Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2011, Article ID 539745, 10 pages
doi:10.1155/2011/539745
Research Article
A New Class of Sequences Related to the lp Spaces
Defined by Sequences of Orlicz Functions
Naim L. Braha
Department of Mathematics and Computer Sciences, University of Prishtina, Avenue Mother Theresa,
5, Prishtinë 10000, Kosovo
Correspondence should be addressed to Naim L. Braha, nbraha@yahoo.com
Received 5 November 2010; Revised 10 February 2011; Accepted 18 February 2011
Academic Editor: S. S. Dragomir
Copyright q 2011 Naim L. Braha. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We introduce new sequence space mM, φ, q, Λ defined by combining an Orlicz function,
seminorms, and λ-sequences. We study its different properties and obtain some inclusion relation
involving the space mM, φ, q, λ. Inclusion relation between statistical convergent sequence
spaces and Cesaro statistical convergent sequence spaces is also given.
1. Introduction
By w, we denote the space of all real or complex valued sequences. If x ∈ w, then we simply
write x xk instead of x xk ∞
k1 . Also, we will use the conventions that e 1, 1, . . ..
Any vector subspace of w is called a sequence space. We will write l∞ , c, and c0 for the
sequence spaces of all bounded, convergent, and null sequences, respectively. Further, by
lp 1 ≤ p < ∞, we denote the sequence space of all p-absolutely convergent series, that is,
p
lp {x xk ∈ w : ∞
k0 |xk | < ∞} for 1 ≤ p < ∞. Throughout the article, wX, l∞ X, and
lp X denote, respectively, the spaces of all, bounded, and p-absolutely summable sequences
with the elements in X, where X, q is a seminormed space. By θ 0, 0, . . ., we denote
the zero element in X. Ps denotes the set of all subsets of N, that do not contain more than s
elements. With φs , we will denote a nondecreasing sequence of positive real numbers such
that s − 1φs−1 ≤ s − 1φs and φs → ∞, as s → ∞. The class of all the sequences φs satisfying this property is denoted by Φ.
In paper 1, the notion of λ-convergent and bounded sequences is introduced as
follows: let λ λk ∞
k0 be a strictly increasing sequence of positive reals tending to infinity,
that is
0 < λ0 < λ1 < · · · ,
λk −→ ∞ as k −→ ∞.
1.1
2
Journal of Inequalities and Applications
We say that a sequence x xk ∈ w is λ-convergent to the number l ∈ C, called as the λ-limit
of x, if Λn x → l as n → ∞, where
Λn x n
1
λk − λk−1 xk ,
λn k0
n ∈ N.
1.2
In particular, we say that x is a λ-null sequence if Λn x → 0 as n → ∞. Further, we
say that x is λ-bounded if sup |Λn x| < ∞. Here and in the sequel, we will use the convention
that any term with a negative subscript is equal to naught, for example, λ−1 0 and x−1 0.
Now, it is well known 1 that if limn xn a in the ordinary sense of convergence, then
lim
n
n
1
λk − λk−1 |xk − a|
λn k0
0.
1.3
This implies that
n
1
lim|Λn x − a| lim
λk − λk−1 xk − a 0,
n
n λn
k0
1.4
which yields that limn Λn x a and hence x is λ-convergent to a. We therefore deduce that
the ordinary convergence implies the λ-convergence to the same limit.
2. Definitions and Background
The space mφ introduced and studied by Sargent 2 is defined as follows:
m φ x xi ∈ s : xmφ
1
sup
|xi | < ∞ .
s≥1,σ∈Ps φs i∈σ
2.1
Sargent 2 studied some of its properties and obtained its relationship with the space
lp . Later on it was investigated from sequence space point of view by Rath 3, Rath and
Tripathy 4, Tripathy and Sen 5, Tripathy and Mahanta 6, and others. Lindenstrauss and
Tzafriri 7 used the idea of Orlicz function to define the following sequence spaces:
lM ∞
|xi |
x xi ∈ s :
M
i1
< ∞, > 0 ,
2.2
which is called an Orlicz sequence space. The space lM is a Banach space with the norm
∞
|xi |
M
x inf > 0 :
i1
≤1 .
2.3
The space lM is closely related to the space lp which is an Orlicz sequence space with
Mx xp , 1 ≤ p < ∞. An Orlicz function is a function M : 0, ∞ → 0, ∞ which is
Journal of Inequalities and Applications
3
continuous, nondecreasing, and convex with M0 0, Mx > 0 for x > 0 and Mx → ∞
as x → ∞. It is well known that if M is a convex function and M0 0, then Mλx ≤
λ · Mx for all λ with 0 < λ ≤ 1.
An Orlicz function M is said to satisfy the Δ2 -condition for all values of u, if there exists
a constant L > 0 such that M2u ≤ LMu, u ≥ 0 see, Krasnoselskii and Rutitsky 8. In the
later stage, different Orlicz sequence spaces were introduced and studied by Bhardwaj and
Singh 9, Güngör et al. 10, Tripathy and Mahanta 6, Esi 11, Esi and Et 12, Parashar
and Choudhary 13, and many others.
The following inequality will be used throughout the paper,
|ai bi |pi ≤ max 1, 2H−1 |ai |pi |bi |pi ,
2.4
where ai and bi are complex numbers, and H sup pi < ∞, h inf pi . Tripathy and Mahanta
6 defined and studied the following sequence space. Let M be an Orlicz function, then
m M, φ 1
|xi |
M
x xi ∈ s : sup
< ∞, for some > 0 .
s≥1,σ∈Ps φs i∈σ
2.5
Recently, Altun and Bilgin 14 defined and studied the following sequence spaces:
1
|Ai x| pi
M
< ∞, for some > 0 ,
x xi ∈ s : sup
s≥1,σ∈Ps φs i∈σ
m M, A, φ, p 2.6
where Ai x ∞
k1 aik xk and converges for each i. In this paper, we will define the following
sequence spaces:
m M, φ, q, Λ pn
1 |Λn x|
Mn q
0, for some > 0 .
x xi ∈ w : lim
n φs
n∈σ,σ∈Ps
2.7
3. Results
Since the proofs of the following theorems are not hard we omit them.
Theorem 3.1. The sequence spaces mM, φ, q, Λ are linear spaces over the complex field C.
Theorem 3.2. The space mM, φ, q, Λ is a linear topological space paranormed by
gx ⎧
⎨
pr /H
⎩
pn
1 |Λn x|
: sup
Mn q
s φs n∈σ,σ∈Ps
1/H
⎫
⎬
≤ 1, r 1, 2, . . . .
⎭
In what follows, we will show inclusion theorems between spacesmM, φ, q, Λ.
3.1
4
Journal of Inequalities and Applications
Theorem 3.3. mM, φ1 , q, Λ ⊂ mM, φ2 , q, Λ if and only if
sup
s≥1
φs1
φs2
< ∞.
3.2
Proof. Let x ∈ mM, φ1 , q, Λ and K sups≥1 φs1 /φs2 < ∞. Then we get
pn
pn
φs1 1 1 |Λn x|
|Λn x|
M
≤
sup
M
q
q
n
n
2 1
φs2 n∈σ,σ∈Ps
s≥1 φs φs n∈σ,σ∈Ps
pn
1 |Λn x|
K· 1
Mn q
,
φs n∈σ,σ∈Ps
3.3
hence x ∈ mM, φ2 , q, Λ. Conversely, let us suppose that mM, φ1 , q, Λ ⊂ mM, φ2 , q, Λ and
x ∈ mM, φ1 , q, Λ. Then there exists a > 0 such that
pn
1 |Λn x|
M
< ,
q
n
φs1 n∈σ,σ∈Ps
3.4
for every > 0. Suppose that sups≥1 φs1 /φs2 ∞, then there exists a sequence of natural
numbers si such that limj → ∞ φs1j /φs2j ∞. Hence we can write
pn
pn
φs1j 1 1 |Λn x|
|Λn x|
M
≥
sup
·
M
∞.
q
q
n
n
2
φs2 n∈σ,σ∈Ps
φs1j n∈σ,σ∈Ps
j≥1 φs
3.5
Therefore, x ∈
/ mM, φ2 , q, Λ, which is contradiction.
Corollary 3.4. Let M be an Orlicz function. Then mM, φ1 , q, Λ mM, φ2 , q, Λ if and only if
sup
s≥1
φs1
φs2
< ∞,
sup
s≥1
φs2
φs1
< ∞.
3.6
Theorem 3.5. Let M, M1 , M2 be Orlicz functions which satisfy the Δ2 -condition and q, q1 , and q2
seminorms. Then
1 mM1 , φ, q, Λ ⊂ mM ◦ M1 , φ, q, Λ,
2 mM1 , φ, q, Λ ∩ mM2 , φ, q, Λ ⊂ mM1 M2 , φ, q, Λ,
3 mM, φ, q1 , Λ ∩ mM, φ, q2 , Λ ⊂ mM, φ, q1 q2 , Λ,
4 If q1 is stronger than q2 , then mM, φ, q1 , Λ ⊂ mM, φ, q2 , Λ, and
5 If q1 is equivalent to q2 , then mM, φ, q1 , Λ mM, φ, q2 , Λ.
Proof. Proof is similar to 14, Theorem 2.5.
Journal of Inequalities and Applications
5
Corollary 3.6. Let M be an Orlicz function which satisfy the Δ2 -condition. Then mφ, q, Λ ⊂
mM, φ, q, Λ.
Theorem 3.7. Let Ω Mi be a sequence of Orlicz functions. Then the sequence space
mM, φ, q, Λ is solid and monotone.
Proof. Let x ∈ mM, φ, q, Λ, then there exists > 0 such that
pn
1 |Λn x|
M q
< ,
φs n∈σ,σ∈Ps
3.7
for every > 0. Let λn be a sequence of scalars with |λn | ≤ 1 for all n ∈ N. Then from
properties of Orlicz functions and seminorm, we get
pn
pn
1 1 |Λn λn x|
|λn ||Λn x|
Mn q
Mn q
φs n∈σ,σ∈Ps
φs n∈σ,σ∈Ps
≤
pn
1 |Λn x|
|λn |Mn q
φs n∈σ,σ∈Ps
≤
pn
1 |Λn x|
Mn q
,
φs n∈σ,σ∈Ps
3.8
which proves that mM, φ, q, Λ is solid space and monotone.
4. Statistical Convergence
In 15, Fast introduced the idea of statistical convergence. This ideas was later studied by
Connor 16, Freedman and Sember 17, and many others. A sequence of positive integers
θ kr is called lacunary if k0 0, 0 < kr < kr1 , and hr kr − kr−1 → ∞ as r → ∞. A
sequence x xi is said to be Sθ φ, Λ statistically convergent to s if for any > 0,
lim
i
1
k
hr
Λi x
0,
− s ≥ i ∈ σ, σ ∈ Pr , r ≥ 1 : 4.1
for some > 0, where kA denotes the cardinality of A. A sequence x xi is said to be
S0θ φ, Λ statistically convergent to s if for any > 0,
lim
i
1
k
hr
Λi x ≥
0,
i ∈ σ, σ ∈ Pr , r ≥ 1 : 4.2
for some > 0.
Theorem 4.1. If M is any Orlicz function, φn strictly increasing sequence, then mM, φ, q, Λ ⊂
S0θ φ, Λ.
6
Journal of Inequalities and Applications
Proof. Let x ∈ mM, φ, q, Λ. Then
pn
1 |Λn x|
Mn q
< 1 ,
φs n∈σ,σ∈Ps
4.3
for every 1 > 0. Let ks sφs be a sequence of positive numbers. Then it follows that ks is
lacunary sequence. Then we get the following relation:
pn
1 |Λn x|
Mn q
φs n∈σ,σ∈Ps
≥
pn
1
|Λn x|
Mn q
sφs − s − 1φs−1 n∈σ,σ∈Ps
pn
1 |Λn x|
Mn q
hs n∈σ,σ∈Ps
pn
1
|Λn x|
≥
Mn q
hs 1
≥
p
1
Mn q n
hs 1
h
H 1
|Λn x|
≥
where the summation
≥
min Mn q , Mn q
is over
hs 1
1
h
H 1
|Λn x|
≥ k n ∈ σ, σ ∈ Ps , s ≥ 1 :
≥ · min Mn q , Mn q
.
hs
4.4
Taking the limit as n → ∞, it follows that x ∈ S0θ M, φ, q, Λ.
Theorem 4.2. If M is any Orlicz bounded function, φs strictly increasing sequence, then
mM, φs , q, Λ·/s S0θ φs , Λ·/s, for every s ≥ 1.
Proof. Inclusion mM, φs , q, Λ·/s ⊂ S0θ φs , Λ·/s, is valid from Theorem 4.1. In what
follows, we will show converse inclusion. Let x ∈ S0θ φs , Λ·/s, since Mn is bounded, there
exists a constant K such that Mn q|Λn x/s|/ < K. Then for every given > 0, we have
pn
pn
1 1 |Λn x/s|
1 |Λn x|
·
Mn q
Mn q
φs n∈σ,σ∈Ps
φs n∈σ,σ∈Ps
s
pn
1 1 |Λn x|
≤ ·
Mn q
.
s φs n∈σ,σ∈Ps
4.5
Journal of Inequalities and Applications
7
Let us denote by ks s · φs , as we know this sequence is lacunary and finally we get the
following relation:
pn
1 |Λn x|
Mn q
ks n∈σ,σ∈Ps
pn
1
|Λn x|
≤
Mn q
ks − ks−1 n∈σ,σ∈Ps
pn
pn
1
1
|Λn x|
|Λn x|
Mn q
Mn q
hs 1
hs 2
h
H 1
|Λn x|
H
≥ max Mn q , Mn q
,
≤ K · k n ∈ σ, σ ∈ Ps , s ≥ 1 :
hs
4.6
where the summation 1 is over |Λn x|/ ≥ and the summation 2 is over |Λn x|/ ≤
. Taking the limit as → 0 and n → ∞, it follows that x ∈ mM, φs , q, Λ·/s.
5. Cesaro Convergence
In this paragraph, we will consider that φs is a nondecreasing sequence of positive real
numbers such that φs ≤ s, φs → ∞, as s → ∞. Let us denote by
mcθ
M, φ, q, Λ pk
n
1 |Λk x|
Mk q
0, for some > 0 .
x xi : lim
n→∞n 1
k1
5.1
Theorem 5.1. If M is an Orlicz function. Then mcθ M, φ, q, Λ ⊂ mM, φ, q, Λ.
Proof. From the definition of the sequences φn , it follows that infn n 1/n 1 − φn ≥ 1.
Then there exist a δ > 0, such that
n1 1δ
.
≤
φn
δ
5.2
Then we get the following relation:
pn
1 |Λn x|
Mn q
φs n∈σ,σ∈Ps
pk
pk
n1
n1 1 1
|Λk x|
|Λk x|
Mk q
−
Mk q
φs n 1 k1
φs k∈{1,2,...,n1}\σ,σ∈P
s
pn0
pk
n1
|Λn0 x|
s1 1 1
|Λk x|
≤
Mk q
− Mn0 q
φs n 1 k1
φs
pk
pn0
n1
|Λn0 x|
1δ 1 1
|Λk x|
≤
Mk q
− Mn0 q
,
δ n 1 k1
φs
5.3
8
Journal of Inequalities and Applications
where n0 ∈ {1, 2, . . . , n 1} \ σ, σ ∈ Ps . Knowing that x ∈ mcθ M, φ, q, Λ and Mi are
continuous, letting n → ∞ on last relation, we obtain
pn
1 |Λn x|
Mn q
−→ 0.
φs n∈σ,σ∈Ps
5.4
Hence x ∈ mM, φ, q, Λ.
Theorem 5.2. Let sups φs /φs−1 < ∞. Then for any Orlicz function, M, mM, φ, q, Λ ⊂
mcθ M, φ, q, Λ.
Proof. Suppose that sups φs /φs−1 < ∞, then there exists B > 0 such that φs /φs−1 < B for all
s ≥ 1. Let x ∈ mM, φ, q, Λ and > 0, there exist R > 0 such that for every k ≥ R
pk
1 |Λk x|
Mk q
< .
φs k∈σ,σ∈P
5.5
s
We can also find a constant K > 0 such that
pk
1 |Λk x|
Mk q
< K,
φs k∈σ,σ∈P
s
for all k ∈ N. Let n be any integer with φs−1 < n 1 ≤ φs , for every s > R. Then
pk
n
1 |Λk x|
Mk q
n 1 k1
pk
φ
s
1 |Λk x|
≤
Mk q
φs−1 k1
⎛
pk φ
pk
φs s
1 ⎝
|Λk x|
|Λk x|
Mk q
Mk q
···
φs−1 k1
φs ⎞
pk
φ
s
|Λk x|
⎠
Mk q
φs−1 ⎛
⎞
pk
φ1 ⎝ 1 |Λk x|
⎠
≤
Mk q
φs−1 φ1 k∈σ,σ∈P 1
⎛
⎞
pk
φ2 ⎝ 1 |Λk x|
⎠ ···
Mk q
φs−1 φ2 k∈σ,σ∈P 2
5.6
Journal of Inequalities and Applications
⎛
⎞
pk
φR ⎝ 1
|Λk x|
⎠ ···
Mk q
φs−1 φR k∈σ,σ∈P R
9
⎛
⎞
pk
φs ⎝ 1 |Λk x|
⎠,
Mk q
φs−1 φs k∈σ,σ∈P s
5.7
where P t are sets of integer numbers which have more than φt elements for t ∈ {1, 2, . . . , s}.
Passing by limit on last relation, where k → ∞ since s → ∞, φs → ∞ and n → ∞, we get
that
pk
n
1 |Λk x|
Mk q
−→ 0;
n 1 k1
5.8
from this, it follows that x ∈ mcθ M, φ, q, Λ.
Theorem 5.3. Let sups φs /φs−1 < ∞. Then for any Orlicz function, M, mM, φ, q, Λ mcθ M, φ, q, Λ.
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