ∈ C. Then there
exists positive numbers ρ1 and ρ2 such that
m
pi
1 X
|∆ xi+s |
lim
M
= 0, unif ormly in s,
r→∞ hr
ρ
1
i∈I
r
and
pi
m
1 X
∆ xi+s
lim
= 0, unif ormly in s.
M
r→∞ hr
ρ2
i∈I
r
Let ρ3 = max(2|α|ρ1 , 2|β|ρ2 ). Since M is non-decreasing convex function,
by using (1.1), we have
m
pi
|∆ (αxi+s + βyi+s )|
1 X
=
M
hr i∈I
ρ3
r
pi
1 X
|α∆m (xi+s )| |β∆m (yi+s )|
=
M
+
hr i∈I
ρ3
ρ3
r
pi
pi
m
m
1 X 1
|∆ (xi+s )|
∆ (yi+s )
1 X 1
+K
M
M
≤K
hr i∈I 2pi
ρ1
hr i∈I 2pi
ρ2
r
r
58
Ayhan Esi
pi
pi
m
m
1 X
∆ (xi+s )
∆ (yi+s )
1 X
≤ +K
M
M
+K
→ 0 as
hr i∈I
ρ1
hr i∈I
ρ2
r
r
r → ∞, uniformly in s.
So, αx + βx ∈ [ĉ, M, p]θ0 (∆m ). Hence [ĉ, M, p]θ0 (∆m ) is a linear space.
The proof for the cases [ĉ, M, p]θ (∆m ) and [ĉ, M, p]θ∞ (∆m ) are routine
work in view of the above proof.
Theorem 2.2. For any Orlicz function M an a bounded sequence p = (pi )
of strictly positive real numbers, [ĉ, M, p]θ0 (∆m ) is a topological linear space
paranormed by

!1/H

X |∆m xi+s | pi
1
≤ 1,
M
h(x) = inf ρpr /H :

hr i∈I
ρ
r
r = 1, 2, . . . , s = 1, 2... , where H = max(1, supi pi < ∞).
Proof. Clearly h(x) ≥ for all x = (xi ) ∈ [ĉ, M ]θ0 (∆m ). Since M (0) = 0, we
get h(0) = 0. Conversely, suppose that h(x) = 0, then


!1/H
p


i
m
1 X |∆ xi+s |
inf ρpr /H :
≤ 1, r = 1, 2, . . . , s = 1, 2, . . . = 0.


hr i∈I
ρ
r
The implies that for a given > 0, there exists some ρ (0 < ρ < ) such
that
m
pi !1/H
∆ xi+s
1 X
M
≤ 1.
hr i∈I
ρ
r
Thus
pi !1/H
m
1 X
|∆ xi+s |
≤
M
hr i∈I
r
pi !1/H
m
1 X
|∆ xi+s |
≤1
M
hr i∈I
ρ
r
for each r and s. Suppose that xi 6= 0 for each i ∈ N . This implies that
m
∆m xi+s 6= 0, for each i, s ∈ N . Let → 0, then |∆ xi+s | → ∞. It follows
Generalized Difference Sequence Spaces ...
that
59
m
pi !1/H
|∆ xi+s |
1 X
M
→∞
hr i∈I
r
which is contradiction. Therefore, ∆m xi+s = 0 for each i and s and thus
xi = 0 for each i ∈ N . Let ρ1 > 0 and ρ2 > 0 be such that
m
pi !1/H
|∆ xi+s |
1 X
M
≤1
hr i∈I
ρ1
r
and
pi !1/H
m
1 X
|∆ xi+s |
≤1
M
hr i∈I
ρ2
r
for each r and s. Let ρ = ρ1 + ρ2 . Then, we have
pi !1/H
m
1 X
|∆ (xi+s + yi+s )|
M
hr i∈I
ρ
r
≤
m
pi !1/H
|∆ (xi+s )| + |∆m (yi+s )|
1 X
M
hr i∈I
ρ1 + ρ 2
r
≤
m
m
pi !1/H
ρ1
|∆ (xi+s )|
|∆ (yi+s )|
ρ2
1 X
M
+
M
hr i∈I ρ1 + ρ2
ρ1
ρ1 + ρ 2
ρ2
r
By Minkowski’s inequality
≤
+
ρ1
ρ1 + ρ 2
ρ2
ρ1 + ρ 2
pi !1/H
m
1 X
∆ (xi+s )
+
M
hr i∈I
ρ1
r
m
pi !1/H
|∆ (yi+s )|
1 X
M
≤1
hr i∈I
ρ2
r
Since the ρ’s are non-negative, so we have

!1/H

X ∆m (xi+s + yi+s ) pi
1
M
≤ 1,
h(x + y) = inf ppr /H :

hr i∈I
ρ
r
60
Ayhan Esi
r = 1, 2, . . . , s = 1, 2, . . .


p /H
≤ inf ρ1r :

,
pi !1/H
m
1 X
|∆ (xi+s )|
≤ 1,
M
hr i∈I
ρ1
r
r = 1, 2, . . . , s = 1, 2, . . . +,


p /H
+ inf ρ2r :

m
pi !1/H
|∆ (yi+s )|
1 X
M
≤ 1,
hr i∈I
ρ2
r
r = 1, 2, . . . , s = 1, 2, . . . .
Therefore h(x + y) ≤ h(x) + h(y). Finally, we prove that the scalar multiplication is continuous. Let λ be any complex number. By definition

!1/H

X |∆m λxi+s | pi
1
h(λx) = inf ρpr /H :
≤ 1,
M

hr i∈I
ρ
r
r = 1, 2, . . . , s = 1, 2, . . .
Then,
h(λx) = inf



(|λ|t)pr /H :
.
pi !1/H
m
|∆ xi+s |
1 X
M
≤ 1,
hr i∈I
t
r
r = 1, 2, . . . , s = 1, 2, . . .
ρ
.
|λ|
Since |λ|pr ≤ max(1, λ|sup pr ), we have

!1/H

X |∆m xi+s | pi
1
h(λx) ≤ max(1, |λ|sup pr ) inf tpr /H :
≤ 1,
M

hr i∈I
t
where t =
r
r = 1, 2, . . . , s = 1, 2, . . .
.
61
Generalized Difference Sequence Spaces ...
So, the fact that scalar multiplication is continuous follows from the above
inequality. This completes the proof of the theorem.
Theorem 2.3. Let M be an Orlicz function. If supi [M (x)]pi < ∞ for all
fixed x > 0, then
[ĉ, M, p]θ0 (∆m ) ⊂ [ĉ, M, p]θ∞ (∆m ).
Proof. Let x = (xi ) ∈ [ĉ, M, p]θ0 (∆m ). There exists some positive ρ1 such
that
pi
m
1 X
|∆ xi+s |
lim
M
= 0, unif ormly in s.
r∈∞ hr
ρ1
i∈I
r
Define ρ = 2ρ1 . Since M is non-decreasing and convex, by using (1.1),
we have
pi
pi
m
m
|∆ xi+s |
∆ xi+s − L + L
1 X
1 X
M
M
= sup
sup
ρ
ρ
r,s hr
r,s hr
i∈I
i∈I
r
r
m
pi
pi
|∆ xi+s − L|
1 X 1
|L|
1 X 1
M
M
+K sup
≤ K sup
p
p
i
i
2
ρ1
2
ρ1
r,s hr
r,s hr
i∈I
i∈I
r
1
≤ K sup
r,s hr
X
i∈Ir
r
M
|∆m xi+s − L|
ρ1
pi
1
+ K sup
r,s hr
X
i∈Ir
M
|L|
ρ1
pi
< ∞.
Hence x = (xi ) ∈ [ĉ, M, p]θ∞ . This completes the proof.
Theorem 2.4. Let 0 < inf pi = h ≤ pi ≤ sup pi = H < ∞ and M, M1
be Orlicz function satisfying ∆2 -condition, then we have [ĉ, M1 , p]θ0 (∆m ) ⊂
[ĉ, M oM1 , p]θ0 (∆m ), [ĉ, M1 , p]θ (∆m ) ⊂ [ĉ, M oM1 , p]θ (∆m ) and [ĉ, M1 , p]θ∞ (∆m )
⊂ [ĉ, M oM1 , p]θ∞ (∆m ).
Proof. Let x = (xi ) ∈ [ĉ, M1 , p]θ0 (∆m ). Then we have
m
pi
1 X
|∆ xi+s − L|
lim
M1
= 0, unif ormly in s, f or some L.
r→∞ hr
ρ
i∈I
r
62
Ayhan Esi
Let > 0 and choose δ with 0 < δ < 1 such that M (t) < for 0 ≤ t ≤ δ.
Let
m
−L|
yi,s = M1 |∆ xi+s
for all i, s ∈ N. We can write
ρ
1 X
1 X
1 X
[M (yi,s )]pi =
[M (yi,s )]pi +
[M (yi,s )]pi
hr i∈I
hr i∈I
hr i∈I
r
r
yi,s ≤δ
r
yi,s >δ
By the Remark, we have
(2.1)
1 X
1 X
1 X
[M (yi,s )]pi ≤ [M (1)]H
[M (yi,s )]pi ≤ [M (2)]H
[M (yi,s )]pi .
hr i∈I
hr i∈I
hr i∈I
r
r
yi,s ≤δ
r
yi,s ≤δ
yi,s ≤δ
For yi,s > δ
yi,s
yi,s
<1+
.
δ
δ
Since M is non-decreasing and convex, it follows that
1
yi,s 1
2yi,s
< M (2) + M
.
M (yi,s ) < M 1 +
δ
2
2
δ
yi,s <
Since M satisfies ∆2 -condition, we can write
1 yi,s
1 yi,s
1 yi,s
yi,s
M (yi,s ) < T
M (2) + T
M (2) + T
M (2) = T
M (2).
2 δ
2 δ
2 δ
δ
Hence,
(2.2)
H
T M (2)
1 X
pi
[M (yi,s )] ≤ max 1,
hr i∈I
δ
r
yi,s >δ
!
1 X
[(yi,s )]pi .
hr i∈I
r
yi,s >δ
By (2.1) and (2.2), we have x = (xi ) ∈ [ĉ, M oM1 , p]θ0 (∆m ).
Following similar arguments we can prove that [ĉ, M1 , p]θ0 (∆m ) ⊂
[ĉ, M oM1 , p]θ0 (∆m ) and [ĉ, M1 , p]θ∞ (∆m ) ⊂ [ĉ, M oM1 , p]θ∞ (∆m ). This completes the proof.
Taking M1 (x) in Theorem 2.4. we have the following result.
63
Generalized Difference Sequence Spaces ...
Corollary 2.5.Let 0 < inf pi = h ≤ pi ≤ sup pi = H < ∞ and M
be an Orlicz function satisfying ∆2 -condition, then we have [ĉ, p]θ0 (δ m ) ⊂
[ĉ, M, p]θ0 (∆m ) and [ĉ, M1 , p]θ∞ (∆m ) ⊂ [ĉ, M, p]θ∞ (∆m ).
Theorem 2.6.Let M be an Orlicz function. Then the following statements
are equivalent:
(a) [ĉ, p]θ∞ (∆m ) ⊂ [ĉ, M, p]θ∞ (∆m ),
θ
m
(b) [ĉ, p]θ0 (∆m ) ⊂ h[ĉ, M,
i∞pi(∆ ),
p]
P
(c) supr h1r i∈Ir M ρt
< ∞ (t, ρ > 0).
Proof. (a)⇒(b): It is obvious, since [ĉ, p]θ0 (∆m ) ⊂ [ĉ, p]θ∞ (∆m ).
(b)⇒(c): Let [ĉ, p]θ0 (∆m ) ⊂ [ĉ, M, p]θ∞ (δ m ). Suppose that (c) does not
hold. Then for some t, ρ > 0
pi
1 X
t
sup
M
=∞
ρ
r hr
i∈I
r
and therefore we cab find a subinterval Ir(j) of the set of interval Ir such
that
−1 pi
1 X
j
M
(2.3)
> j, j = 1, 2, . . .
hr(j) i∈I
ρ
r(j)
Define the sequence x = x(i) by
(
j −1 , i ∈ Ir(j)
∆m xi+s =
0, i 6∈ Ir(j)
f or all s ∈ N.
Then x = (xi ) ∈ [ĉ, p]θ0 (∆m ) but by (2.3) x = (xi ) 6∈ [ĉ, M, p]θ∞ (δ m ), which
contradicts (b).
Hence (c) must hold.
(c)⇒(a): Let (c) hold and x = (xi ) ∈ [ĉ, p]θ∞ (∆m ). Suppose that
x = (xi ) 6∈ [ĉ, M, p]θ∞ (∆m ). Then
m
pi
1 X
|∆ xi+s |
(2.4)
sup
M
=∞
ρ
r,s hr
i∈I
r
64
Ayhan Esi
Let t = |∆m xi+s | for each i and fixed s, then by (2.4)
t
1 X
sup
=∞
M
ρ
r hr
i∈I
r
which contradicts (c). Hence (a) must hold.
Theorem 2.7. Let 1 ≤ pi ≤ sup pi < ∞ and M be an Orlicz function.
Then the following statement are equivalent:
(a) [ĉ, M, p]θ0 (δ m ) ⊂ [ĉ, p]θ0 (∆m ),
(b) [ĉ, M, p]θ0 (∆m ) ⊂ [ĉ, p]θ∞ (∆m ),
h ipi
P
1
> 0 (t, ρ > 0).
(c) inf r hr i∈Ir M ρt
Proof. (a)⇒(b): It is obvious.
(b)⇒(c): Let (b) hold. Suppose that (c) does not hold. Then
pi
1 X
t
inf
= 0 (t, ρ > 0),
M
r hr
ρ
i∈I
r
so we can find a subinterval Ir(j) of the set of interval Ir such that
1
(2.5)
hr(j)
X i∈Ir(j)
pi
j
M
< j −1 , j = 1, 2, ...
ρ
Define the sequence x = (xi ) by
(
j, i ∈ Ir(j)
∆m xi+s =
0, i ∈
6 Ir(j)
for all s ∈ N
Thus, by (2.5), x = (xi ) ∈ [ĉ, M, p]θ0 (∆m ) but by (2.3) x = (xi ) 6∈ [ĉ, p]θ∞ (∆m ),
which contradicts (b). Hence (c) must hold.
(c)⇒(a) Let (c)hold and suppose that x = (xi ) ∈ [ĉ, M, p]θ0 (∆m ), i.e.,
(2.6)
m
pi
1 X
|∆ xi+s |
lim
M
= 0, unif ormly in s, f orsomeρ > 0.
r→∞ hr
ρ
i∈I
r
Generalized Difference Sequence Spaces ...
65
Again, suppose that x = (xi ) 6∈ [ĉ, p]θ0 (∆m ). Then, for some number > 0
and a subinterval Ir(j) of the set of interval Ir , we have |∆m xi+s | ≥ for all
i ∈ N and some s ≥ s0 . Then, from the properties of the Orlicz function,
we can write
m
pi
p
|∆ xi+s | i
M
≥M
ρ
ρ
and consequently by (2.6)
pi
1 X
lim
M
= 0,
r→∞ hr
ρ
i∈I
r
which contradicts (c). Hence (a) must hold.
Finally, we consider that p = pi and q = (qi ) are any bounded sequences
of strictly positive real numbers. We are able to prove below theorem only
under additional conditions.
Theorem 2.8. Let 0 < pi ≤ qi for all i ∈ N and pqii be bounded. Then,
[ĉ, M, q]θ (∆m ) ⊂ [ĉ, M, p]θ (∆m ).
Proof. Using the same technique of Theorem 2 of Nanda [11], it is easy to
prove the theorem.
By using Theorem 2.8., it is easy to prove the following result.
Corollary 2.9. (a) If 0 < inf pi ≤ pi ≤ 1 for all i ∈ N, then
[ĉ, M, p]θ (∆m ) ⊂ [ĉ, M ]θ (∆m ).
(b) If 1 ≤ pi ≤ sup pi < ∞ for all i ∈ N, then [ĉ, M ]θ (∆m ) ⊂ [ĉ, M, p]θ (∆m ).
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Ayhan Esi
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Adiyaman University,
Science and Art Faculty
Department of Mathematics
Adiyaman, 02040, Turkey
E-mail: aesi23@gmail.com