ON ORLICZ FUNCTIONS OF GENERALIZED DIFFERENCE SEQUENCE SPACES
Ahmad H. A. Bataineh and Alaa A. Al-Smadi
Abstract. In this paper, we define the sequence spaces : [V, M, p,∆n u, s], [V, M, p,∆nu, s]0 and [V, M, p,∆un, s]∞, where for any sequence x = (xn), the
dif-ference sequence ∆x is given by ∆x= (∆xn)∞n=1 = (xn−xn−1)∞n=1. We also
exam-ine some inclusion relations between these spaces and discuss some properties and results related to them.
2000Mathematics Subject Classification:40D05, 40A05. 1.Introduction and definitions
LetXbe a linear space. A functionp:X →Ris called paranorm if the following are satisfied :
(i) p(0)≥0
(ii)p(x)≥0 for all x∈X
(iii) p(x) =p(−x) for all x∈X
(iv)p(x+y)≤p(x) +p(y) for all x∈X ( triangle inequality )
(v) if (λn) is a sequence of scalars withλn→λ(n→ ∞) and (xn) is a sequence of vectors withp(xn−x)→0 (n→ ∞),thenp(λnxn−λx)→0 (n→ ∞) ( continuity of multiplication by scalars ).
A paranorm pfor which p(x) = 0 impliesx= 0 is called total. It is well known that the metric of any linear metric space is given by some total paranorm (cf.[11]). Let Λ = (λn) a nondecreasing sequence of positive reals tending to infinity and
λ1 = 1 andλn+1≤λn+ 1.
The generalized de la Vall˙ee-Poussin means is defined by :
tn(x) = 1
λn X
k∈In
xk,
We write
[V, λ]0 = {x= (xk) : lim n
1
λn X
k∈In
|xk|= 0}
[V, λ] = {x= (xk) : lim n
1
λn X
k∈In
|xk−le|= 0, for somel∈C}
and
[V, λ]∞={x= (xk) : sup n
1
λn X
k∈In
|xk |<∞}.
For the set of sequences that are strongly summable to zero, strongly summable and strongly bounded by the de la Vall˙ee-Poussin method. If λn = n for n = 1,2,3,· · ·, then these sets reduce toω0, ωandω∞introduced and studied by Maddox
[5].
Following Lidenstrauss and Tzafriri [4], we recall that an Orlicz function M is continuous, convex, nondecreasing function defined for x ≥ 0 such that M(0) = 0 and M(x)≥0 forx >0.
If convexity of M is replaced by M(x+y) ≤M(x) +M(y),then it is called a modulus function, defined and studied by Nakano [8], Ruckle [10], Maddox [6] and others.
An Orlicz function M is said to satisfy the ∆2−condition for all values of u, if
there exist a constant K >0 such that
M(2u)≤KM(u) (u≥0).
It is easy to see that always K > 2. The ∆2−condition is equivalent to the
satisfaction of the inequality
M(lu)≤KlM(u),
for all values of uand for l >1.
Lidenstrauss and Tzafriri [4] used the idea of Orlicz function to construct the Orlicz sequence space :
lM :={x= (xk) :
∞
X
k=1
M(|xk|
ρ )<∞, for someρ >0},
kxkM= inf{ρ >0 :
∞
X
k=1
M(|xk|
ρ )≤1}.
If M(x) = xp,1 ≤ p < ∞, the space lM coincide with the classical sequence space lp.
Parashar and Choudhary [9] have introduced and examined some properties of four sequence spaces defined by using an Orlicz function M, which generalized the well-known Orlicz sequence space lM and strongly summable sequence spaces [C,1, p],[C,1, p]0 and [C,1, p]∞.
Let M be an Orlicz function, p = (pk) be any sequence of strictly positive real numbers and u = (uk) be any sequence such that uk 6= 0(k = 1,2,· · ·) . Then Alsaedi and Bataineh [1] defined the following sequence spaces :
[V, M, p, u,∆] ={x= (xk) : limnλ1n
P∞ k∈In[M(
|uk∆xk−le|
ρ )]p
k = 0,
for somel andρ >0},
[V, M, p, u,∆]0 = {x = (xk) : limnλ1n
P∞ k∈In[M(
|uk∆xk|
ρ )]p
k = 0, for someρ >
0},
and
[V, M, p, u,∆]∞={x= (xk) : sup n
1
λn
∞
X
k∈In
[M(|uk∆xk|
ρ )]
pk <
∞, for someρ >0}.
Now, ifnis a nonnegative integer andsis any real number such thats≥0,then we define the following sequence spaces :
[V, M, p,∆nu, s] ={x= (xk) : limnλ1n
P∞ k∈Ink
−s[M(|∆n uxk−le|
ρ )]p
k = 0,
for somel, ρ >0 and s≥0},
[V, M, p∆nu, s]0 ={x= (xk) : limnλ1n
P∞ k∈Ink
−s[M(|∆n uxk|
ρ )]p
k = 0,
for someρ >0 ands≥0},
and
[V, M, p,∆nu, s]∞={x= (xk) : supnλ1n
P∞ k∈Ink
−s[M(|∆n uxk|
ρ )]p
k <
∞,
for someρ >0 ands≥0},
∆1ux=ukxk−uk+1xk+1,
∆2ux= ∆(∆1ux),
.. .
∆nux= ∆(∆nu−1x),
so that
∆nux= ∆nukxk= n X
r=0
(−1)r
n r
uk+rxk+r.
Ifn= 0 ands= 0,then these gives the spaces of Alsaedi and Bataineh [1]. 2.Main results
We prove the following theorems :
Theorem 1. For any Orlicz function M and any sequence p= (pk) of strictly
positive real numbers, [V, M, p,∆nu, s],[V, M, p,∆un, s]0 and [V, M, p,∆nu, s]∞ are
lin-ear spaces over the set of complex numbers.
Proof. We shall prove only for [V, M, p,∆nu, s]0.The others can be treated simi-larly. Letx, y∈[V, M, p,∆nu, s]0 and α, β∈C. In order to prove the result, we need
to find some ρ >0 such that : lim
n 1
λn X
k∈In
k−s[M(|α∆ n
uxk+β∆nuyk|
ρ )]
pk
= 0.
Since x, y∈[V, M, p,∆n
u, s]0,there exists some positiveρ1 and ρ2 such that :
lim n
1
λn X
k∈In
k−s[M(|∆ n uxk|
ρ1
)]pk
= 0 and lim n
1
λn X
k∈In
k−s[M(|∆ n uyk|
ρ2
)]pk
Define ρ= max(2|α|ρ1,2|β |ρ2).SinceM is nondecreasing and convex, This completes the proof.
Suppose that xnm 6= 0 for some m∈In,then (
which is a contradiction. Therefor xnm = 0 for all m. Finally we prove that scalar
multiplication is continuous. Letµ be any complex number, then by definition,
g(µx) = inf{ρpn/H
is continuous at zero. So there exists 1 > δ > 0 such that | f(t) |< (ǫ/2)H for
Theorem 3. For any Orlicz function M which satisfies the ∆2−condition, we have [V, λ,∆nu, s]⊆[V, M,∆nu, s],where and convex, it follows that
Hence
X
2
M(yk)≤Kδ−1M(2)λnTn which together with P
1 ≤ ǫλn yields [V, λ,∆nu, s] ⊆ [V, M,∆nu, s]. This completes the proof.
The method of the proof of Theorem 3 shows that for any Orlicz function
M which satisfies the ∆2−condition, we have [V, λ,∆nu, s]0 ⊆ [V, M,∆nu, s]0 and
[V, λ,∆nu, s]∞⊆[V, M,∆nu, s]∞,where
[V, λ,∆nu, s]0 ={x= (xk) : lim n
1
λn X
k∈In
k−s|∆nuxk|= 0}, and
[V, λ,∆nu, s]∞={x= (xk) : sup n
1
λn X
k∈In
k−s|∆nuxk|<∞}.
Theorem 4. Let 0≤pk≤qk and (qk/pk) be bounded. Then [V, M, q,∆nu, s]⊂ [V, M, p,∆nu, s].
Proof. The proof of Theorem 4 used the ideas similar to those used in proving Theorem 7 of Parashar and Choudhary [9].Mursaleen [7] introduced the concept of statistical convergence as follows :
A sequence x= (xk) is said to beλ−statistically convergent or sλ−statistically convergent to L if for everyǫ >0,
lim n
1
λn | {
k∈In:|xk−L|≥ǫ} |= 0,
where the vertical bars indicates the number of elements in the enclosed set. In this case we writesλ−limx=Lorxk →L(sλ) andsλ={x:∃L∈R:sλ−limx=L}.In a similar way, we say that a sequence x = (xk) is said to be (λ,∆nu)−statistically convergent or sλ(∆nu)−statistically convergent toL if for everyǫ >0,
lim n
1
λn | {
k∈In:|∆nuxk−Le|≥ǫ} |= 0,
where the vertical bars indicates the number of elements in the enclosed set. In this case we write sλ(∆nu)−limx = Le or ∆nuxk → Le (sλ) and sλ(∆nu) = {x : ∃L ∈ R:sλ(∆n
u)−limx=Le}.
Proof. Let x∈[V, M,∆nu, s] and ǫ >0.Then 1
λn X
k∈In
k−sM(|∆ n
uxk−le|
ρ ) ≥
1
λn
X
k∈In,|∆nuxk−le|≥ǫ
M(|∆ n
uxk−le|
ρ )
≥ 1
λn
M(ǫ/ρ).| {k∈In:|∆nuxk−le|≥ǫ} | from which it follows thatx∈sλ(∆nu).
To show that sλ(∆nu) strictly contain [V, M,∆nu, s], we proceed as in [7]. We define x= (xk) by (xk) =kifn−[√λn] + 1≤k≤n and (xk) = 0 otherwise. Then
x /∈l∞(∆nu, s) and for everyǫ (0< ǫ≤1), 1
λn | {
k∈In:|∆unxk−0|≥ǫ} |= [√λn]
λn →
0 as n→ ∞
i.e. x → 0 (sλ(∆nu)), where [ ] denotes the greatest integer function. On the other hand,
1
λn X
k∈In
k−sM(|∆ n
uxk−0|
ρ )→ ∞ asn→ ∞
i.e. xk90 [V, M,∆nu, s].This completes the proof.
References
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Ahmad H. A. Bataineh and Alaa A. Al-Smadi Faculty of Science
Department of Mathematics Al al-Bayt University
